Finite Group Databasebeta

← #33056865

Subgroups — #33056865

Order 18688.

748 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619929C211yesmaximal, characteristic
2#22618670C211yesmaximal, characteristic, complemented
3D4×#11924011C211yesmaximal, characteristic, complemented
4#22608817C211yesmaximal, characteristic, complemented
5#22608813C211yesmaximal, characteristic, complemented
6#22608761C211yesmaximal, characteristic, complemented
7#22608760C211yesmaximal, characteristic, complemented
8#22608760C211yesmaximal, characteristic, complemented
9#22608759C211yesmaximal, characteristic, complemented
10#22608723C211yesmaximal, characteristic, complemented
11#22608704C211yesmaximal, characteristic, complemented
12#22608380C211yesmaximal, characteristic, complemented
13D4⋊C8×D73C211yesmaximal, characteristic, complemented
14C8×#11924011C211yesmaximal, characteristic, complemented
15C8×D8×C73C211yesmaximal, characteristic, complemented
16C2×#14556597—21no—
17C22×#11924011—21no—
18#17989134C2211yescharacteristic
19C4×D4×D73C2211yescharacteristic
20#17988902C411yescharacteristic
21#17988899C411yescharacteristic
22#17988866C411yescharacteristic
23#17988865C411yescharacteristic
24#17988781C411yescharacteristic
25#17988780C411yescharacteristic
26#17988722C411yescharacteristic
27#17988719C411yescharacteristic
28C4×D584C2211yescharacteristic
29C4×#11924013C2211yescharacteristic
30C4×#11924013C2211yescharacteristic
31#17988666C2211yescharacteristic
32#17988666C2211yescharacteristic, complemented
33#17988665C2211yescharacteristic, complemented
34#17988664C2211yescharacteristic
35#17988661—21no—
36#17988658—21no—
37#17988658—21no—
38#17988655—21no—
39D4×#10596831C2211yescharacteristic
40#17988652C2211yescharacteristic, complemented
41#17988638C2211yescharacteristic
42#17988609C2211yescharacteristic
43C4⋊C8×D73C2211yescharacteristic
44#17988552—21no—
45#17988549—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic, complemented
49#17988339C2211yescharacteristic, complemented
50#17988339C2211yescharacteristic, complemented
51#17988316C2211yescharacteristic
52#17988316C2211yescharacteristic, complemented
53D4⋊#10596831C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic, complemented
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988308C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61C8×D4×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic, complemented
63#335×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66C82×C73C2211yescharacteristic, complemented
67C2×#11924028—21no—
68C2×D4×D73C2×C411yescharacteristic
69C22×C4×D73—21no—
70C2×D584C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73#14556597—44no—
74C2×#11924012—21no—
75C2×#11924011—21no—
76C2×#11924011—21no—
77C2×#11924011—21no—
78C2×#11924011—21no—
79C2×#11924011—44no—
80C2×#11924011D411yescharacteristic
81C2×#11924011D411yescharacteristic
82C22×#10596831—21no—
83C2×C8×D73—21no—
84C2×C8×D73—21no—
85C2×C8×D73—21no—
86#14556590C2×C411yescharacteristic
87#14556587C2×C411yescharacteristic
88#14556577—21no—
89#14556566C2×C411yescharacteristic
90#14556564—21no—
