Finite Group Databasebeta

← #27180927

Subgroups — #27180927

Order 12928.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265733C211yesmaximal, characteristic, complemented
2C4×D4×D101C211yesmaximal, characteristic
3#20265478C211yesmaximal, characteristic, complemented
4#20265477C211yesmaximal, characteristic, complemented
5#20265474C211yesmaximal, characteristic, complemented
6#20265466C211yesmaximal, characteristic, complemented
7#20265465C211yesmaximal, characteristic, complemented
8#20265456C211yesmaximal, characteristic, complemented
9#20265455C211yesmaximal, characteristic, complemented
10#20265218C211yesmaximal, characteristic, complemented
11#20265216C211yesmaximal, characteristic, complemented
12#20265199C211yesmaximal, characteristic, complemented
13#20265180C211yesmaximal, characteristic
14#20265180C211yesmaximal, characteristic
15#478×C101C211yesmaximal, characteristic, complemented
16C2×D4×D101C2211yescharacteristic
17C22×C4×D101—21no—
18C2×#13108057C2211yescharacteristic
19C2×#13108057C2211yescharacteristic
20#15692254C2211yescharacteristic, complemented
21#15692254C2211yescharacteristic, complemented
22#15692247C2211yescharacteristic, complemented
23#15692233—21no—
24#15692233—21no—
25#15692220—21no—
26#15692220—21no—
27C4×D404C2211yescharacteristic
28D4×Dic101C2211yescharacteristic
29#15692217C2211yescharacteristic
30C4×C22⋊D101—21no—
31#15692215—21no—
32#15692213C2211yescharacteristic
33#15692213C2211yescharacteristic
34C4⋊C4×D101C2211yescharacteristic
35C22⋊C4×D101—21no—
36C42×D101C2211yescharacteristic
37#15692189C2211yescharacteristic, complemented
38#15692188C2211yescharacteristic, complemented
39#15692188C2211yescharacteristic, complemented
40#15692187C2211yescharacteristic, complemented
41#15692186C2211yescharacteristic, complemented
42#15692186C2211yescharacteristic, complemented
43C4⋊#11717579C2211yescharacteristic, complemented
44#15692184C2211yescharacteristic, complemented
45#15692184C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic
50#15692175C2211yescharacteristic
51#15692175C2211yescharacteristic
52#125×C101C2211yescharacteristic, complemented
53C4×D4×C101C2211yescharacteristic
54#107×C101C2211yescharacteristic, complemented
55#106×C101C2211yescharacteristic, complemented
56C4⋊C8×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C23×D101—21no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64D4×D101—22no—
65D4×D101Q814yescomplemented
66C22×Dic101—21no—
67C2×C4×D101—21no—
68C2×C4×D101—21no—
69C2×C4×D101—21no—
70C2×C4×D101—44no—
71C2×C4×D101D411yescharacteristic
72C2×C4×D101D411yescharacteristic
73C2×C4×D101C2311yescharacteristic
74#13108057—22no—
75#13108057—22no—
76C2×#11717579C2311yescharacteristic
77C2×#11717579C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86C22⋊Dic101—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101D411yescharacteristic
94C4×Dic101D411yescharacteristic
95C4×Dic101C2311yescharacteristic
96C2×D4×C101C2311yescharacteristic
97C22×C4×C101—21no—
98C2×C8×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C4⋊C4×C101—21no—
101C4⋊C4×C101—21no—
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C22⋊C4×C101—21no—
106C42×C101C2311yescharacteristic
107C23×C101—21no—
108C22×D101—21no—
109C22×D101—24no—
110C22×D101—42no—
111C22×D101—42no—
112C22×D101C2×D411yescharacteristic
113D4×C101—21no—
114D4×C101C2×Q812yes—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—42no—
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2×D411yescharacteristic
121C2×C4×C101C2411yescharacteristic
122C22⋊D101—24no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—42no—
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129C2×Dic101C2×D411yescharacteristic
130D404—21no—
131D404C2×Q812yes—
132C4×D101—22no—
133C4×D101—41no—
134C4×D101—42no—
135C4×D101—42no—
136C4×D101—42no—
137C4×D101C2×Q812yes—
138C4×D101#4112yes—
139Dic202—42no—
140Dic202—42no—
141C8×C101—21no—
142C8×C101—21no—
143#11717579—21no—
144#11717579—21no—
145C22×C101—22no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101C22×D411yescharacteristic
149C2×D101—21no—
150C2×D101—21no—
151C2×D101—22no—
152C2×D101—48no—
153C2×D101—41no—
154C2×D101—41no—
155C2×D101C22⋊Q812yes—
156C4×C101—21no—
157C4×C101—21no—
158C4×C101—41no—
159C4×C101—41no—
160C4×C101—41no—
161C4×C101C22×Q811yescharacteristic
162C4×C101C2×#4111yescharacteristic
163Dic101—21no—
164Dic101—41no—
165Dic101—41no—
166Dic101—41no—
167Dic101C22⋊Q812yes—
168Dic101AGL(1,ℤ/8)12yes—
169C2×C101—42no—
170C2×C101C2×C22⋊Q811yescharacteristic
171C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C101#57511yescharacteristic
