Finite Group Databasebeta

← #46848057

Subgroups — #46848057

Order 34688.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16864643C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3#31774560C211yesmaximal, characteristic, complemented
4#31774531C211yesmaximal, characteristic
5#31774268C211yesmaximal, characteristic, complemented
6#31774127C211yesmaximal, characteristic, complemented
7#31774101C211yesmaximal, characteristic, complemented
8#31774096C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774047C211yesmaximal, characteristic, complemented
11#31774037C211yesmaximal, characteristic, complemented
12#31774035C211yesmaximal, characteristic, complemented
13#31773984C211yesmaximal, characteristic, complemented
14C8×C22⋊D271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C22×Dic542C2211yescharacteristic, complemented
17C2×Q8×D271C2211yescharacteristic, complemented
18C22×D1084C2211yescharacteristic
19C2×D4×D271—21no—
20C2×D4×D271C2211yescharacteristic
21C2×#16864643—24no—
22C2×#16864643C2211yescharacteristic
23C2×#16864643C2211yescharacteristic
24C2×#16864643C2211yescharacteristic, complemented
25C2×#16864643C2211yescharacteristic, complemented
26C2×#16864642C2211yescharacteristic, complemented
27C2×#16864642C2211yescharacteristic, complemented
28C2×#16864641C2211yescharacteristic
29C2×#16864641C2211yescharacteristic, complemented
30SD16×D271—24no—
31C2×C8×D271C2211yescharacteristic, complemented
32#22576159—21no—
33#22576158C2211yescharacteristic, complemented
34#22576156C2211yescharacteristic
35#22576140C2211yescharacteristic, complemented
36#22576138—21no—
37#22576137—21no—
38#22576135C2211yescharacteristic
39#22576133—21no—
40#22576130—21no—
41#22576128—21no—
42#22576125C2211yescharacteristic, complemented
43#22576122C2211yescharacteristic
44C4×C22⋊D271C2211yescharacteristic
45(C22⋊D271)⋊C4—21no—
46Dic271⋊C8C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic
50#22576080C2211yescharacteristic, complemented
51C22⋊(C271⋊C8)C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53C8×Dic271C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C22×C8×C271C2211yescharacteristic, complemented
56C22⋊Q8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#106×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63C2×Dic542—21no—
64C2×Dic542—22no—
65C2×Dic542—21no—
66C2×Dic542—41no—
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69Q8×D271—22no—
70Q8×D271—41no—
71C2×D1084—22no—
72C2×D1084—21no—
73C2×D1084—21no—
74C2×D1084—41no—
75C2×D1084C2311yescharacteristic
76C2×D1084C2311yescharacteristic
77C2×C22⋊D271—21no—
78C2×C22⋊D271—21no—
79C2×C22⋊D271—21no—
80C2×C22⋊D271D411yescharacteristic
81D4×D271—22no—
82D4×D271—41no—
83D4×D271—44no—
84C22×Dic271—21no—
85C2×C4×D271—21no—
86C2×C4×D271—21no—
87C2×C4×D271C2311yescharacteristic
88#16864643—22no—
89#16864643—22no—
90#16864643—22no—
91#16864643D414yescomplemented
92#16864642—22no—
93#16864642—41no—
94#16864641—22no—
95#16864641—41no—
96C2×C271⋊C8C2311yescharacteristic
97C2×C271⋊C8C2311yescharacteristic
98C8×D271—22no—
99C4⋊Dic271—21no—
100Dic271⋊C4—21no—
101Dic271⋊C4—21no—
102Dic271⋊C4—21no—
103Dic271⋊C4D411yescharacteristic
104C22⋊Dic271D411yescharacteristic
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4D411yescharacteristic
109C4×Dic271—21no—
110C4×Dic271C2311yescharacteristic
111C2×Q8×C271C2311yescharacteristic
112C2×D4×C271—21no—
113C2×D4×C271C2311yescharacteristic
114C22×C4×C271C2311yescharacteristic
115SD16×C271—22no—
116C2×C8×C271—22no—
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C4⋊C4×C271—21no—
120C4⋊C4×C271C2311yescharacteristic
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22⋊C4×C271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—42no—
130C22×D271—81no—
131C22×D271—81no—
