Finite Group Databasebeta

← #46848324

Subgroups — #46848324

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#31774550C211yesmaximal, characteristic, complemented
2#31774540C211yesmaximal, characteristic, complemented
3#31774226C211yesmaximal, characteristic, complemented
4#31774214C211yesmaximal, characteristic, complemented
5#31774184C211yesmaximal, characteristic, complemented
6#31774149C211yesmaximal, characteristic, complemented
7#31774129C211yesmaximal, characteristic, complemented
8#31774109C211yesmaximal, characteristic, complemented
9#31774045C211yesmaximal, characteristic, complemented
10#31774032C211yesmaximal, characteristic, complemented
11#31774010C211yesmaximal, characteristic, complemented
12#31773970C211yesmaximal, characteristic, complemented
13#31773897C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#489×C271C211yesmaximal, characteristic, complemented
16C2×#16864655—21no—
17C2×#16864655C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19C2×#16864641C2211yescharacteristic, complemented
20C2×D2168C2211yescharacteristic
21C2×#16864639C2211yescharacteristic
22C2×#16864637C2211yescharacteristic, complemented
23C2×C8×D271C2211yescharacteristic
24#22576145—21no—
25#22576145—21no—
26#22576142—21no—
27#22576136—21no—
28#22576136C2211yescharacteristic
29#22576135C2211yescharacteristic
30#22576135C2211yescharacteristic, complemented
31#22576133—21no—
32#22576131C2211yescharacteristic
33#22576130—21no—
34C4×D1084C2211yescharacteristic
35C4×D1084C2211yescharacteristic
36C4×D1084C2211yescharacteristic, complemented
37C4⋊C4×D271C2211yescharacteristic
38C4⋊C4×D271C2211yescharacteristic, complemented
39C22⋊C4×D271—21no—
40#22576092C2211yescharacteristic, complemented
41#22576091C2211yescharacteristic, complemented
42#22576091C2211yescharacteristic, complemented
43C4⋊(C271⋊C8)C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52C4⋊Q8×C271C2211yescharacteristic, complemented
53#124×C271C2211yescharacteristic
54#103×C271C2211yescharacteristic, complemented
55#103×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864655—22no—
61#16864655—41no—
62#16864655—44no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084D411yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71D4×D271—22no—
72D4×D271—41no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271C2311yescharacteristic
79C2×C4×D271C2311yescharacteristic
80#16864641—41no—
81D2168—22no—
82#16864639—41no—
83#16864637—41no—
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C8×D271—41no—
87C4⋊Dic271D411yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89C4⋊Dic271C2311yescharacteristic
90Dic271⋊C4—21no—
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93C22⋊Dic271—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—
99(C2×D271)⋊C4D411yescharacteristic
100(C2×D271)⋊C4D411yescharacteristic
101C2×Q8×C271—21no—
102C2×Q8×C271C2311yescharacteristic
103C2×D4×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271—21no—
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C22⋊C4×C271—21no—
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—42no—
115C22×D271—42no—
116C22×D271—81no—
117C22×D271C2×D411yescharacteristic
118C22×D271C2×D411yescharacteristic
119D1084—22no—
120D1084—21no—
121D1084—21no—
122D1084—21no—
123D1084—42no—
124D1084—42no—
125D1084—42no—
126D1084#4112yescomplemented
127C22⋊D271—42no—
128C22⋊D271—81no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—21no—
136C4×D271—21no—
137C4×D271—42no—
138C4×D271—42no—
139C4×D271—42no—
140C4×D271—42no—
141C4×D271—81no—
142C271⋊C8—21no—
143C271⋊C8—21no—
144C23×C271—21no—
145Q8×C271—21no—
146Q8×C271—21no—
147Q8×C271—42no—
148D4×C271—21no—
149D4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271C2×D411yescharacteristic
154C2×C4×C271C2×D411yescharacteristic
155C2×C4×C271C2411yescharacteristic
156C8×C271—21no—
157C8×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—21no—
165C2×D271—22no—
166C2×D271—21no—
167C2×D271—21no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—84no—
174C2×D271AGL(1,ℤ/8)12yes—
175C4×C271—21no—
176C4×C271—21no—
177C4×C271—41no—
178C4×C271—41no—
179C4×C271—41no—
180C4×C271C2×#4111yescharacteristic
181C4×C271H(5,𝔽2)11yescharacteristic
182Dic271—21no—
183Dic271—21no—
184Dic271—41no—
185Dic271—41no—
186C2×C271—81no—
187C2×C271#54611yescharacteristic
188C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C271#57711yescharacteristic
190D271—22no—
191D271—41no—
192D271—42no—
193D271—81no—
194C271#330611yescharacteristic, complemented, Sylow
