Finite Group Databasebeta

← #27179591

Subgroups — #27179591

Order 12928.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692301C211yesmaximal, characteristic
2#20265651C212yesmaximal, complemented
3#20265151C211yesmaximal, characteristic, complemented
4#20265150C212yesmaximal, complemented
5#99×D101C218yesmaximal, complemented
6#410×C101C211yesmaximal, characteristic, complemented
7#15692301—24no—
8#15692301—24no—
9C2×#13108075C412yescomplemented
10C2×#13108074C412yescomplemented
11C2×#13108074C2211yescharacteristic
12C22×D404C412yescomplemented
13C22×D404C2211yescharacteristic
14C2×D4×D101C412yescomplemented
15C2×D4×D101C2214yes—
16C4×D404—24no—
17#15692217—24no—
18#15692214C2212yescomplemented
19C22⋊C4×D101C2218yescomplemented
20#15692170C2214yescomplemented
21#15692169C2218yescomplemented
22C2×#41×C101C2211yescharacteristic
23#117×C101C2212yescomplemented
24#99×C101C2214yescomplemented
25C23×D101C2×C414yescomplemented
26C23×D101C2312yes—
27#13108075—28no—
28#13108074—22no—
29#13108074—22no—
30#13108074—24no—
31#13108074—24no—
32C2×Dic202C2×C411yescharacteristic, complemented
33C2×D404—24no—
34C2×D404—24no—
35C2×D404C2×C412yes—
36C2×D404C2×C412yes—
37C2×D404C2×C414yescomplemented
38C2×D404C2×C411yescharacteristic, complemented
39C2×D404D414yes—
40C2×D404D414yes—
41C2×C22⋊D101C2×C414yescomplemented
42C2×C22⋊D101C2312yes—
43D4×D101—216no—
44D4×D101—28no—
45C2×C4×D101—28no—
46C2×C4×D101C2×C412yescomplemented
47C2×C4×D101C2×C414yescomplemented
48C4⋊Dic101—22no—
49C22⋊Dic101C2314yes—
50#13108051—28no—
51C4×Dic101—22no—
52#41×C101—22no—
53#41×C101—22no—
54C2×Q8×C101C2×C411yescharacteristic, complemented
55C2×D4×C101C2×C411yescharacteristic, complemented
56C2×D4×C101C2312yes—
57C22×C4×C101C2×C412yescomplemented
58C22×C4×C101C2311yescharacteristic
59C4⋊C4×C101—22no—
60C22⋊C4×C101C2314yes—
61C42×C101—22no—
62C23×C101C22×C412yes—
63C23×C101C2411yescharacteristic
64C22×D101—28no—
65C22×D101—82no—
66C22×D101—84no—
67C22×D101C22×C412yes—
68C22×D101C22×C414yes—
69C22×D101C2×D418yes—
70Q8×C101—22no—
71D4×C101—24no—
72D4×C101—22no—
73C2×C4×C101—22no—
74C2×C4×C101—22no—
75C2×C4×C101—24no—
76C2×C4×C101C22×C412yes—
77C2×C4×C101C22×C411yescharacteristic
78C2×C4×C101C22×C411yescharacteristic
79C2×C4×C101C2×D412yes—
80C2×C4×C101C2×D412yes—
81C22⋊D101—24no—
82C22⋊D101—28no—
83C2×Dic101—24no—
84C2×Dic101C22×C412yes—
85D404—28no—
86D404—21no—
87D404—21no—
88D404C22⋊C418yes—
89D404C22⋊C414yes—
90D404C22⋊C414yes—
91C4×D101—28no—
92C4×D101—24no—
93C4×D101—88no—
94Dic202—21no—
95Dic202—21no—
96C22×C101—22no—
97C22×C101—41no—
98C22×C101—42no—
99C22×C101C23×C411yescharacteristic
100C22×C101C22×D412yes—
101C2×D101—22no—
102C2×D101—48no—
103C2×D101—44no—
104C2×D101—42no—
105C2×D101—88no—
106C2×D101C2×C22⋊C418yes—
107C4×C101—22no—
108C4×C101—44no—
109C4×C101C2×C22⋊C411yescharacteristic
110C4×C101C2×C22⋊C411yescharacteristic
111C4×C101C2×C22⋊C412yes—
112Dic101—22no—
113Dic101—44no—
114C2×C101—22no—
115C2×C101—21no—
116C2×C101—42no—
117C2×C101C22×C22⋊C411yescharacteristic
118D101—28no—
119D101—42no—
120#2883—1011nomaximal, Sylow
121C101#288311yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—1011no—
123#518—1012no—
124#410—1011no—
125#410—1012no—
126#410D10111yescharacteristic, complemented
127C2×#99—1018no—
128H(5,𝔽2)—2024no—
129H(5,𝔽2)—2024no—
130C2×#41—1012no—
131C2×#41—1012no—