91#14556562C2×C411yescharacteristic
92C4×D292C2311yescharacteristic
93C4×D292C2311yescharacteristic
94D4×Dic73C2×C411yescharacteristic
95#14556553C2×C411yescharacteristic
96C4×C22⋊D73—21no—
97C4×C22⋊D73—21no—
98#14556551—21no—
99C4⋊C4×D73C2×C411yescharacteristic
100C22⋊C4×D73—21no—
101C42×D73C2311yescharacteristic
102#14556525C2×C411yescharacteristic
103#14556524C2×C411yescharacteristic
104#14556524C2×C411yescharacteristic
105C4⋊#10596831C2311yescharacteristic
106C4⋊#10596831C2311yescharacteristic
107#14556520—21no—
108#14556520—21no—
109#14556520C2×C411yescharacteristic
110#14556520C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556513C2×C411yescharacteristic
113#14556512C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556511C2×C411yescharacteristic
116#14556511C2×C411yescharacteristic
117C22⋊#10596831—21no—
118C22⋊#10596831—21no—
119#14556503—21no—
120#14556503—21no—
121#14556503—21no—
122#14556503—21no—
123#14556503C2×C411yescharacteristic
124#14556503C2×C411yescharacteristic
125#14556501D411yescharacteristic
126#14556501D411yescharacteristic
127C4×#10596831C2311yescharacteristic
128C4×#10596831C2311yescharacteristic
129C4×#10596831C2311yescharacteristic
130C8×Dic73—21no—
131C2×D8×C73C2×C411yescharacteristic
132C22×C8×C73—21no—
133C22×C8×C73—21no—
134C4×D4×C73C2311yescharacteristic
135C4×D4×C73C2311yescharacteristic
136#107×C73C2×C411yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138C4⋊C8×C73C2311yescharacteristic
139#102×C73C2×C411yescharacteristic
140#102×C73C2×C411yescharacteristic
141C22⋊C8×C73—21no—
142C22⋊C8×C73—21no—
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C4×C8×C73C2311yescharacteristic
146C23×D73—21no—
147#11924028—42no—
148#11924028—42no—
149C2×Dic146—21no—
150C2×D292—21no—
151C2×D292C22×C411yescharacteristic
152C2×D292C22×C411yescharacteristic
153C2×C22⋊D73—21no—
154C2×C22⋊D73—21no—
155C2×C22⋊D73—21no—
156C2×C22⋊D73—21no—
157D4×D73—22no—
158D4×D73—41no—
159C22×Dic73—21no—
160C2×C4×D73—21no—
161C2×C4×D73—21no—
162C2×C4×D73—21no—
163C2×C4×D73—21no—
164C2×C4×D73—42no—
165C2×C4×D73—42no—
166C2×C4×D73C22×C411yescharacteristic
167C2×C4×D73C2×D411yescharacteristic
168C2×C4×D73#4111yescharacteristic
169D584—41no—
170#11924013—22no—
171#11924013—41no—
172#11924012—42no—
173#11924011—22no—
174#11924011—42no—
175#11924011—42no—
176#11924011—42no—
177#11924011—42no—
178#11924011D814yescomplemented
179C2×#10596831—21no—
180C2×#10596831—21no—
181C2×#10596831—21no—
182C2×#10596831—42no—
183C2×#10596831C22×C411yescharacteristic
184C2×#10596831C22×C411yescharacteristic
185C2×#10596831C2×D411yescharacteristic
186C2×#10596831C2×D411yescharacteristic
187C8×D73—42no—
188C8×D73—42no—
189C8×D73—42no—
190C4⋊Dic73C22×C411yescharacteristic
191C4⋊Dic73C22×C411yescharacteristic
192#11924007—21no—
193#11924007—21no—
194C22⋊Dic73—21no—
195C22⋊Dic73—21no—
196#11924005—21no—
197#11924005—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73C22×C411yescharacteristic
201C4×Dic73C2×D411yescharacteristic
202C4×Dic73#4111yescharacteristic
203C2×D4×C73C22×C411yescharacteristic
204C2×D4×C73C22×C411yescharacteristic
