Finite Group Databasebeta

← #46848277

Subgroups — #46848277

Order 34688.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774615C211yesmaximal, characteristic, complemented
2#31774546C211yesmaximal, characteristic
3#31774246C211yesmaximal, characteristic, complemented
4#31774238C211yesmaximal, characteristic, complemented
5#31774237C211yesmaximal, characteristic, complemented
6#31774175C211yesmaximal, characteristic, complemented
7#31774138C211yesmaximal, characteristic, complemented
8#31774137C211yesmaximal, characteristic, complemented
9#31774106C211yesmaximal, characteristic, complemented
10#31774060C211yesmaximal, characteristic, complemented
11#31774059C211yesmaximal, characteristic, complemented
12#31773988C211yesmaximal, characteristic, complemented
13C8×C22⋊D271C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#416×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×Dic1084C2211yescharacteristic
18C2×#16864644C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576158C2211yescharacteristic
23#22576154—21no—
24#22576153—21no—
25#22576146—21no—
26#22576143—21no—
27#22576141—21no—
28#22576140C2211yescharacteristic
29#22576139C2211yescharacteristic
30#22576139C2211yescharacteristic
31#22576136C2211yescharacteristic
32#22576132—21no—
33C4×Dic542—21no—
34#22576125C2211yescharacteristic
35D4×Dic271C2211yescharacteristic
36C4×C22⋊D271C2211yescharacteristic
37#22576118C2211yescharacteristic
38C2×Dic271⋊C4—21no—
39#22576094C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576091C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576083C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54#117×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—22no—
60#16864653—41no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63C2×Dic542C2311yescharacteristic
64C2×C22⋊D271—21no—
65C2×C22⋊D271D411yescharacteristic
66C22×Dic271—21no—
67C22×Dic271—21no—
68C2×C4×D271C2311yescharacteristic
69Dic1084—41no—
70#16864644—41no—
71#16864642—22no—
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C8×D271—41no—
75C4⋊Dic271—21no—
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78Dic271⋊C4—21no—
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271D411yescharacteristic
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94C4×Dic271—21no—
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×D271C2×D411yescharacteristic
112Dic542—21no—
113Dic542—21no—
114Dic542—21no—
115Dic542—21no—
116Dic542—42no—
117C22⋊D271—22no—
118C22⋊D271—42no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—42no—
124C2×Dic271—42no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132C23×C271—21no—
133C23×C271C2×D411yescharacteristic
134Q8×C271—21no—
135Q8×C271—21no—
136D4×C271—21no—
137D4×C271—42no—
138D4×C271#4112yes—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—42no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2411yescharacteristic
147C8×C271—21no—
148C8×C271—21no—
149C22×C271—21no—
150C22×C271—22no—
151C22×C271—21no—
152C22×C271—21no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271C22×D411yescharacteristic
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—21no—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271C2×#4111yescharacteristic
164C4×C271H(5,𝔽2)11yescharacteristic
165Dic271—21no—
166Dic271—21no—
167Dic271—41no—
168Dic271—41no—
169Dic271—41no—
170Dic271—41no—
171Dic271#13512yes—
172C2×C271—41no—
173C2×C271—42no—
174C2×C271#54611yescharacteristic
175C2×C271C2×#13511yescharacteristic
176C2×C271#57811yescharacteristic
177D271—41no—
178C271#330211yescharacteristic, complemented, Sylow
179#3302—2711nomaximal, Sylow
180#556—2711no—
181#547—2711no—
182#487—2711no—
183#485—2711no—
