Finite Group Databasebeta

← #27179416

Subgroups — #27179416

Order 12928.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D101C211yesmaximal, characteristic
2C2×#15692231C212yesmaximal, complemented
3#20265049C212yesmaximal, complemented
4#20265048C214yesmaximal, complemented
5#20265037C211yesmaximal, characteristic, complemented
6#20265034C212yesmaximal, complemented
7#20265032C212yesmaximal
8C2×C22⋊Q8×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic
10C23×Dic101C2211yescharacteristic
11C22×C4×D101—28no—
12C22×C4×D101—24no—
13C22×C4×D101C2212yes—
14C22×C4×D101C2212yes—
15#15692231—28no—
16C2×C4⋊Dic101C2212yescomplemented
17C2×#13108053C2214yescomplemented
18C2×C22⋊Dic101C2212yes—
19C2×#13108051C2214yes—
20C2×#13108051C2214yes—
21#15692162C2214yescomplemented
22#15692162C2212yescomplemented
23C22×Q8×C101C2211yescharacteristic, complemented
24C23×C4×C101C2211yescharacteristic
25C22⋊Q8×C101—24no—
26C2×C4⋊C4×C101C2211yescharacteristic, complemented
27C2×C4⋊C4×C101C2212yescomplemented
28C2×C22⋊C4×C101C2212yes—
29C23×D101—22no—
30C23×D101—22no—
31C23×D101—24no—
32C23×D101—24no—
33C23×D101Q814yescomplemented
34C23×D101C2312yes—
35C22×Dic101—22no—
36C22×Dic101—24no—
37C22×Dic101C2314yes—
38C22×Dic101C2312yes—
39C2×C4×D101—232no—
40C2×C4×D101—24no—
41C2×C4×D101—24no—
42C2×C4×D101—24no—
43C2×C4×D101—24no—
44C2×C4×D101—24no—
45C2×C4×D101D418yes—
46C4⋊Dic101—24no—
47#13108053—28no—
48C22⋊Dic101—24no—
49#13108051—28no—
50#13108051—28no—
51C24×C101C2311yescharacteristic
52C2×Q8×C101—22no—
53C2×Q8×C101—24no—
54C22×C4×C101—24no—
55C22×C4×C101D414yes—
56C22×C4×C101C2312yes—
57C22×C4×C101C2312yes—
58C22×C4×C101C2311yescharacteristic
59C22×C4×C101C2311yescharacteristic
60C4⋊C4×C101—22no—
61C4⋊C4×C101—24no—
62C22⋊C4×C101—24no—
63C23×C101—22no—
64C23×C101—22no—
65C23×C101C2×D412yes—
66C23×C101C2×Q812yes—
67C23×C101#4112yes—
68C23×C101C2411yescharacteristic
69C22×D101—28no—
70C22×D101—216no—
71C22×D101—28no—
72C22×D101—28no—
73C22×D101—22no—
74C22×D101—22no—
75C22×D101—22no—
76C22×D101—22no—
77C22×D101—216no—
78C22×D101C2×D414yes—
79C22×D101C2×Q814yes—
80C22×D101#4114yes—
81Q8×C101—44no—
82C2×C4×C101—22no—
83C2×C4×C101—28no—
84C2×C4×C101—22no—
85C2×C4×C101—22no—
86C2×C4×C101—24no—
87C2×C4×C101—24no—
88C2×C4×C101C2×D414yes—
89C2×C4×C101C2×D412yes—
90C2×C4×C101C2×D412yes—
91C2×Dic101—28no—
92C2×Dic101—22no—
93C2×Dic101—22no—
94C2×Dic101—24no—
95C2×Dic101—24no—
96C2×Dic101—24no—
97C2×Dic101C2×D414yes—
98C4×D101—216no—
99C4×D101—216no—
100C22×C101—22no—
101C22×C101—24no—
102C22×C101—22no—
103C22×C101—24no—
104C22×C101C22⋊Q814yescomplemented
105C22×C101C22×D412yes—
106C22×C101C22×D411yescharacteristic
107C22×C101C22×Q811yescharacteristic
108C22×C101C2×#4112yes—
109C22×C101C2×#4111yescharacteristic
110C2×D101—232no—
111C2×D101—24no—
112C2×D101—24no—
113C2×D101—24no—
114C2×D101—24no—
115C2×D101—24no—
116C2×D101—24no—
117C2×D101C22⋊Q818yescomplemented
118C4×C101—22no—
119C4×C101—42no—
120C4×C101—42no—
121C4×C101#12014yes—
122Dic101—24no—
123Dic101—44no—
124C2×C101—24no—
125C2×C101C2×#12011yescharacteristic
126C2×C101C2×C22⋊Q812yes—
127C2×C101C2×C22⋊Q811yescharacteristic
128C2×C101#52511yescharacteristic
129C2×C101#52512yes—
130D101—28no—
131#2848—1011nomaximal, Sylow
132C101#284811yescharacteristic, complemented, Sylow
133C24×C4—1011no—
134C2×C22⋊Q8—1012no—
135C2×C22⋊Q8D10111yescharacteristic, complemented
