Finite Group Databasebeta

← #46848209

Subgroups — #46848209

Order 34688.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774594C211yesmaximal, characteristic, complemented
2#31774544C211yesmaximal, characteristic, complemented
3#31774250C211yesmaximal, characteristic, complemented
4#31774234C211yesmaximal, characteristic, complemented
5#31774218C211yesmaximal, characteristic, complemented
6#31774200C211yesmaximal, characteristic, complemented
7#31774125C211yesmaximal, characteristic, complemented
8#31774110C211yesmaximal, characteristic, complemented
9#31774040C211yesmaximal, characteristic, complemented
10#31774029C211yesmaximal, characteristic, complemented
11#31773977C211yesmaximal, characteristic, complemented
12C8×C22⋊D271C211yesmaximal, characteristic, complemented
13#31773894C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#466×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864642C2211yescharacteristic, complemented
19C2×#16864639C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576153—21no—
22#22576151C2211yescharacteristic, complemented
23#22576147C2211yescharacteristic
24#22576142—21no—
25#22576141—21no—
26#22576140C2211yescharacteristic
27#22576131—21no—
28#22576131—21no—
29#22576131C2211yescharacteristic
30#22576128—21no—
31D4×Dic271C2211yescharacteristic, complemented
32#22576122C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34(C22⋊D271)⋊C4—21no—
35#22576119C2211yescharacteristic, complemented
36C4⋊C4×D271C2211yescharacteristic
37C22⋊C4×D271—21no—
38#22576093C2211yescharacteristic, complemented
39#22576091C2211yescharacteristic, complemented
40Dic271⋊C8C2211yescharacteristic, complemented
41#22576085C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic, complemented
52C2×D8×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic, complemented
54#121×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic, complemented
56#107×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62C2×Dic542C2311yescharacteristic
63C2×D1084C2311yescharacteristic
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271D411yescharacteristic
68D4×D271—22no—
69D4×D271—41no—
70C22×Dic271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271C2311yescharacteristic
75#16864642—41no—
76#16864639—22no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C8×D271—41no—
80C4⋊Dic271—21no—
81C4⋊Dic271—21no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271D411yescharacteristic
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
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×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116Dic542—21no—
117Dic542—21no—
118D1084—21no—
119D1084#4112yes—
120C22⋊D271—22no—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C22⋊D271—42no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—81no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C2×Dic271C2×D411yescharacteristic
131C4×D271—21no—
132C4×D271—21no—
133C4×D271—42no—
134C4×D271—81no—
135C4×D271—81no—
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—22no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—84no—
171C2×D271#13512yes—
172C4×C271—21no—
173C4×C271—41no—
174C4×C271—41no—
175C4×C271C2×#4111yescharacteristic
176C4×C271H(5,𝔽2)11yescharacteristic
177Dic271—21no—
178Dic271—21no—
179Dic271—21no—
180Dic271—41no—
181Dic271—41no—
182Dic271—41no—
183C2×C271—41no—
184C2×C271—81no—
185C2×C271—81no—
186C2×C271#54611yescharacteristic
187C2×C271C2×#13511yescharacteristic
188C2×C271#57611yescharacteristic
189D271—22no—
190D271—42no—
191C271#329711yescharacteristic, complemented, Sylow
192#3297—2711nomaximal, Sylow
193#553—2711no—
194#546—2711no—
195#488—2711no—
196#484—2711no—
197#482—2711no—
198#478—2711no—
199#466D27111yescharacteristic, complemented
200#465—2711no—
201#463—2711no—
202#449—2711no—
203#447—2711no—
204C4×D8—2711no—
205C8×D4—2711no—
206#416—2711no—
207C2×#102—2711no—
208C2×#41—2711no—
209C22×D4—2711no—
210C2×SD16—2711no—
211C2×D8—2711no—
212C2×D8C2×D27111yescharacteristic
213C22×C8—2711no—
