Finite Group Databasebeta

← #27187732

Subgroups — #27187732

Order 12928.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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
1#20266244C212yesmaximal, complemented
2#20266158C212yesmaximal, complemented
3#20266156C212yesmaximal, complemented
4C2×#15692314C214yesmaximal, complemented
5D101⋊#99C214yesmaximal, complemented
6C2×#15692301C211yesmaximal, characteristic
7D4×#87938—28no—
8#15692334C2214yescomplemented
9C2×C22⋊#87938C2218yescomplemented
10#15692314C2218yescomplemented
11#15692314C2214yescomplemented
12#15692313C2214yescomplemented
13#15692301—28no—
14C2×#13108075C411yescharacteristic, complemented
15C2×#13108075C2211yescharacteristic
16C2×#13108074C412yescomplemented
17C2×#13108074C2211yescharacteristic
18C22×D404C411yescharacteristic, complemented
19C22×D404C2212yes—
20C2×D4×D101C411yescharacteristic, complemented
21C2×D4×D101C412yescomplemented
22C2×D4×D101C2211yescharacteristic
23C2×D4×D101C2212yes—
24C2×#41×C101C411yescharacteristic, complemented
25C22×#87938—28no—
26C4⋊#87938—24no—
27C22⋊#87938—28no—
28C22⋊#87938C2318yes—
29C4×#87938—24no—
30C23×D101C2×C412yescomplemented
31C23×D101C2×C412yescomplemented
32C23×D101C2312yes—
33#13108075—24no—
34#13108075—24no—
35#13108074—28no—
36#13108074—24no—
37C2×Dic202C2×C411yescharacteristic, complemented
38C2×D404—28no—
39C2×D404—24no—
40C2×D404C2×C411yescharacteristic
41C2×D404C2×C411yescharacteristic, complemented
42C2×D404C2×C412yescomplemented
43C2×D404C2×C412yescomplemented
44C2×D404C2312yes—
45C2×D404C2311yescharacteristic
46C2×C22⋊D101C2×C412yes—
47C2×C22⋊D101C2×C411yescharacteristic, complemented
48C2×C22⋊D101C2×C412yescomplemented
49C2×C22⋊D101C2311yescharacteristic
50D4×D101—28no—
51D4×D101—24no—
52D4×D101—28no—
53D4×D101D418yes—
54C2×C4×D101C2×C412yescomplemented
55C2×C4×D101C2×C411yescharacteristic, complemented
56C2×C4×D101C2×C412yescomplemented
57C2×C4×D101C2311yescharacteristic
58#41×C101—24no—
59C2×Q8×C101C2×C411yescharacteristic, complemented
60C2×D4×C101C2×C411yescharacteristic
61C2×D4×C101C2×C412yescomplemented
62C22×C4×C101C2×C411yescharacteristic, complemented
63C22×C4×C101C2×C412yescomplemented
64C23×C101C22×C411yescharacteristic
65C23×C101C22×C412yes—
66C22×D101—24no—
67C22×D101—24no—
68C22×D101—82no—
69C22×D101—82no—
70C22×D101—82no—
71C22×D101C22×C412yes—
72C22×D101C22×C411yescharacteristic
73C22×D101C22×C412yes—
74C22×D101C2×D418yes—
75C22×D101C2411yescharacteristic
76C2×#87938—28no—
77C2×#87938—88no—
78Q8×C101—22no—
79D4×C101—24no—
80D4×C101C22⋊C414yes—
81C2×C4×C101—24no—
82C2×C4×C101—22no—
83C2×C4×C101C22×C411yescharacteristic
84C2×C4×C101C22×C412yes—
85C2×C4×C101C22×C411yescharacteristic
86C22⋊D101—24no—
87C22⋊D101—22no—
88C22⋊D101—22no—
89C22⋊D101C22⋊C418yes—
90C2×Dic101C22×C411yescharacteristic
91C2×Dic101C22×C411yescharacteristic
92D404—24no—
93D404—24no—
94D404—24no—
95D404—22no—
96D404—22no—
97D404C22⋊C414yes—
98C4×D101—24no—
99C4×D101—22no—
100C4×D101—24no—
101C4×D101C2×D414yes—
102Dic202—22no—
103C22×C101—21no—
104C22×C101—42no—
105C22×C101—41no—
106C22×C101C2×C22⋊C414yes—
107C22×C101C23×C411yescharacteristic
108C2×D101—21no—
109C2×D101—22no—
110C2×D101—48no—
111C2×D101—42no—
112C2×D101—41no—
113C2×D101—42no—
114C2×D101—41no—
115C2×D101—84no—
116C2×D101—84no—
117C2×D101C2×C22⋊C414yes—
118C2×D101C22×D412yes—
119#87938—48no—
120C4×C101—21no—
121C4×C101—22no—
122C4×C101C2×C22⋊C412yes—
123Dic101—21no—
124Dic101C2×C22⋊C412yes—
125C2×C101—21no—
126C2×C101—24no—
127C2×C101—41no—
128C2×C101C22×C22⋊C411yescharacteristic
129D101—22no—
130D101—24no—
131D101—42no—
132D101—41no—
133#2883—1011nomaximal, Sylow
134C101#288311yescharacteristic, complemented, Sylow
135C2×H(5,𝔽2)—1011no—
136#518—1012no—
137#410—1012no—
138#410—1012no—
139C2×#99—1014no—
140C2×#99—1014no—
141H(5,𝔽2)—2028no—
