Finite Group Databasebeta

← #33243009

Subgroups — #33243009

Order 18688.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D584C211yesmaximal, characteristic
2#22612174C211yesmaximal, characteristic, complemented
3#22612122C211yesmaximal, characteristic, complemented
4#22612111C211yesmaximal, characteristic, complemented
5#22612085C211yesmaximal, characteristic, complemented
6#22612046C211yesmaximal, characteristic, complemented
7#22612029C211yesmaximal, characteristic, complemented
8#22611971C211yesmaximal, characteristic, complemented
9#22611859C211yesmaximal, characteristic, complemented
10#22611859C211yesmaximal, characteristic, complemented
11#22611793C211yesmaximal, characteristic, complemented
12#22611757C211yesmaximal, characteristic, complemented
13#22611646C211yesmaximal, characteristic, complemented
14C2×#17988387C211yesmaximal, characteristic, complemented
15C73×#3336C211yesmaximal, characteristic, complemented
16C22×D584C2211yescharacteristic
17C22×D584C2211yescharacteristic
18D4×D292—21no—
19C2×#14556534C2211yescharacteristic
20C2×#14556534C2211yescharacteristic, complemented
21C2×#14556534C2211yescharacteristic, complemented
22C2×#14556531C2211yescharacteristic, complemented
23C22×#11924003C2211yescharacteristic, complemented
24#17988964C2211yescharacteristic
25#17988823C2211yescharacteristic
26#17988823C2211yescharacteristic
27#17988783—21no—
28#17988723—21no—
29#17988723—21no—
30C4×D584C2211yescharacteristic
31#17988414C2211yescharacteristic, complemented
32#17988412C2211yescharacteristic, complemented
33#17988412C2211yescharacteristic, complemented
34C4⋊#11924003C2211yescharacteristic, complemented
35#17988391C2211yescharacteristic, complemented
36#17988387C2211yescharacteristic
37#17988387C2211yescharacteristic
38#17988387C2211yescharacteristic
39#17988387C2211yescharacteristic, complemented
40#17988387C2211yescharacteristic, complemented
41#17988387C2211yescharacteristic, complemented
42#17988387C2211yescharacteristic, complemented
43#17988387C2211yescharacteristic, complemented
44#17988387C2211yescharacteristic, complemented
45C22⋊#11924003C2211yescharacteristic, complemented
46C22⋊#11924003C2211yescharacteristic, complemented
47#17988363C2211yescharacteristic, complemented
48#552×C73—21no—
49#499×C73C2211yescharacteristic, complemented
50#485×C73—21no—
51#483×C73—21no—
52#479×C73—21no—
53#467×C73C2211yescharacteristic, complemented
54#467×C73C2211yescharacteristic, complemented
55C4×Q16×C73C2211yescharacteristic, complemented
56C8×D4×C73C2211yescharacteristic
57#427×C73C2211yescharacteristic, complemented
58C2×#107×C73C2211yescharacteristic, complemented
59C22×D292—21no—
60C22×D292—21no—
61C2×D4×D73—41no—
62C2×D584—22no—
63C2×D584—22no—
64C2×D584—21no—
65C2×D584—21no—
66C2×D584—21no—
67C2×D584—41no—
68C2×D584—41no—
69C2×D584C2311yescharacteristic
70C2×D584C2311yescharacteristic
71C2×D584C2311yescharacteristic
72#14556588—21no—
73#14556566—41no—
74#14556565—21no—
75#14556565—21no—
76#14556561—41no—
77#14556561—41no—
78C4×D292—21no—
79#14556534—22no—
80#14556534—41no—
81#14556534—41no—
82#14556531—41no—
83C2×#11924003—41no—
84C2×#11924003C2311yescharacteristic
85C2×#11924003C2311yescharacteristic
86C2×#11924003C2311yescharacteristic
87C2×#11924003C2311yescharacteristic
88#14556525C2311yescharacteristic
89#14556513—21no—
90#14556513—21no—
91#14556513—21no—
