Finite Group Databasebeta

← #33243011

Subgroups — #33243011

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
1#22622295C211yesmaximal, characteristic, complemented
2D8×C22⋊D73C211yesmaximal, characteristic
3#22612171C211yesmaximal, characteristic, complemented
4#22612121C211yesmaximal, characteristic, complemented
5#22612111C211yesmaximal, characteristic, complemented
6#22612090C211yesmaximal, characteristic, complemented
7#22612043C211yesmaximal, characteristic, complemented
8#22612027C211yesmaximal, characteristic, complemented
9#22611970C211yesmaximal, characteristic, complemented
10#22611861C211yesmaximal, characteristic, complemented
11#22611860C211yesmaximal, characteristic, complemented
12#22611792C211yesmaximal, characteristic, complemented
13#22611756C211yesmaximal, characteristic, complemented
14#22611645C211yesmaximal, characteristic, complemented
15C2×#357×C73C211yesmaximal, characteristic, complemented
16C2×D8×D73C2211yescharacteristic
17#17989272—21no—
18D4×C22⋊D73—21no—
19C2×#14556536C2211yescharacteristic, complemented
20C2×#14556535C2211yescharacteristic
21C2×#14556535C2211yescharacteristic, complemented
22C2×#14556530C2211yescharacteristic, complemented
23#17988988C2211yescharacteristic, complemented
24#17988962C2211yescharacteristic
25#17988930—21no—
26#17988917—21no—
27#17988893—21no—
28#17988829C2211yescharacteristic, complemented
29#17988828C2211yescharacteristic, complemented
30#17988823C2211yescharacteristic
31#17988820C2211yescharacteristic
32#17988780—21no—
33#17988721—21no—
34#17988719—21no—
35#17988681C2211yescharacteristic, complemented
36D8×Dic73C2211yescharacteristic
37C8×C22⋊D73C2211yescharacteristic
38#17988624C2211yescharacteristic, complemented
39#17988415C2211yescharacteristic, complemented
40#17988412C2211yescharacteristic, complemented
41#17988412C2211yescharacteristic, complemented
42#17988405C2211yescharacteristic, complemented
43#17988392C2211yescharacteristic, complemented
44#17988389C2211yescharacteristic, complemented
45#17988388C2211yescharacteristic
46#17988388C2211yescharacteristic, complemented
47#17988387C2211yescharacteristic, complemented
48#17988387C2211yescharacteristic, complemented
49C22⋊#11924003C2211yescharacteristic, complemented
50#17988366C2211yescharacteristic, complemented
51#17988362C2211yescharacteristic, complemented
52C22×D8×C73C2211yescharacteristic
53C22×C16×C73C2211yescharacteristic, complemented
54C2×#107×C73C2211yescharacteristic, complemented
55#357×C73C2211yescharacteristic
56#357×C73C2211yescharacteristic
57#357×C73C2211yescharacteristic, complemented
58#357×C73C2211yescharacteristic, complemented
59C22×C22⋊D73—41no—
60C2×D4×D73—21no—
61C2×Dic292C2311yescharacteristic
62C2×D584C2311yescharacteristic
63C2×#11924013—21no—
64C2×#11924013—21no—
65D8×D73—22no—
66D8×D73—41no—
67C2×C8×D73C2311yescharacteristic
68#14556587—21no—
69#14556586—41no—
70#14556582—21no—
71#14556581—41no—
72#14556574—41no—
73#14556571—21no—
74#14556570—21no—
75#14556565—21no—
76#14556563—41no—
77#14556562—21no—
78#14556560—41no—
79#14556559—41no—
80#14556556—21no—
81D4×Dic73—21no—
82C4×C22⋊D73D411yescharacteristic
83#14556550—41no—
84#14556549—21no—
85#14556536—41no—
86#14556535—22no—
87#14556535—41no—
88#14556530—41no—
89C2×#11924003C2311yescharacteristic
90C2×#11924003C2311yescharacteristic
