Finite Group Databasebeta

← #33242473

Subgroups — #33242473

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989028C211yesmaximal, characteristic, complemented
2C22×#14556534C211yesmaximal, characteristic, complemented
3#22621618C211yesmaximal, characteristic
4#22612200C211yesmaximal, characteristic, complemented
5#22612046C211yesmaximal, characteristic, complemented
6#22612020C211yesmaximal, characteristic, complemented
7#22611993C211yesmaximal, characteristic, complemented
8#22611973C211yesmaximal, characteristic, complemented
9#22611869C211yesmaximal, characteristic, complemented
10#22611865C211yesmaximal, characteristic, complemented
11#22611859C211yesmaximal, characteristic, complemented
12#22611848C211yesmaximal, characteristic, complemented
13#22611793C211yesmaximal, characteristic, complemented
14#22611757C211yesmaximal, characteristic, complemented
15D4×Q16×C73C211yesmaximal, characteristic, complemented
16C2×#14556613C2211yescharacteristic
17C22×D584C2211yescharacteristic
18#17989233—21no—
19#17989028—24no—
20C2×#14556537C2211yescharacteristic, complemented
21C2×#14556537C2211yescharacteristic, complemented
22C2×#14556537C2211yescharacteristic, complemented
23C2×#14556534—24no—
24C2×#14556534C2211yescharacteristic
25C2×#14556534C2211yescharacteristic
26C2×#14556534C2211yescharacteristic
27C2×#14556534C2211yescharacteristic
28C2×#14556534C2211yescharacteristic
29C2×#14556534C2211yescharacteristic
30C2×#14556531C2211yescharacteristic, complemented
31C22×#11924003C2211yescharacteristic
32#17988971C2211yescharacteristic
33#17988828C2211yescharacteristic
34#17988823C2211yescharacteristic
35#17988803—21no—
36#17988740—21no—
37#17988734—21no—
38C4×D584C2211yescharacteristic
39#17988412C2211yescharacteristic, complemented
40C4⋊#11924003C2211yescharacteristic, complemented
41#17988391C2211yescharacteristic, complemented
42#17988391C2211yescharacteristic, complemented
43#17988391C2211yescharacteristic, complemented
44#17988390C2211yescharacteristic, complemented
45#17988387C2211yescharacteristic
46#17988387C2211yescharacteristic
47C22⋊#11924003C2211yescharacteristic
48C22⋊#11924003C2211yescharacteristic, complemented
49#17988363C2211yescharacteristic, complemented
50C22×Q16×C73C2211yescharacteristic
51C22×Q16×C73C2211yescharacteristic, complemented
52D4×Q8×C73—21no—
53#494×C73C2211yescharacteristic, complemented
54#467×C73C2211yescharacteristic
55#467×C73C2211yescharacteristic, complemented
56#462×C73—21no—
57#451×C73—21no—
58#451×C73—21no—
59C4×Q16×C73C2211yescharacteristic, complemented
60C8×D4×C73C2211yescharacteristic
61C2×#11924028—21no—
62C22×D292—21no—
63#14556613—22no—
64#14556613—41no—
65C2×Dic292C2311yescharacteristic
66C2×#11924017—21no—
67C2×#11924017—21no—
68C2×D584—22no—
69C2×D584—21no—
70C2×D584—41no—
71C2×D584C2311yescharacteristic
72C2×D584C2311yescharacteristic
73#14556579—21no—
74#14556576—41no—
75#14556570—21no—
76#14556565—21no—
77#14556564—41no—
78#14556559—41no—
79C4×D292—21no—
80C22⋊C4×D73—41no—
81#14556537—22no—
82#14556537—22no—
83#14556537—41no—
84#14556534—22no—
85#14556534—22no—
86#14556534—22no—
87#14556534—22no—
88#14556534—41no—
89#14556534D414yescomplemented
90#14556531—22no—
91C2×#11924003—22no—
92C2×#11924003C2311yescharacteristic
93C2×#11924003C2311yescharacteristic
94C2×#11924003C2311yescharacteristic
95C2×#11924003C2311yescharacteristic
96#14556525C2311yescharacteristic
97#14556516—21no—
98#14556513—21no—
99#14556513—21no—
100C22×Q8×C73—21no—
101C22×Q8×C73—21no—
