Finite Group Databasebeta

← #27180917

Subgroups — #27180917

Order 12928.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265733C211yesmaximal, characteristic, complemented
2C4×#13108073C211yesmaximal, characteristic
3#20265485C211yesmaximal, characteristic, complemented
4#20265481C211yesmaximal, characteristic, complemented
5#20265477C211yesmaximal, characteristic, complemented
6#20265471C211yesmaximal, characteristic, complemented
7#20265465C211yesmaximal, characteristic, complemented
8#20265461C211yesmaximal, characteristic, complemented
9#20265455C211yesmaximal, characteristic, complemented
10#20265214C211yesmaximal, characteristic, complemented
11#20265212C211yesmaximal, characteristic, complemented
12C4⋊C8×D101C211yesmaximal, characteristic, complemented
13#20265170C211yesmaximal, characteristic
14#20265170C211yesmaximal, characteristic
15#478×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×C8×D101C2211yescharacteristic
18C2×C8×D101C2211yescharacteristic
19#15692254C2211yescharacteristic
20#15692254C2211yescharacteristic
21#15692247C2211yescharacteristic
22#15692233—21no—
23#15692233—21no—
24C4×Dic202C2211yescharacteristic
25#15692220—21no—
26#15692220—21no—
27#15692220C2211yescharacteristic
28D4×Dic101C2211yescharacteristic
29C4×C22⋊D101—21no—
30#15692215—21no—
31#15692213C2211yescharacteristic
32#15692213C2211yescharacteristic
33#15692213C2211yescharacteristic
34#15692211—21no—
35C2×C4×Dic101—21no—
36C42×D101C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692186C2211yescharacteristic
43C4⋊#11717579C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692180C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692178C2211yescharacteristic
50#15692176C2211yescharacteristic
51#15692176C2211yescharacteristic
52#125×C101C2211yescharacteristic
53C4×D4×C101C2211yescharacteristic
54#107×C101C2211yescharacteristic
55#106×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#13108073—22no—
60#13108073Q814yescomplemented
61C2×Dic202—21no—
62C2×Dic202—21no—
63C2×Dic202C2311yescharacteristic
64C2×C22⋊D101—21no—
65C22×Dic101—21no—
66C22×Dic101—21no—
67C2×C4×D101D411yescharacteristic
68C2×C4×D101D411yescharacteristic
69C2×C4×D101C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C8×D101—22no—
73C8×D101—22no—
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83C22⋊Dic101—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87C4×Dic101—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101D411yescharacteristic
93C4×Dic101D411yescharacteristic
94C4×Dic101C2311yescharacteristic
95C2×D4×C101C2311yescharacteristic
96C22×C4×C101—21no—
97C2×C8×C101C2311yescharacteristic
98C2×C8×C101C2311yescharacteristic
99C4⋊C4×C101—21no—
100C4⋊C4×C101—21no—
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C42×C101C2311yescharacteristic
106C23×C101—21no—
107C22×D101C2×D411yescharacteristic
108D4×C101—21no—
109D4×C101C2×Q812yes—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—42no—
114C2×C4×C101C2×D411yescharacteristic
115C2×C4×C101C2×D411yescharacteristic
116C2×C4×C101C2411yescharacteristic
117C22⋊D101—24no—
118C2×Dic101—21no—
119C2×Dic101—24no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—42no—
124C2×Dic101C2×D411yescharacteristic
125C2×Dic101C2×D411yescharacteristic
126C2×Dic101C2×D411yescharacteristic
127C4×D101—22no—
128C4×D101—41no—
129C4×D101C2×Q812yes—
130C4×D101#4112yes—
131Dic202—21no—
132Dic202—42no—
133Dic202—42no—
134Dic202C2×Q812yes—
135C8×C101—21no—
136C8×C101—21no—
137#11717579—21no—
138#11717579—21no—
139C22×C101—22no—
140C22×C101—41no—
141C22×C101—41no—
142C22×C101C22×D411yescharacteristic
143C2×D101—21no—
144C2×D101—21no—
145C2×D101C22⋊Q812yes—
146C4×C101—21no—
147C4×C101—21no—
148C4×C101—41no—
149C4×C101—41no—
150C4×C101—41no—
151C4×C101C22×Q811yescharacteristic
152C4×C101C2×#4111yescharacteristic
153Dic101—22no—
154Dic101—21no—
155Dic101—41no—
156Dic101—41no—
157Dic101—41no—
158Dic101C22⋊Q812yes—
159Dic101#13512yes—
160C2×C101—42no—
161C2×C101C2×C22⋊Q811yescharacteristic
