Finite Group Databasebeta

← #46846551

Subgroups — #46846551

Order 34688.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774520C211yesmaximal, characteristic, complemented
2#31774371C211yesmaximal, characteristic, complemented
3#31774221C211yesmaximal, characteristic, complemented
4#31774186C211yesmaximal, characteristic, complemented
5#31774176C211yesmaximal, characteristic, complemented
6#31774115C211yesmaximal, characteristic, complemented
7#31774115C211yesmaximal, characteristic, complemented
8#31774047C211yesmaximal, characteristic, complemented
9#31774033C211yesmaximal, characteristic
10#31774008C211yesmaximal, characteristic, complemented
11#31774000C211yesmaximal, characteristic, complemented
12#31773982C211yesmaximal, characteristic, complemented
13#31773979C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#485×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×D2168C2211yescharacteristic
20C2×D2168C2211yescharacteristic
21#22576159C2211yescharacteristic, complemented
22#22576155—21no—
23#22576149C2211yescharacteristic, complemented
24#22576146—21no—
25#22576142—21no—
26#22576140—21no—
27#22576134C2211yescharacteristic
28Q8×Dic271C2211yescharacteristic, complemented
29C4×D1084—21no—
30C4×D1084—21no—
31#22576122C2211yescharacteristic
32#22576122C2211yescharacteristic
33#22576122C2211yescharacteristic
34#22576122C2211yescharacteristic
35#22576122C2211yescharacteristic, complemented
36#22576119C2211yescharacteristic
37C22⋊C4×D271—21no—
38C22⋊C4×D271—21no—
39#22576093C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576080C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic, complemented
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22⋊Q8×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic
59C23×D271—21no—
60C2×Dic542—21no—
61C2×D1084—22no—
62C2×D1084—22no—
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73#16864641—22no—
74#16864641—22no—
75D2168—22no—
76D2168—22no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C4⋊Dic271—21no—
80C4⋊Dic271D411yescharacteristic
81C4⋊Dic271D411yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84C22⋊Dic271D411yescharacteristic
85C22⋊Dic271D411yescharacteristic
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C2×C8×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C4⋊C4×C271—21no—
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C22×D271—21no—
109C22×D271—21no—
110C22×D271—24no—
111C22×D271—21no—
112C22×D271—81no—
113C22×D271—81no—
114Dic542—81no—
115D1084—24no—
116D1084—21no—
117D1084—21no—
118D1084—21no—
119D1084—21no—
120D1084#4112yes—
121D1084#4112yes—
122C22⋊D271—81no—
123C2×Dic271—21no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—81no—
129C4×D271—81no—
130C4×D271—81no—
131C4×D271—81no—
132C4×D271—81no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271C2×D411yescharacteristic
136Q8×C271—21no—
137Q8×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271—21no—
148C22×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271C22×D411yescharacteristic
152C2×D271—22no—
153C2×D271—22no—
154C2×D271—41no—
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—84no—
162C4×C271—21no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271—41no—
166C4×C271—41no—
167C4×C271C2×#4111yescharacteristic
168C4×C271C2×#4111yescharacteristic
169Dic271—21no—
170Dic271—21no—
171Dic271—21no—
172Dic271—21no—
173Dic271—41no—
174C2×C271—41no—
175C2×C271#52511yescharacteristic
176C2×C271#57611yescharacteristic
177C2×C271#57711yescharacteristic
178D271—42no—
179D271—42no—
180D271—81no—
181C271#311811yescharacteristic, complemented, Sylow
182#3118—2711nomaximal, Sylow
183#542—2711no—
184#518—2711no—
185#485D27111yescharacteristic, complemented
186#482—2711no—
187#476—2711no—
188#474—2711no—
189#463—2711no—
190#463—2711no—
191#450—2711no—
192#447—2711no—
193#442—2711no—
194#440—2711no—
195C4×SD16—2711no—
196C4×D8—2711no—
197#433—2711no—
198C22×D4—2711no—
199C2×SD16—2711no—
200C2×SD16—2711no—
201C2×D8—2711no—
202C2×D8—2711no—
