Finite Group Databasebeta

← #33055849

Subgroups — #33055849

Order 18688.

774 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619929C211yesmaximal, characteristic
2#22618640C211yesmaximal, characteristic, complemented
3C8×D4×D73C211yesmaximal, characteristic, complemented
4#22608762C211yesmaximal, characteristic, complemented
5#22608760C211yesmaximal, characteristic, complemented
6#22608758C211yesmaximal, characteristic, complemented
7D8×#10596831C211yesmaximal, characteristic, complemented
8#22608725C211yesmaximal, characteristic, complemented
9#22608723C211yesmaximal, characteristic, complemented
10#22608723C211yesmaximal, characteristic, complemented
11#22608704C211yesmaximal, characteristic, complemented
12#22608354C211yesmaximal, characteristic, complemented
13#22608328C211yesmaximal, characteristic, complemented
14#22608216C211yesmaximal, characteristic, complemented
15C73×#1581C211yesmaximal, characteristic, complemented
16C2×#14556599—21no—
17C22×C8×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#17988663—21no—
33C8×D292C2211yescharacteristic
34D4×#10596831C2211yescharacteristic
35D4×#10596831C2211yescharacteristic, complemented
36#17988652C2211yescharacteristic
37#17988652C2211yescharacteristic, complemented
38C8×C22⋊D73—21no—
39#17988650—21no—
40#17988650—21no—
41#17988638C2211yescharacteristic
42#17988638C2211yescharacteristic
43#17988609C2211yescharacteristic
44C4⋊C8×D73C2211yescharacteristic
45#17988552—21no—
46C22⋊C8×D73—21no—
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#17988308C2211yescharacteristic, complemented
58C8×#10596831C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61#436×C73C2211yescharacteristic, complemented
62C8×D4×C73C2211yescharacteristic
63#335×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#14556599—44no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C22×#10596831—21no—
77C22×#10596831—21no—
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80C2×C8×D73—21no—
81C2×C8×D73—21no—
82C2×C8×D73—44no—
83C2×C8×D73D411yescharacteristic
84C2×C8×D73D411yescharacteristic
85#14556590C2×C411yescharacteristic
86#14556587C2×C411yescharacteristic
87#14556577—21no—
88#14556566C2×C411yescharacteristic
89#14556564—21no—
90#14556562C2×C411yescharacteristic
91C4×D292C2311yescharacteristic
92C4×D292C2311yescharacteristic
93D4×Dic73C2×C411yescharacteristic
94#14556553C2×C411yescharacteristic
95C4×C22⋊D73—21no—
96C4×C22⋊D73—21no—
97#14556551—21no—
98C4⋊C4×D73C2×C411yescharacteristic
99C22⋊C4×D73—21no—
100C42×D73C2311yescharacteristic
101#14556525C2×C411yescharacteristic
102#14556524C2×C411yescharacteristic
103#14556524C2×C411yescharacteristic
104C4⋊#10596831C2311yescharacteristic
105C4⋊#10596831C2311yescharacteristic
106#14556520—21no—
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#14556503—21no—
124#14556503C2×C411yescharacteristic
125#14556503C2×C411yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C4×#10596831C2311yescharacteristic
129C8×Dic73D411yescharacteristic
130C8×Dic73D411yescharacteristic
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—22no—
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—22no—
172#11924011—42no—
173#11924011—42no—
174C2×#10596831—21no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—42no—
178C2×#10596831—42no—
179C2×#10596831C22×C411yescharacteristic
180C2×#10596831C22×C411yescharacteristic