102C2×Q16×C73—22no—
103C2×Q16×C73—22no—
104C2×Q16×C73—21no—
105C2×Q16×C73—21no—
106C2×Q16×C73—21no—
107C2×Q16×C73—41no—
108C2×Q16×C73—41no—
109C2×Q16×C73C2311yescharacteristic
110C2×Q16×C73C2311yescharacteristic
111C2×Q16×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C4⋊Q8×C73—21no—
115C4⋊Q8×C73—41no—
116C22⋊Q8×C73—21no—
117C22⋊Q8×C73—21no—
118C22⋊Q8×C73—41no—
119C22⋊Q8×C73—41no—
120C4×Q8×C73—21no—
121C4×D4×C73—41no—
122C4×D4×C73D411yescharacteristic
123#107×C73C2311yescharacteristic
124C4⋊C8×C73D411yescharacteristic
125#103×C73—21no—
126#103×C73—21no—
127#103×C73—21no—
128C22⋊C8×C73D411yescharacteristic
129C22⋊C8×C73D411yescharacteristic
130C4×C8×C73C2311yescharacteristic
131C23×D73—41no—
132#11924028—42no—
133#11924028—81no—
134C2×Dic146—21no—
135C2×D292—21no—
136C2×D292—21no—
137C2×D292—42no—
138C2×D292—41no—
139C2×D292—81no—
140C2×C22⋊D73—41no—
141C2×C22⋊D73—41no—
142C2×C4×D73—41no—
143C2×C4×D73—41no—
144Dic292—21no—
145Dic292—21no—
146#11924017—42no—
147#11924017—81no—
148D584—21no—
149D584—21no—
150D584—21no—
151D584—42no—
152D584C2×D412yes—
153C4⋊Dic73—21no—
154#11924007—41no—
155C22⋊Dic73—41no—
156#11924005—41no—
157#11924005—41no—
158#11924005—41no—
159#11924005—41no—
160#11924003—21no—
161#11924003—21no—
162#11924003—21no—
163#11924003C2×D412yes—
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—42no—
168C2×Q8×C73—42no—
169C2×Q8×C73—41no—
170C2×Q8×C73—41no—
171C2×Q8×C73—81no—
172C2×Q8×C73—81no—
173C2×D4×C73D811yescharacteristic
174C22×C4×C73—41no—
175C22×C4×C73—41no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178Q16×C73—21no—
179Q16×C73—21no—
180Q16×C73—21no—
181Q16×C73—21no—
182Q16×C73—21no—
183Q16×C73—42no—
184Q16×C73—42no—
185Q16×C73—81no—
186Q16×C73C2×D412yes—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73C2×D411yescharacteristic
192C2×C8×C73C2×D411yescharacteristic
193C2×C8×C73C2×D411yescharacteristic
194C2×C8×C73C2411yescharacteristic
195C4⋊C4×C73—21no—
196C4⋊C4×C73—41no—
197C4⋊C4×C73—41no—
198C4⋊C4×C73—41no—
199C4⋊C4×C73—41no—
200C4⋊C4×C73—41no—
201C4⋊C4×C73D811yescharacteristic
202C22⋊C4×C73—41no—
203C22⋊C4×C73—41no—
204C22⋊C4×C73D811yescharacteristic
205C22⋊C4×C73D811yescharacteristic
206C42×C73—41no—
207C42×C73C2×D411yescharacteristic
208C23×C73C2×D811yescharacteristic
209C23×C73C2×D811yescharacteristic
210C22×D73—41no—
211C22×D73—41no—
212C22×D73—82no—
213C22×D73—82no—
214C22×D73—161no—
215Q8×C73—22no—
216Q8×C73—41no—
217Q8×C73—41no—
218Q8×C73—41no—
219Q8×C73—41no—
220Q8×C73—41no—
221Q8×C73—82no—
222Q8×C73—82no—
223D4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—82no—
233C2×C4×C73—82no—
234C2×C4×C73C2×D811yescharacteristic
235C2×C4×C73C2×D811yescharacteristic
236C2×C4×C73C2×D811yescharacteristic
237C2×C4×C73C2×D811yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C22⋊D73—82no—
240C22⋊D73—161no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243D292—22no—
244D292—41no—
245D292—41no—
246D292—41no—
247D292—82no—
248C4×D73—82no—
249C4×D73—161no—
250Dic146—41no—
251Dic146—41no—
252C8×C73—21no—
253C8×C73—21no—
254C8×C73—21no—
255C8×C73C22×D411yescharacteristic
256C8×C73H(5,𝔽2)11yescharacteristic
257C22×C73—21no—
258C22×C73—21no—
259C22×C73—21no—
260C22×C73—21no—
261C22×C73—21no—
262C22×C73#51012yescomplemented
263C22×C73C22×D811yescharacteristic
264C2×D73—42no—
265C2×D73—81no—
266C2×D73—81no—
267C2×D73—81no—
268C2×D73—81no—
269C2×D73—81no—
270C2×D73—164no—
271C4×C73—21no—
272C4×C73—21no—
273C4×C73—21no—
274C4×C73—21no—
275C4×C73—42no—