173D101—22no—
174D101—42no—
175#3077—1011nomaximal, Sylow
176C101#307711yescharacteristic, complemented, Sylow
177#533—1011no—
178C2×C4×D4—1011no—
179#478—1011no—
180#478—1011no—
181#478—1011no—
182#478D10111yescharacteristic, complemented
183#476—1011no—
184#476—1011no—
185#474—1011no—
186#474—1011no—
187#428—1011no—
188#428—1011no—
189#423—1011no—
190#418—1011no—
191#418—1011no—
192C22×D4—1011no—
193C23×C4—2021no—
194C2×M16—1011no—
195C2×M16—1011no—
196C4⋊Q8—1011no—
197C4⋊Q8—1011no—
198#125—1011no—
199#125C2×D10111yescharacteristic, complemented
200C22⋊Q8—2021no—
201C22⋊Q8—2021no—
202C4×Q8—2021no—
203C4×Q8—2021no—
204C4×D4—1011no—
205C4×D4—1011no—
206C4×D4—1011no—
207C4×D4—2021no—
208C4×D4—2021no—
209C4×D4C2×D10111yescharacteristic
210#117—1011no—
211#117—1011no—
212C2×C4⋊C4—1011no—
213C2×C22⋊C4—2021no—
214C2×C42—1011no—
215#107—1011no—
216#107—1011no—
217#107—1011no—
218#107C2×D10111yescharacteristic, complemented
219#106—1011no—
220#106—1011no—
221#106—1011no—
222#106C2×D10111yescharacteristic, complemented
223C4⋊C8—1011no—
224C4⋊C8—1011no—
225C4⋊C8—1011no—
226C4⋊C8C2×D10111yescharacteristic, complemented
227#102—1011no—
228#102—1011no—
229#102—1011no—
230#102—1011no—
231#102—1011no—
232#102—1011no—
233#102C2×D10111yescharacteristic
234#102C2×D10111yescharacteristic
235C24—2021no—
236C2×Q8—2021no—
237C2×Q8—2021no—
238C2×D4—1014no—
239C2×D4—1011no—
240C2×D4—2022no—
241C2×D4—2021no—
242C2×D4C22×D10111yescharacteristic
243C22×C4—21no—
244C22×C4—1011no—
245C22×C4—1011no—
246C22×C4—1011no—
247C22×C4—4044no—
248C22×C4—2021no—
249C22×C4—2021no—
250C22×C4—2021no—
251C22×C4—2021no—
252M16—2022no—
253M16—2022no—
254C2×C8—1011no—
255C2×C8—1011no—
256C2×C8C22×D10111yescharacteristic
257C2×C8C22×D10111yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—1011no—
261C4⋊C4—1011no—
262C4⋊C4—1011no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4C22×D10111yescharacteristic
269C4⋊C4C22×D10111yescharacteristic
270C4⋊C4C22×D10111yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4—2021no—
277C42—1011no—
278C42—1011no—
279C42—1011no—
280C42—2021no—
281C42—2021no—
282C42C22×D10111yescharacteristic
283C23—21no—
284C23—1011no—
285C23—4042no—
286C23—4042no—
287C23—2024no—
288C23—2021no—
289Q8—4042no—
290Q8—4042no—
291D4—21no—
292D4—1012no—
293D4—2024no—
294D4—2021no—
295D4Q8×D10112yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—1011no—
301C2×C4—1011no—
302C2×C4—1012no—
303C2×C4—1012no—
304C2×C4—1011no—
305C2×C4—4042no—
306C2×C4—4041no—
307C2×C4—4042no—
308C2×C4—4042no—
309C2×C4—4042no—
310C2×C4—2022no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4D4×D10111yescharacteristic
315C2×C4D4×D10111yescharacteristic
316C2×C4C23×D10111yescharacteristic
317C8—21no—
318C8—21no—
319C8—2021no—
320C8—2021no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—1012no—
325C22—4048no—
326C22—4041no—
327C22—4041no—
328C22—2022no—
329C22—2021no—
330C22—2021no—
331C22C2×D4×D10111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—1012no—
338C4—1012no—
339C4—4041no—
340C4—4041no—
341C4—4041no—
342C4—2021no—
343C4C2×Q8×D10111yescharacteristic
344C4#41×D10111yescharacteristic
345C2—42no—
346C2—4042no—
347C2—2022no—
348C2C22⋊Q8×D10111yescharacteristic
349C2AGL(1,ℤ/8)×D10111yescharacteristic
350C2#2026597411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#478×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
hypercenter#478D101
hypocenterC101#3077
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20265733C2semidirect
3. #20265478C2semidirect
4. #20265477C2semidirect
5. #20265474C2semidirect
6. #20265466C2semidirect
7. #20265465C2semidirect
8. #20265456C2semidirect
9. #20265455C2semidirect
10. #20265218C2semidirect
11. #20265216C2semidirect
12. #20265199C2semidirect
15. #478×C101C2semidirect
20. #15692254C22semidirect
21. #15692254C22semidirect
22. #15692247C22semidirect
37. #15692189C22semidirect
38. #15692188C22semidirect
39. #15692188C22semidirect
40. #15692187C22semidirect
41. #15692186C22semidirect
42. #15692186C22semidirect
43. C4⋊#11717579C22semidirect
44. #15692184C22semidirect
45. #15692184C22semidirect
52. #125×C101C22semidirect
54. #107×C101C22semidirect
55. #106×C101C22semidirect
56. C4⋊C8×C101C22semidirect
65. D4×D101Q8semidirect
176. C101#3077semidirect
182. #478D101semidirect
199. #125C2×D101semidirect
218. #107C2×D101semidirect
222. #106C2×D101semidirect
226. C4⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657332
C4×D4×D1012
#202654782
#202654772
#202654742
#202654662
#202654652
#202654562
#202654552
#202652182
#202652162
#202651992
#202651802
#478×C1012
#3077101