181C2×#10596831C2×D411yescharacteristic
182C2×#10596831C2×D411yescharacteristic
183C8×D73—22no—
184C8×D73—42no—
185C8×D73—42no—
186C8×D73—42no—
187C8×D73—42no—
188C8×D73D814yescomplemented
189C4⋊Dic73C22×C411yescharacteristic
190C4⋊Dic73C22×C411yescharacteristic
191#11924007—21no—
192#11924007—21no—
193C22⋊Dic73—21no—
194C22⋊Dic73—21no—
195#11924005—21no—
196#11924005—21no—
197#11924005—21no—
198#11924005—21no—
199C4×Dic73C22×C411yescharacteristic
200C4×Dic73C2×D411yescharacteristic
201C4×Dic73#4111yescharacteristic
202C2×D4×C73C22×C411yescharacteristic
203C2×D4×C73C22×C411yescharacteristic
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206D8×C73—22no—
207M16×C73—42no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—42no—
212C2×C8×C73C22×C411yescharacteristic
213C2×C8×C73C22×C411yescharacteristic
214C2×C8×C73C2×D411yescharacteristic
215C2×C8×C73C2×D411yescharacteristic
216C4⋊C4×C73C22×C411yescharacteristic
217C4⋊C4×C73C22×C411yescharacteristic
218C22⋊C4×C73—21no—
219C22⋊C4×C73—21no—
220C42×C73C2411yescharacteristic
221C23×C73—21no—
222C23×C73—21no—
223C22×D73—24no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—42no—
227C22×D73—42no—
228C22×D73C4×D411yescharacteristic
229D4×C73—21no—
230D4×C73—21no—
231D4×C73—21no—
232D4×C73#13112yescomplemented
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73C23×C411yescharacteristic
240C2×C4×C73C22×D411yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C22⋊D73—24no—
243C22⋊D73—42no—
244C22⋊D73—42no—
245C22⋊D73—42no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—42no—
249C2×Dic73C4×D411yescharacteristic
250C2×Dic73C4×D411yescharacteristic
251C2×Dic73C4×D411yescharacteristic
252D292—21no—
253D292—21no—
254D292—21no—
255D292—42no—
256D292#13112yescomplemented
257C4×D73—21no—
258C4×D73—21no—
259C4×D73—41no—
260C4×D73—42no—
261C4×D73—41no—
262C4×D73—41no—
263C4×D73—41no—
264C4×D73—41no—
265C4×D73—42no—
266C4×D73C2×D812yes—
267C4×D73#13512yes—
268Dic146—42no—
269C8×C73—21no—
270C8×C73—21no—
271C8×C73—21no—
272C8×C73—41no—
273C8×C73—41no—
274C8×C73—41no—
275C8×C73C2×D812yes—
276#10596831—21no—
277#10596831—21no—
278#10596831—21no—
279#10596831—41no—
280#10596831—41no—
281#10596831—41no—
282#10596831C2×D812yes—
283C22×C73—22no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73C2×C4×D411yescharacteristic
290C2×D73—21no—
291C2×D73—21no—
292C2×D73—22no—
293C2×D73—48no—
294C2×D73—41no—
295C2×D73—41no—
296C2×D73—41no—
297C2×D73—41no—
298C2×D73—41no—
299C2×D73C4×D812yes—
300C4×C73—21no—
301C4×C73—41no—
302C4×C73—41no—
303C4×C73C2×#13111yescharacteristic
304C4×C73C2×#13111yescharacteristic
305C4×C73C22×D811yescharacteristic
306C4×C73C2×#13511yescharacteristic
307Dic73—21no—
308Dic73—21no—
309Dic73—41no—
310Dic73—41no—
311Dic73C4×D812yes—
312#17900—731nomaximal, Sylow
313C2×C73—42no—
314C2×C73—81no—
315C2×C73#292911yescharacteristic
316C2×C73C2×C4×D811yescharacteristic
317C2×C73#295511yescharacteristic
318D73—22no—
319D73—42no—
320D73—81no—
321#3100—731no—
322#2931—731no—
323C2×C8×D4—731no—
324#1581—731no—
325#1581—731no—
326#1581—731no—
327#1581D7311yescharacteristic, complemented
328C8×D8—731no—
329C8×D8—731no—
330C8×D8—731no—
331C8×D8—731no—
332#1571—731no—
333#1486—731no—
334#1480—731no—
335#1450—731no—
336C73#1790011yescharacteristic, complemented, Sylow
337C2×#131—1461no—
338C23×C8—1461no—
339#532—731no—
340C2×C4×D4—731no—
