Finite Group Databasebeta

← #46848200

Subgroups — #46848200

Order 34688.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774586C211yesmaximal, characteristic, complemented
2#31774535C211yesmaximal, characteristic
3#31774253C211yesmaximal, characteristic, complemented
4#31774226C211yesmaximal, characteristic, complemented
5#31774221C211yesmaximal, characteristic, complemented
6#31774199C211yesmaximal, characteristic, complemented
7#31774137C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774122C211yesmaximal, characteristic, complemented
10#31774059C211yesmaximal, characteristic, complemented
11#31774049C211yesmaximal, characteristic, complemented
12#31773988C211yesmaximal, characteristic, complemented
13C8×C22⋊D271C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#416×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C2×#16864644C2211yescharacteristic, complemented
18C2×#16864643C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576153—21no—
22#22576152—21no—
23#22576151C2211yescharacteristic, complemented
24#22576149C2211yescharacteristic
25#22576143—21no—
26#22576142—21no—
27#22576140C2211yescharacteristic
28#22576139C2211yescharacteristic, complemented
29#22576136C2211yescharacteristic
30#22576134C2211yescharacteristic
31#22576132—21no—
32#22576128—21no—
33#22576125C2211yescharacteristic, complemented
34D4×Dic271C2211yescharacteristic
35C4×C22⋊D271C2211yescharacteristic
36#22576117—21no—
37C4⋊C4×D271C2211yescharacteristic
38C2×(C2×D271)⋊C4—21no—
39#22576093C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41#22576091C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic, complemented
52C2×#41×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic, complemented
54#117×C271C2211yescharacteristic, complemented
55#103×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C23×D271—21no—
60C2×Dic542C2311yescharacteristic
61C2×D1084C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C2×C22⋊D271D411yescharacteristic
65D4×D271—22no—
66D4×D271—41no—
67C22×Dic271—21no—
68C2×C4×D271—21no—
69C2×C4×D271C2311yescharacteristic
70#16864644—41no—
71#16864643—41no—
72#16864642—22no—
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C8×D271—41no—
76C4⋊Dic271—21no—
77C4⋊Dic271—21no—
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271D411yescharacteristic
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104C2×C8×C271—41no—
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C42×C271—21no—
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116Dic542—21no—
117Dic542—21no—
118D1084—21no—
119D1084—21no—
120C22⋊D271—22no—
121C22⋊D271—42no—
122C22⋊D271—81no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—81no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136C23×C271C2×D411yescharacteristic
137Q8×C271—21no—
138Q8×C271—21no—
139D4×C271—21no—
140D4×C271—42no—
141D4×C271#4112yes—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—42no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—22no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—84no—
165C2×D271#13512yes—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271—41no—
170C4×C271C2×#4111yescharacteristic
171C4×C271H(5,𝔽2)11yescharacteristic
172Dic271—21no—
173Dic271—21no—
174Dic271—21no—
175Dic271—41no—
176Dic271—41no—
177Dic271—41no—
178C2×C271—42no—
179C2×C271—41no—
180C2×C271#54611yescharacteristic
181C2×C271C2×#13511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—22no—
184D271—81no—
185C271#329611yescharacteristic, complemented, Sylow
186#3296—2711nomaximal, Sylow
187#552—2711no—
188#546—2711no—
189#488—2711no—
190#483—2711no—
191#482—2711no—
192#478—2711no—
193#467—2711no—
194#465—2711no—
195#464—2711no—
196#452—2711no—
197#450—2711no—
198C4×SD16—2711no—
199C8×D4—2711no—
200#416D27111yescharacteristic, complemented
201C2×#102—2711no—
202C2×#41C2×D27111yescharacteristic
203C22×D4—2711no—
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C22×C8—2711no—