195#3306—2711nomaximal, Sylow
196#547—2711no—
197#546—2711no—
198#489D27111yescharacteristic, complemented
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#435—2711no—
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×M16—2711no—
218C22×C8—2711no—
219C4⋊Q8C2×D27111yescharacteristic, complemented
220#124C2×D27111yescharacteristic
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×D4—2711no—
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C22⋊C4—5421no—
237#106—2711no—
238#106—2711no—
239#106—2711no—
240C4⋊C8—2711no—
241#103—2711no—
242#103C2×D27111yescharacteristic, complemented
243#103C2×D27111yescharacteristic, complemented
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102C2×D27111yescharacteristic
250#102C2×D27111yescharacteristic
251C22⋊C8—2711no—
252C22⋊C8—2711no—
253#97C2×D27111yescharacteristic, complemented
254C24—5421no—
255#41—5422no—
256#41—10841no—
257#41—10844no—
258C2×Q8—21no—
259C2×Q8C22×D27111yescharacteristic
260C2×D4—2711no—
261C2×D4—2711no—
262C2×D4—2711no—
263C2×D4—5421no—
264C2×D4—5421no—
265C2×D4—5421no—
266C2×D4—5421no—
267C2×D4—5422no—
268C2×D4—5421no—
269C2×D4—10841no—
270C2×D4C22×D27111yescharacteristic
271C22×C4—2711no—
272C22×C4—2711no—
273C22×C4—5421no—
274C22×C4—5421no—
275C22×C4—5421no—
276C22×C4—5421no—
277C22×C4—5421no—
278SD16—10841no—
279D8—5422no—
280D8—10841no—
281M16—10841no—
282C2×C8—2711no—
283C2×C8—2711no—
284C2×C8—10841no—
285C2×C8C22×D27111yescharacteristic
286C2×C8C22×D27111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—2711no—
289C4⋊C4—2711no—
290C4⋊C4—2711no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4C22×D27111yescharacteristic
295C4⋊C4C22×D27111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—2711no—
299C22⋊C4—2711no—
300C22⋊C4—5421no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C22⋊C4—5421no—
306C42C22×D27111yescharacteristic
307C23—21no—
308C23—2711no—
309C23—2711no—
310C23—5421no—
311C23—5421no—
312C23—10842no—
313C23—10842no—
314C23—21681no—
315Q8—21no—
316Q8—21no—
317Q8—42no—
318D4—21no—
319D4—21no—
320D4—2712no—
321D4—5422no—
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—2711no—
334C2×C4—2711no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—5421no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—10842no—
342C2×C4—10842no—
343C2×C4—10842no—
344C2×C4—10842no—
345C2×C4—21681no—
346C2×C4D4×D27111yescharacteristic
347C2×C4D4×D27111yescharacteristic
348C2×C4C23×D27111yescharacteristic
349C8—21no—
350C8—21no—
351C8—5421no—
352C8—5421no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2712no—
357C22—5421no—
358C22—5421no—
359C22—5421no—
360C22—5422no—
361C22—5421no—
362C22—5421no—
363C22—10841no—
364C22—10841no—
365C22—10841no—
366C22—10841no—
367C22—10841no—
368C22—21684no—
369C22C2×D4×D27111yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—5421no—
376C4—5421no—
377C4—10841no—
378C4—10841no—
379C4#41×D27111yescharacteristic
380C4#2257620611yescharacteristic
381C2—81no—
382C2—5422no—
383C2—10841no—
384C2—10842no—
385C2—21681no—
386C2#3177454011yescharacteristic
387C2AGL(1,ℤ/8)×D27111yescharacteristic
388C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#489×C271C2
socleC22×C271C22×D4
hypercenter#489D271
hypocenterC271#3306
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #31774550C2semidirect
2. #31774540C2semidirect
3. #31774226C2semidirect
4. #31774214C2semidirect
5. #31774184C2semidirect
6. #31774149C2semidirect
7. #31774129C2semidirect
8. #31774109C2semidirect
9. #31774045C2semidirect
10. #31774032C2semidirect
11. #31774010C2semidirect
12. #31773970C2semidirect
13. #31773897C2semidirect
14. #102×D271C2semidirect
15. #489×C271C2semidirect
19. C2×#16864641C22semidirect
22. C2×#16864637C22semidirect
30. #22576135C22semidirect
36. C4×D1084C22semidirect
38. C4⋊C4×D271C22semidirect
40. #22576092C22semidirect
41. #22576091C22semidirect
42. #22576091C22semidirect
43. C4⋊(C271⋊C8)C22semidirect
44. #22576084C22semidirect
46. #22576082C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. C4⋊Q8×C271C22semidirect
54. #103×C271C22semidirect
55. #103×C271C22semidirect
58. #97×C271C22semidirect
126. D1084#41semidirect
194. C271#3306semidirect
198. #489D271semidirect
219. C4⋊Q8C2×D271semidirect
242. #103C2×D271semidirect
243. #103C2×D271semidirect
253. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745502
#317745402
#317742262
#317742142
#317741842
#317741492
#317741292
#317741092
#317740452
#317740322
#317740102
#317739702
#317738972
#102×D2712
#489×C2712
#3306271