341#480—731no—
342#480—731no—
343#474—731no—
344#474—731no—
345#459—731no—
346#459—731no—
347#447—731no—
348#447—731no—
349C4×D8—731no—
350C4×D8—731no—
351C4×D8—731no—
352C4×D8C2×D7311yescharacteristic
353#436—731no—
354#436—1461no—
355#436C2×D7311yescharacteristic, complemented
356C8×D4—731no—
357C8×D4—731no—
358C8×D4—731no—
359C8×D4—731no—
360C8×D4—731no—
361C8×D4—1461no—
362C8×D4—1461no—
363C8×D4—1461no—
364C8×D4C2×D7311yescharacteristic
365#431—731no—
366#431—731no—
367#423—731no—
368C2×C4⋊C8—731no—
369#408—1461no—
370C2×C22⋊C8—1461no—
371C2×C4×C8—731no—
372#335—731no—
373#335—731no—
374#335—731no—
375#335C2×D7311yescharacteristic, complemented
376D4⋊C8—731no—
377D4⋊C8—731no—
378D4⋊C8—731no—
379D4⋊C8—731no—
380D4⋊C8—731no—
381D4⋊C8—731no—
382D4⋊C8C2×D7311yescharacteristic
383D4⋊C8C2×D7311yescharacteristic, complemented
384#322—731no—
385#322C2×D7311yescharacteristic, complemented
386C82—731no—
387C82—731no—
388C2×#41—1461no—
389C22×D4—731no—
390C23×C4—1461no—
391C2×D8—731no—
392C2×D8—731no—
393C2×D8—731no—
394C2×D8C4×D7311yescharacteristic
395#131—2924no—
396C2×M16—21no—
397C2×M16—1461no—
398C2×M16—1461no—
399C22×C8—21no—
400C22×C8—2924no—
401C22×C8—731no—
402C22×C8—731no—
403C22×C8—1461no—
404C22×C8—1461no—
405C22×C8—1461no—
406C22×C8—1461no—
407C22×C8—1461no—
408C22×C8—1461no—
409C4⋊Q8—731no—
410#127—731no—
411#124—1461no—
412#121—731no—
413#121—731no—
414#121—1461no—
415C4×D4—731no—
416C4×D4—731no—
417C4×D4—731no—
418C4×D4—731no—
419C4×D4—1461no—
420C4×D4—1461no—
421C4×D4—1461no—
422C4×D4C22×D7311yescharacteristic
423C4×D4C22×D7311yescharacteristic
424C2×C4⋊C4—731no—
425C2×C22⋊C4—1461no—
426C2×C42—731no—
427#107—731no—
428#107—731no—
429#107—731no—
430#107C4×D7311yescharacteristic
431C4⋊C8—731no—
432C4⋊C8—731no—
433C4⋊C8—731no—
434C4⋊C8—731no—
435C4⋊C8—1461no—
436C4⋊C8—1461no—
437C4⋊C8—1461no—
438C4⋊C8C22×D7311yescharacteristic
439C4⋊C8C22×D7311yescharacteristic
440#102—731no—
441#102—731no—
442#102—731no—
443#102—731no—
444#102—731no—
445#102—731no—
446#102C4×D7311yescharacteristic
447#102C4×D7311yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—731no—
451C22⋊C8—731no—
452C22⋊C8—1461no—
453C22⋊C8—1461no—
454C22⋊C8—1461no—
455C22⋊C8—1461no—
456C22⋊C8—1461no—
457C22⋊C8—1461no—
458C22⋊C8—1461no—
459C4×C8—731no—
460C4×C8—731no—
461C4×C8—731no—
462C4×C8—731no—
463C4×C8—731no—
464C4×C8C22×D7311yescharacteristic
465C4×C8C22×D7311yescharacteristic
466C4×C8C22×D7311yescharacteristic
467C24—1461no—
468#41—2922no—
469#41—2922no—
470C2×Q8—1461no—
471C2×D4—731no—
472C2×D4—731no—
473C2×D4—1461no—
474C2×D4—1461no—
475C2×D4—1461no—
476C2×D4—1461no—
477C2×D4—1462no—
478C2×D4—1461no—
479C2×D4—1462no—
480C2×D4C2×C4×D7311yescharacteristic
481C2×D4C2×C4×D7311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—731no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494D8—22no—
495D8—1462no—
496D8—1462no—
497D8—1462no—
498M16—42no—
499M16—2922no—
500M16—2922no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—42no—
505C2×C8—2922no—
506C2×C8—2922no—
507C2×C8—2922no—
508C2×C8—2922no—
509C2×C8—2922no—
510C2×C8—2922no—
511C2×C8—731no—
512C2×C8—734no—
513C2×C8—731no—
514C2×C8—731no—
515C2×C8—731no—
516C2×C8—1462no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8C2×C4×D7311yescharacteristic
521C2×C8C2×C4×D7311yescharacteristic
522C2×C8D4×D7311yescharacteristic
523C2×C8D4×D7311yescharacteristic
