Finite Group Databasebeta

← #46848383

Subgroups — #46848383

Order 34688.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774582C211yesmaximal, characteristic, complemented
2#31774531C211yesmaximal, characteristic
3#31774260C211yesmaximal, characteristic, complemented
4#31774224C211yesmaximal, characteristic, complemented
5#31774200C211yesmaximal, characteristic, complemented
6#31774144C211yesmaximal, characteristic, complemented
7#31774124C211yesmaximal, characteristic, complemented
8#31774096C211yesmaximal, characteristic, complemented
9#31774050C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31773996C211yesmaximal, characteristic, complemented
12#31773965C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14C2×#22576080C211yesmaximal, characteristic, complemented
15#480×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C2×D4×D271—21no—
18C2×D4×D271C2211yescharacteristic
19C2×#16864643C2211yescharacteristic
20C2×#16864643C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic, complemented
22C2×#16864637C2211yescharacteristic, complemented
23C22×C271⋊C8C2211yescharacteristic, complemented
24#22576156C2211yescharacteristic
25#22576155—21no—
26#22576151C2211yescharacteristic, complemented
27#22576150—21no—
28#22576143—21no—
29#22576140C2211yescharacteristic, complemented
30#22576139C2211yescharacteristic, complemented
31#22576135C2211yescharacteristic
32#22576133—21no—
33#22576130—21no—
34#22576128—21no—
35#22576125C2211yescharacteristic, complemented
36#22576122C2211yescharacteristic
37C4×C22⋊D271C2211yescharacteristic
38#22576119—21no—
39#22576118C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic, complemented
45#22576081C2211yescharacteristic, complemented
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51#22576070C2211yescharacteristic, complemented
52#121×C271C2211yescharacteristic
53C2×C4⋊C4×C271C2211yescharacteristic, complemented
54#107×C271C2211yescharacteristic, complemented
55#107×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C23×D271—21no—
61C2×Dic542C2311yescharacteristic
62C2×D1084—22no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—41no—
66C2×D1084C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271D411yescharacteristic
72D4×D271—22no—
73D4×D271—41no—
74D4×D271—44no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77#16864643—22no—
78#16864643—41no—
79#16864642—41no—
80#16864637—41no—
81C2×C271⋊C8—41no—
82C2×C271⋊C8C2311yescharacteristic
83C2×C271⋊C8C2311yescharacteristic
84C4⋊Dic271—21no—
85C4⋊Dic271—21no—
86C4⋊Dic271C2311yescharacteristic
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4—21no—
90Dic271⋊C4D411yescharacteristic
91C22⋊Dic271—21no—
92C22⋊Dic271D411yescharacteristic
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4D411yescharacteristic
98C4×Dic271—21no—
99C4×Dic271—21no—
100C4×Dic271C2311yescharacteristic
101C2×D4×C271—21no—
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271—21no—
108C4⋊C4×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C4⋊C4×C271C2311yescharacteristic
111C22⋊C4×C271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—42no—
115C22×D271—42no—
116C22×D271—42no—
117C22×D271—42no—
118C22×D271—81no—
119C22×D271—81no—
120C22×D271C2×D411yescharacteristic
121Dic542—21no—
122Dic542—21no—
123D1084—21no—
124D1084—21no—
125D1084—21no—
126D1084—42no—
127D1084—42no—
128D1084#4112yes—
129C22⋊D271—22no—
130C22⋊D271—42no—
131C22⋊D271—42no—
132C22⋊D271—81no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271C2×D411yescharacteristic
136C2×Dic271C2×D411yescharacteristic
137C2×Dic271C2×D411yescharacteristic
138C4×D271—21no—
139C4×D271—21no—
140C4×D271—42no—
141C271⋊C8—21no—
142C271⋊C8—21no—
143C23×C271—21no—
144C23×C271C2×D411yescharacteristic
145D4×C271—21no—
146D4×C271—21no—
147D4×C271—42no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—42no—
154C2×C4×C271C2×D411yescharacteristic
155C2×C4×C271C2411yescharacteristic
156C8×C271—21no—
157C8×C271—21no—
158C22×C271—21no—
159C22×C271—21no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271AGL(1,ℤ/8)12yes—
164C22×C271C22×D411yescharacteristic
165C2×D271—21no—
166C2×D271—22no—
167C2×D271—21no—
168C2×D271—21no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—41no—
174C2×D271—84no—
175C2×D271—84no—
176C4×C271—21no—
177C4×C271—41no—
178C4×C271—41no—
179C4×C271—41no—
180C4×C271C2×#4111yescharacteristic
181C4×C271H(5,𝔽2)11yescharacteristic
182Dic271—21no—
183Dic271—21no—
184Dic271—21no—
185Dic271—41no—
186Dic271—41no—
187C2×C271—22no—
188C2×C271—81no—
189C2×C271#54611yescharacteristic
190C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C271#57711yescharacteristic
192D271—41no—
193D271—42no—
194D271—81no—
195C271#331011yescharacteristic, complemented, Sylow
196#3310—2711nomaximal, Sylow
197#552—2711no—
198D42—2711no—
199#490—2711no—
200#482—2711no—
201#480D27111yescharacteristic, complemented
202#478—2711no—
203#468—2711no—
204#465—2711no—
205#460—2711no—
206#450—2711no—
