Finite Group Databasebeta

← #27174447

Subgroups — #27174447

Order 12928.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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×#15692223C212yesmaximal, complemented
2C22×#13108051C211yesmaximal, characteristic, complemented
3#20265088C212yesmaximal, complemented
4#20265085C212yesmaximal, complemented
5#20265057C212yesmaximal, complemented
6#20265056C214yesmaximal, complemented
7#20265040C211yesmaximal, characteristic, complemented
8C22×#41×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic
10C22×C22⋊D101C2214yes—
11C23×Dic101C2211yescharacteristic
12#15692223—28no—
13C2×C22⋊Dic101C2212yes—
14C2×C22⋊Dic101C2214yescomplemented
15C2×#13108051—24no—
16C2×#13108051C2212yes—
17C2×#13108051C2214yes—
18C2×#13108051C2212yes—
19#15692162C2212yescomplemented
20#15692162C2212yescomplemented
21#15692162C2214yescomplemented
22C2×#41×C101—28no—
23C2×#41×C101—24no—
24C22×Q8×C101C2211yescharacteristic, complemented
25C22×D4×C101C2212yes—
26C22×D4×C101C2211yescharacteristic, complemented
27C23×C4×C101C2211yescharacteristic
28C23×C4×C101C2212yescomplemented
29C23×D101—22no—
30C23×D101—24no—
31C23×D101—24no—
32C23×D101—42no—
33C23×D101C2312yes—
34C2×C22⋊D101—28no—
35C2×C22⋊D101—28no—
36C2×C22⋊D101—44no—
37C22×Dic101—22no—
38C22×Dic101—42no—
39C22×Dic101C2312yes—
40C22×Dic101C2314yes—
41C22⋊Dic101—24no—
42C22⋊Dic101—44no—
43#13108051—24no—
44#13108051—24no—
45#13108051—28no—
46C24×C101C2312yes—
47C24×C101C2311yescharacteristic
48#41×C101—232no—
49C2×Q8×C101—24no—
50C2×Q8×C101—22no—
51C2×D4×C101—24no—
52C2×D4×C101—24no—
53C2×D4×C101—24no—
54C2×D4×C101—22no—
55C2×D4×C101D418yescomplemented
56C22×C4×C101—24no—
57C22×C4×C101—24no—
58C22×C4×C101—24no—
59C22×C4×C101—24no—
60C22×C4×C101D414yescomplemented
61C22×C4×C101C2311yescharacteristic
62C22×C4×C101C2311yescharacteristic
63C22×C4×C101C2312yes—
64C23×C101—22no—
65C23×C101—22no—
66C23×C101—22no—
67C23×C101—22no—
68C23×C101—24no—
69C23×C101—24no—
70C23×C101C2×D414yes—
71C23×C101C2×D414yes—
72C23×C101C2×D412yes—
73C23×C101C2411yescharacteristic
74C22×D101—28no—
75C22×D101—22no—
76C22×D101—24no—
77C22×D101—22no—
78C22×D101—24no—
79C22×D101—416no—
80C22×D101—44no—
81C22×D101—44no—
82C22×D101#4114yescomplemented
83Q8×C101—28no—
84D4×C101—28no—
85D4×C101—216no—
86C2×C4×C101—216no—
87C2×C4×C101—28no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101C2×D412yes—
93C2×C4×C101C2×D414yes—
94C2×C4×C101C2×D412yes—
95C22⋊D101—416no—
96C2×Dic101—22no—
97C2×Dic101—24no—
98C2×Dic101—24no—
99C2×Dic101—24no—
100C2×Dic101—24no—
101C2×Dic101—44no—
102C22×C101—22no—
103C22×C101—22no—
104C22×C101—28no—
105C22×C101—24no—
106C22×C101—28no—
107C22×C101—42no—
108C22×C101—42no—
109C22×C101—44no—
110C22×C101#12014yes—
111C22×C101C22×D411yescharacteristic
112C22×C101C22×D412yes—
113C22×C101C2×#4112yes—
114C22×C101H(5,𝔽2)12yes—
115C2×D101—24no—
116C2×D101—28no—
117C2×D101—416no—
118C2×D101—42no—
119C2×D101—42no—
120C2×D101—44no—
121C4×C101—22no—
122C4×C101—24no—
123C4×C101—22no—
124Dic101—42no—
125Dic101—44no—
126C2×C101—24no—
127C2×C101—44no—
128C2×C101C2×#12011yescharacteristic
129C2×C101#52511yescharacteristic
130C2×C101#53511yescharacteristic
131C2×C101#54614yes—
132D101—44no—
133#2404—1011nomaximal, Sylow
134C101#240411yescharacteristic, complemented, Sylow