92C2×Q16×C73C2311yescharacteristic
93C22×C8×C73C2311yescharacteristic
94C22×C8×C73C2311yescharacteristic
95#126×C73—41no—
96#125×C73—21no—
97#125×C73—41no—
98#123×C73—41no—
99C22⋊Q8×C73—21no—
100C22⋊Q8×C73—21no—
101C4×Q8×C73—21no—
102C4×D4×C73D411yescharacteristic
103#117×C73—21no—
104#117×C73—41no—
105C2×C4⋊C4×C73—21no—
106#107×C73—21no—
107#107×C73C2311yescharacteristic
108#107×C73C2311yescharacteristic
109#107×C73C2311yescharacteristic
110#106×C73—21no—
111#106×C73—21no—
112C4⋊C8×C73D411yescharacteristic
113#103×C73—21no—
114#103×C73—21no—
115#103×C73—21no—
116C22⋊C8×C73D411yescharacteristic
117C22⋊C8×C73D411yescharacteristic
118C4×C8×C73C2311yescharacteristic
119C23×D73—41no—
120C23×D73—41no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—42no—
125C2×D292—42no—
126C2×D292—41no—
127C2×D292—41no—
128C2×D292—41no—
129C2×D292—41no—
130C2×D292—81no—
131C2×D292—81no—
132C2×C22⋊D73—41no—
133C2×C22⋊D73—41no—
134D4×D73—84no—
135C2×C4×D73—41no—
136D584—21no—
137D584—21no—
138D584—21no—
139D584—21no—
140D584—21no—
141D584—42no—
142D584—42no—
143D584—81no—
144D584#4112yes—
145C4⋊Dic73—21no—
146#11924005—41no—
147#11924005—41no—
148#11924005—41no—
149#11924003—21no—
150#11924003—21no—
151#11924003—21no—
152#11924003—21no—
153C2×Q8×C73—21no—
154C2×D4×C73D811yescharacteristic
155C22×C4×C73—41no—
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158Q16×C73—21no—
159Q16×C73—21no—
160C2×C8×C73—21no—
161C2×C8×C73—21no—
162C2×C8×C73—21no—
163C2×C8×C73—21no—
164C2×C8×C73C2×D411yescharacteristic
165C2×C8×C73C2×D411yescharacteristic
166C2×C8×C73C2×D411yescharacteristic
167C2×C8×C73C2411yescharacteristic
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73—41no—
172C4⋊C4×C73—41no—
173C4⋊C4×C73—41no—
174C4⋊C4×C73—41no—
175C4⋊C4×C73—41no—
176C4⋊C4×C73—41no—
177C4⋊C4×C73D811yescharacteristic
178C22⋊C4×C73—41no—
179C22⋊C4×C73—41no—
180C22⋊C4×C73—41no—
181C22⋊C4×C73D811yescharacteristic
182C22⋊C4×C73D811yescharacteristic
183C42×C73—41no—
184C42×C73—41no—
185C42×C73C2×D411yescharacteristic
186C23×C73C2×D811yescharacteristic
187C23×C73C2×D811yescharacteristic
188C22×D73—41no—
189C22×D73—41no—
190C22×D73—41no—
191C22×D73—82no—
192C22×D73—82no—
193C22×D73—82no—
194C22×D73—82no—
195C22×D73—161no—
196C22×D73—161no—
197Q8×C73—41no—
198Q8×C73—41no—
199D4×C73—22no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—82no—
209C2×C4×C73C2×D811yescharacteristic
210C2×C4×C73C2×D811yescharacteristic
211C2×C4×C73C2×D811yescharacteristic
212C2×C4×C73C2×D811yescharacteristic
213C2×C4×C73C22×D411yescharacteristic
214C22⋊D73—82no—
215C22⋊D73—82no—
216C2×Dic73—41no—
217D292—22no—
218D292—41no—
219D292—41no—
220D292—41no—
221D292—41no—
222D292—41no—
223D292—82no—
224D292—82no—
225D292—82no—
226D292—161no—
227C4×D73—82no—
228C8×C73—21no—
229C8×C73—21no—
230C8×C73—21no—
231C8×C73C2×#4111yescharacteristic
232C8×C73H(5,𝔽2)11yescharacteristic
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—21no—
238C22×C73#50912yes—
239C22×C73C22×D811yescharacteristic
240C2×D73—42no—
241C2×D73—81no—
242C2×D73—81no—
243C2×D73—81no—
244C2×D73—81no—
245C2×D73—81no—
246C2×D73—81no—
247C2×D73—81no—
248C2×D73—81no—
249C2×D73—164no—
250C2×D73—164no—
251C4×C73—21no—
252C4×C73—21no—
253C4×C73—21no—
254C4×C73—21no—
255C4×C73—81no—
256C4×C73—81no—
257C4×C73—81no—
258C4×C73—81no—
259C4×C73#54611yescharacteristic
260C4×C73#57711yescharacteristic
261Dic73—81no—
