Finite Group Databasebeta

← #33242453

Subgroups — #33242453

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989023C211yesmaximal, characteristic, complemented
2C22×#14556535C211yesmaximal, characteristic, complemented
3#22621674C211yesmaximal, characteristic, complemented
4#22612195C211yesmaximal, characteristic, complemented
5#22612042C211yesmaximal, characteristic, complemented
6#22612015C211yesmaximal, characteristic, complemented
7#22611986C211yesmaximal, characteristic, complemented
8#22611968C211yesmaximal, characteristic, complemented
9#22611860C211yesmaximal, characteristic, complemented
10#22611852C211yesmaximal, characteristic, complemented
11#22611847C211yesmaximal, characteristic, complemented
12#22611843C211yesmaximal, characteristic, complemented
13#22611795C211yesmaximal, characteristic, complemented
14#22611757C211yesmaximal, characteristic, complemented
15D4×D8×C73C211yesmaximal, characteristic, complemented
16C2×#14556613C2211yescharacteristic, complemented
17C22×Dic292C2211yescharacteristic, complemented
18#17989246—21no—
19#17989023—24no—
20C2×#14556535—24no—
21C2×#14556535C2211yescharacteristic
22C2×#14556535C2211yescharacteristic
23C2×#14556535C2211yescharacteristic, complemented
24C2×#14556535C2211yescharacteristic, complemented
25C2×#14556535C2211yescharacteristic, complemented
26C2×#14556535C2211yescharacteristic, complemented
27C2×#14556532C2211yescharacteristic
28C2×#14556532C2211yescharacteristic, complemented
29C2×#14556532C2211yescharacteristic, complemented
30C2×#14556531C2211yescharacteristic, complemented
31C22×#11924003C2211yescharacteristic, complemented
32#17988971C2211yescharacteristic, complemented
33#17988828C2211yescharacteristic, complemented
34#17988823C2211yescharacteristic, complemented
35#17988810—21no—
36#17988751—21no—
37#17988728—21no—
38C4×Dic292C2211yescharacteristic, complemented
39#17988412C2211yescharacteristic, complemented
40C4⋊#11924003C2211yescharacteristic, complemented
41#17988390C2211yescharacteristic, complemented
42#17988390C2211yescharacteristic, complemented
43#17988388C2211yescharacteristic
44#17988388C2211yescharacteristic, complemented
45#17988388C2211yescharacteristic, complemented
46#17988387C2211yescharacteristic, complemented
47C22⋊#11924003C2211yescharacteristic, complemented
48C22⋊#11924003C2211yescharacteristic, complemented
49#17988363C2211yescharacteristic, complemented
50C22×D8×C73C2211yescharacteristic
51C22×D8×C73C2211yescharacteristic
52D42×C73—21no—
53#493×C73C2211yescharacteristic
54#466×C73C2211yescharacteristic
55#466×C73C2211yescharacteristic
56#459×C73—21no—
57#447×C73—21no—
58#447×C73—21no—
59C4×D8×C73C2211yescharacteristic
60C8×D4×C73C2211yescharacteristic
61C2×#11924028—21no—
62C22×Dic146—21no—
63#14556613—22no—
64#14556613—41no—
65C2×Dic292—22no—
66C2×Dic292—21no—
67C2×Dic292—41no—
68C2×Dic292C2311yescharacteristic
69C2×Dic292C2311yescharacteristic
70C2×#11924017—21no—
71C2×#11924017—21no—
72C2×D584C2311yescharacteristic
73#14556579—21no—
74#14556577—41no—
75#14556570—21no—
76#14556569—41no—
77#14556565—21no—
78#14556563—41no—
79C4×Dic146—21no—
80#14556551—41no—
81#14556535—22no—
82#14556535—22no—
83#14556535—22no—
84#14556535—22no—
85#14556535—41no—
86#14556535D414yescomplemented
87#14556532—22no—
88#14556532—22no—
89#14556532—41no—
90#14556531—22no—
91C2×#11924003—22no—
92C2×#11924003C2311yescharacteristic
93C2×#11924003C2311yescharacteristic
94C2×#11924003C2311yescharacteristic
95C2×#11924003C2311yescharacteristic
96#14556525C2311yescharacteristic
97#14556516—21no—
98#14556516—21no—
99#14556513—21no—
100C22×D4×C73—21no—
101C22×D4×C73—21no—
102C22×D4×C73—41no—
103C2×D8×C73—22no—
