Finite Group Databasebeta

← #46848309

Subgroups — #46848309

Order 34688.

408 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774611C211yesmaximal, characteristic, complemented
2#31774565C211yesmaximal, characteristic, complemented
3#31774242C211yesmaximal, characteristic, complemented
4#31774224C211yesmaximal, characteristic, complemented
5#31774218C211yesmaximal, characteristic, complemented
6#31774191C211yesmaximal, characteristic, complemented
7#31774132C211yesmaximal, characteristic, complemented
8#31774090C211yesmaximal, characteristic, complemented
9#31774040C211yesmaximal, characteristic, complemented
10#31774039C211yesmaximal, characteristic, complemented
11#31773988C211yesmaximal, characteristic, complemented
12C8×C22⋊D271C211yesmaximal, characteristic, complemented
13#31773894C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#466×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576156—21no—
23#22576156C2211yescharacteristic
24#22576147—21no—
25#22576146—21no—
26#22576135C2211yescharacteristic
27#22576133—21no—
28#22576132—21no—
29#22576132—21no—
30#22576131C2211yescharacteristic
31#22576125C2211yescharacteristic
32D4×Dic271C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34(C22⋊D271)⋊C4—21no—
35#22576119C2211yescharacteristic
36#22576118C2211yescharacteristic
37#22576116—21no—
38#22576093C2211yescharacteristic
39#22576091C2211yescharacteristic
40Dic271⋊C8C2211yescharacteristic
41#22576085C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576083C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×D8×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54#121×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#107×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864654—22no—
60#16864654—41no—
61#16864653—22no—
62#16864653—41no—
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65C2×D1084—21no—
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271D411yescharacteristic
73C22×Dic271—21no—
74C22×Dic271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77#16864642—22no—
78#16864639—41no—
79C2×C271⋊C8C2311yescharacteristic
80C2×C271⋊C8C2311yescharacteristic
81C8×D271—41no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271D411yescharacteristic
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271—21no—
97C4×Dic271C2311yescharacteristic
98C2×D4×C271—21no—
99C2×D4×C271—21no—
100C2×D4×C271C2311yescharacteristic
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103D8×C271—41no—
104C2×C8×C271—41no—
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—81no—
115Dic542#4112yes—
116D1084—21no—
117D1084—21no—
118D1084—81no—
119C22⋊D271—22no—
120C22⋊D271—42no—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C22⋊D271—81no—
124C22⋊D271—81no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—42no—
129C2×Dic271—42no—
130C2×Dic271C2×D411yescharacteristic
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—42no—
136C271⋊C8—21no—
137C271⋊C8—21no—
138C23×C271—21no—
139C23×C271—21no—
140C23×C271C2×D411yescharacteristic
141D4×C271—21no—
142D4×C271—21no—
143D4×C271—21no—
144D4×C271—21no—
145D4×C271—81no—
146D4×C271—81no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×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—41no—
169C2×D271—41no—
170C2×D271—41no—
171C4×C271—21no—
172C4×C271—41no—
173C4×C271—41no—
174C4×C271C2×#4111yescharacteristic
175C4×C271H(5,𝔽2)11yescharacteristic
176Dic271—21no—
177Dic271—22no—
178Dic271—21no—
179Dic271—41no—
180Dic271—41no—
181Dic271#13512yes—
182C2×C271—41no—
183C2×C271—81no—
184C2×C271—81no—
185C2×C271#54611yescharacteristic
186C2×C271C2×#13511yescharacteristic
187C2×C271#57611yescharacteristic
188D271—41no—
189D271—81no—
190C271#330511yescharacteristic, complemented, Sylow
191#3305—2711nomaximal, Sylow
192#555—2711no—
193#550—2711no—
194#486—2711no—
195#482—2711no—
196#482—2711no—
197#477—2711no—
198#466—2711no—