184#485—2711no—
185#474—2711no—
186#467—2711no—
187#467—2711no—
188#462—2711no—
189#452—2711no—
190#452—2711no—
191C4×SD16—2711no—
192C8×D4—2711no—
193#416—2711no—
194#416D27111yescharacteristic, complemented
195C2×#41—2711no—
196C2×#41C2×D27111yescharacteristic
197C2×Q16—2711no—
198C2×Q16—2711no—
199C2×SD16—2711no—
200C22×C8—2711no—
201C22×C8C2×D27111yescharacteristic
202C4⋊Q8—2711no—
203C4⋊Q8—2711no—
204#126—5421no—
205#126—5421no—
206#124—5421no—
207#123—5421no—
208#123—5421no—
209C22⋊Q8—2711no—
210C22⋊Q8—2711no—
211C22⋊Q8—2711no—
212C22⋊Q8—2711no—
213#121—5421no—
214C4×Q8—2711no—
215C4×Q8—5421no—
216C4×D4—2711no—
217C4×D4—2711no—
218#117—2711no—
219#117C2×D27111yescharacteristic
220C2×C4⋊C4—5421no—
221#107—2711no—
222#107—2711no—
223#106—2711no—
224C4⋊C8—2711no—
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103C2×D27111yescharacteristic
231#103C2×D27111yescharacteristic
232#102—2711no—
233#102C2×D27111yescharacteristic
234#102C2×D27111yescharacteristic
235C22⋊C8—2711no—
236C22⋊C8—2711no—
237C4×C8—2711no—
238#41—22no—
239#41—41no—
240#41—5422no—
241#41—10841no—
242C2×Q8—2711no—
243C2×Q8—2711no—
244C2×Q8—5421no—
245C2×Q8C22×D27111yescharacteristic
246C2×D4—21no—
247C2×D4—2711no—
248C2×D4—5421no—
249C2×D4C22×D27111yescharacteristic
250C22×C4—21no—
251C22×C4—2711no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4C22×D27111yescharacteristic
255Q16—10841no—
256Q16—10841no—
257SD16—5422no—
258C2×C8—41no—
259C2×C8—2711no—
260C2×C8—2711no—
261C2×C8—10841no—
262C2×C8C22×D27111yescharacteristic
263C2×C8C22×D27111yescharacteristic
264C4⋊C4—2711no—
265C4⋊C4—2711no—
266C4⋊C4—2711no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4C22×D27111yescharacteristic
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2711no—
279C22⋊C4—2711no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—21no—
287C42—2711no—
288C42—5421no—
289C42—5421no—
290C23—21no—
291C23—2711no—
292C23D4×D27111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—5421no—
296Q8—5421no—
297Q8—5421no—
298Q8—5421no—
299Q8—10842no—
300D4—21no—
301D4—42no—
302D4—5422no—
303D4—10842no—
304D4#1686465412yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—2711no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—10842no—
321C2×C4—10842no—
322C2×C4D4×D27111yescharacteristic
323C2×C4C23×D27111yescharacteristic
324C8—21no—
325C8—21no—
326C8—5421no—
327C8—5421no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—5421no—
335C22—5421no—
336C22—5421no—
337C22C2×D4×D27111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—2712no—
343C4—5421no—
344C4—5421no—
345C4—10841no—
346C4—10841no—
347C4—10841no—
348C4—10841no—
349C4C2×#1686465411yescharacteristic
350C4#2257620511yescharacteristic
351C2—41no—
352C2—42no—
353C2—10841no—
354C2#3177453711yescharacteristic
355C2#135×D27111yescharacteristic
356C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#416×C271C2
socleC22×C271C22×D4
prime radical#31774546C2
hypercenter#416D271
hypocenterC271#3302
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. #31774615C2semidirect
3. #31774246C2semidirect
4. #31774238C2semidirect
5. #31774237C2semidirect
6. #31774175C2semidirect
7. #31774138C2semidirect
8. #31774137C2semidirect
9. #31774106C2semidirect
10. #31774060C2semidirect
11. #31774059C2semidirect
12. #31773988C2semidirect
13. C8×C22⋊D271C2semidirect
14. #31773890C2semidirect
15. #416×C271C2semidirect
178. C271#3302semidirect
194. #416D271semidirect

Maximal subgroups

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

subgroupindex
#317746152
#317745462
#317742462
#317742382
#317742372
#317741752
#317741382
#317741372
#317741062
#317740602
#317740592
#317739882
C8×C22⋊D2712
#317738902
#416×C2712
#3302271