91#14556525C2311yescharacteristic
92#14556525C2311yescharacteristic
93#14556522—21no—
94#14556522—21no—
95#14556520D411yescharacteristic
96#14556516—21no—
97#14556514—21no—
98#14556514—21no—
99#14556513—21no—
100#14556512—21no—
101#14556512—21no—
102C22⋊#10596831D411yescharacteristic
103#14556503D411yescharacteristic
104C8×Dic73C2311yescharacteristic
105C22×D4×C73—21no—
106C2×D8×C73—22no—
107C2×D8×C73—21no—
108C2×D8×C73—41no—
109C2×D8×C73C2311yescharacteristic
110C2×D8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C2×C4⋊C4×C73—21no—
113C2×C16×C73—41no—
114C2×C16×C73C2311yescharacteristic
115C2×C16×C73C2311yescharacteristic
116#107×C73—21no—
117#107×C73C2311yescharacteristic
118#107×C73C2311yescharacteristic
119C23×D73—41no—
120C2×Dic146—21no—
121C2×D292—21no—
122C2×C22⋊D73—41no—
123C2×C22⋊D73—41no—
124C2×C22⋊D73—41no—
125C2×C22⋊D73—41no—
126C2×C22⋊D73—41no—
127C2×C22⋊D73—84no—
128C2×C22⋊D73D811yescharacteristic
129D4×D73—42no—
130D4×D73—81no—
131C22×Dic73—41no—
132C2×C4×D73C2×D411yescharacteristic
133Dic292—21no—
134Dic292—21no—
135D584—21no—
136D584—21no—
137#11924013—42no—
138#11924013—81no—
139C2×#10596831C2×D411yescharacteristic
140C2×#10596831C2×D411yescharacteristic
141C8×D73—21no—
142C8×D73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—41no—
146C4⋊Dic73—41no—
147#11924007—41no—
148#11924007—41no—
149#11924007—41no—
150#11924007D811yescharacteristic
151C22⋊Dic73—41no—
152C22⋊Dic73—41no—
153C22⋊Dic73—41no—
154C22⋊Dic73D811yescharacteristic
155#11924005—41no—
156#11924005—41no—
157#11924005—41no—
158#11924005D811yescharacteristic
159C4×Dic73—41no—
160C4×Dic73—41no—
161C4×Dic73C2×D411yescharacteristic
162#11924003—21no—
163#11924003—21no—
164C24×C73—41no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—42no—
168C2×D4×C73—41no—
169C2×D4×C73—81no—
170C22×C4×C73—41no—
171C22×C4×C73C2×D411yescharacteristic
172D8×C73—21no—
173D8×C73—21no—
174D8×C73—21no—
175D8×C73—42no—
176D8×C73#4112yes—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73C2×D411yescharacteristic
180C2×C8×C73C2411yescharacteristic
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—41no—
184C16×C73—21no—
185C16×C73—21no—
186C23×C73—41no—
187C23×C73—41no—
188C23×C73—82no—
189C23×C73—82no—
190C23×C73—161no—
191C23×C73C2×D811yescharacteristic
192C22×D73—41no—
193C22×D73—82no—
194C22×D73—82no—
195C22×D73—161no—
196C22×D73C2×D811yescharacteristic
197D4×C73—22no—
198D4×C73—41no—
199D4×C73—41no—
200D4×C73—41no—
201D4×C73—82no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—82no—
207C2×C4×C73C2×D811yescharacteristic
208C2×C4×C73C22×D411yescharacteristic
209C22⋊D73—22no—
210C22⋊D73—82no—
211C22⋊D73—82no—
212C22⋊D73—82no—
213C22⋊D73—82no—
214C22⋊D73—161no—
215C2×Dic73—41no—
216C2×Dic73—41no—
217C2×Dic73—41no—
218C2×Dic73—82no—
219C2×Dic73C2×D811yescharacteristic
220C2×Dic73C2×D811yescharacteristic
221C2×Dic73C2×D811yescharacteristic
222D292—41no—
223D292—41no—
224C4×D73—21no—
225C4×D73—21no—
226Dic146—41no—
227Dic146—41no—
228C8×C73—21no—
229C8×C73C2×#4111yescharacteristic
230C8×C73H(5,𝔽2)11yescharacteristic
231#10596831—21no—
232#10596831—21no—
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—42no—
236C22×C73—81no—
237C22×C73—81no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73—81no—
241C22×C73—164no—
242C22×C73#50912yes—
243C22×C73C22×D811yescharacteristic