207#450—2711no—
208#440—2711no—
209#435—2711no—
210#418—2711no—
211C2×#102—2711no—
212C22×D4—2711no—
213C22×D4—2711no—
214C22×D4—5421no—
215C2×SD16—2711no—
216C2×SD16—2711no—
217C2×SD16—2711no—
218C2×M16—2711no—
219C22×C8—2711no—
220#127—2711no—
221#126—5421no—
222#125—2711no—
223#125—5421no—
224#123—5421no—
225C22⋊Q8—2711no—
226C22⋊Q8—2711no—
227#121—2711no—
228#121—5421no—
229#121C2×D27111yescharacteristic
230#120—5421no—
231#120—5421no—
232C4×Q8—2711no—
233C4×D4—2711no—
234C4×D4—2711no—
235#117—2711no—
236#117—5421no—
237C2×C4⋊C4C2×D27111yescharacteristic, complemented
238#107C2×D27111yescharacteristic, complemented
239#107C2×D27111yescharacteristic, complemented
240#106—2711no—
241C4⋊C8—2711no—
242#103—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102—2711no—
250#102C2×D27111yescharacteristic, complemented
251#102C2×D27111yescharacteristic, complemented
252C22⋊C8—2711no—
253C22⋊C8C2×D27111yescharacteristic, complemented
254#97—2711no—
255C24—5421no—
256C24—5421no—
257C2×Q8—2711no—
258C2×D4—21no—
259C2×D4—2711no—
260C2×D4—2711no—
261C2×D4—2711no—
262C2×D4—5422no—
263C2×D4—5421no—
264C2×D4—5421no—
265C2×D4—5421no—
266C2×D4—5421no—
267C2×D4—5422no—
268C2×D4—5421no—
269C2×D4—10841no—
270C2×D4—10841no—
271C2×D4—10844no—
272C2×D4C22×D27111yescharacteristic
273C22×C4—21no—
274C22×C4—2711no—
275C22×C4—5421no—
276C22×C4C22×D27111yescharacteristic
277SD16—5422no—
278SD16—10841no—
279SD16—10841no—
280M16—10841no—
281C2×C8—2711no—
282C2×C8—2711no—
283C2×C8—10841no—
284C2×C8C22×D27111yescharacteristic
285C2×C8C22×D27111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—2711no—
288C4⋊C4—2711no—
289C4⋊C4—5421no—
290C4⋊C4—5421no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4C22×D27111yescharacteristic
295C4⋊C4C22×D27111yescharacteristic
296C4⋊C4C22×D27111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—2711no—
299C22⋊C4—2711no—
300C22⋊C4—5421no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C42—2711no—
306C42—5421no—
307C42—5421no—
308C23—21no—
309C23—2711no—
310C23—5421no—
311C23—5421no—
312C23—10842no—
313C23—10842no—
314C23—10842no—
315C23—10842no—
316C23—21681no—
317C23—21681no—
318C23D4×D27111yescharacteristic
319Q8—5421no—
320Q8—5421no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—2712no—
325D4—5422no—
326D4—5421no—
327D4—5421no—
328D4—5421no—
329D4—10842no—
330D4—10842no—
331D4—10842no—
332D4—10842no—
333D4—21681no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—2711no—
341C2×C4—2711no—
342C2×C4—2711no—
343C2×C4—5421no—
344C2×C4—5421no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—10842no—
348C2×C4D4×D27111yescharacteristic
349C2×C4C23×D27111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5421no—
353C8—5421no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—5421no—
360C22—5422no—
361C22—5421no—
362C22—5421no—
363C22—10841no—
364C22—10841no—
365C22—10841no—
366C22—10841no—
367C22—10841no—
368C22—21684no—
369C22—21684no—
370C22#2257618412yes—
371C22C2×D4×D27111yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—5421no—
377C4—5421no—
378C4—5421no—
379C4—10841no—
380C4—10841no—
381C4#41×D27111yescharacteristic
382C4#2257620611yescharacteristic
383C2—22no—
384C2—81no—
385C2—10841no—
386C2—10842no—
387C2—21681no—
388C2#3177453911yescharacteristic
389C2C2×#2257618411yescharacteristic
390C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#480×C271C2
socleC22×C271C22×D4
prime radical#31774531C2
hypercenter#480D271
hypocenterC271#3310
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #31774582C2semidirect
3. #31774260C2semidirect
4. #31774224C2semidirect
5. #31774200C2semidirect
6. #31774144C2semidirect
7. #31774124C2semidirect
8. #31774096C2semidirect
9. #31774050C2semidirect
10. #31774049C2semidirect
11. #31773996C2semidirect
12. #31773965C2semidirect
13. #31773900C2semidirect
14. C2×#22576080C2semidirect
15. #480×C271C2semidirect
20. C2×#16864643C22semidirect
21. C2×#16864642C22semidirect
22. C2×#16864637C22semidirect
23. C22×C271⋊C8C22semidirect
26. #22576151C22semidirect
29. #22576140C22semidirect
30. #22576139C22semidirect
35. #22576125C22semidirect
39. #22576118C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576083C22semidirect
44. #22576082C22semidirect
45. #22576081C22semidirect
48. #22576080C22semidirect
49. #22576080C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. #22576070C22semidirect
53. C2×C4⋊C4×C271C22semidirect
54. #107×C271C22semidirect
55. #107×C271C22semidirect
56. #102×C271C22semidirect
57. #102×C271C22semidirect
58. C22⋊C8×C271C22semidirect
195. C271#3310semidirect
201. #480D271semidirect
237. C2×C4⋊C4C2×D271semidirect
238. #107C2×D271semidirect
239. #107C2×D271semidirect
250. #102C2×D271semidirect
251. #102C2×D271semidirect
253. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745822
#317745312
#317742602
#317742242
#317742002
#317741442
#317741242
#317740962
#317740502
#317740492
#317739962
#317739652
#317739002
C2×#225760802
#480×C2712
#3310271