162C2×C101C2×#13511yescharacteristic
163C2×C101#57511yescharacteristic
164D101—22no—
165#3076—1011nomaximal, Sylow
166C101#307611yescharacteristic, complemented, Sylow
167#533—1011no—
168C4×#41—1011no—
169#479—1011no—
170#479—1011no—
171#478—1011no—
172#478D10111yescharacteristic, complemented
173#477—1011no—
174#476—1011no—
175#475—1011no—
176#474—1011no—
177#427—1011no—
178#427—1011no—
179C2×C4⋊C8—1011no—
180#416—1011no—
181#416—1011no—
182C2×#41—1011no—
183C22×C8—1011no—
184C22×C8—1011no—
185C4⋊Q8—1011no—
186C4⋊Q8—1011no—
187#125—1011no—
188#125C2×D10111yescharacteristic
189C22⋊Q8—2021no—
190C22⋊Q8—2021no—
191C4×Q8—1011no—
192C4×Q8—1011no—
193C4×Q8—2021no—
194C4×Q8—2021no—
195C4×D4—1011no—
196C4×D4—2021no—
197C4×D4—2021no—
198C4×D4C2×D10111yescharacteristic
199#117—1011no—
200#117—1011no—
201#117—1011no—
202#117—2021no—
203C2×C42—1011no—
204C2×C42—2021no—
205#107—1011no—
206#107—1011no—
207#107—1011no—
208#107C2×D10111yescharacteristic
209#106—1011no—
210#106—1011no—
211#106—1011no—
212#106C2×D10111yescharacteristic
213C4⋊C8—1011no—
214C4⋊C8—1011no—
215C4⋊C8—1011no—
216C4⋊C8C2×D10111yescharacteristic
217#103—1011no—
218#103—1011no—
219#103—1011no—
220#103—1011no—
221#102—1011no—
222#102—1011no—
223#102C2×D10111yescharacteristic
224#102C2×D10111yescharacteristic
225#41—1014no—
226#41—2022no—
227C2×Q8—1011no—
228C2×Q8—2021no—
229C2×Q8—2021no—
230C2×D4—2021no—
231C2×D4C22×D10111yescharacteristic
232C22×C4—21no—
233C22×C4—1011no—
234C22×C4—1011no—
235C22×C4—1011no—
236C22×C4—2021no—
237C22×C4—2021no—
238C2×C8—1011no—
239C2×C8—1011no—
240C2×C8—2022no—
241C2×C8—2022no—
242C2×C8C22×D10111yescharacteristic
243C2×C8C22×D10111yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—1011no—
247C4⋊C4—1011no—
248C4⋊C4—1011no—
249C4⋊C4—2021no—
250C4⋊C4—2021no—
251C4⋊C4—2021no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4C22×D10111yescharacteristic
256C4⋊C4C22×D10111yescharacteristic
257C4⋊C4C22×D10111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—2021no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C42—1011no—
264C42—1011no—
265C42—1011no—
266C42—2021no—
267C42—2021no—
268C42—2021no—
269C42—2021no—
270C42—2021no—
271C42C22×D10111yescharacteristic
272C23—21no—
273C23—1011no—
274Q8—1012no—
275Q8—4042no—
276Q8—4042no—
277Q8—2021no—
278D4—21no—
279D4—2024no—
280D4Q8×D10112yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—1011no—
286C2×C4—1011no—
287C2×C4—1012no—
288C2×C4—1012no—
289C2×C4—1011no—
290C2×C4—4041no—
291C2×C4—4042no—
292C2×C4—2022no—
293C2×C4—2024no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4—2021no—
298C2×C4D4×D10111yescharacteristic
299C2×C4D4×D10111yescharacteristic
300C2×C4C23×D10111yescharacteristic
301C8—21no—
302C8—21no—
303C8—2021no—
304C8—2021no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—1012no—
309C22—2021no—
310C22—2021no—
311C22C2×D4×D10111yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—1012no—
318C4—1012no—
319C4—4041no—
320C4—4041no—
321C4—4041no—
322C4—2022no—
323C4—2021no—
324C4C2×Q8×D10111yescharacteristic
325C4#41×D10111yescharacteristic
326C2—42no—
327C2—2022no—
328C2C22⋊Q8×D10111yescharacteristic
329C2#135×D10111yescharacteristic
330C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#478×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108073C22
hypercenter#478D101
hypocenterC101#3076
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. #20265485C2semidirect
4. #20265481C2semidirect
5. #20265477C2semidirect
6. #20265471C2semidirect
7. #20265465C2semidirect
8. #20265461C2semidirect
9. #20265455C2semidirect
10. #20265214C2semidirect
11. #20265212C2semidirect
12. C4⋊C8×D101C2semidirect
15. #478×C101C2semidirect
60. #13108073Q8semidirect
166. C101#3076semidirect
172. #478D101semidirect

Maximal subgroups

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

subgroupindex
#202657332
C4×#131080732
#202654852
#202654812
#202654772
#202654712
#202654652
#202654612
#202654552
#202652142
#202652122
C4⋊C8×D1012
#202651702
#478×C1012
#3076101