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207D8×C73—22no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—42no—
212C2×C8×C73—42no—
213C2×C8×C73C22×C411yescharacteristic
214C2×C8×C73C22×C411yescharacteristic
215C2×C8×C73C2×D411yescharacteristic
216C2×C8×C73C2×D411yescharacteristic
217C4⋊C4×C73C22×C411yescharacteristic
218C4⋊C4×C73C22×C411yescharacteristic
219C22⋊C4×C73—21no—
220C22⋊C4×C73—21no—
221C42×C73C2411yescharacteristic
222C23×C73—21no—
223C23×C73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—42no—
227C22×D73—42no—
228C22×D73—81no—
229C22×D73C4×D411yescharacteristic
230D4×C73—21no—
231D4×C73—21no—
232D4×C73—21no—
233D4×C73#13112yescomplemented
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73C23×C411yescharacteristic
241C2×C4×C73C22×D411yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C22⋊D73—42no—
244C22⋊D73—42no—
245C22⋊D73—42no—
246C22⋊D73—81no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—42no—
250C2×Dic73C4×D411yescharacteristic
251C2×Dic73C4×D411yescharacteristic
252C2×Dic73C4×D411yescharacteristic
253D292—21no—
254D292—21no—
255D292—21no—
256D292—21no—
257D292—42no—
258C4×D73—21no—
259C4×D73—21no—
260C4×D73—41no—
261C4×D73—42no—
262C4×D73—41no—
263C4×D73—41no—
264C4×D73—41no—
265C4×D73—41no—
266C4×D73—42no—
267C4×D73C2×D812yes—
268C4×D73#13512yes—
269Dic146—42no—
270C8×C73—22no—
271C8×C73—21no—
272C8×C73—21no—
273C8×C73—21no—
274C8×C73—41no—
275C8×C73—41no—
276C8×C73C2×D812yes—
277#10596831—21no—
278#10596831—21no—
279#10596831—21no—
280#10596831—41no—
281#10596831—41no—
282#10596831—41no—
283#10596831C2×D812yes—
284C22×C73—22no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73C2×C4×D411yescharacteristic
291C2×D73—21no—
292C2×D73—21no—
293C2×D73—41no—
294C2×D73—41no—
295C2×D73—41no—
296C2×D73—41no—
297C2×D73—41no—
298C2×D73—41no—
299C2×D73—84no—
300C2×D73C4×D812yes—
301C4×C73—21no—
302C4×C73—41no—
303C4×C73—41no—
304C4×C73C2×#13111yescharacteristic
305C4×C73#56811yescharacteristic
306C4×C73C22×D811yescharacteristic
307C4×C73C2×#13511yescharacteristic
308Dic73—21no—
309Dic73—21no—
310Dic73—41no—
311Dic73—41no—
312Dic73C4×D812yes—
313#17994—731nomaximal, Sylow
314C2×C73—42no—
315C2×C73—81no—
316C2×C73#293311yescharacteristic
317C2×C73C2×C4×D811yescharacteristic
318C2×C73#295611yescharacteristic
319D73—22no—
320D73—81no—
321D73—81no—
322#3100—731no—
323#2935—731no—
324D4×M16—731no—
325#1589—731no—
326#1589—731no—
327#1581—731no—
328#1581—731no—
329#1581—731no—
330#1581—731no—
331C8×D8—731no—
332C8×D8D7311yescharacteristic, complemented
333#1571—731no—
334#1492—731no—
335C2×D4⋊C8—731no—
336C8×M16—731no—
337C73#1799411yescharacteristic, complemented, Sylow
338C2×#131—1461no—
339C22×M16—1461no—
340#532—731no—
341C2×C4×D4—731no—
342#480—731no—
343#480—731no—
344#474—731no—
345#474—731no—
346#459—731no—
347#459—731no—
348#447—731no—
349#447—731no—
350C4×D8—731no—
351C4×D8—731no—
352C4×D8—731no—
353C4×D8C2×D7311yescharacteristic
354#436—731no—
355#436—731no—
356#436—731no—
357#436—731no—
358#435—1461no—
359#435—1461no—
360#435—1461no—
361#435—1461no—
362C8×D4—731no—
363C8×D4—731no—
