Finite Group Databasebeta

← #33056953

Subgroups — #33056953

Order 18688.

740 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
3#22618667C211yesmaximal, characteristic, complemented
4#22608845C211yesmaximal, characteristic, complemented
5#22608843C211yesmaximal, characteristic, complemented
6#22608798C211yesmaximal, characteristic, complemented
7#22608798C211yesmaximal, characteristic, complemented
8#22608797C211yesmaximal, characteristic, complemented
9#22608754C211yesmaximal, characteristic, complemented
10#22608753C211yesmaximal, characteristic, complemented
11#22608700C211yesmaximal, characteristic, complemented
12#22608380C211yesmaximal, characteristic, complemented
13D4⋊C8×D73C211yesmaximal, characteristic, complemented
14#22608217C211yesmaximal, characteristic, complemented
15C73×#1586C211yesmaximal, characteristic, complemented
16C2×#14556597—21no—
17C2×M16×D73—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, complemented
32#17988665C2211yescharacteristic
33#17988665C2211yescharacteristic, complemented
34#17988661—21no—
35#17988661—21no—
36#17988658—21no—
37#17988656—21no—
38C8×D292C2211yescharacteristic
39#17988652C2211yescharacteristic
40#17988652C2211yescharacteristic, complemented
41#17988638C2211yescharacteristic
42#17988609C2211yescharacteristic
43C4⋊C8×D73C2211yescharacteristic
44#17988552—21no—
45#17988549—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988338C2211yescharacteristic, complemented
49#17988337C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988316C2211yescharacteristic
52#17988316C2211yescharacteristic, complemented
53D4⋊#10596831C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic, complemented
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988308C2211yescharacteristic, complemented
59#17988308C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61#436×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic, complemented
63#334×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66#322×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×#11924012—21no—
76C2×#11924011—21no—
77C2×#11924011—21no—
78C2×#11924011D411yescharacteristic
79C2×#11924011D411yescharacteristic
80M16×D73—44no—
81C2×C8×D73—21no—
82C2×C8×D73—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
132C2×M16×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—22no—
170#11924013—22no—
171#11924013—41no—
172#11924012—42no—
173#11924012—42no—
174#11924011—22no—
175#11924011—22no—
176#11924011—42no—
177#11924011—42no—
178C2×#10596831—21no—
179C2×#10596831—21no—
180C2×#10596831—21no—
181C2×#10596831C22×C411yescharacteristic
182C2×#10596831C22×C411yescharacteristic
183C2×#10596831C2×D411yescharacteristic
184C2×#10596831C2×D411yescharacteristic
185C8×D73—42no—
186C8×D73—42no—
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—41no—
208M16×C73—42no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
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—21no—
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—42no—
257D292#13112yescomplemented
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×D73AGL(1,ℤ/8)12yes—
268C4×D73AGL(1,ℤ/8)12yes—
269Dic146—42no—
270C8×C73—22no—
271C8×C73—21no—
272C8×C73—21no—
273C8×C73—21no—
274C8×C73—21no—
275C8×C73—41no—
276C8×C73—41no—
277#10596831—21no—
278#10596831—21no—
279#10596831—21no—
280#10596831—21no—
281#10596831—41no—
282#10596831—41no—
283#10596831—41no—
284C22×C73—41no—
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—22no—
294C2×D73—41no—
295C2×D73—41no—
296C2×D73—41no—
297C2×D73—41no—
298C2×D73—41no—
299C2×D73—84no—
300C2×D73#44212yes—
301C4×C73—21no—
302C4×C73—41no—
303C4×C73—41no—
304C4×C73C2×#13111yescharacteristic
305C4×C73#56811yescharacteristic
306C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
307C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
308Dic73—21no—
309Dic73—21no—
310Dic73—41no—
311Dic73—41no—