104C2×D8×C73—22no—
105C2×D8×C73—21no—
106C2×D8×C73—21no—
107C2×D8×C73—21no—
108C2×D8×C73—41no—
109C2×D8×C73—41no—
110C2×D8×C73C2311yescharacteristic
111C2×D8×C73C2311yescharacteristic
112C2×D8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115#127×C73—21no—
116#121×C73—21no—
117#121×C73—21no—
118#121×C73—41no—
119#120×C73—41no—
120#120×C73—41no—
121C4×D4×C73—21no—
122C4×D4×C73D411yescharacteristic
123#107×C73C2311yescharacteristic
124C4⋊C8×C73D411yescharacteristic
125#102×C73—21no—
126#102×C73—21no—
127#102×C73—21no—
128C22⋊C8×C73D411yescharacteristic
129C22⋊C8×C73D411yescharacteristic
130C4×C8×C73C2311yescharacteristic
131#11924028—42no—
132#11924028—81no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—42no—
136C2×Dic146—41no—
137C2×Dic146—81no—
138C2×D292—21no—
139C2×C22⋊D73—41no—
140C2×C22⋊D73—41no—
141C22×Dic73—41no—
142C2×C4×D73—41no—
143Dic292—21no—
144Dic292—21no—
145Dic292—21no—
146Dic292—42no—
147Dic292C2×D412yescomplemented
148#11924017—42no—
149#11924017—81no—
150D584—21no—
151D584—21no—
152C4⋊Dic73—21no—
153#11924007—41no—
154#11924007—41no—
155C22⋊Dic73—41no—
156#11924005—41no—
157#11924005—41no—
158#11924005—41no—
159C4×Dic73—41no—
160#11924003—21no—
161#11924003—21no—
162#11924003—21no—
163#11924003C2×D412yescomplemented
164C24×C73—41no—
165C24×C73—41no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—42no—
170C2×D4×C73—42no—
171C2×D4×C73—41no—
172C2×D4×C73—41no—
173C2×D4×C73—41no—
174C2×D4×C73—41no—
175C2×D4×C73—41no—
176C2×D4×C73—41no—
177C2×D4×C73—81no—
178C2×D4×C73—81no—
179C2×D4×C73—84no—
180C2×D4×C73D811yescharacteristic
181C22×C4×C73—41no—
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184D8×C73—21no—
185D8×C73—21no—
186D8×C73—21no—
187D8×C73—21no—
188D8×C73—21no—
189D8×C73—42no—
190D8×C73—42no—
191D8×C73—81no—
192D8×C73C2×D412yes—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73C2×D411yescharacteristic
198C2×C8×C73C2×D411yescharacteristic
199C2×C8×C73C2×D411yescharacteristic
200C2×C8×C73C2411yescharacteristic
201C4⋊C4×C73—21no—
202C4⋊C4×C73D811yescharacteristic
203C22⋊C4×C73—41no—
204C22⋊C4×C73—41no—
205C22⋊C4×C73—41no—
206C22⋊C4×C73D811yescharacteristic
207C22⋊C4×C73D811yescharacteristic
208C42×C73C2×D411yescharacteristic
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—82no—
213C23×C73—82no—
214C23×C73—82no—
215C23×C73—82no—
216C23×C73—161no—
217C23×C73—161no—
218C23×C73C2×D811yescharacteristic
219C23×C73C2×D811yescharacteristic
220C22×D73—41no—
221D4×C73—22no—
222D4×C73—22no—
223D4×C73—41no—
224D4×C73—41no—
225D4×C73—41no—
226D4×C73—41no—
227D4×C73—41no—
228D4×C73—82no—
229D4×C73—82no—
230D4×C73—82no—
231D4×C73—82no—
232D4×C73—82no—
233D4×C73—161no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—41no—
239C2×C4×C73—82no—
240C2×C4×C73C2×D811yescharacteristic
241C2×C4×C73C2×D811yescharacteristic
242C2×C4×C73C2×D811yescharacteristic
243C2×C4×C73C2×D811yescharacteristic
244C2×C4×C73C22×D411yescharacteristic
245C22⋊D73—82no—
246C22⋊D73—82no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—82no—
251D292—41no—
252D292—41no—
253C4×D73—82no—
254Dic146—22no—
255Dic146—41no—
256Dic146—41no—
257Dic146—41no—
258Dic146—82no—
259C8×C73—21no—
260C8×C73—21no—
261C8×C73—21no—
262C8×C73C22×D411yescharacteristic
263C8×C73H(5,𝔽2)11yescharacteristic
264C22×C73—21no—
265C22×C73—21no—
266C22×C73—21no—
267C22×C73—21no—
268C22×C73—21no—
269C22×C73—42no—
270C22×C73—81no—
271C22×C73—81no—
272C22×C73—81no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73—81no—
278C22×C73—164no—
279C22×C73—164no—
280C22×C73#50912yescomplemented
281C22×C73C22×D811yescharacteristic
282C2×D73—81no—
283C2×D73—81no—
284C2×D73—81no—
285C4×C73—21no—
286C4×C73—21no—
