Finite Group Databasebeta

← #46848392

Subgroups — #46848392

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#31774588C211yesmaximal, characteristic, complemented
2#31774532C211yesmaximal, characteristic
3#31774261C211yesmaximal, characteristic, complemented
4#31774220C211yesmaximal, characteristic, complemented
5#31774209C211yesmaximal, characteristic, complemented
6#31774198C211yesmaximal, characteristic, complemented
7#31774140C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774046C211yesmaximal, characteristic, complemented
10#31774046C211yesmaximal, characteristic, complemented
11#31773995C211yesmaximal, characteristic, complemented
12#31773967C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#460×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271—21no—
17C2×D4×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×#16864637C2211yescharacteristic, complemented
21C2×C8×D271C2211yescharacteristic, complemented
22#22576154—21no—
23#22576152C2211yescharacteristic, complemented
24#22576150—21no—
25#22576145—21no—
26#22576136C2211yescharacteristic, complemented
27#22576136C2211yescharacteristic, complemented
28#22576131—21no—
29#22576131C2211yescharacteristic
30#22576131C2211yescharacteristic
31#22576128—21no—
32#22576128—21no—
33C4×D1084C2211yescharacteristic
34C4×D1084C2211yescharacteristic
35#22576118C2211yescharacteristic, complemented
36#22576117—21no—
37C4⋊C4×D271C2211yescharacteristic, complemented
38#22576094C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576082C2211yescharacteristic
43#22576082C2211yescharacteristic
44#22576081C2211yescharacteristic, complemented
45#22576081C2211yescharacteristic, complemented
46#22576081C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51#22576071C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic, complemented
53C2×SD16×C271C2211yescharacteristic, complemented
54#127×C271C2211yescharacteristic
55C4×Q8×C271C2211yescharacteristic, complemented
56C4⋊C8×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C23×D271—21no—
61C2×D1084D411yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67D4×D271—22no—
68D4×D271—22no—
69D4×D271—41no—
70D4×D271—41no—
71D4×D271—44no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271C2311yescharacteristic
75C2×C4×D271C2311yescharacteristic
76#16864641—22no—
77#16864637—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—41no—
81C4⋊Dic271D411yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97(C2×D271)⋊C4D411yescharacteristic
98C4×Dic271—21no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271—21no—
102C2×D4×C271C2311yescharacteristic
103C2×D4×C271C2311yescharacteristic
104SD16×C271—41no—
105SD16×C271—41no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C42×C271—21no—
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—42no—
116C22×D271—42no—
117C22×D271—81no—
118C22×D271—81no—
119C22×D271C2×D411yescharacteristic
120C22×D271C2×D411yescharacteristic
121D1084—22no—
122D1084—21no—
123D1084#4112yes—
124C22⋊D271—42no—
125C22⋊D271—42no—
126C22⋊D271—42no—
127C22⋊D271—42no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—21no—
136C4×D271—21no—
137C4×D271—42no—
138C4×D271—42no—
139C271⋊C8—21no—
140C271⋊C8—21no—
141C23×C271—21no—
142C23×C271—21no—
143Q8×C271—21no—
144Q8×C271—21no—
145D4×C271—21no—
146D4×C271—21no—
147D4×C271—21no—
148D4×C271—21no—
149D4×C271—42no—
150D4×C271—81no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271C2×D411yescharacteristic
154C2×C4×C271C2×D411yescharacteristic
155C2×C4×C271C2411yescharacteristic
156C8×C271—21no—
157C8×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271C22×D411yescharacteristic
165C2×D271—21no—
166C2×D271—21no—
167C2×D271—21no—
168C2×D271—22no—
169C2×D271—21no—
170C2×D271—21no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—84no—
174C2×D271—84no—
175C2×D271AGL(1,ℤ/8)12yes—
176C4×C271—21no—
177C4×C271—21no—
178C4×C271—41no—
179C4×C271—41no—
180C4×C271C2×#4111yescharacteristic
181C4×C271H(5,𝔽2)11yescharacteristic
182Dic271—21no—
183Dic271—21no—
184Dic271—41no—
185Dic271—41no—
186Dic271—41no—
187C2×C271—81no—
188C2×C271—81no—
189C2×C271#54611yescharacteristic
190C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C271#57711yescharacteristic
192D271—22no—
193D271—41no—
194D271—42no—
195C271#331011yescharacteristic, complemented, Sylow
196#3310—2711nomaximal, Sylow
197#552—2711no—
198D42—2711no—
199#490—2711no—
200#482—2711no—
201#480—2711no—
202#478—2711no—
203#468—2711no—
204#465—2711no—