364C8×D4C2×D7311yescharacteristic
365C8×D4C2×D7311yescharacteristic, complemented
366#431—731no—
367#423—731no—
368C2×C4⋊C8—731no—
369#408—1461no—
370#407—1461no—
371C4×M16—731no—
372C2×C4×C8—731no—
373#335—731no—
374#335—731no—
375#335—731no—
376#335C2×D7311yescharacteristic, complemented
377D4⋊C8—731no—
378D4⋊C8—731no—
379D4⋊C8—731no—
380D4⋊C8—731no—
381D4⋊C8—731no—
382D4⋊C8—731no—
383D4⋊C8C2×D7311yescharacteristic
384D4⋊C8C2×D7311yescharacteristic, complemented
385#322—731no—
386#322—731no—
387C82—731no—
388C82C2×D7311yescharacteristic, complemented
389C2×#41—1461no—
390C22×D4—731no—
391C23×C4—1461no—
392C2×D8—731no—
393C2×D8—731no—
394C2×D8—731no—
395C2×D8C4×D7311yescharacteristic
396#131—2924no—
397C2×M16—2924no—
398C2×M16—731no—
399C2×M16—731no—
400C2×M16—1461no—
401C2×M16—1461no—
402C2×M16—1461no—
403C2×M16—1461no—
404C2×M16—1461no—
405C22×C8—21no—
406C22×C8—21no—
407C22×C8—1461no—
408C22×C8—1461no—
409C22×C8—1461no—
410C22×C8—1461no—
411C4⋊Q8—731no—
412#127—731no—
413#124—1461no—
414#121—731no—
415#121—731no—
416#121—1461no—
417C4×D4—731no—
418C4×D4—731no—
419C4×D4—731no—
420C4×D4—731no—
421C4×D4—1461no—
422C4×D4—1461no—
423C4×D4—1461no—
424C4×D4C22×D7311yescharacteristic
425C4×D4C22×D7311yescharacteristic
426C2×C4⋊C4—731no—
427C2×C22⋊C4—1461no—
428C2×C42—731no—
429#107—731no—
430#107—731no—
431#107—731no—
432#107C4×D7311yescharacteristic
433C4⋊C8—731no—
434C4⋊C8—731no—
435C4⋊C8—731no—
436C4⋊C8—731no—
437C4⋊C8—1461no—
438C4⋊C8—1461no—
439C4⋊C8C22×D7311yescharacteristic
440C4⋊C8C22×D7311yescharacteristic
441#102—731no—
442#102—731no—
443#102—731no—
444#102—731no—
445#102—731no—
446#102—731no—
447#102C4×D7311yescharacteristic
448#102C4×D7311yescharacteristic
449C22⋊C8—21no—
450C22⋊C8—21no—
451C22⋊C8—731no—
452C22⋊C8—731no—
453C22⋊C8—1461no—
454C22⋊C8—1461no—
455C22⋊C8—1461no—
456C22⋊C8—1461no—
457C22⋊C8—1461no—
458C22⋊C8—1461no—
459#97—731no—
460#97—731no—
461C4×C8—731no—
462C4×C8—731no—
463C4×C8—731no—
464C4×C8—1461no—
465C4×C8C22×D7311yescharacteristic
466C4×C8C22×D7311yescharacteristic
467C4×C8C22×D7311yescharacteristic
468C24—1461no—
469#41—2922no—
470#41—2922no—
471C2×Q8—1461no—
472C2×D4—2921no—
473C2×D4—731no—
474C2×D4—731no—
475C2×D4—1461no—
476C2×D4—1461no—
477C2×D4—1461no—
478C2×D4—1461no—
479C2×D4—1461no—
480C2×D4—1462no—
481C2×D4C2×C4×D7311yescharacteristic
482C2×D4C2×C4×D7311yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495D8—22no—
496D8—2921no—
497D8—2921no—
498D8—1462no—
499M16—2922no—
500M16—2922no—
501M16—2922no—
502M16—2922no—
503M16—2922no—
504M16—734no—
505M16—1462no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—2922no—
512C2×C8—2922no—
513C2×C8—2922no—
514C2×C8—2922no—
515C2×C8—731no—
516C2×C8—731no—
517C2×C8—731no—
518C2×C8—731no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8C2×C4×D7311yescharacteristic
523C2×C8C2×C4×D7311yescharacteristic
524C2×C8D4×D7311yescharacteristic
525C2×C8D4×D7311yescharacteristic
526C4⋊C4—731no—
527C4⋊C4—731no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4C2×C4×D7311yescharacteristic
531C4⋊C4C2×C4×D7311yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—1461no—