135C22×#41D10111yescharacteristic, complemented
136C2×#120—1012no—
137C22×C22⋊C4—1011no—
138#393—1012no—
139#392—1012no—
140#386—1012no—
141#386—1014no—
142#381—1011no—
143C25—1011no—
144C2×#41—28no—
145C2×#41—24no—
146C22×Q8C2×D10111yescharacteristic, complemented
147C22×D4—1014no—
148C22×D4C2×D10112yes—
149C22×D4C2×D10111yescharacteristic, complemented
150C23×C4—1011no—
151C23×C4C2×D10111yescharacteristic
152C23×C4C2×D10112yescomplemented
153#120—2028no—
154C2×C22⋊C4—1014no—
155C2×C22⋊C4—1012no—
156C2×C22⋊C4—1012no—
157C2×C22⋊C4—1012no—
158C2×C22⋊C4—1014no—
159C2×C22⋊C4—2024no—
160#95—1012no—
161#95—1012no—
162#95—1014no—
163C24—1012no—
164C24—4042no—
165C24—2024no—
166C24—2024no—
167C24—2022no—
168C24C22×D10112yes—
169C24C22×D10111yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—4044no—
178C2×D4—2028no—
179C2×D4—2028no—
180C2×D4C22⋊D10118yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—1014no—
186C22×C4—1012no—
187C22×C4—4042no—
188C22×C4—2022no—
189C22×C4C22⋊D10114yescomplemented
190C22×C4C22×D10111yescharacteristic
191C22×C4C22×D10111yescharacteristic
192C22×C4C22×D10112yes—
193C22⋊C4—4044no—
194C22⋊C4—2024no—
195C22⋊C4—2024no—
196C22⋊C4—2024no—
197C22⋊C4—2028no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—1014no—
205C23—4044no—
206C23—40416no—
207C23—4044no—
208C23—2028no—
209C23—2022no—
210C23—2024no—
211C23—2022no—
212C23—2024no—
213C23C2×C22⋊D10114yes—
214C23C2×C22⋊D10114yes—
215C23C2×C22⋊D10112yes—
216C23C23×D10111yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—40416no—
221C2×C4—216no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—28no—
227C2×C4—4044no—
228C2×C4—2022no—
229C2×C4—2024no—
230C2×C4—2024no—
231C2×C4—2024no—
232C2×C4—2024no—
233C2×C4C2×C22⋊D10112yes—
234C2×C4C2×C22⋊D10114yes—
235C2×C4C2×C22⋊D10112yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—40416no—
245C22—4042no—
246C22—4044no—
247C22—4042no—
248C22—2028no—
249C22—2024no—
250C22#1569222414yes—
251C22C22×C22⋊D10111yescharacteristic
252C22C22×C22⋊D10112yes—
253C22#41×D10112yes—
254C22#1569230112yes—
255C4—22no—
256C4—24no—
257C4—22no—
258C4—4042no—
259C4—4044no—
260C2—24no—
261C2—44no—
262C2—4044no—
263C2C2×#1569222411yescharacteristic
264C2#2026568411yescharacteristic
265C2#2026574111yescharacteristic
266C2#2026581514yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
FittingC22×#41×C101C2
hypercenterC22×#41D101
hypocenterC101#2404
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15692223C2semidirect
2. C22×#13108051C2semidirect
3. #20265088C2semidirect
4. #20265085C2semidirect
5. #20265057C2semidirect
6. #20265056C2semidirect
7. #20265040C2semidirect
8. C22×#41×C101C2semidirect
14. C2×C22⋊Dic101C22semidirect
19. #15692162C22semidirect
20. #15692162C22semidirect
21. #15692162C22semidirect
24. C22×Q8×C101C22semidirect
26. C22×D4×C101C22semidirect
28. C23×C4×C101C22semidirect
55. C2×D4×C101D4semidirect
60. C22×C4×C101D4semidirect
82. C22×D101#41semidirect
134. C101#2404semidirect
135. C22×#41D101semidirect
146. C22×Q8C2×D101semidirect
149. C22×D4C2×D101semidirect
152. C23×C4C2×D101semidirect
180. C2×D4C22⋊D101semidirect
189. C22×C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922232
C22×#131080512
#202650882
#202650852
#202650572
#202650562
#202650402
C22×#41×C1012
#2404101