199#466D27111yescharacteristic, complemented
200#459—2711no—
201#449—2711no—
202#449—2711no—
203C4×SD16—2711no—
204C8×D4—2711no—
205#416—2711no—
206#416—2711no—
207C2×#41—2711no—
208C2×#41—2711no—
209C2×SD16—2711no—
210C2×D8—2711no—
211C2×D8C2×D27111yescharacteristic
212C22×C8—2711no—
213C22×C8C2×D27111yescharacteristic
214C4⋊Q8—2711no—
215#127—2711no—
216#127—5421no—
217#124—5421no—
218#124—5421no—
219#121—2711no—
220#121—2711no—
221#121—5421no—
222#121—5421no—
223#121—5421no—
224#121C2×D27111yescharacteristic
225#121C2×D27111yescharacteristic
226C4×Q8—2711no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—5421no—
230#117—2711no—
231#117—2711no—
232#117—5421no—
233#107—2711no—
234#107C2×D27111yescharacteristic
235#106—2711no—
236C4⋊C8—2711no—
237#103—2711no—
238#103—2711no—
239#103—2711no—
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102C2×D27111yescharacteristic
246#102C2×D27111yescharacteristic
247C22⋊C8—2711no—
248C22⋊C8—2711no—
249C4×C8—2711no—
250#41—5422no—
251#41—5422no—
252#41—10841no—
253#41—10841no—
254C2×Q8—2711no—
255C2×Q8—5421no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2711no—
259C2×D4—2711no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4—5421no—
265C2×D4—5421no—
266C2×D4C22×D27111yescharacteristic
267C2×D4C22×D27111yescharacteristic
268C22×C4—2711no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4—5421no—
272C22×C4C22×D27111yescharacteristic
273SD16—5422no—
274D8—41no—
275D8—10841no—
276C2×C8—41no—
277C2×C8—2711no—
278C2×C8—2711no—
279C2×C8—10841no—
280C2×C8C22×D27111yescharacteristic
281C2×C8C22×D27111yescharacteristic
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4C22×D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—2711no—
292C22⋊C4—2711no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C42—2711no—
299C42—5421no—
300C42—5421no—
301C42—5421no—
302C23—21no—
303C23—21no—
304C23—2711no—
305C23—5421no—
306C23D4×D27111yescharacteristic
307Q8—2712no—
308Q8—5421no—
309Q8—21681no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—81no—
315D4—81no—
316D4—5422no—
317D4—5421no—
318D4—5421no—
319D4—10842no—
320D4—10842no—
321D4—10842no—
322D4—21681no—
323D4—21681no—
324D4—21681no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—2711no—
330C2×C4—2711no—
331C2×C4—2711no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—10842no—
338C2×C4—10842no—
339C2×C4—10842no—
340C2×C4D4×D27111yescharacteristic
341C2×C4C23×D27111yescharacteristic
342C8—21no—
343C8—21no—
344C8—5421no—
345C8—5421no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—5421no—
356C22—5421no—
357C22—5421no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22C2×D4×D27111yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—2712no—
366C4—5421no—
367C4—5422no—
368C4—5421no—
369C4—10841no—
370C4—10841no—
371C4#41×D27111yescharacteristic
372C4#2257620511yescharacteristic
373C2—41no—
374C2—81no—
375C2—81no—
376C2—10841no—
377C2—21681no—
378C2#3177453611yescharacteristic
379C2#135×D27111yescharacteristic
380C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#466×C271C2
socleC22×C271C22×D4
hypercenter#466D271
hypocenterC271#3305
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #31774611C2semidirect
2. #31774565C2semidirect
3. #31774242C2semidirect
4. #31774224C2semidirect
5. #31774218C2semidirect
6. #31774191C2semidirect
7. #31774132C2semidirect
8. #31774090C2semidirect
9. #31774040C2semidirect
10. #31774039C2semidirect
11. #31773988C2semidirect
12. C8×C22⋊D271C2semidirect
13. #31773894C2semidirect
14. #31773890C2semidirect
15. #466×C271C2semidirect
190. C271#3305semidirect
199. #466D271semidirect

Maximal subgroups

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

subgroupindex
#317746112
#317745652
#317742422
#317742242
#317742182
#317741912
#317741322
#317740902
#317740402
#317740392
#317739882
C8×C22⋊D2712
#317738942
#317738902
#466×C2712
#3305271