262#33877—731nomaximal, Sylow
263C2×C73—22no—
264C2×C73—41no—
265C2×C73#329411yescharacteristic
266C2×C73C2×#50911yescharacteristic
267C2×C73#341611yescharacteristic
268D73—82no—
269D73—161no—
270D73—161no—
271#3336D7311yescharacteristic, complemented
272D4×D8—731no—
273#2244—731no—
274#2234—731no—
275#2232—731no—
276#2228—731no—
277#2220—731no—
278#2217—731no—
279#2207—731no—
280#2187—731no—
281#2187—731no—
282#2175—731no—
283#2168—731no—
284#2140—731no—
285C2×#357—731no—
286C73#3387711yescharacteristic, complemented, Sylow
287C22×D8—731no—
288C22×D8—731no—
289#552—21no—
290D42—1461no—
291C2×SD32—731no—
292C2×SD32—731no—
293C2×SD32—731no—
294C2×M32—731no—
295C22×C16—731no—
296#499C2×D7311yescharacteristic, complemented
297#493—731no—
298#485—21no—
299#483—21no—
300#479—21no—
301#467C2×D7311yescharacteristic, complemented
302#467C2×D7311yescharacteristic, complemented
303#466—731no—
304#466—731no—
305#459—1461no—
306#447—1461no—
307#447—1461no—
308C4×Q16C2×D7311yescharacteristic, complemented
309C4×D8—731no—
310C8×D4C2×D7311yescharacteristic
311#427C2×D7311yescharacteristic, complemented
312C2×#107C2×D7311yescharacteristic, complemented
313#367—731no—
314#366—731no—
315#366—731no—
316C4⋊C16—731no—
317#358—731no—
318#357—731no—
319#357—731no—
320#357—731no—
321#357—731no—
322#357—731no—
323#357—731no—
324#357—731no—
325#357—731no—
326#357—731no—
327C22⋊C16—731no—
328C22⋊C16—731no—
329#346—731no—
330C22×D4—2921no—
331C22×D4—1461no—
332C22×D4—1461no—
333C2×Q16C22×D7311yescharacteristic
334C2×D8—2921no—
335C2×D8—2921no—
336C2×D8—731no—
337C2×D8—731no—
338C2×D8—731no—
339C2×D8—1462no—
340C2×D8—1462no—
341C2×D8—1461no—
342C2×D8—1461no—
343C2×D8—1461no—
344C22×C8C22×D7311yescharacteristic
345C22×C8C22×D7311yescharacteristic
346#127—1461no—
347#126—41no—
348#125—21no—
349#125—41no—
350#123—41no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353#121—2921no—
354#121—1461no—
355#121—1461no—
356#120—2921no—
357#120—2921no—
358C4×Q8—21no—
359C4×D4—1461no—
360C4×D4C22⋊D7311yescharacteristic
361#117—21no—
362#117—41no—
363C2×C4⋊C4—21no—
364SD32—2921no—
365SD32—2921no—
366SD32—1462no—
367M32—2921no—
368C2×C16—2921no—
369C2×C16—731no—
370C2×C16—731no—
371C2×C16—731no—
372C2×C16—731no—
373#107—21no—
374#107—731no—
375#107C22×D7311yescharacteristic
376#107C22×D7311yescharacteristic
377#107C22×D7311yescharacteristic
378#106—21no—
379#106—21no—
380C4⋊C8C22⋊D7311yescharacteristic
381#103—21no—
382#103—21no—
383#103—21no—
384#102—1461no—
385#102—1461no—
386#102—1461no—
387C22⋊C8C22⋊D7311yescharacteristic
388C22⋊C8C22⋊D7311yescharacteristic
389C4×C8C22×D7311yescharacteristic
390C24—2921no—
391C24—2921no—
392C2×Q8—21no—
393C2×D4—2922no—
394C2×D4—2922no—
395C2×D4—2921no—
396C2×D4—2921no—
397C2×D4—2921no—
398C2×D4—2921no—
399C2×D4—2921no—
400C2×D4—2921no—
401C2×D4—5841no—
402C2×D4—5841no—
403C2×D4—5844no—
404C2×D4—1461no—
405C2×D4—1461no—
406C2×D4—1461no—
407C2×D4#1192401311yescharacteristic
408C22×C4—41no—
409C22×C4—2921no—
410C22×C4C2×C22⋊D7311yescharacteristic
411C22×C4C2×C22⋊D7311yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—2922no—
415D8—2922no—
416D8—5841no—
417D8—732no—
418D8—1461no—
419D8—1461no—
420D8—1461no—
421D8—1461no—
422D8—1461no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D7311yescharacteristic
428C2×C8C2×C22⋊D7311yescharacteristic
429C2×C8C2×C22⋊D7311yescharacteristic
