Finite Group Databasebeta

← #33239299

Subgroups — #33239299

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989024C211yesmaximal, characteristic
2C4×#14556614C211yesmaximal, characteristic, complemented
3C4×D8×D73C211yesmaximal, characteristic, complemented
4#22611808C211yesmaximal, characteristic, complemented
5#22611805C211yesmaximal, characteristic, complemented
6#22611793C211yesmaximal, characteristic, complemented
7SD32×Dic73C211yesmaximal, characteristic, complemented
8#22611780C211yesmaximal, characteristic, complemented
9#22611777C211yesmaximal, characteristic, complemented
10#22611775C211yesmaximal, characteristic, complemented
11#22611716C211yesmaximal, characteristic, complemented
12#22611645C211yesmaximal, characteristic, complemented
13#22611642C211yesmaximal, characteristic, complemented
14#22611524C211yesmaximal, characteristic, complemented
15C73×#2175C211yesmaximal, characteristic, complemented
16C2×#14556614C2211yescharacteristic
17C2×D8×D73C2211yescharacteristic
18C4×#11924029—21no—
19C4×D4×D73—21no—
20#17989024C418yescomplemented
21C2×#14556534C2211yescharacteristic
22C2×D1168C2211yescharacteristic
23C2×#14556532C2211yescharacteristic
24C2×#14556530C2211yescharacteristic
25C2×#14556530C2211yescharacteristic
26Q16×Dic73C2211yescharacteristic, complemented
27C4×#11924015—21no—
28#17988672—21no—
29C4×D584C2211yescharacteristic
30C4×D584C2211yescharacteristic, complemented
31D8×Dic73C2211yescharacteristic
32#17988669C2211yescharacteristic
33#17988669C2211yescharacteristic, complemented
34C4×#11924013—21no—
35#17988667—21no—
36#17988624C2211yescharacteristic
37#107×D73C2211yescharacteristic
38#17988582—21no—
39#102×D73—21no—
40C4×C8×D73C2211yescharacteristic
41#17988415C2211yescharacteristic, complemented
42#17988414C2211yescharacteristic, complemented
43#17988412C2211yescharacteristic, complemented
44#17988391C2211yescharacteristic, complemented
45#17988389C2211yescharacteristic
46#17988389C2211yescharacteristic, complemented
47#17988388C2211yescharacteristic
48#17988387C2211yescharacteristic
49#17988387C2211yescharacteristic, complemented
50#17988363C2211yescharacteristic, complemented
51C16×Dic73C2211yescharacteristic, complemented
52C16×Dic73C2211yescharacteristic, complemented
53C2×SD32×C73C2211yescharacteristic
54C4×Q16×C73C2211yescharacteristic, complemented
55C4×D8×C73C2211yescharacteristic
56#366×C73C2211yescharacteristic, complemented
57#358×C73C2211yescharacteristic, complemented
58#357×C73C2211yescharacteristic
59#346×C73C2211yescharacteristic, complemented
60C2×#11924029—21no—
61C2×D4×D73—21no—
62C22×C4×D73—41no—
63#14556614—22no—
64#14556614C2×C414yescomplemented
65C2×#11924015—21no—
66C2×D584C2311yescharacteristic
67C2×D584C2311yescharacteristic
68C2×#11924013—21no—
69D8×D73—22no—
70D8×D73C2×C414yescomplemented
71C2×C8×D73D411yescharacteristic
72C2×C8×D73C2311yescharacteristic
73Q8×Dic73—21no—
74C4×D292—21no—
75C4×D292—21no—
76C4×D292—41no—
77D4×Dic73—21no—
78#14556553—21no—
79#14556553—21no—
80#14556553—41no—
81C4×C22⋊D73—41no—
82#14556551—41no—
83#14556549—21no—
84#14556549—41no—
85C4⋊C4×D73—21no—
86C22⋊C4×D73—41no—
87C42×D73—41no—
88C42×D73D411yescharacteristic
89#14556534C2×C414yescomplemented
90D1168C2×C414yescomplemented
91#14556532C2×C414yescomplemented
92#14556530—22no—
93#14556530C2×C414yescomplemented
94C2×#11924003C2311yescharacteristic
95C2×#11924003C2311yescharacteristic
96#14556525C2311yescharacteristic
97#14556525C2311yescharacteristic