244C2×D73—21no—
245C2×D73—21no—
246C2×D73—21no—
247C2×D73—81no—
248C2×D73—81no—
249C2×D73—81no—
250C2×D73—164no—
251C4×C73—21no—
252C4×C73—81no—
253C4×C73—81no—
254C4×C73#54611yescharacteristic
255C4×C73#57711yescharacteristic
256Dic73—21no—
257Dic73—21no—
258Dic73—21no—
259Dic73—81no—
260Dic73—81no—
261Dic73—81no—
262#33877—731nomaximal, Sylow
263C2×C73—22no—
264C2×C73—82no—
265C2×C73—161no—
266C2×C73#329411yescharacteristic
267C2×C73C2×#50911yescharacteristic
268C2×C73#341611yescharacteristic
269D73—41no—
270D73—161no—
271#3336—731no—
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×#357D7311yescharacteristic, complemented
286C73#3387711yescharacteristic, complemented, Sylow
287C22×D8—731no—
288C22×D8C2×D7311yescharacteristic
289#552—1461no—
290D42—1461no—
291C2×SD32—731no—
292C2×SD32—731no—
293C2×SD32—731no—
294C2×M32—731no—
295C22×C16C2×D7311yescharacteristic, complemented
296#499—731no—
297#493—731no—
298#485—1461no—
299#483—1461no—
300#479—1461no—
301#467—731no—
302#467—731no—
303#466—731no—
304#466—731no—
305#459—1461no—
306#447—1461no—
307#447—1461no—
308C4×Q16—731no—
309C4×D8—731no—
310C8×D4—731no—
311#427—731no—
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#357C2×D7311yescharacteristic
324#357C2×D7311yescharacteristic
325#357C2×D7311yescharacteristic, complemented
326#357C2×D7311yescharacteristic, complemented
327C22⋊C16—731no—
328C22⋊C16—731no—
329#346—731no—
330C22×D4—21no—
331C22×D4—2921no—
332C22×D4—1461no—
333C2×Q16—731no—
334C2×D8—22no—
335C2×D8—21no—
336C2×D8—41no—
337C2×D8—2921no—
338C2×D8—731no—
339C2×D8—1461no—
340C2×D8—1461no—
341C2×D8—1462no—
342C2×D8C22×D7311yescharacteristic
343C2×D8C22×D7311yescharacteristic
344C22×C8—731no—
345C22×C8C22×D7311yescharacteristic
346#127—1461no—
347#126—2921no—
348#125—2921no—
349#125—1461no—
350#123—2921no—
351C22⋊Q8—1461no—
352C22⋊Q8—1461no—
353#121—2921no—
354#121—1461no—
355#121—1461no—
356#120—2921no—
357#120—2921no—
358C4×Q8—1461no—
359C4×D4—731no—
360C4×D4—1461no—
361#117—2921no—
362#117—1461no—
363C2×C4⋊C4—21no—
364SD32—2921no—
365SD32—2921no—
366SD32—1462no—
367M32—2921no—
368C2×C16—41no—
369C2×C16—731no—
370C2×C16—731no—
371C2×C16C22×D7311yescharacteristic
372C2×C16C22×D7311yescharacteristic
373#107—21no—
374#107—731no—
375#107—731no—
376#107C22×D7311yescharacteristic
377#107C22×D7311yescharacteristic
378#106—1461no—
379#106—1461no—
380C4⋊C8—731no—
381#103—1461no—
382#103—1461no—
383#103—1461no—
384#102—1461no—
385#102—1461no—
386#102—1461no—
387C22⋊C8—731no—
388C22⋊C8—731no—
389C4×C8—731no—
390C24—41no—
391C24—2921no—
392C2×Q8—1461no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—42no—
396C2×D4—41no—
397C2×D4—81no—
398C2×D4—2921no—
399C2×D4—2921no—
400C2×D4—2921no—
401C2×D4—2921no—
402C2×D4—2921no—
403C2×D4—2922no—
404C2×D4—5844no—
405C2×D4—5841no—
406C2×D4—731no—
407C2×D4—1461no—
408C22×C4—41no—
409C22×C4—2921no—
410C22×C4—731no—
411C22×C4D4×D7311yescharacteristic
412Q16—1461no—
413Q16—1461no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—42no—
418D8—2922no—
419D8—5841no—
420D8—1461no—
421D8—1461no—
422D8#1192402812yes—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—731no—
426C2×C8—731no—
427C2×C8—1461no—
428C2×C8—1461no—
429C2×C8D4×D7311yescharacteristic
430C2×C8C23×D7311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—2921no—
435C4⋊C4—2921no—