430C2×C8C23×D7311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—1461no—
441C4⋊C4#1192401311yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—2921no—
446C22⋊C4—2921no—
447C22⋊C4—2921no—
448C22⋊C4#1192401311yescharacteristic
449C22⋊C4#1192401311yescharacteristic
450C42—41no—
451C42—41no—
452C42C2×C22⋊D7311yescharacteristic
453C16—1461no—
454C16—1461no—
455C16—1461no—
456C16—1461no—
457C23—2921no—
458C23—2921no—
459C23—2921no—
460C23—5842no—
461C23—5842no—
462C23—5842no—
463C23—5842no—
464C23—11681no—
465C23—11681no—
466C23C2×#1192401311yescharacteristic
467C23C2×#1192401311yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—2921no—
472D4—2921no—
473D4—2921no—
474D4—2921no—
475D4—2921no—
476D4—5842no—
477D4—5842no—
478D4—5842no—
479D4—5842no—
480D4—5842no—
481D4—11681no—
482D4—1462no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—2921no—
493C2×C4—5842no—
494C2×C4C2×#1192401311yescharacteristic
495C2×C4C2×#1192401311yescharacteristic
496C2×C4C2×#1192401311yescharacteristic
497C2×C4C2×#1192401311yescharacteristic
498C2×C4C22×C22⋊D7311yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8#41×D7311yescharacteristic
503C8#1455663711yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—2922no—
510C22—5841no—
511C22—5841no—
512C22—5841no—
513C22—5841no—
514C22—5841no—
515C22—5841no—
516C22—5841no—
517C22—5841no—
518C22—11684no—
519C22—11684no—
520C22#1798902212yes—
521C22C22×#1192401311yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—5841no—
531C4#1798922511yescharacteristic
532C4#1798939511yescharacteristic
533C2—22no—
534C2—41no—
535C2—5842no—
536C2—11681no—
537C2—11681no—
538C2#2262173811yescharacteristic
539C2C2×#1798902211yescharacteristic
540C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3336C2
socleC22×C73C22×D8
prime radicalD4×D584C2
hypercenter#3336D73
hypocenterC73#33877
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #22612174C2semidirect
3. #22612122C2semidirect
4. #22612111C2semidirect
5. #22612085C2semidirect
6. #22612046C2semidirect
7. #22612029C2semidirect
8. #22611971C2semidirect
9. #22611859C2semidirect
10. #22611859C2semidirect
11. #22611793C2semidirect
12. #22611757C2semidirect
13. #22611646C2semidirect
14. C2×#17988387C2semidirect
15. C73×#3336C2semidirect
20. C2×#14556534C22semidirect
21. C2×#14556534C22semidirect
22. C2×#14556531C22semidirect
23. C22×#11924003C22semidirect
31. #17988414C22semidirect
32. #17988412C22semidirect
33. #17988412C22semidirect
34. C4⋊#11924003C22semidirect
35. #17988391C22semidirect
39. #17988387C22semidirect
40. #17988387C22semidirect
41. #17988387C22semidirect
42. #17988387C22semidirect
43. #17988387C22semidirect
44. #17988387C22semidirect
45. C22⋊#11924003C22semidirect
46. C22⋊#11924003C22semidirect
47. #17988363C22semidirect
49. #499×C73C22semidirect
53. #467×C73C22semidirect
54. #467×C73C22semidirect
55. C4×Q16×C73C22semidirect
57. #427×C73C22semidirect
58. C2×#107×C73C22semidirect
271. #3336D73semidirect
286. C73#33877semidirect
296. #499C2×D73semidirect
301. #467C2×D73semidirect
302. #467C2×D73semidirect
308. C4×Q16C2×D73semidirect
311. #427C2×D73semidirect
312. C2×#107C2×D73semidirect

Maximal subgroups

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

subgroupindex
D4×D5842
#226121742
#226121222
#226121112
#226120852
#226120462
#226120292
#226119712
#226118592
#226117932
#226117572
#226116462
C2×#179883872
C73×#33362
#3387773