287C4×C73—21no—
288C4×C73—21no—
289C4×C73—81no—
290C4×C73D4211yescharacteristic
291C4×C73#57611yescharacteristic
292Dic73—42no—
293Dic73—81no—
294Dic73—81no—
295#33831—731nomaximal, Sylow
296C2×C73—22no—
297C2×C73—41no—
298C2×C73—82no—
299C2×C73—161no—
300C2×C73—161no—
301C2×C73D4×D811yescharacteristic
302C2×C73C2×#50911yescharacteristic
303C2×C73#341611yescharacteristic
304D73—161no—
305C2×#509—731no—
306C22×SD32—731no—
307#3285—731no—
308D4×D8D7311yescharacteristic, complemented
309#2250—731no—
310#2220—731no—
311#2215—731no—
312#2210—731no—
313#2207—731no—
314#2187—731no—
315#2186—731no—
316#2185—731no—
317#2184—731no—
318#2175—731no—
319#2168—731no—
320C73#3383111yescharacteristic, complemented, Sylow
321C2×#135—731no—
322C22×Q16—731no—
323C22×D8C2×D7311yescharacteristic
324C22×D8C2×D7311yescharacteristic
325#548—1461no—
326D42—21no—
327#509—1464no—
328C2×SD32—731no—
329C2×SD32—731no—
330C2×SD32—731no—
331C2×SD32—731no—
332C2×SD32—731no—
333C2×SD32—731no—
334C2×SD32—1464no—
335C2×D16—731no—
336C2×D16—731no—
337C2×D16—731no—
338C2×M32—731no—
339C22×C16—731no—
340#495—731no—
341#493C2×D7311yescharacteristic
342#467—731no—
343#466—731no—
344#466C2×D7311yescharacteristic
345#466C2×D7311yescharacteristic
346#464—1461no—
347#459—21no—
348#452—1461no—
349#448—1461no—
350#447—21no—
351#447—21no—
352C4×Q16—731no—
353C4×D8C2×D7311yescharacteristic
354C8×D4C2×D7311yescharacteristic
355#366—731no—
356C4⋊C16—731no—
357#358—731no—
358#358—731no—
359#357—731no—
360#357—731no—
361#357—731no—
362#357—731no—
363C22⋊C16—731no—
364C22⋊C16—731no—
365#346—731no—
366C2×#41—1461no—
367C22×Q8—1461no—
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—41no—
371#135—2921no—
372#135—1462no—
373C2×Q16—2921no—
374C2×Q16—731no—
375C2×Q16—731no—
376C2×Q16—1462no—
377C2×Q16—1461no—
378C2×SD16—1461no—
379C2×SD16—1461no—
380C2×D8—22no—
381C2×D8—22no—
382C2×D8—21no—
383C2×D8—21no—
384C2×D8—21no—
385C2×D8—41no—
386C2×D8—41no—
387C2×D8—731no—
388C2×D8C22×D7311yescharacteristic
389C2×D8C22×D7311yescharacteristic
390C2×D8C22×D7311yescharacteristic
391C22×C8C22×D7311yescharacteristic
392C22×C8C22×D7311yescharacteristic
393#127—21no—
394#124—2921no—
395#124—1461no—
396C22⋊Q8—2921no—
397C22⋊Q8—1461no—
398#121—21no—
399#121—21no—
400#121—41no—
401#121—2921no—
402#121—1461no—
403#120—41no—
404#120—41no—
405C4×Q8—1461no—
406C4×D4—21no—
407C4×D4—2921no—
408C4×D4C22⋊D7311yescharacteristic
409SD32—2921no—
410SD32—734no—
411SD32—1462no—
412SD32—1462no—
413SD32—1462no—
414SD32—1462no—
415D16—2921no—
416D16—1462no—
417D16—1462no—
418M32—1462no—
419C2×C16—731no—
420C2×C16—731no—
421C2×C16—731no—
422C2×C16—731no—
423C2×C16—1462no—
424#107—731no—
425#107C22×D7311yescharacteristic
426C4⋊C8C22⋊D7311yescharacteristic
427#103—1461no—
428#103—1461no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—1461no—
433C22⋊C8C22⋊D7311yescharacteristic
434C22⋊C8C22⋊D7311yescharacteristic
435C4×C8C22×D7311yescharacteristic
436C24—41no—
437C24—41no—
438#41—2922no—
439#41—5841no—
440C2×Q8—2922no—
441C2×Q8—2921no—
442C2×Q8—5841no—
443C2×Q8—1461no—
444C2×Q8—1461no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—81no—
457C2×D4—81no—
458C2×D4—84no—
459C2×D4—2921no—
460C2×D4—2921no—
461C2×D4—1461no—
462C2×D4#1192401311yescharacteristic
463C22×C4—41no—
464C22×C4—2921no—
465C22×C4—2921no—
466C22×C4C2×C22⋊D7311yescharacteristic
467C22×C4C2×C22⋊D7311yescharacteristic
468Q16—2922no—
469Q16—732no—
470Q16—1461no—
471Q16—1461no—
472Q16—1461no—
473SD16—2922no—
474SD16—5841no—
475D8—21no—
476D8—21no—
477D8—21no—
478D8—21no—
479D8—21no—
480D8—42no—
481D8—42no—
482D8—81no—
483D8—1461no—
484D8—1461no—
485D8D4×D7312yes—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8C2×C22⋊D7311yescharacteristic
