Finite Group Databasebeta

← #46848216

Subgroups — #46848216

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#31774593C211yesmaximal, characteristic, complemented
2#31774543C211yesmaximal, characteristic, complemented
3#31774252C211yesmaximal, characteristic, complemented
4#31774233C211yesmaximal, characteristic, complemented
5#31774222C211yesmaximal, characteristic, complemented
6#31774198C211yesmaximal, characteristic, complemented
7#31774133C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774113C211yesmaximal, characteristic, complemented
10#31774044C211yesmaximal, characteristic, complemented
11#31774033C211yesmaximal, characteristic, complemented
12C4×D2168C211yesmaximal, characteristic, complemented
13#31773894C211yesmaximal, characteristic, complemented
14C2×#22576082C211yesmaximal, characteristic, complemented
15C8×D4×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×#16864643C2211yescharacteristic, complemented
19C2×D2168C2211yescharacteristic
20C2×D2168C2211yescharacteristic
21#22576155—21no—
22#22576152C2211yescharacteristic, complemented
23#22576148C2211yescharacteristic
24#22576145—21no—
25#22576142—21no—
26#22576139C2211yescharacteristic
27#22576134C2211yescharacteristic
28#22576134C2211yescharacteristic
29#22576134C2211yescharacteristic, complemented
30#22576133—21no—
31#22576132—21no—
32#22576128—21no—
33C4×D1084C2211yescharacteristic
34C4×D1084C2211yescharacteristic, complemented
35(C22⋊D271)⋊C4—21no—
36#22576119C2211yescharacteristic, complemented
37C2×C4⋊Dic271C2211yescharacteristic
38C22⋊C4×D271—21no—
39#22576094C2211yescharacteristic, complemented
40#22576094C2211yescharacteristic, complemented
41#22576092C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576082C2211yescharacteristic
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic
51#22576082C2211yescharacteristic, complemented
52C22×C8×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic, complemented
54C4×D4×C271C2211yescharacteristic
55C4⋊C8×C271C2211yescharacteristic, complemented
56C22⋊C8×C271C2211yescharacteristic
57C22⋊C8×C271C2211yescharacteristic, complemented
58C4×C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—22no—
64C2×D1084—21no—
65C2×D1084—41no—
66C2×D1084C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C22×Dic271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77#16864643—41no—
78D2168—22no—
79D2168—41no—
80C4⋊Dic271—21no—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97C4×Dic271—21no—
98C2×D4×C271D411yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C2×C8×C271—41no—
102C2×C8×C271—41no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271D411yescharacteristic
108C22⋊C4×C271D411yescharacteristic
109C22⋊C4×C271D411yescharacteristic
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—42no—
115C22×D271—42no—
116C22×D271—81no—
117Dic542—21no—
118Dic542—21no—
119D1084—21no—
120D1084—21no—
121D1084—21no—
122D1084—21no—
123D1084—21no—
124D1084—42no—
125D1084#4112yes—
126C22⋊D271—42no—
127C22⋊D271—42no—
128C22⋊D271—81no—
129C22⋊D271—81no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—81no—
136C4×D271—42no—
137C4×D271—81no—
138C4×D271—81no—
139C23×C271C2×D411yescharacteristic
140C23×C271C2×D411yescharacteristic
141D4×C271—22no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2×D411yescharacteristic
150C2×C4×C271C2411yescharacteristic
151C8×C271—21no—
152C8×C271—21no—
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—21no—
159C22×C271—21no—
160C22×C271#13512yes—
161C22×C271C22×D411yescharacteristic
162C2×D271—22no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—84no—
172C4×C271—21no—
173C4×C271—21no—
174C4×C271—21no—
175C4×C271—21no—
176C4×C271C2×#4111yescharacteristic
177C4×C271H(5,𝔽2)11yescharacteristic
178Dic271—41no—
179Dic271—41no—
180Dic271—41no—
181Dic271—41no—
182Dic271—41no—
183C2×C271—22no—
184C2×C271—41no—
185C2×C271#54611yescharacteristic
186C2×C271C2×#13511yescharacteristic
187C2×C271#57611yescharacteristic
188D271—42no—
189D271—81no—
190D271—81no—
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#466—2711no—
200#465—2711no—
201#463—2711no—
202#449—2711no—
203#447—2711no—
204C4×D8—2711no—
205C8×D4D27111yescharacteristic, complemented
206#416—2711no—