312Dic73#44212yes—
313#18001—731nomaximal, Sylow
314C2×C73—81no—
315C2×C73—81no—
316C2×C73#293311yescharacteristic
317C2×C73C2×#44211yescharacteristic
318C2×C73#295611yescharacteristic
319D73—22no—
320D73—42no—
321D73—81no—
322#3100—731no—
323#2935—731no—
324D4×M16—731no—
325#1594—731no—
326#1594—731no—
327#1586—731no—
328#1586—731no—
329#1586—731no—
330#1586D7311yescharacteristic, complemented
331#1580—731no—
332#1580—731no—
333#1570—731no—
334#1492—731no—
335C2×D4⋊C8—731no—
336#1451—731no—
337C73#1800111yescharacteristic, 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#436C2×D7311yescharacteristic
358#435—1461no—
359#435—1461no—
360#435—1461no—
361#435—1461no—
362C8×D4—731no—
363C8×D4—731no—
364C8×D4—731no—
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#334—731no—
374#334—731no—
375#334—731no—
376#334C2×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—
387#322—731no—
388#322C2×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—21no—
398C2×M16—2924no—
399C2×M16—731no—
400C2×M16—731no—
401C2×M16—1461no—
402C2×M16—1461no—
403C2×M16—1461no—
404C2×M16—1461no—
405C22×C8—21no—
406C22×C8—1461no—
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—1462no—
480C2×D4—1461no—
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—41no—
496D8—2921no—
497D8—1462no—
498D8—1462no—
499M16—42no—
500M16—2922no—
501M16—2922no—
502M16—2922no—
503M16—2922no—
504M16—1462no—
505M16—1462no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—42no—
510C2×C8—2922no—
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—21no—
557D4—2922no—
558D4—2922no—
559D4—2922no—
560D4—2922no—
561D4—5841no—
562D4—732no—
563D4—1461no—
564D4—1461no—
565D4—1461no—
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—21no—
597C8—41no—
598C8—41no—
599C8—2921no—
600C8—2921no—
601C8—2921no—
602C8—1461no—
603C8—1461no—
604C8—1461no—
605C8—1461no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—2921no—
613C22—2921no—
614C22—2921no—
615C22—2921no—
616C22—2921no—
617C22—5844no—
618C22—732no—
619C22—1462no—
620C22—1461no—
621C22—1461no—
622C22C4×D4×D7311yescharacteristic
623C4—21no—
624C4—41no—
625C4—41no—
626C4—2921no—
627C4—2921no—
628C4—732no—
629C4—1461no—
630C4—1461no—
631C4#131×D7311yescharacteristic
632C4#1798935311yescharacteristic
633C4AGL(1,ℤ/8)×D7311yescharacteristic
634C4AGL(1,ℤ/8)×D7311yescharacteristic
635C2—81no—
636C2—81no—
637C2—2922no—
638C2—5841no—
639C2—1462no—
640C2#2261865411yescharacteristic
641C2#442×D7311yescharacteristic
642C2#2261883611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1586C2
socleC22×C73C2×C4×D4
prime radical#22619929C2
cosocleC42×C73C24
hypercenter#1586D73
hypocenterC73#18001
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #22618670C2semidirect
3. #22618667C2semidirect
4. #22608845C2semidirect
5. #22608843C2semidirect
6. #22608798C2semidirect
7. #22608798C2semidirect
8. #22608797C2semidirect
9. #22608754C2semidirect
10. #22608753C2semidirect
11. #22608700C2semidirect
12. #22608380C2semidirect
13. D4⋊C8×D73C2semidirect
14. #22608217C2semidirect
15. C73×#1586C2semidirect
31. #17988666C22semidirect
33. #17988665C22semidirect
40. #17988652C22semidirect
48. #17988338C22semidirect
49. #17988337C22semidirect
50. #17988337C22semidirect
52. #17988316C22semidirect
54. D4⋊#10596831C22semidirect
56. #17988314C22semidirect
57. #17988309C22semidirect
58. #17988308C22semidirect
59. #17988308C22semidirect
62. C8×D4×C73C22semidirect
63. #334×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. #322×C73C22semidirect
257. D292#131semidirect
330. #1586D73semidirect
337. C73#18001semidirect
365. C8×D4C2×D73semidirect
376. #334C2×D73semidirect
384. D4⋊C8C2×D73semidirect
388. #322C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199292
#226186702
#226186672
#226088452
#226088432
#226087982
#226087972
#226087542
#226087532
#226087002
#226083802
D4⋊C8×D732
#226082172
C73×#15862
#1800173