535C22⋊C4—1461no—
536C22⋊C4—1461no—
537C22⋊C4—1461no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C42—731no—
541C42—731no—
542C42—731no—
543C42C23×D7311yescharacteristic
544C23—21no—
545C23—21no—
546C23—2922no—
547C23—2922no—
548C23—5841no—
549C23—731no—
550C23—1461no—
551C23—1461no—
552Q8—2922no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—2922no—
557D4—2922no—
558D4—2922no—
559D4—2922no—
560D4—5841no—
561D4—1461no—
562D4—1461no—
563D4—1461no—
564D4—1461no—
565D4#1455659712yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—2921no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—731no—
581C2×C4—732no—
582C2×C4—732no—
583C2×C4—731no—
584C2×C4—731no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4C22×C4×D7311yescharacteristic
590C2×C4C2×D4×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C8—22no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—2921no—
599C8—2921no—
600C8—2921no—
601C8—732no—
602C8—1461no—
603C8—1461no—
604C8—1461no—
605C8D8×D7312yes—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—2921no—
613C22—2921no—
614C22—2921no—
615C22—2921no—
616C22—2921no—
617C22—2921no—
618C22—5844no—
619C22—732no—
620C22—1461no—
621C22—1461no—
622C22C4×D4×D7311yescharacteristic
623C4—21no—
624C4—41no—
625C4—41no—
626C4—2921no—
627C4—2921no—
628C4—732no—
629C4—1461no—
630C4—1461no—
631C4C2×#1455659711yescharacteristic
632C4#1798935211yescharacteristic
633C4C2×D8×D7311yescharacteristic
634C4#135×D7311yescharacteristic
635C2—42no—
636C2—81no—
637C2—5841no—
638C2—5841no—
639C2—1462no—
640C2#2261865111yescharacteristic
641C2C4×D8×D7311yescharacteristic
642C2#2261883611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×D8×C73C2
socleC22×C73C2×C4×D4
prime radical#22619929C2
cosocleC42×C73C24
hypercenterC8×D8D73
hypocenterC73#17994
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22618670C2semidirect
3. D4×#11924011C2semidirect
4. #22608817C2semidirect
5. #22608813C2semidirect
6. #22608761C2semidirect
7. #22608760C2semidirect
8. #22608760C2semidirect
9. #22608759C2semidirect
10. #22608723C2semidirect
11. #22608704C2semidirect
12. #22608380C2semidirect
13. D4⋊C8×D73C2semidirect
14. C8×#11924011C2semidirect
15. C8×D8×C73C2semidirect
32. #17988666C22semidirect
33. #17988665C22semidirect
40. #17988652C22semidirect
48. #17988340C22semidirect
49. #17988339C22semidirect
50. #17988339C22semidirect
52. #17988316C22semidirect
54. D4⋊#10596831C22semidirect
56. #17988314C22semidirect
57. #17988309C22semidirect
58. #17988308C22semidirect
59. C8×#10596831C22semidirect
62. C8×D4×C73C22semidirect
63. #335×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. C82×C73C22semidirect
178. #11924011D8semidirect
233. D4×C73#131semidirect
332. C8×D8D73semidirect
337. C73#17994semidirect
365. C8×D4C2×D73semidirect
376. #335C2×D73semidirect
384. D4⋊C8C2×D73semidirect
388. C82C2×D73semidirect
565. D4#14556597semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226199292
#226186702
D4×#119240112
#226088172
#226088132
#226087612
#226087602
#226087592
#226087232
#226087042
#226083802
D4⋊C8×D732
C8×#119240112
C8×D8×C732
#1799473