132C22×D271C2×D411yescharacteristic
133Dic542—21no—
134Dic542—21no—
135Dic542—21no—
136Dic542—42no—
137Dic542—42no—
138Dic542C2×D412yescomplemented
139D1084—21no—
140D1084—21no—
141D1084—21no—
142D1084—42no—
143D1084—42no—
144D1084C2×D412yes—
145C22⋊D271—42no—
146C22⋊D271—42no—
147C22⋊D271—81no—
148C22⋊D271SD1614yescomplemented
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—42no—
152C2×Dic271C2×D411yescharacteristic
153C2×Dic271C2×D411yescharacteristic
154C2×Dic271C2×D411yescharacteristic
155C4×D271—21no—
156C4×D271—21no—
157C4×D271—42no—
158C4×D271—42no—
159C271⋊C8—21no—
160C271⋊C8—21no—
161C23×C271—21no—
162C23×C271C2×D411yescharacteristic
163Q8×C271—21no—
164Q8×C271—21no—
165D4×C271—21no—
166D4×C271—21no—
167D4×C271—42no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2411yescharacteristic
175C8×C271—21no—
176C8×C271C2×D412yescomplemented
177C22×C271—21no—
178C22×C271—21no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271C2×SD1612yescomplemented
183C22×C271C22×D411yescharacteristic
184C2×D271—21no—
185C2×D271—21no—
186C2×D271—22no—
187C2×D271—41no—
188C2×D271—41no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—84no—
193C2×D271—84no—
194C2×D271C2×SD1612yescomplemented
195C4×C271—21no—
196C4×C271—41no—
197C4×C271—41no—
198C4×C271—41no—
199C4×C271C22×D411yescharacteristic
200C4×C271H(5,𝔽2)11yescharacteristic
201Dic271—21no—
202Dic271—22no—
203Dic271—21no—
204Dic271—41no—
205Dic271C2×SD1612yescomplemented
206C2×C271—22no—
207C2×C271—81no—
208C2×C271D4211yescharacteristic
209C2×C271C22×SD1611yescharacteristic
210C2×C271#57711yescharacteristic
211D271—22no—
212D271—42no—
213D271—81no—
214C271D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—2711nomaximal, Sylow
216C22×SD16—2711no—
217C22×SD16—2711no—
218D4×Q8—2711no—
219D42—2711no—
220#492—2711no—
221#465—2711no—
222#465D27111yescharacteristic, complemented
223#461—2711no—
224#460—2711no—
225#450—2711no—
226#450—2711no—
227#448—2711no—
228#448—2711no—
229C4×SD16—2711no—
230C8×D4—2711no—
231C22×Q8—2711no—
232C22×Q8—2711no—
233C22×D4—2711no—
234C22×D4—2711no—
235C22×D4—5421no—
236C2×SD16—2711no—
237C2×SD16—2711no—
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16—2711no—
242C2×SD16—2711no—
243C2×SD16—2711no—
244C2×SD16—5424no—
245C2×SD16—5424no—
246C2×SD16C2×D27111yescharacteristic, complemented
247C22×C8—2711no—
248C22×C8C2×D27111yescharacteristic, complemented
249C4⋊Q8—2711no—
250C4⋊Q8—5421no—
251#127—2711no—
252C22⋊Q8—2711no—
253C22⋊Q8—5421no—
254C22⋊Q8—5421no—
255C22⋊Q8C2×D27111yescharacteristic, complemented
256#121—2711no—
257#121—5421no—
258#121C2×D27111yescharacteristic
259#120—5421no—
260#120—5421no—
261C4×Q8—2711no—
262C4×D4—2711no—
263C4×D4—2711no—
264C4×D4—5421no—
265#106C2×D27111yescharacteristic, complemented
266C4⋊C8—2711no—
267#103—2711no—
268#103—2711no—
269#103C2×D27111yescharacteristic, complemented
270#102—2711no—
271#102—2711no—
272#102C2×D27111yescharacteristic, complemented
273C22⋊C8—2711no—
274C22⋊C8—2711no—
275C4×C8—2711no—
276C24—5421no—
277C24—5421no—
278C2×Q8—2711no—
279C2×Q8—2711no—
280C2×Q8—5421no—
281C2×Q8—5422no—
282C2×Q8—5421no—
283C2×Q8—5422no—
284C2×Q8—10841no—
285C2×Q8—10841no—
286C2×Q8C22×D27111yescharacteristic
287C2×D4—21no—
288C2×D4—2711no—
289C2×D4—2711no—
290C2×D4—2711no—
291C2×D4—5422no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5422no—
297C2×D4—5421no—
298C2×D4—10841no—
299C2×D4—10841no—
300C2×D4—10844no—
301C2×D4C22×D27111yescharacteristic
302C22×C4—2711no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4—5421no—
306C22×C4C22×D27111yescharacteristic
307SD16—22no—
308SD16—2714no—
309SD16—5422no—
310SD16—5422no—
311SD16—5422no—
312SD16—5422no—
313SD16—5422no—
314SD16—10841no—
315SD16—10841no—
316C2×C8—22no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8—5422no—
