Finite Group Databasebeta

← #46848327

Subgroups — #46848327

Order 34688.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774551C211yesmaximal, characteristic, complemented
2#31774543C211yesmaximal, characteristic, complemented
3#31774257C211yesmaximal, characteristic, complemented
4#31774228C211yesmaximal, characteristic, complemented
5#31774216C211yesmaximal, characteristic, complemented
6#31774188C211yesmaximal, characteristic, complemented
7#31774150C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774113C211yesmaximal, characteristic, complemented
10#31774044C211yesmaximal, characteristic, complemented
11#31774033C211yesmaximal, characteristic, complemented
12#31774010C211yesmaximal, characteristic, complemented
13#31773899C211yesmaximal, characteristic, complemented
14C2×#22576082C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864653—21no—
18C22×D1084C2211yescharacteristic
19C2×#16864643C2211yescharacteristic, complemented
20C2×D2168C2211yescharacteristic
21C2×D2168C2211yescharacteristic
22#22576160C2211yescharacteristic, complemented
23#22576148C2211yescharacteristic
24#22576145—21no—
25#22576144—21no—
26#22576144—21no—
27#22576140—21no—
28#22576139C2211yescharacteristic
29#22576134C2211yescharacteristic
30#22576134C2211yescharacteristic
31#22576134C2211yescharacteristic, complemented
32#22576133—21no—
33#22576128—21no—
34C4×D1084C2211yescharacteristic
35C4×D1084C2211yescharacteristic, complemented
36C2×C4⋊Dic271C2211yescharacteristic
37C2×C4⋊Dic271C2211yescharacteristic, complemented
38C22⋊C4×D271—21no—
39#22576092C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, 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
52C2×M16×C271C2211yescharacteristic, complemented
53C22×C8×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55C4⋊C8×C271C2211yescharacteristic, complemented
56C22⋊C8×C271C2211yescharacteristic
57C22⋊C8×C271C2211yescharacteristic, complemented
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62#16864653—44no—
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65C2×D1084—22no—
66C2×D1084—21no—
67C2×D1084—41no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×D1084C2311yescharacteristic
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C22×Dic271—21no—
76C22×Dic271—21no—
77C2×C4×D271—21no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80#16864643—41no—
81D2168—22no—
82D2168—41no—
83C4⋊Dic271—21no—
84C4⋊Dic271—21no—
85C4⋊Dic271—21no—
86C4⋊Dic271C2311yescharacteristic
87C4⋊Dic271C2311yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89C4⋊Dic271C2311yescharacteristic
90Dic271⋊C4—21no—
91C22⋊Dic271—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—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99C2×D4×C271D411yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102M16×C271—41no—
103C2×C8×C271—41no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271D411yescharacteristic
109C22⋊C4×C271D411yescharacteristic
110C22⋊C4×C271D411yescharacteristic
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—42no—
116C22×D271—42no—
117C22×D271—81no—
118Dic542—21no—
119Dic542—21no—
120Dic542—42no—
121D1084—21no—
122D1084—21no—
123D1084—21no—
124D1084—21no—
125D1084—21no—
126D1084—42no—
127D1084#4112yescomplemented
128C22⋊D271—42no—
129C22⋊D271—42no—
130C22⋊D271—42no—
131C22⋊D271—81no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—42no—
138C2×Dic271—42no—
139C4×D271—42no—
140C4×D271—42no—
141C4×D271—81no—
142C23×C271C2×D411yescharacteristic
143C23×C271C2×D411yescharacteristic
144D4×C271—22no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271C2×D411yescharacteristic
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2×D411yescharacteristic
153C2×C4×C271C2411yescharacteristic
154C8×C271—21no—
155C8×C271—21no—
156C8×C271—21no—
157C8×C271—21no—
158C22×C271—21no—
159C22×C271—21no—
160C22×C271—21no—
161C22×C271—21no—
162C22×C271—21no—
163C22×C271AGL(1,ℤ/8)12yes—
164C22×C271C22×D411yescharacteristic
165C2×D271—22no—
166C2×D271—41no—
167C2×D271—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—41no—
174C2×D271—84no—
175C4×C271—21no—
176C4×C271—21no—
177C4×C271—21no—
178C4×C271—21no—
179C4×C271C2×#4111yescharacteristic
180C4×C271H(5,𝔽2)11yescharacteristic
181Dic271—41no—
182Dic271—41no—
183Dic271—41no—
184Dic271—41no—
185Dic271—41no—
186C2×C271—22no—
187C2×C271—41no—
188C2×C271#54611yescharacteristic
189C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C271#57711yescharacteristic
191D271—42no—
192D271—81no—
193D271—81no—
194C271#330611yescharacteristic, complemented, Sylow
195#3306—2711nomaximal, Sylow
196#547—2711no—
197#546—2711no—
198#489—2711no—
199#483—2711no—
200#481—2711no—
201#476—2711no—
202#469—2711no—
203#465—2711no—