205#460D27111yescharacteristic, complemented
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×SD16C2×D27111yescharacteristic, complemented
217C2×SD16C2×D27111yescharacteristic, complemented
218C2×M16—2711no—
219C22×C8—2711no—
220#127C2×D27111yescharacteristic
221#126—5421no—
222#125—2711no—
223#125—5421no—
224#123—5421no—
225C22⋊Q8—2711no—
226C22⋊Q8—2711no—
227#121—2711no—
228#121—2711no—
229#121—5421no—
230#120—5421no—
231#120—5421no—
232C4×Q8C2×D27111yescharacteristic, complemented
233C4×D4—2711no—
234C4×D4—2711no—
235#117—2711no—
236#117—5421no—
237C2×C4⋊C4—2711no—
238#107—2711no—
239#107—2711no—
240#106—2711no—
241C4⋊C8C2×D27111yescharacteristic, complemented
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⋊C8—2711no—
254#97—2711no—
255C24—5421no—
256C24—5421no—
257C2×Q8C22×D27111yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—2711no—
261C2×D4—2711no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4—5422no—
265C2×D4—5422no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4—10841no—
269C2×D4—10841no—
270C2×D4—10844no—
271C2×D4C22×D27111yescharacteristic
272C2×D4C22×D27111yescharacteristic
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—5421no—
276C22×C4—5421no—
277SD16—41no—
278SD16—41no—
279SD16—5422no—
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—2711no—
290C4⋊C4—2711no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4—5421no—
295C4⋊C4—5421no—
296C4⋊C4C22×D27111yescharacteristic
297C22⋊C4—2711no—
298C22⋊C4—2711no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C42—21no—
306C42—5421no—
307C42C22×D27111yescharacteristic
308C23—21no—
309C23—21no—
310C23—2711no—
311C23—2711no—
312C23—5421no—
313C23—10842no—
314C23—10842no—
315C23—10842no—
316C23—10842no—
317C23—21681no—
318C23—21681no—
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—2712no—
328D4—5422no—
329D4—5421no—
330D4—10842no—
331D4—10842no—
332D4—10842no—
333D4—10842no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—2711no—
337C2×C4—2711no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—5421no—
344C2×C4—5421no—
345C2×C4—10842no—
346C2×C4—10842no—
347C2×C4D4×D27111yescharacteristic
348C2×C4D4×D27111yescharacteristic
349C2×C4C23×D27111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5421no—
353C8—5421no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2712no—
361C22—5421no—
362C22—5421no—
363C22—5421no—
364C22—5422no—
365C22—5421no—
366C22—5421no—
367C22—10841no—
368C22—10841no—
369C22—21684no—
370C22—21684no—
371C22C2×D4×D27111yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—5421no—
377C4—5421no—
378C4—10841no—
379C4—10841no—
380C4—10841no—
381C4#41×D27111yescharacteristic
382C4#2257620511yescharacteristic
383C2—81no—
384C2—81no—
385C2—5422no—
386C2—10841no—
387C2—10842no—
388C2#3177454411yescharacteristic
389C2AGL(1,ℤ/8)×D27111yescharacteristic
390C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#460×C271C2
socleC22×C271C22×D4
prime radical#31774532C2
hypercenter#460D271
hypocenterC271#3310
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #31774588C2semidirect
3. #31774261C2semidirect
4. #31774220C2semidirect
5. #31774209C2semidirect
6. #31774198C2semidirect
7. #31774140C2semidirect
8. #31774123C2semidirect
9. #31774046C2semidirect
10. #31774046C2semidirect
11. #31773995C2semidirect
12. #31773967C2semidirect
13. #31773900C2semidirect
14. #102×D271C2semidirect
15. #460×C271C2semidirect
20. C2×#16864637C22semidirect
21. C2×C8×D271C22semidirect
23. #22576152C22semidirect
26. #22576136C22semidirect
27. #22576136C22semidirect
35. #22576118C22semidirect
37. C4⋊C4×D271C22semidirect
38. #22576094C22semidirect
39. #22576093C22semidirect
40. #22576092C22semidirect
41. #22576084C22semidirect
44. #22576081C22semidirect
45. #22576081C22semidirect
46. #22576081C22semidirect
47. #22576081C22semidirect
49. (C2×D271)⋊C8C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. #22576071C22semidirect
52. C2×SD16×C271C22semidirect
53. C2×SD16×C271C22semidirect
55. C4×Q8×C271C22semidirect
56. C4⋊C8×C271C22semidirect
57. #102×C271C22semidirect
58. #102×C271C22semidirect
195. C271#3310semidirect
205. #460D271semidirect
216. C2×SD16C2×D271semidirect
217. C2×SD16C2×D271semidirect
232. C4×Q8C2×D271semidirect
241. C4⋊C8C2×D271semidirect
250. #102C2×D271semidirect
251. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745882
#317745322
#317742612
#317742202
#317742092
#317741982
#317741402
#317741232
#317740462
#317739952
#317739672
#317739002
#102×D2712
#460×C2712
#3310271