320C2×C8C22×D27111yescharacteristic
321C2×C8C22×D27111yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—2711no—
324C4⋊C4—5421no—
325C4⋊C4—5421no—
326C4⋊C4—5421no—
327C4⋊C4—5421no—
328C4⋊C4C22×D27111yescharacteristic
329C4⋊C4C22×D27111yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—2711no—
333C22⋊C4—2711no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C42—2711no—
338C42—5421no—
339C23—21no—
340C23—2711no—
341C23—5421no—
342C23—5421no—
343C23—10842no—
344C23—10842no—
345C23—10842no—
346C23—10842no—
347C23—21681no—
348C23—21681no—
349C23D4×D27111yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—2712no—
353Q8—5421no—
354Q8—5421no—
355Q8—5421no—
356Q8—10842no—
357Q8—10842no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—2714no—
362D4—2712no—
363D4—5421no—
364D4—5421no—
365D4—5421no—
366D4—10842no—
367D4—10842no—
368D4—10842no—
369D4—10842no—
370D4—21681no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—2711no—
377C2×C4—2711no—
378C2×C4—2711no—
379C2×C4—5421no—
380C2×C4—5421no—
381C2×C4—5421no—
382C2×C4—5421no—
383C2×C4—10842no—
384C2×C4—10842no—
385C2×C4—10842no—
386C2×C4D4×D27111yescharacteristic
387C2×C4C23×D27111yescharacteristic
388C8—21no—
389C8—5421no—
390C8—5421no—
391C8D4×D27112yescomplemented
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—2712no—
398C22—5421no—
399C22—5421no—
400C22—5422no—
401C22—10841no—
402C22—10841no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—21684no—
407C22—21684no—
408C22SD16×D27112yescomplemented
409C22C2×D4×D27111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—2712no—
415C4—5421no—
416C4—5422no—
417C4—5421no—
418C4—10841no—
419C4C2×D4×D27111yescharacteristic
420C4#2257620611yescharacteristic
421C2—22no—
422C2—81no—
423C2—5422no—
424C2—10842no—
425C2—21681no—
426C2#3177453111yescharacteristic
427C2C2×SD16×D27111yescharacteristic
428C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
prime radical#31774531C2
hypercenter#465D271
hypocenterC271D4×SD16
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C22×#16864643C2semidirect
2. C2×SD16×D271C2semidirect
3. #31774560C2semidirect
5. #31774268C2semidirect
6. #31774127C2semidirect
7. #31774101C2semidirect
8. #31774096C2semidirect
9. #31774049C2semidirect
10. #31774047C2semidirect
11. #31774037C2semidirect
12. #31774035C2semidirect
13. #31773984C2semidirect
14. C8×C22⋊D271C2semidirect
15. #465×C271C2semidirect
16. C22×Dic542C22semidirect
17. C2×Q8×D271C22semidirect
24. C2×#16864643C22semidirect
25. C2×#16864643C22semidirect
26. C2×#16864642C22semidirect
27. C2×#16864642C22semidirect
29. C2×#16864641C22semidirect
31. C2×C8×D271C22semidirect
33. #22576158C22semidirect
35. #22576140C22semidirect
42. #22576125C22semidirect
46. Dic271⋊C8C22semidirect
47. #22576083C22semidirect
48. #22576083C22semidirect
50. #22576080C22semidirect
51. C22⋊(C271⋊C8)C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. C8×Dic271C22semidirect
54. C2×SD16×C271C22semidirect
55. C22×C8×C271C22semidirect
56. C22⋊Q8×C271C22semidirect
58. #106×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
91. #16864643D4semidirect
138. Dic542C2×D4semidirect
148. C22⋊D271SD16semidirect
176. C8×C271C2×D4semidirect
182. C22×C271C2×SD16semidirect
194. C2×D271C2×SD16semidirect
205. Dic271C2×SD16semidirect
214. C271D4×SD16semidirect
222. #465D271semidirect
246. C2×SD16C2×D271semidirect
248. C22×C8C2×D271semidirect
255. C22⋊Q8C2×D271semidirect
265. #106C2×D271semidirect
269. #103C2×D271semidirect
272. #102C2×D271semidirect
391. C8D4×D271semidirect
408. C22SD16×D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646432
C2×SD16×D2712
#317745602
#317745312
#317742682
#317741272
#317741012
#317740962
#317740492
#317740472
#317740372
#317740352
#317739842
C8×C22⋊D2712
#465×C2712
D4×SD16271