214C22×C8C2×D27111yescharacteristic, complemented
215#126—5421no—
216#125—2711no—
217#124—2711no—
218#123—5421no—
219#123—5421no—
220C22⋊Q8—2711no—
221#121—2711no—
222#121—5421no—
223#121—5421no—
224#121C2×D27111yescharacteristic
225#121C2×D27111yescharacteristic, complemented
226#120—5421no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—2711no—
230C4×D4—5421no—
231#117—2711no—
232C2×C4⋊C4—2711no—
233C2×C22⋊C4—5421no—
234#107—2711no—
235#107C2×D27111yescharacteristic, complemented
236#106—2711no—
237C4⋊C8—2711no—
238#103—2711no—
239#103—2711no—
240#103—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102C2×D27111yescharacteristic
247#102C2×D27111yescharacteristic, complemented
248C22⋊C8—2711no—
249C22⋊C8—2711no—
250C4×C8—2711no—
251C24—5421no—
252#41—5422no—
253#41—10841no—
254C2×Q8—2711no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2711no—
258C2×D4—2711no—
259C2×D4—5421no—
260C2×D4—5422no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4—10841no—
264C2×D4C22×D27111yescharacteristic
265C2×D4C22×D27111yescharacteristic
266C22×C4—2711no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4C22×D27111yescharacteristic
272SD16—10841no—
273D8—41no—
274D8—5422no—
275C2×C8—41no—
276C2×C8—2711no—
277C2×C8—2711no—
278C2×C8—10841no—
279C2×C8C22×D27111yescharacteristic
280C2×C8C22×D27111yescharacteristic
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4C22×D27111yescharacteristic
289C4⋊C4C22×D27111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2711no—
293C22⋊C4—2711no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C42—2711no—
302C42—5421no—
303C23—21no—
304C23—21no—
305C23—2711no—
306C23—5421no—
307C23—10842no—
308C23—10842no—
309C23—21681no—
310C23D4×D27111yescharacteristic
311Q8—5421no—
312Q8—5421no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—81no—
318D4—81no—
319D4—2712no—
320D4—5422no—
321D4—5421no—
322D4—10842no—
323D4—10842no—
324D4—10842no—
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—21681no—
339C2×C4—21681no—
340C2×C4—21681no—
341C2×C4D4×D27111yescharacteristic
342C2×C4C23×D27111yescharacteristic
343C8—21no—
344C8—21no—
345C8—5421no—
346C8—5421no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2712no—
357C22—5421no—
358C22—5421no—
359C22—5422no—
360C22—10841no—
361C22—10841no—
362C22—21684no—
363C22C2×D4×D27111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—5421no—
368C4—5421no—
369C4—5421no—
370C4—10841no—
371C4—10841no—
372C4—10841no—
373C4#41×D27111yescharacteristic
374C4#2257620511yescharacteristic
375C2—41no—
376C2—81no—
377C2—81no—
378C2—5422no—
379C2—10842no—
380C2#3177453611yescharacteristic
381C2#135×D27111yescharacteristic
382C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#466×C271C2
socleC22×C271C22×D4
hypercenter#466D271
hypocenterC271#3297
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. #31774594C2semidirect
2. #31774544C2semidirect
3. #31774250C2semidirect
4. #31774234C2semidirect
5. #31774218C2semidirect
6. #31774200C2semidirect
7. #31774125C2semidirect
8. #31774110C2semidirect
9. #31774040C2semidirect
10. #31774029C2semidirect
11. #31773977C2semidirect
12. C8×C22⋊D271C2semidirect
13. #31773894C2semidirect
14. #102×D271C2semidirect
15. #466×C271C2semidirect
18. C2×#16864642C22semidirect
22. #22576151C22semidirect
31. D4×Dic271C22semidirect
35. #22576119C22semidirect
38. #22576093C22semidirect
39. #22576091C22semidirect
40. Dic271⋊C8C22semidirect
41. #22576085C22semidirect
42. #22576085C22semidirect
43. #22576083C22semidirect
49. C22⋊(C271⋊C8)C22semidirect
51. C8×Dic271C22semidirect
53. C22×C8×C271C22semidirect
55. #121×C271C22semidirect
56. #107×C271C22semidirect
58. #102×C271C22semidirect
191. C271#3297semidirect
199. #466D271semidirect
214. C22×C8C2×D271semidirect
225. #121C2×D271semidirect
235. #107C2×D271semidirect
247. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745942
#317745442
#317742502
#317742342
#317742182
#317742002
#317741252
#317741102
#317740402
#317740292
#317739772
C8×C22⋊D2712
#317738942
#102×D2712
#466×C2712
#3297271