491C2×C8C2×C22⋊D7311yescharacteristic
492C2×C8C2×C22⋊D7311yescharacteristic
493C2×C8C23×D7311yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—2921no—
496C4⋊C4—2921no—
497C4⋊C4—1461no—
498C4⋊C4#1192401311yescharacteristic
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—2921no—
503C22⋊C4—2921no—
504C22⋊C4—2921no—
505C22⋊C4—2921no—
506C22⋊C4#1192401311yescharacteristic
507C22⋊C4#1192401311yescharacteristic
508C42—2921no—
509C42C2×C22⋊D7311yescharacteristic
510C16—732no—
511C16—1461no—
512C16—1461no—
513C16—1461no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—161no—
522C23—161no—
523C23—2921no—
524C23C2×#1192401311yescharacteristic
525C23C2×#1192401311yescharacteristic
526Q8—2921no—
527Q8—2921no—
528Q8—2921no—
529Q8—5842no—
530Q8—1462no—
531D4—22no—
532D4—22no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—82no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—161no—
544D4—2921no—
545D4—2921no—
546D4—5842no—
547D4—5842no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—82no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—5842no—
558C2×C4—5842no—
559C2×C4C2×#1192401311yescharacteristic
560C2×C4C2×#1192401311yescharacteristic
561C2×C4C2×#1192401311yescharacteristic
562C2×C4C2×#1192401311yescharacteristic
563C2×C4C22×C22⋊D7311yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8C2×D4×D7311yescharacteristic
568C8#1455663611yescharacteristic
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—164no—
584C22—164no—
585C22—5841no—
586C22—5841no—
587C22—5841no—
588C22#1798902312yescomplemented
589C22C22×#1192401311yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—81no—
595C4—2922no—
596C4—5841no—
597C4—5841no—
598C4D4×C22⋊D7311yescharacteristic
599C4#1798939011yescharacteristic
600C2—22no—
601C2—41no—
602C2—82no—
603C2—161no—
604C2—161no—
605C2—11681no—
606C2D4×#1192401311yescharacteristic
607C2C2×#1798902311yescharacteristic
608C2#2262307811yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingD4×D8×C73C2
socleC22×C73C22×D8
hypercenterD4×D8D73
hypocenterC73#33831
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#17989023C2semidirect
2. C22×#14556535C2semidirect
3. #22621674C2semidirect
4. #22612195C2semidirect
5. #22612042C2semidirect
6. #22612015C2semidirect
7. #22611986C2semidirect
8. #22611968C2semidirect
9. #22611860C2semidirect
10. #22611852C2semidirect
11. #22611847C2semidirect
12. #22611843C2semidirect
13. #22611795C2semidirect
14. #22611757C2semidirect
15. D4×D8×C73C2semidirect
16. C2×#14556613C22semidirect
17. C22×Dic292C22semidirect
23. C2×#14556535C22semidirect
24. C2×#14556535C22semidirect
25. C2×#14556535C22semidirect
26. C2×#14556535C22semidirect
28. C2×#14556532C22semidirect
29. C2×#14556532C22semidirect
30. C2×#14556531C22semidirect
31. C22×#11924003C22semidirect
32. #17988971C22semidirect
33. #17988828C22semidirect
34. #17988823C22semidirect
38. C4×Dic292C22semidirect
39. #17988412C22semidirect
40. C4⋊#11924003C22semidirect
41. #17988390C22semidirect
42. #17988390C22semidirect
44. #17988388C22semidirect
45. #17988388C22semidirect
46. #17988387C22semidirect
47. C22⋊#11924003C22semidirect
48. C22⋊#11924003C22semidirect
49. #17988363C22semidirect
86. #14556535D4semidirect
147. Dic292C2×D4semidirect
163. #11924003C2×D4semidirect
280. C22×C73#509semidirect
308. D4×D8D73semidirect
320. C73#33831semidirect
588. C22#17989023semidirect

Maximal subgroups

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

subgroupindex
C2×#179890232
C22×#145565352
#226216742
#226121952
#226120422
#226120152
#226119862
#226119682
#226118602
#226118522
#226118472
#226118432
#226117952
#226117572
D4×D8×C732
#3383173