524C4⋊C4—731no—
525C4⋊C4—731no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4C2×C4×D7311yescharacteristic
529C4⋊C4C2×C4×D7311yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—1461no—
533C22⋊C4—1461no—
534C22⋊C4—1461no—
535C22⋊C4—1461no—
536C22⋊C4—1461no—
537C22⋊C4—1461no—
538C42—731no—
539C42—731no—
540C42—731no—
541C42C23×D7311yescharacteristic
542C23—21no—
543C23—21no—
544C23—2922no—
545C23—2922no—
546C23—731no—
547C23—1464no—
548C23—1461no—
549C23—1461no—
550Q8—2922no—
551D4—21no—
552D4—21no—
553D4—21no—
554D4—2922no—
555D4—2922no—
556D4—2922no—
557D4—2922no—
558D4—732no—
559D4—1464no—
560D4—1461no—
561D4—1461no—
562D4—1461no—
563D4#1455659912yescomplemented
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—2921no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—731no—
579C2×C4—732no—
580C2×C4—732no—
581C2×C4—731no—
582C2×C4—731no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4C22×C4×D7311yescharacteristic
588C2×C4C2×D4×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C8—21no—
591C8—21no—
592C8—21no—
593C8—41no—
594C8—41no—
595C8—41no—
596C8—2921no—
597C8—2921no—
598C8—2921no—
599C8—732no—
600C8—1461no—
601C8—1461no—
602C8—1461no—
603C8D8×D7312yes—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—2928no—
611C22—2921no—
612C22—2921no—
613C22—2921no—
614C22—2921no—
615C22—2921no—
616C22—732no—
617C22—1462no—
618C22—1461no—
619C22—1461no—
620C22C4×D4×D7311yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—2921no—
625C4—2921no—
626C4—732no—
627C4—1461no—
628C4—1461no—
629C4#131×D7311yescharacteristic
630C4C2×#1455659911yescharacteristic
631C4C2×D8×D7311yescharacteristic
632C4#135×D7311yescharacteristic
633C2—42no—
634C2—81no—
635C2—2922no—
636C2—5841no—
637C2—1462no—
638C2#2261862211yescharacteristic
639C2C4×D8×D7311yescharacteristic
640C2#2261883111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1581C2
socleC22×C73C2×C4×D4
prime radical#22619929C2
cosocleC42×C73C24
hypercenter#1581D73
hypocenterC73#17900
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #22618640C2semidirect
3. C8×D4×D73C2semidirect
4. #22608762C2semidirect
5. #22608760C2semidirect
6. #22608758C2semidirect
7. D8×#10596831C2semidirect
8. #22608725C2semidirect
9. #22608723C2semidirect
10. #22608723C2semidirect
11. #22608704C2semidirect
12. #22608354C2semidirect
13. #22608328C2semidirect
14. #22608216C2semidirect
15. C73×#1581C2semidirect
31. #17988666C22semidirect
35. D4×#10596831C22semidirect
37. #17988652C22semidirect
48. #17988340C22semidirect
49. #17988339C22semidirect
50. #17988339C22semidirect
52. #17988316C22semidirect
54. D4⋊#10596831C22semidirect
56. #17988314C22semidirect
57. #17988308C22semidirect
58. C8×#10596831C22semidirect
59. C8×#10596831C22semidirect
61. #436×C73C22semidirect
63. #335×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. #322×C73C22semidirect
188. C8×D73D8semidirect
232. D4×C73#131semidirect
256. D292#131semidirect
327. #1581D73semidirect
336. C73#17900semidirect
355. #436C2×D73semidirect
375. #335C2×D73semidirect
383. D4⋊C8C2×D73semidirect
385. #322C2×D73semidirect
563. D4#14556599semidirect

Maximal subgroups

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

subgroupindex
#226199292
#226186402
C8×D4×D732
#226087622
#226087602
#226087582
D8×#105968312
#226087252
#226087232
#226087042
#226083542
#226083282
#226082162
C73×#15812
#1790073