142C2×#41—1011no—
143C2×#41—1011no—
144C2×#41—1012no—
145C2×#41—1011no—
146C2×#41#8793811yescharacteristic, complemented
147C22×D4—1011no—
148C22×D4—1011no—
149C22×D4—1012no—
150C22×D4—1012no—
151C22×D4—1011no—
152C22×D4—1012no—
153C4×D4—2028no—
154#117—1014no—
155C2×C22⋊C4—1018no—
156#99—1014no—
157#99—1018no—
158#99—1014no—
159C24—1012no—
160C24—1012no—
161C24—1012no—
162#41—24no—
163#41—2024no—
164#41—2028no—
165#41—2024no—
166#41—2024no—
167C2×Q8—1011no—
168C2×Q8C2×#8793811yescharacteristic, complemented
169C2×D4—1012no—
170C2×D4—1018no—
171C2×D4—1011no—
172C2×D4—1011no—
173C2×D4—1012no—
174C2×D4—1011no—
175C2×D4—1012no—
176C2×D4—1011no—
177C2×D4—1012no—
178C2×D4—1012no—
179C2×D4—1011no—
180C2×D4—2024no—
181C2×D4—2028no—
182C2×D4—2028no—
183C2×D4—2028no—
184C2×D4—2024no—
185C2×D4C2×#8793811yescharacteristic
186C2×D4C2×#8793812yescomplemented
187C22×C4—1011no—
188C22×C4—1012no—
189C22×C4—1012no—
190C22×C4—1011no—
191C22×C4—2028no—
192C22×C4C2×#8793812yescomplemented
193C22×C4C2×#8793811yescharacteristic, complemented
194C4⋊C4—2024no—
195C22⋊C4—1018no—
196C22⋊C4—2028no—
197C42—2024no—
198C23—8082no—
199C23—8082no—
200C23—8082no—
201C23—1018no—
202C23—1011no—
203C23—1012no—
204C23—1012no—
205C23—1011no—
206C23—2024no—
207C23—2024no—
208C23C22×#8793811yescharacteristic
209C23C22×#8793812yes—
210Q8—22no—
211Q8—2022no—
212D4—24no—
213D4—1018no—
214D4—1014no—
215D4—2022no—
216D4—2024no—
217D4—2024no—
218D4—2024no—
219D4—2022no—
220D4—2022no—
221D4—2024no—
222D4—2022no—
223D4C22⋊#8793814yes—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—8088no—
227C2×C4—1011no—
228C2×C4—1014no—
229C2×C4—1011no—
230C2×C4—2024no—
231C2×C4—2024no—
232C2×C4—2028no—
233C2×C4—2022no—
234C2×C4C22×#8793812yes—
235C2×C4C22×#8793811yescharacteristic
236C2×C4C22×#8793811yescharacteristic
237C22—21no—
238C22—42no—
239C22—41no—
240C22—8084no—
241C22—8084no—
242C22—1012no—
243C22—1014no—
244C22—4041no—
245C22—4042no—
246C22—4048no—
247C22—4041no—
248C22—4042no—
249C22—2022no—
250C22—2021no—
251C22C2×C22⋊#8793814yes—
252C22C23×#8793811yescharacteristic
253C4—21no—
254C4—22no—
255C4—1012no—
256C4—4048no—
257C4—2021no—
258C4C2×C22⋊#8793812yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—4041no—
263C2—4042no—
264C2—2022no—
265C2—2024no—
266C2C22×C22⋊#8793811yescharacteristic
subgroupG/H
centerC2C22×C22⋊#87938
commutatorC22×C101C23×C4
FrattiniC22C23×#87938
FittingC2×#41×C101C4
socleC2×C101C22×C22⋊C4
prime radicalC2×#15692301C2
cosocleC22×D101C24
hypercenterC2×#41#87938
hypocenterC101#2883
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20266244C2semidirect
2. #20266158C2semidirect
3. #20266156C2semidirect
4. C2×#15692314C2semidirect
5. D101⋊#99C2semidirect
8. #15692334C22semidirect
9. C2×C22⋊#87938C22semidirect
10. #15692314C22semidirect
11. #15692314C22semidirect
12. #15692313C22semidirect
14. C2×#13108075C4semidirect
16. C2×#13108074C4semidirect
18. C22×D404C4semidirect
20. C2×D4×D101C4semidirect
21. C2×D4×D101C4semidirect
24. C2×#41×C101C4semidirect
30. C23×D101C2×C4semidirect
31. C23×D101C2×C4semidirect
37. C2×Dic202C2×C4semidirect
41. C2×D404C2×C4semidirect
42. C2×D404C2×C4semidirect
43. C2×D404C2×C4semidirect
47. C2×C22⋊D101C2×C4semidirect
48. C2×C22⋊D101C2×C4semidirect
54. C2×C4×D101C2×C4semidirect
55. C2×C4×D101C2×C4semidirect
56. C2×C4×D101C2×C4semidirect
59. C2×Q8×C101C2×C4semidirect
61. C2×D4×C101C2×C4semidirect
62. C22×C4×C101C2×C4semidirect
63. C22×C4×C101C2×C4semidirect
134. C101#2883semidirect
146. C2×#41#87938semidirect
168. C2×Q8C2×#87938semidirect
186. C2×D4C2×#87938semidirect
192. C22×C4C2×#87938semidirect
193. C22×C4C2×#87938semidirect

Maximal subgroups

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

subgroupindex
#202662442
#202661582
#202661562
C2×#156923142
D101⋊#992
C2×#156923012
#2883101