98#14556524—21no—
99#14556522—21no—
100#14556515—21no—
101#14556513—21no—
102#14556513—21no—
103#14556512—21no—
104#14556511—21no—
105#14556511—21no—
106C4×#10596831D411yescharacteristic
107C8×Dic73D411yescharacteristic
108C8×Dic73C2311yescharacteristic
109C2×Q16×C73C2311yescharacteristic
110C2×D8×C73C2311yescharacteristic
111C4×Q8×C73—21no—
112C4×D4×C73—21no—
113SD32×C73C2×C414yescomplemented
114C2×C16×C73C2311yescharacteristic
115C2×C16×C73C2311yescharacteristic
116#107×C73C2311yescharacteristic
117#107×C73C2311yescharacteristic
118#103×C73—21no—
119#102×C73—21no—
120C4×C8×C73C2311yescharacteristic
121C23×D73—41no—
122#11924029—24no—
123#11924029—42no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—41no—
127C2×C22⋊D73—41no—
128D4×D73—24no—
129D4×D73—42no—
130C22×Dic73—41no—
131C2×C4×D73—41no—
132C2×C4×D73—41no—
133C2×C4×D73—41no—
134C2×C4×D73—41no—
135C2×C4×D73—41no—
136C2×C4×D73—41no—
137C2×C4×D73—84no—
138C2×C4×D73D811yescharacteristic
139C2×C4×D73D811yescharacteristic
140C2×C4×D73C2×D411yescharacteristic
141#11924015—24no—
142D584—21no—
143D584—21no—
144D584C22×C412yes—
145D584C22×C412yescomplemented
146#11924013—24no—
147C2×#10596831C2×D411yescharacteristic
148C2×#10596831C2×D411yescharacteristic
149C8×D73—22no—
150C8×D73C22×C412yes—
151C8×D73#4112yes—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—41no—
155#11924007—41no—
156C22⋊Dic73—41no—
157#11924005—41no—
158#11924005—41no—
159#11924005—41no—
160#11924005—41no—
161#11924005—41no—
162C4×Dic73—41no—
163C4×Dic73—41no—
164C4×Dic73D811yescharacteristic
165C4×Dic73D811yescharacteristic
166C4×Dic73C2×D411yescharacteristic
167#11924003—21no—
168#11924003C22×C412yescomplemented
169C2×Q8×C73—21no—
170C2×D4×C73—21no—
171C22×C4×C73—41no—
172Q16×C73—21no—
173Q16×C73C22×C412yescomplemented
174D8×C73—21no—
175D8×C73C22×C412yes—
176C2×C8×C73C2×D411yescharacteristic
177C2×C8×C73C2411yescharacteristic
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—41no—
181C22⋊C4×C73—41no—
182C42×C73—41no—
183C42×C73C2×D411yescharacteristic
184C16×C73—21no—
185C16×C73C22×C412yescomplemented
186C23×C73—41no—
187C22×D73—41no—
188C22×D73—41no—
189C22×D73—44no—
190C22×D73—82no—
191C22×D73—82no—
192C22×D73C2×D811yescharacteristic
193Q8×C73—22no—
194Q8×C73—41no—
195D4×C73—22no—
196D4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—41no—
200C2×C4×C73—82no—
201C2×C4×C73C2×D811yescharacteristic
202C2×C4×C73C2×D811yescharacteristic
203C2×C4×C73C22×D411yescharacteristic
204C22⋊D73—44no—
205C2×Dic73—41no—
206C2×Dic73—41no—
207C2×Dic73—82no—
208C2×Dic73C2×D811yescharacteristic
209C2×Dic73C2×D811yescharacteristic
210C2×Dic73C2×D811yescharacteristic
211D292—22no—
212D292—22no—
213D292—44no—
214D292—41no—
215D292—41no—
216C4×D73—22no—
217C4×D73—41no—
218C4×D73—44no—
219C4×D73—82no—
220C4×D73—82no—
221C4×D73—82no—
222C4×D73—82no—
223C4×D73—82no—
224C4×D73C4×D412yes—
225C4×D73#13512yes—
226C8×C73—21no—
227C8×C73C23×C411yescharacteristic
228C8×C73C2×#4111yescharacteristic
229#10596831—21no—
230#10596831C4×D412yes—
231C22×C73—42no—
232C22×C73—81no—
233C22×C73—81no—
234C22×C73C22×D811yescharacteristic
235C2×D73—21no—
236C2×D73—21no—
237C2×D73—42no—
238C2×D73—42no—
239C2×D73—88no—
240C2×D73—81no—
241C2×D73—81no—
242C2×D73—81no—
243C2×D73—81no—
244C2×D73C4×D812yes—
245C4×C73—21no—
246C4×C73—21no—
247C4×C73—42no—
248C4×C73—81no—
249C4×C73—81no—
250C4×C73C2×C4×D411yescharacteristic
251C4×C73C2×#13511yescharacteristic
252Dic73—21no—
253Dic73—81no—
254Dic73—81no—