204#463—2711no—
205#449—2711no—
206#447—2711no—
207#442—2711no—
208#435D27111yescharacteristic, complemented
209#417—2711no—
210C2×#102—2711no—
211C2×#41—2711no—
212C2×#41—5421no—
213C22×D4—2711no—
214C2×SD16—2711no—
215C2×D8—2711no—
216C2×D8—2711no—
217C2×M16C2×D27111yescharacteristic, complemented
218C22×C8C2×D27111yescharacteristic
219C4⋊Q8—2711no—
220#124—2711no—
221#123—5421no—
222#123—5421no—
223#123—5421no—
224C22⋊Q8—2711no—
225C22⋊Q8—5421no—
226#121—2711no—
227#121—2711no—
228#121—2711no—
229#121—5421no—
230#120—5421no—
231C4×D4—2711no—
232C4×D4—2711no—
233C4×D4C2×D27111yescharacteristic
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C22⋊C4—5421no—
237#106—2711no—
238#106—2711no—
239#106—2711no—
240C4⋊C8C2×D27111yescharacteristic, complemented
241#103—2711no—
242#103—2711no—
243#103—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102—2711no—
250#102—2711no—
251C22⋊C8C2×D27111yescharacteristic
252C22⋊C8C2×D27111yescharacteristic, complemented
253#97C2×D27111yescharacteristic, complemented
254C24—5421no—
255#41—5422no—
256#41—10841no—
257#41—10844no—
258C2×Q8—2711no—
259C2×Q8—5421no—
260C2×D4—2711no—
261C2×D4—2711no—
262C2×D4—2711no—
263C2×D4—5422no—
264C2×D4—5421no—
265C2×D4—5421no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4—5421no—
269C2×D4—10841no—
270C2×D4D108411yescharacteristic
271C22×C4—5421no—
272C22×C4—5421no—
273C22×C4—5421no—
274C22×C4—5421no—
275C22×C4—5421no—
276C22×C4C22×D27111yescharacteristic
277C22×C4C22×D27111yescharacteristic
278SD16—10841no—
279D8—5422no—
280D8—10841no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D27111yescharacteristic
284C2×C8C22×D27111yescharacteristic
285C2×C8C22×D27111yescharacteristic
286C2×C8C22×D27111yescharacteristic
287C4⋊C4—2711no—
288C4⋊C4—2711no—
289C4⋊C4—2711no—
290C4⋊C4—2711no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4—5421no—
295C4⋊C4D108411yescharacteristic
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4D108411yescharacteristic
305C22⋊C4D108411yescharacteristic
306C42C22×D27111yescharacteristic
307C23—5421no—
308C23—5421no—
309C23—5421no—
310C23—10842no—
311C23—10842no—
312C23—21681no—
313C23C2×D108411yescharacteristic
314C23C2×D108411yescharacteristic
315Q8—5421no—
316Q8—5421no—
317Q8—10842no—
318D4—22no—
319D4—2712no—
320D4—5421no—
321D4—5421no—
322D4—5421no—
323D4—5421no—
324D4—5421no—
325D4—10842no—
326D4—10842no—
327D4—10842no—
328D4—10842no—
329D4—21681no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—5421no—
338C2×C4—5421no—
339C2×C4—10842no—
340C2×C4—10842no—
341C2×C4—10842no—
342C2×C4—10842no—
343C2×C4—21681no—
344C2×C4C2×D108411yescharacteristic
345C2×C4C2×D108411yescharacteristic
346C2×C4C2×D108411yescharacteristic
347C2×C4C2×D108411yescharacteristic
348C2×C4C23×D27111yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—5422no—
359C22—10841no—
360C22—10841no—
361C22—10841no—
362C22—10841no—
363C22—10841no—
364C22—10841no—
365C22—10841no—
366C22—10841no—
367C22—21684no—
368C22#2257618812yes—
369C22C22×D108411yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—10841no—
375C4—10841no—
376C4—10841no—
377C4—10841no—
378C4—10841no—
379C4#41×D27111yescharacteristic
380C4#2257620511yescharacteristic
381C2—22no—
382C2—41no—
383C2—10842no—
384C2—21681no—
385C2—21681no—
386C2#3177454211yescharacteristic
387C2C2×#2257618811yescharacteristic
388C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
hypercenter#435D271
hypocenterC271#3306
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #31774551C2semidirect
2. #31774543C2semidirect
3. #31774257C2semidirect
4. #31774228C2semidirect
5. #31774216C2semidirect
6. #31774188C2semidirect
7. #31774150C2semidirect
8. #31774126C2semidirect
9. #31774113C2semidirect
10. #31774044C2semidirect
11. #31774033C2semidirect
12. #31774010C2semidirect
13. #31773899C2semidirect
14. C2×#22576082C2semidirect
15. #435×C271C2semidirect
19. C2×#16864643C22semidirect
22. #22576160C22semidirect
31. #22576134C22semidirect
35. C4×D1084C22semidirect
37. C2×C4⋊Dic271C22semidirect
39. #22576092C22semidirect
40. #22576092C22semidirect
41. #22576092C22semidirect
42. #22576085C22semidirect
43. #22576085C22semidirect
44. #22576085C22semidirect
51. #22576082C22semidirect
52. C2×M16×C271C22semidirect
55. C4⋊C8×C271C22semidirect
57. C22⋊C8×C271C22semidirect
58. #97×C271C22semidirect
127. D1084#41semidirect
194. C271#3306semidirect
208. #435D271semidirect
217. C2×M16C2×D271semidirect
240. C4⋊C8C2×D271semidirect
252. C22⋊C8C2×D271semidirect
253. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745512
#317745432
#317742572
#317742282
#317742162
#317741882
#317741502
#317741262
#317741132
#317740442
#317740332
#317740102
#317738992
C2×#225760822
#435×C2712
#3306271