208C22×C8C2×D27111yescharacteristic, complemented
209#126—5421no—
210#125—2711no—
211#125—5421no—
212#124—2711no—
213#123—5421no—
214#123—5421no—
215C22⋊Q8—2711no—
216C22⋊Q8—2711no—
217C22⋊Q8—2711no—
218#121—2711no—
219#121—5421no—
220#120—5421no—
221C4×Q8—2711no—
222C4×D4—2711no—
223C4×D4—2711no—
224#117—5421no—
225#117C2×D27111yescharacteristic, complemented
226C2×C4⋊C4—2711no—
227C2×C22⋊C4—5421no—
228#107—2711no—
229#106—2711no—
230#106—2711no—
231C4⋊C8—2711no—
232#103—2711no—
233#103—2711no—
234#103C2×D27111yescharacteristic, complemented
235#103C2×D27111yescharacteristic, complemented
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102C2×D27111yescharacteristic
241#102C2×D27111yescharacteristic
242C22⋊C8—2711no—
243C22⋊C8—2711no—
244C4×C8—2711no—
245C24—5421no—
246#41—22no—
247#41—41no—
248C2×Q8—2711no—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—21no—
251C2×D4—2711no—
252C2×D4—2711no—
253C2×D4—5421no—
254C2×D4—5422no—
255C2×D4—5421no—
256C2×D4—10841no—
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4C22×D27111yescharacteristic
263Q16—10841no—
264SD16—5422no—
265SD16—10841no—
266C2×C8—41no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—10841no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4C22×D27111yescharacteristic
283C4⋊C4C22×D27111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2711no—
286C22⋊C4—2711no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C42—21no—
295C42—2711no—
296C42—5421no—
297C23—21no—
298C23—2711no—
299C23—5421no—
300C23—10842no—
301C23—10842no—
302C23—21681no—
303C23D4×D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5421no—
307Q8—5421no—
308D4—21no—
309D4—42no—
310D4—5422no—
311D4—5421no—
312D4—5421no—
313D4—10842no—
314D4—21681no—
315D4#1686465412yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—2711no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—10842no—
331C2×C4—21681no—
332C2×C4D4×D27111yescharacteristic
333C2×C4C23×D27111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5421no—
337C8—5421no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—2712no—
345C22—5421no—
346C22—5421no—
347C22—10841no—
348C22—10841no—
349C22—10841no—
350C22—21684no—
351C22C2×D4×D27111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—5421no—
357C4—5421no—
358C4—5421no—
359C4—10841no—
360C4—10841no—
361C4—10841no—
362C4C2×#1686465411yescharacteristic
363C4#2257620511yescharacteristic
364C2—41no—
365C2—42no—
366C2—5422no—
367C2—21681no—
368C2#3177453711yescharacteristic
369C2#135×D27111yescharacteristic
370C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#416×C271C2
socleC22×C271C22×D4
prime radical#31774535C2
hypercenter#416D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #31774586C2semidirect
3. #31774253C2semidirect
4. #31774226C2semidirect
5. #31774221C2semidirect
6. #31774199C2semidirect
7. #31774137C2semidirect
8. #31774126C2semidirect
9. #31774122C2semidirect
10. #31774059C2semidirect
11. #31774049C2semidirect
12. #31773988C2semidirect
13. C8×C22⋊D271C2semidirect
14. #102×D271C2semidirect
15. #416×C271C2semidirect
17. C2×#16864644C22semidirect
23. #22576151C22semidirect
28. #22576139C22semidirect
33. #22576125C22semidirect
39. #22576093C22semidirect
40. #22576092C22semidirect
41. #22576091C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576085C22semidirect
44. #22576084C22semidirect
49. C22⋊(C271⋊C8)C22semidirect
51. C8×Dic271C22semidirect
53. C22×C8×C271C22semidirect
54. #117×C271C22semidirect
55. #103×C271C22semidirect
56. #103×C271C22semidirect
185. C271#3296semidirect
200. #416D271semidirect
208. C22×C8C2×D271semidirect
225. #117C2×D271semidirect
234. #103C2×D271semidirect
235. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745862
#317745352
#317742532
#317742262
#317742212
#317741992
#317741372
#317741262
#317741222
#317740592
#317740492
#317739882
C8×C22⋊D2712
#102×D2712
#416×C2712
#3296271