255Dic73C4×D812yes—
256Dic73#50912yescomplemented
257C4×#509—731nomaximal, Sylow
258C2×C73—82no—
259C2×C73C2×C4×D811yescharacteristic
260C2×C73C2×#50911yescharacteristic
261C2×C73#341411yescharacteristic
262D73—22no—
263D73—82no—
264D73—82no—
265C2×#509—731no—
266C4×#135—731no—
267C2×C4×D8—731no—
268#2177—731no—
269#2177—731no—
270#2175—731no—
271#2175D7311yescharacteristic, complemented
272C4×SD32—731no—
273C4×SD32—731no—
274C4×D16—731no—
275C4×D16—731no—
276#2159—731no—
277#2140—731no—
278#2139—731no—
279C4×M32—731no—
280C73C4×#50911yescharacteristic, complemented, Sylow
281C2×#135—731no—
282C22×D8—731no—
283C4×#41—1461no—
284C2×C4×D4—1461no—
285#509—738no—
286C2×SD32—731no—
287C2×SD32C2×D7311yescharacteristic
288C2×D16—731no—
289C2×D16—731no—
290C2×M32—731no—
291C2×M32—731no—
292C4×Q16—731no—
293C4×Q16C2×D7311yescharacteristic, complemented
294C4×SD16—1461no—
295C4×SD16—1461no—
296C4×D8—731no—
297C4×D8—731no—
298C4×D8—731no—
299C4×D8—731no—
300C4×D8—731no—
301C4×D8—1461no—
302C4×D8—1461no—
303C4×D8C2×D7311yescharacteristic
304#427—731no—
305C2×#107—731no—
306#416—1461no—
307C2×#102—1461no—
308C2×C4×C8—731no—
309#367—731no—
310#367—731no—
311#366—731no—
312#366C2×D7311yescharacteristic, complemented
313#358—731no—
314#358C2×D7311yescharacteristic, complemented
315#357—731no—
316#357—731no—
317#357—731no—
318#357—731no—
319#357—731no—
320#357C2×D7311yescharacteristic
321#346—731no—
322#346C2×D7311yescharacteristic, complemented
323C4×C16—731no—
324C4×C16—731no—
325C2×#41—1461no—
326C22×D4—1461no—
327C23×C4—2921no—
328#135—734no—
329#135—1462no—
330C2×Q16C22×D7311yescharacteristic
331C2×SD16—1461no—
332C2×D8—734no—
333C2×D8—731no—
334C2×D8—731no—
335C2×D8—1462no—
336C2×D8—1461no—
337C2×D8C22×D7311yescharacteristic
338C22×C8—731no—
339C22×C8—731no—
340C4×Q8—21no—
341C4×Q8—1461no—
342C4×D4—21no—
343C4×D4—2921no—
344C4×D4—2921no—
345C4×D4—2921no—
346C4×D4—2921no—
347C4×D4—1461no—
348C4×D4—1461no—
349C4×D4—1461no—
350C4×D4—1461no—
351C4×D4—1461no—
352#117—2921no—
353#117—1461no—
354C2×C4⋊C4—1461no—
355C2×C22⋊C4—2921no—
356C2×C42—2921no—
357C2×C42—731no—
358SD32—734no—
359SD32C4×D7314yescomplemented
360D16—734no—
361D16—734no—
362M32—734no—
363M32—1462no—
364C2×C16—731no—
365C2×C16—731no—
366C2×C16C22×D7311yescharacteristic
367C2×C16C22×D7311yescharacteristic
368#107—731no—
369#107—731no—
370#107—1461no—
371#107C22×D7311yescharacteristic
372#107C22×D7311yescharacteristic
373#106—1461no—
374#103—21no—
375#103—1461no—
376#102—21no—
377#102—1461no—
378#102—1461no—
379#102—1461no—
380#102—1461no—
381#102—1461no—
382C4×C8—731no—
383C4×C8—731no—
384C4×C8—731no—
385C4×C8C22×D7311yescharacteristic
386C24—2921no—
387#41—2922no—
388#41—1464no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—2921no—
392C2×D4—2922no—
393C2×D4—2921no—
394C2×D4—1464no—
395C2×D4—1461no—
396C2×D4—1461no—
397C22×C4—41no—
398C22×C4—2921no—
399C22×C4—2921no—
400C22×C4—2921no—
401C22×C4—2921no—
402C22×C4—2921no—
403C22×C4—2921no—
404C22×C4—2921no—
405C22×C4—5844no—
406C22×C4—731no—
407C22×C4—731no—
408C22×C4—731no—
409Q16—21no—
410Q16C2×C4×D7312yescomplemented
411SD16—1464no—
412D8—21no—
413D8—732no—
414D8—732no—
415D8—1464no—
416D8—1461no—
417D8—1461no—
418D8C2×C4×D7312yes—
419C2×C8—731no—
420C2×C8—732no—
421C2×C8—732no—
422C2×C8—731no—
423C2×C8—1462no—
424C2×C8D4×D7311yescharacteristic
425C2×C8C23×D7311yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—2921no—
430C4⋊C4—2921no—
431C4⋊C4—1461no—
432C4⋊C4—1461no—