276C4×C73—81no—
277C4×C73—81no—
278C4×C73—81no—
279C4×C73D4211yescharacteristic
280C4×C73#57611yescharacteristic
281Dic73—81no—
282Dic73—81no—
283#33832—731nomaximal, Sylow
284C2×C73—22no—
285C2×C73—41no—
286C2×C73D4×D811yescharacteristic
287C2×C73C2×#51011yescharacteristic
288C2×C73#341611yescharacteristic
289D73—82no—
290D73—161no—
291C2×#510—731no—
292C22×SD32—731no—
293D4×Q16D7311yescharacteristic, complemented
294#3280—731no—
295#2251—731no—
296#2220—731no—
297#2216—731no—
298#2211—731no—
299#2208—731no—
300#2189—731no—
301#2188—731no—
302#2187—731no—
303#2185—731no—
304#2175—731no—
305#2168—731no—
306C73#3383211yescharacteristic, complemented, Sylow
307C2×#135—731no—
308C22×Q16C2×D7311yescharacteristic
309C22×Q16C2×D7311yescharacteristic, complemented
310C22×D8—731no—
311D4×Q8—21no—
312#546—1461no—
313#510—1464no—
314C2×Q32—731no—
315C2×Q32—731no—
316C2×Q32—731no—
317C2×SD32—731no—
318C2×SD32—731no—
319C2×SD32—731no—
320C2×SD32—731no—
321C2×SD32—731no—
322C2×SD32—731no—
323C2×SD32—1464no—
324C2×M32—731no—
325C22×C16—731no—
326#495—731no—
327#494C2×D7311yescharacteristic, complemented
328#467—731no—
329#467C2×D7311yescharacteristic
330#467C2×D7311yescharacteristic, complemented
331#466—731no—
332#463—1461no—
333#462—21no—
334#451—21no—
335#451—21no—
336#450—1461no—
337#449—1461no—
338C4×Q16C2×D7311yescharacteristic, complemented
339C4×D8—731no—
340C8×D4C2×D7311yescharacteristic
341#366—731no—
342C4⋊C16—731no—
343#358—731no—
344#358—731no—
345#358—731no—
346#358—731no—
347#357—731no—
348#357—731no—
349C22⋊C16—731no—
350C22⋊C16—731no—
351#346—731no—
352C2×#41—1461no—
353C22×Q8—21no—
354C22×Q8—21no—
355C22×D4—1461no—
356#135—2921no—
357#135—1462no—
358C2×Q16—22no—
359C2×Q16—22no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—41no—
364C2×Q16—41no—
365C2×Q16—731no—
366C2×Q16C22×D7311yescharacteristic
367C2×Q16C22×D7311yescharacteristic
368C2×Q16C22×D7311yescharacteristic
369C2×SD16—1461no—
370C2×SD16—1461no—
371C2×D8—2921no—
372C2×D8—731no—
373C2×D8—731no—
374C2×D8—1462no—
375C2×D8—1461no—
376C22×C8C22×D7311yescharacteristic
377C22×C8C22×D7311yescharacteristic
378C4⋊Q8—21no—
379C4⋊Q8—41no—
380#124—1461no—
381#123—2921no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—41no—
385C22⋊Q8—41no—
386C22⋊Q8—1461no—
387#121—2921no—
388#121—1461no—
389#120—2921no—
390C4×Q8—21no—
391C4×D4—41no—
392C4×D4—1461no—
393C4×D4C22⋊D7311yescharacteristic
394C2×C22⋊C4—2921no—
395Q32—2921no—
396Q32—1462no—
397Q32—1462no—
398SD32—2921no—
399SD32—734no—
400SD32—1462no—
401SD32—1462no—
402SD32—1462no—
403SD32—1462no—
404M32—1462no—
405C2×C16—731no—
406C2×C16—731no—
407C2×C16—731no—
408C2×C16—731no—
409C2×C16—1462no—
410#107—731no—
411#107C22×D7311yescharacteristic
412C4⋊C8C22⋊D7311yescharacteristic
413#103—21no—
414#103—21no—
415#103—21no—
416#103—1461no—
417#102—1461no—
418#102—1461no—
419C22⋊C8C22⋊D7311yescharacteristic
420C22⋊C8C22⋊D7311yescharacteristic
421C4×C8C22×D7311yescharacteristic
422C24—2921no—
423#41—2922no—
424#41—5841no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—1461no—
435C2×D4—2922no—
436C2×D4—2921no—
437C2×D4—2921no—
438C2×D4—2921no—
439C2×D4—5841no—
440C2×D4—1461no—
441C2×D4—1461no—
442C2×D4#1192401311yescharacteristic
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—2921no—
446C22×C4—2921no—
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7311yescharacteristic
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—21no—
454Q16—42no—
455Q16—42no—
456Q16—81no—
457Q16—1461no—