136#385—1014no—
137#385—1012no—
138#380—1011no—
139#380—1012no—
140#379—1012no—
141C25—1011no—
142C22×Q8C2×D10111yescharacteristic, complemented
143C23×C4—1012no—
144C23×C4—1011no—
145C23×C4—1012no—
146C23×C4—2028no—
147C23×C4—2024no—
148C23×C4C2×D10111yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—2028no—
151C2×C4⋊C4—1012no—
152C2×C4⋊C4—1014no—
153C2×C4⋊C4C2×D10111yescharacteristic, complemented
154C2×C4⋊C4C2×D10112yescomplemented
155C2×C22⋊C4—1012no—
156C2×C22⋊C4—1014no—
157C2×C22⋊C4—1014no—
158C2×C22⋊C4C2×D10112yes—
159#95—1014no—
160#95—1012no—
161C24—1012no—
162C24—1014no—
163C24—2024no—
164C24—2024no—
165C24—2022no—
166C24—2022no—
167C24C22×D10111yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—1012no—
172C22×C4—1018no—
173C22×C4—1014no—
174C22×C4—2024no—
175C22×C4—2024no—
176C22×C4—20232no—
177C22×C4—2024no—
178C22×C4—2022no—
179C22×C4—2024no—
180C22×C4—2024no—
181C22×C4—2024no—
182C22×C4C22⋊D10114yes—
183C22×C4C22×D10112yes—
184C22×C4C22×D10112yes—
185C22×C4C22×D10111yescharacteristic
186C22×C4C22×D10111yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—2028no—
190C4⋊C4—2024no—
191C22⋊C4—24no—
192C22⋊C4—2028no—
193C22⋊C4—2028no—
194C22⋊C4—2024no—
195C23—22no—
196C23—22no—
197C23—1014no—
198C23—1014no—
199C23—1014no—
200C23—20216no—
201C23—2028no—
202C23—2028no—
203C23—20216no—
204C23—2022no—
205C23—2022no—
206C23—2022no—
207C23—2022no—
208C23—2028no—
209C23C2×C22⋊D10112yes—
210C23Q8×D10112yes—
211C23#1310807512yes—
212C23C23×D10111yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—1014no—
221C2×C4—20216no—
222C2×C4—2024no—
223C2×C4—2024no—
224C2×C4—2024no—
225C2×C4—2028no—
226C2×C4—2022no—
227C2×C4—2022no—
228C2×C4—20216no—
229C2×C4D4×D10114yes—
230C2×C4C2×C22⋊D10112yes—
231C2×C4C2×C22⋊D10112yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—1018no—
237C22—20232no—
238C22—2024no—
239C22—2024no—
240C22—2024no—
241C22—2024no—
242C22—2024no—
243C22—2024no—
244C22#1569223114yescomplemented
245C22C2×D4×D10112yes—
246C22C22×C22⋊D10111yescharacteristic
247C22C2×Q8×D10111yescharacteristic
248C22#41×D10112yes—
249C22C2×#1310807511yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—4044no—
254C4—2024no—
255C4#1569222314yes—
256C2—24no—
257C2—2028no—
258C2C2×#1569222311yescharacteristic
259C2C22⋊Q8×D10112yes—
260C2C2×#1569223111yescharacteristic
261C2#2026568411yescharacteristic
262C2#2026569012yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
FittingC2×C22⋊Q8×C101C2
prime radicalC24×D101C22
hypercenterC2×C22⋊Q8D101
hypocenterC101#2848
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15692231C2semidirect
3. #20265049C2semidirect
4. #20265048C2semidirect
5. #20265037C2semidirect
6. #20265034C2semidirect
8. C2×C22⋊Q8×C101C2semidirect
16. C2×C4⋊Dic101C22semidirect
17. C2×#13108053C22semidirect
21. #15692162C22semidirect
22. #15692162C22semidirect
23. C22×Q8×C101C22semidirect
26. C2×C4⋊C4×C101C22semidirect
27. C2×C4⋊C4×C101C22semidirect
33. C23×D101Q8semidirect
104. C22×C101C22⋊Q8semidirect
117. C2×D101C22⋊Q8semidirect
132. C101#2848semidirect
135. C2×C22⋊Q8D101semidirect
142. C22×Q8C2×D101semidirect
153. C2×C4⋊C4C2×D101semidirect
154. C2×C4⋊C4C2×D101semidirect
244. C22#15692231semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1012
C2×#156922312
#202650492
#202650482
#202650372
#202650342
#202650322
C2×C22⋊Q8×C1012
#2848101