132C2×#41—1011no—
133C2×#41C2×D10111yescharacteristic
134C22×D4—1012no—
135C22×D4—1014no—
136C22×D4—1012no—
137C22×D4—1011no—
138C4×D4—2024no—
139C4×D4—2024no—
140#117—1012no—
141#117C2×D10112yescomplemented
142C2×C22⋊C4—1018no—
143#99—1014no—
144#99—1018no—
145#99C2×D10114yescomplemented
146C24—1012no—
147C24—1014no—
148#41—22no—
149#41—22no—
150#41—2024no—
151#41—2024no—
152#41—2028no—
153#41—2022no—
154#41—2022no—
155C2×Q8—1011no—
156C2×Q8C4×D10111yescharacteristic, complemented
157C2×D4—1014no—
158C2×D4—1012no—
159C2×D4—1012no—
160C2×D4—1012no—
161C2×D4—1014no—
162C2×D4—1011no—
163C2×D4—1014no—
164C2×D4—1014no—
165C2×D4—2024no—
166C2×D4—2024no—
167C2×D4—2028no—
168C2×D4—20216no—
169C2×D4C4×D10111yescharacteristic, complemented
170C2×D4C22×D10112yes—
171C22×C4—1014no—
172C22×C4—1012no—
173C22×C4—2028no—
174C22×C4C4×D10112yescomplemented
175C22×C4C22×D10111yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—2022no—
178C22⋊C4—1014no—
179C22⋊C4—2028no—
180C22⋊C4C22×D10114yes—
181C42—22no—
182C42—2022no—
183C23—8084no—
184C23—8082no—
185C23—1014no—
186C23—1018no—
187C23—1012no—
188C23—2028no—
189C23C2×C4×D10112yes—
190C23C23×D10111yescharacteristic
191Q8—22no—
192Q8—2021no—
193Q8—2021no—
194D4—24no—
195D4—22no—
196D4—1014no—
197D4—1014no—
198D4—1018no—
199D4—2028no—
200D4—2028no—
201D4—2024no—
202D4—2021no—
203D4—2021no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—8088no—
208C2×C4—1012no—
209C2×C4—2028no—
210C2×C4—2024no—
211C2×C4—2024no—
212C2×C4C2×C4×D10112yes—
213C2×C4C2×C4×D10111yescharacteristic
214C2×C4C2×C4×D10111yescharacteristic
215C2×C4D4×D10112yes—
216C2×C4D4×D10112yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—8088no—
221C22—1018no—
222C22—4042no—
223C22—4048no—
224C22—4044no—
225C22—2022no—
226C22C22×C4×D10111yescharacteristic
227C22C2×D4×D10112yes—
228C4—22no—
229C4—44no—
230C4—4044no—
231C4—2022no—
232C4C22⋊C4×D10111yescharacteristic
233C4C22⋊C4×D10111yescharacteristic
234C4C22⋊C4×D10112yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—4042no—
239C2—2028no—
240C2C2×C22⋊C4×D10111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D101
commutatorC22×C101C23×C4
FrattiniC23C23×D101
Fitting#410×C101C2
socleC2×C101C22×C22⋊C4
prime radicalC2×#15692301C2
cosocleC23×C101C24
hypercenter#410D101
hypocenterC101#2883
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20265651C2semidirect
3. #20265151C2semidirect
4. #20265150C2semidirect
5. #99×D101C2semidirect
6. #410×C101C2semidirect
9. C2×#13108075C4semidirect
10. C2×#13108074C4semidirect
12. C22×D404C4semidirect
14. C2×D4×D101C4semidirect
18. #15692214C22semidirect
19. C22⋊C4×D101C22semidirect
20. #15692170C22semidirect
21. #15692169C22semidirect
23. #117×C101C22semidirect
24. #99×C101C22semidirect
25. C23×D101C2×C4semidirect
32. C2×Dic202C2×C4semidirect
37. C2×D404C2×C4semidirect
38. C2×D404C2×C4semidirect
41. C2×C22⋊D101C2×C4semidirect
46. C2×C4×D101C2×C4semidirect
47. C2×C4×D101C2×C4semidirect
54. C2×Q8×C101C2×C4semidirect
55. C2×D4×C101C2×C4semidirect
57. C22×C4×C101C2×C4semidirect
121. C101#2883semidirect
126. #410D101semidirect
141. #117C2×D101semidirect
145. #99C2×D101semidirect
156. C2×Q8C4×D101semidirect
169. C2×D4C4×D101semidirect
174. C22×C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923012
#202656512
#202651512
#202651502
#99×D1012
#410×C1012
#2883101