458Q16—1461no—
459Q16D4×D7312yes—
460SD16—2922no—
461SD16—5841no—
462D8—2922no—
463D8—732no—
464D8—1461no—
465D8—1461no—
466D8—1461no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8C2×C22⋊D7311yescharacteristic
472C2×C8C2×C22⋊D7311yescharacteristic
473C2×C8C2×C22⋊D7311yescharacteristic
474C2×C8C23×D7311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—2921no—
482C4⋊C4—1461no—
483C4⋊C4#1192401311yescharacteristic
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—2921no—
487C22⋊C4—2921no—
488C22⋊C4—2921no—
489C22⋊C4—2921no—
490C22⋊C4—2921no—
491C22⋊C4#1192401311yescharacteristic
492C22⋊C4#1192401311yescharacteristic
493C42—41no—
494C42C2×C22⋊D7311yescharacteristic
495C16—732no—
496C16—1461no—
497C16—1461no—
498C16—1461no—
499C23—2921no—
500C23—2921no—
501C23—5842no—
502C23—5842no—
503C23—11681no—
504C23C2×#1192401311yescharacteristic
505C23C2×#1192401311yescharacteristic
506Q8—22no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—82no—
514Q8—2921no—
515Q8—2921no—
516D4—22no—
517D4—2921no—
518D4—2921no—
519D4—2921no—
520D4—5842no—
521D4—5842no—
522D4—11681no—
523D4—1462no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—5842no—
537C2×C4—11681no—
538C2×C4C2×#1192401311yescharacteristic
539C2×C4C2×#1192401311yescharacteristic
540C2×C4C2×#1192401311yescharacteristic
541C2×C4C2×#1192401311yescharacteristic
542C2×C4C22×C22⋊D7311yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D7311yescharacteristic
547C8#1455663611yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—2922no—
554C22—5841no—
555C22—5841no—
556C22—5841no—
557C22—5841no—
558C22—5841no—
559C22—11684no—
560C22#1798902812yescomplemented
561C22C22×#1192401311yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—5841no—
571C4—5841no—
572C4D4×C22⋊D7311yescharacteristic
573C4#1798939011yescharacteristic
574C2—22no—
575C2—41no—
576C2—5842no—
577C2—11681no—
578C2D4×#1192401311yescharacteristic
579C2C2×#1798902811yescharacteristic
580C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingD4×Q16×C73C2
socleC22×C73C22×D8
prime radical#22621618C2
hypercenterD4×Q16D73
hypocenterC73#33832
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#17989028C2semidirect
2. C22×#14556534C2semidirect
4. #22612200C2semidirect
5. #22612046C2semidirect
6. #22612020C2semidirect
7. #22611993C2semidirect
8. #22611973C2semidirect
9. #22611869C2semidirect
10. #22611865C2semidirect
11. #22611859C2semidirect
12. #22611848C2semidirect
13. #22611793C2semidirect
14. #22611757C2semidirect
15. D4×Q16×C73C2semidirect
20. C2×#14556537C22semidirect
21. C2×#14556537C22semidirect
22. C2×#14556537C22semidirect
30. C2×#14556531C22semidirect
39. #17988412C22semidirect
40. C4⋊#11924003C22semidirect
41. #17988391C22semidirect
42. #17988391C22semidirect
43. #17988391C22semidirect
44. #17988390C22semidirect
48. C22⋊#11924003C22semidirect
49. #17988363C22semidirect
51. C22×Q16×C73C22semidirect
53. #494×C73C22semidirect
55. #467×C73C22semidirect
59. C4×Q16×C73C22semidirect
89. #14556534D4semidirect
262. C22×C73#510semidirect
293. D4×Q16D73semidirect
306. C73#33832semidirect
309. C22×Q16C2×D73semidirect
327. #494C2×D73semidirect
330. #467C2×D73semidirect
338. C4×Q16C2×D73semidirect
560. C22#17989028semidirect

Maximal subgroups

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

subgroupindex
C2×#179890282
C22×#145565342
#226216182
#226122002
#226120462
#226120202
#226119932
#226119732
#226118692
#226118652
#226118592
#226118482
#226117932
#226117572
D4×Q16×C732
#3383273