207C2×#102—2711no—
208C2×#41—2711no—
209C22×D4—2711no—
210C2×SD16—2711no—
211C2×D8—2711no—
212C2×D8—2711no—
213C22×C8C2×D27111yescharacteristic
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—2711no—
223#121—2711no—
224#121—5421no—
225#121—5421no—
226#120—5421no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—5421no—
230C4×D4C2×D27111yescharacteristic
231#117—2711no—
232C2×C4⋊C4—2711no—
233C2×C22⋊C4—5421no—
234#107—2711no—
235#107—2711no—
236#106—2711no—
237C4⋊C8C2×D27111yescharacteristic, complemented
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#102—2711no—
247#102—2711no—
248C22⋊C8C2×D27111yescharacteristic
249C22⋊C8C2×D27111yescharacteristic, complemented
250C4×C8C2×D27111yescharacteristic, complemented
251C24—5421no—
252#41—5422no—
253#41—10841no—
254C2×Q8—2711no—
255C2×D4—2711no—
256C2×D4—2711no—
257C2×D4—2711no—
258C2×D4—5422no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4—10841no—
265C2×D4D108411yescharacteristic
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4—5421no—
270C22×C4C22×D27111yescharacteristic
271C22×C4C22×D27111yescharacteristic
272SD16—10841no—
273D8—5422no—
274D8—10841no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D27111yescharacteristic
278C2×C8C22×D27111yescharacteristic
279C2×C8C22×D27111yescharacteristic
280C2×C8C22×D27111yescharacteristic
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4—5421no—
289C4⋊C4D108411yescharacteristic
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4D108411yescharacteristic
300C22⋊C4D108411yescharacteristic
301C42—5421no—
302C42C22×D27111yescharacteristic
303C23—5421no—
304C23—5421no—
305C23—5421no—
306C23—10842no—
307C23—10842no—
308C23—21681no—
309C23C2×D108411yescharacteristic
310C23C2×D108411yescharacteristic
311Q8—5421no—
312Q8—5421no—
313D4—22no—
314D4—2712no—
315D4—5421no—
316D4—5421no—
317D4—5421no—
318D4—5421no—
319D4—5421no—
320D4—10842no—
321D4—10842no—
322D4—10842no—
323D4—21681no—
324D4—21681no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—10842no—
335C2×C4—21681no—
336C2×C4—21681no—
337C2×C4—21681no—
338C2×C4C2×D108411yescharacteristic
339C2×C4C2×D108411yescharacteristic
340C2×C4C2×D108411yescharacteristic
341C2×C4C2×D108411yescharacteristic
342C2×C4C23×D27111yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—5422no—
353C22—10841no—
354C22—10841no—
355C22—10841no—
356C22—10841no—
357C22—10841no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22—21684no—
362C22#2257618212yes—
363C22C22×D108411yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—10841no—
369C4—10841no—
370C4—10841no—
371C4—10841no—
372C4—10841no—
373C4#41×D27111yescharacteristic
374C4#2257620511yescharacteristic
375C2—22no—
376C2—41no—
377C2—10842no—
378C2—21681no—
379C2—21681no—
380C2#3177454211yescharacteristic
381C2C2×#2257618211yescharacteristic
382C2#3177470211yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC8×D4×C271C2
socleC22×C271C22×D4
hypercenterC8×D4D271
hypocenterC271#3297
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #31774593C2semidirect
2. #31774543C2semidirect
3. #31774252C2semidirect
4. #31774233C2semidirect
5. #31774222C2semidirect
6. #31774198C2semidirect
7. #31774133C2semidirect
8. #31774126C2semidirect
9. #31774113C2semidirect
10. #31774044C2semidirect
11. #31774033C2semidirect
12. C4×D2168C2semidirect
13. #31773894C2semidirect
14. C2×#22576082C2semidirect
15. C8×D4×C271C2semidirect
18. C2×#16864643C22semidirect
22. #22576152C22semidirect
29. #22576134C22semidirect
34. C4×D1084C22semidirect
36. #22576119C22semidirect
39. #22576094C22semidirect
40. #22576094C22semidirect
41. #22576092C22semidirect
42. #22576085C22semidirect
43. #22576085C22semidirect
44. #22576085C22semidirect
51. #22576082C22semidirect
53. C22×C8×C271C22semidirect
55. C4⋊C8×C271C22semidirect
57. C22⋊C8×C271C22semidirect
58. C4×C8×C271C22semidirect
191. C271#3297semidirect
205. C8×D4D271semidirect
214. C22×C8C2×D271semidirect
237. C4⋊C8C2×D271semidirect
249. C22⋊C8C2×D271semidirect
250. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745932
#317745432
#317742522
#317742332
#317742222
#317741982
#317741332
#317741262
#317741132
#317740442
#317740332
C4×D21682
#317738942
C2×#225760822
C8×D4×C2712
#3297271