433C22⋊C4—41no—
434C22⋊C4—2921no—
435C22⋊C4—2921no—
436C22⋊C4—2921no—
437C22⋊C4—2921no—
438C22⋊C4—2921no—
439C22⋊C4—2921no—
440C42—41no—
441C42—2921no—
442C42—2921no—
443C42—731no—
444C42—731no—
445C42—731no—
446C42D4×D7311yescharacteristic
447C16—21no—
448C16—732no—
449C16—1461no—
450C16C2×C4×D7312yescomplemented
451C23—41no—
452C23—2924no—
453C23—2921no—
454C23—2921no—
455C23—5842no—
456C23—5842no—
457C23—731no—
458Q8—22no—
459Q8—41no—
460D4—22no—
461D4—41no—
462D4—2924no—
463D4—2924no—
464D4—2921no—
465D4—2921no—
466D4—1462no—
467D4—1462no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—2921no—
473C2×C4—2924no—
474C2×C4—2921no—
475C2×C4—2921no—
476C2×C4—5842no—
477C2×C4—5842no—
478C2×C4—5842no—
479C2×C4—5842no—
480C2×C4—5842no—
481C2×C4—5842no—
482C2×C4—731no—
483C2×C4—731no—
484C2×C4—732no—
485C2×C4—732no—
486C2×C4—731no—
487C2×C4—1462no—
488C2×C4D8×D7311yescharacteristic
489C2×C4D8×D7311yescharacteristic
490C2×C4C2×D4×D7311yescharacteristic
491C8—21no—
492C8—732no—
493C8—1461no—
494C8C22×C4×D7311yescharacteristic
495C8#41×D7311yescharacteristic
496C22—42no—
497C22—81no—
498C22—81no—
499C22—2922no—
500C22—2922no—
501C22—5848no—
502C22—5841no—
503C22—5841no—
504C22—5841no—
505C22—5841no—
506C22—732no—
507C22—1461no—
508C22—1461no—
509C22C2×D8×D7311yescharacteristic
510C4—21no—
511C4—21no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—5841no—
516C4—5841no—
517C4—732no—
518C4—732no—
519C4—1461no—
520C4C4×D4×D7311yescharacteristic
521C4#135×D7311yescharacteristic
522C2—82no—
523C2—5842no—
524C2—5842no—
525C2—1462no—
526C2C4×D8×D7311yescharacteristic
527C2#509×D7311yescharacteristic
528C2#2262306611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2175C2
socleC22×C73C22×D8
prime radicalC2×#17989024C2
cosocleC2×C8×C73C24
hypercenter#2175D73
hypocenterC73C4×#509
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. C4×#14556614C2semidirect
3. C4×D8×D73C2semidirect
4. #22611808C2semidirect
5. #22611805C2semidirect
6. #22611793C2semidirect
7. SD32×Dic73C2semidirect
8. #22611780C2semidirect
9. #22611777C2semidirect
10. #22611775C2semidirect
11. #22611716C2semidirect
12. #22611645C2semidirect
13. #22611642C2semidirect
14. #22611524C2semidirect
15. C73×#2175C2semidirect
20. #17989024C4semidirect
26. Q16×Dic73C22semidirect
30. C4×D584C22semidirect
33. #17988669C22semidirect
41. #17988415C22semidirect
42. #17988414C22semidirect
43. #17988412C22semidirect
44. #17988391C22semidirect
46. #17988389C22semidirect
49. #17988387C22semidirect
50. #17988363C22semidirect
51. C16×Dic73C22semidirect
52. C16×Dic73C22semidirect
54. C4×Q16×C73C22semidirect
56. #366×C73C22semidirect
57. #358×C73C22semidirect
59. #346×C73C22semidirect
64. #14556614C2×C4semidirect
70. D8×D73C2×C4semidirect
89. #14556534C2×C4semidirect
90. D1168C2×C4semidirect
91. #14556532C2×C4semidirect
93. #14556530C2×C4semidirect
113. SD32×C73C2×C4semidirect
145. D584C22×C4semidirect
168. #11924003C22×C4semidirect
173. Q16×C73C22×C4semidirect
185. C16×C73C22×C4semidirect
256. Dic73#509semidirect
271. #2175D73semidirect
280. C73C4×#509semidirect
293. C4×Q16C2×D73semidirect
312. #366C2×D73semidirect
314. #358C2×D73semidirect
322. #346C2×D73semidirect
359. SD32C4×D73semidirect
410. Q16C2×C4×D73semidirect
450. C16C2×C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890242
C4×#145566142
C4×D8×D732
#226118082
#226118052
#226117932
SD32×Dic732
#226117802
#226117772
#226117752
#226117162
#226116452
#226116422
#226115242
C73×#21752
C4×#50973