203C4⋊Q8—2711no—
204#126—5421no—
205#124—2711no—
206#124—5421no—
207#123—5421no—
208C22⋊Q8—5421no—
209C22⋊Q8C2×D27111yescharacteristic
210#121—2711no—
211C4×Q8—2711no—
212C4×D4—2711no—
213C4×D4—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—2711no—
217C4×D4—5421no—
218C4×D4—5421no—
219#117—2711no—
220#117C2×D27111yescharacteristic
221C2×C22⋊C4—5421no—
222C2×C22⋊C4—5421no—
223#107—2711no—
224#107C2×D27111yescharacteristic, complemented
225#106—2711no—
226#106C2×D27111yescharacteristic, complemented
227C4⋊C8—2711no—
228C4⋊C8—2711no—
229#103C2×D27111yescharacteristic, complemented
230#103C2×D27111yescharacteristic, complemented
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102—2711no—
236#102—2711no—
237C22⋊C8—2711no—
238C22⋊C8C2×D27111yescharacteristic
239#97—2711no—
240C4×C8—2711no—
241C24—5421no—
242C2×Q8—5421no—
243C2×Q8C22×D27111yescharacteristic
244C2×D4—2711no—
245C2×D4—2711no—
246C2×D4—2711no—
247C2×D4—5422no—
248C2×D4—5422no—
249C2×D4—5421no—
250C2×D4—5421no—
251C22×C4—5421no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4C22×D27111yescharacteristic
257SD16—5422no—
258SD16—5422no—
259D8—5422no—
260D8—5422no—
261C2×C8—2711no—
262C2×C8—2711no—
263C2×C8C22×D27111yescharacteristic
264C2×C8C22×D27111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4C22×D27111yescharacteristic
272C4⋊C4C22×D27111yescharacteristic
273C4⋊C4C22×D27111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—2711no—
277C22⋊C4—2711no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—21no—
287C42—2711no—
288C42—2711no—
289C42—5421no—
290C23—5424no—
291C23—5421no—
292C23—5421no—
293C23—5421no—
294C23—21681no—
295C23—21681no—
296C23D4×D27111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—21681no—
300D4—2712no—
301D4—2712no—
302D4—5424no—
303D4—5421no—
304D4—5421no—
305D4—5421no—
306D4—5421no—
307D4—21681no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—2711no—
315C2×C4—2711no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—5421no—
319C2×C4—21681no—
320C2×C4—21681no—
321C2×C4—21681no—
322C2×C4—21681no—
323C2×C4—21681no—
324C2×C4D4×D27111yescharacteristic
325C2×C4C23×D27111yescharacteristic
326C8—21no—
327C8—21no—
328C8—5421no—
329C8—5421no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—5422no—
334C22—5422no—
335C22—10841no—
336C22—10841no—
337C22—10841no—
338C22—10841no—
339C22—10841no—
340C22—10841no—
341C22—10841no—
342C22—21684no—
343C22C2×D4×D27111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—5421no—
350C4—5421no—
351C4—5421no—
352C4—5421no—
353C4—10841no—
354C4#41×D27111yescharacteristic
355C4#41×D27111yescharacteristic
356C2—41no—
357C2—10842no—
358C2—10842no—
359C2—21681no—
360C2#3177440511yescharacteristic
361C2#3177469911yescharacteristic
362C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#485×C271C2
socleC22×C271C22×D4
prime radical#31774033C2
hypercenter#485D271
hypocenterC271#3118
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #31774520C2semidirect
2. #31774371C2semidirect
3. #31774221C2semidirect
4. #31774186C2semidirect
5. #31774176C2semidirect
6. #31774115C2semidirect
7. #31774115C2semidirect
8. #31774047C2semidirect
10. #31774008C2semidirect
11. #31774000C2semidirect
12. #31773982C2semidirect
13. #31773979C2semidirect
14. #31773955C2semidirect
15. #485×C271C2semidirect
21. #22576159C22semidirect
23. #22576149C22semidirect
28. Q8×Dic271C22semidirect
35. #22576122C22semidirect
39. #22576093C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. Dic271⋊C8C22semidirect
47. #22576080C22semidirect
48. #22576080C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
56. #103×C271C22semidirect
57. #103×C271C22semidirect
181. C271#3118semidirect
185. #485D271semidirect
224. #107C2×D271semidirect
226. #106C2×D271semidirect
229. #103C2×D271semidirect
230. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745202
#317743712
#317742212
#317741862
#317741762
#317741152
#317740472
#317740332
#317740082
#317740002
#317739822
#317739792
#317739552
#485×C2712
#3118271