436C4⋊C4—2921no—
437C4⋊C4—2921no—
438C4⋊C4—2921no—
439C4⋊C4—731no—
440C4⋊C4—1461no—
441C4⋊C4—1461no—
442C22⋊C4—2921no—
443C22⋊C4—2921no—
444C22⋊C4—2921no—
445C22⋊C4—2921no—
446C22⋊C4—2921no—
447C22⋊C4—2921no—
448C22⋊C4—731no—
449C22⋊C4—731no—
450C42—2921no—
451C42—2921no—
452C42—731no—
453C16—21no—
454C16—21no—
455C16—1461no—
456C16—1461no—
457C23—41no—
458C23—41no—
459C23—82no—
460C23—82no—
461C23—161no—
462C23—2921no—
463C23—5842no—
464C23—5842no—
465C23—731no—
466C23—11681no—
467C23D8×D7311yescharacteristic
468Q8—2921no—
469Q8—2921no—
470D4—22no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—82no—
475D4—2921no—
476D4—2921no—
477D4—5842no—
478D4—5842no—
479D4—5842no—
480D4—5842no—
481D4—11681no—
482D4—1462no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—82no—
488C2×C4—2921no—
489C2×C4—2921no—
490C2×C4—2921no—
491C2×C4—5842no—
492C2×C4—731no—
493C2×C4—731no—
494C2×C4—731no—
495C2×C4—1461no—
496C2×C4—1461no—
497C2×C4D8×D7311yescharacteristic
498C2×C4C2×D4×D7311yescharacteristic
499C8—21no—
500C8—1461no—
501C8—1461no—
502C8C2×#1192402811yescharacteristic
503C8#1455663611yescharacteristic
504C22—21no—
505C22—21no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—164no—
513C22—5841no—
514C22—5841no—
515C22—5841no—
516C22—11684no—
517C22—1461no—
518C22—1461no—
519C22—1461no—
520C22#1798902112yes—
521C22C2×D8×D7311yescharacteristic
522C4—21no—
523C4—81no—
524C4—81no—
525C4—5841no—
526C4—5841no—
527C4—5841no—
528C4—1461no—
529C4—1461no—
530C4—1461no—
531C4#1798922711yescharacteristic
532C4#1798939311yescharacteristic
533C2—22no—
534C2—82no—
535C2—161no—
536C2—2921no—
537C2—11681no—
538C2#2262174011yescharacteristic
539C2C2×#1798902111yescharacteristic
540C2#2262307311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC2×#357×C73C2
socleC22×C73C22×D8
prime radicalD8×C22⋊D73C2
hypercenterC2×#357D73
hypocenterC73#33877
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. #22622295C2semidirect
3. #22612171C2semidirect
4. #22612121C2semidirect
5. #22612111C2semidirect
6. #22612090C2semidirect
7. #22612043C2semidirect
8. #22612027C2semidirect
9. #22611970C2semidirect
10. #22611861C2semidirect
11. #22611860C2semidirect
12. #22611792C2semidirect
13. #22611756C2semidirect
14. #22611645C2semidirect
15. C2×#357×C73C2semidirect
19. C2×#14556536C22semidirect
21. C2×#14556535C22semidirect
22. C2×#14556530C22semidirect
23. #17988988C22semidirect
28. #17988829C22semidirect
29. #17988828C22semidirect
35. #17988681C22semidirect
38. #17988624C22semidirect
39. #17988415C22semidirect
40. #17988412C22semidirect
41. #17988412C22semidirect
42. #17988405C22semidirect
43. #17988392C22semidirect
44. #17988389C22semidirect
46. #17988388C22semidirect
47. #17988387C22semidirect
48. #17988387C22semidirect
49. C22⋊#11924003C22semidirect
50. #17988366C22semidirect
51. #17988362C22semidirect
53. C22×C16×C73C22semidirect
54. C2×#107×C73C22semidirect
57. #357×C73C22semidirect
58. #357×C73C22semidirect
285. C2×#357D73semidirect
286. C73#33877semidirect
295. C22×C16C2×D73semidirect
312. C2×#107C2×D73semidirect
325. #357C2×D73semidirect
326. #357C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226222952
D8×C22⋊D732
#226121712
#226121212
#226121112
#226120902
#226120432
#226120272
#226119702
#226118612
#226118602
#226117922
#226117562
#226116452
C2×#357×C732
#3387773