Finite Group Databasebeta

← C2×#27161272 #38752122

Subgroups — C2×#27161272 #38752122

Order 25216.

648 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#15691343C211yesmaximal, characteristic, complemented
2C2×#20263322C211yesmaximal, characteristic, complemented
3C2×D4×Dic197C211yesmaximal, characteristic, complemented
4#27161272C218yesmaximal, direct product factor
5C2×#20263255C211yesmaximal, characteristic, complemented
6C2×#20263248C211yesmaximal, characteristic, complemented
7C2×#20263247C211yesmaximal, characteristic, complemented
8C22×D8×C197C211yesmaximal, characteristic, complemented
9C22×D788C2211yescharacteristic
10C22×C22⋊D197—21no—
11C22×C22⋊D197—21no—
12C23×Dic197—21no—
13C2×#15691343—24no—
14C2×#15691343C2214yescomplemented
15C22×#13096802C2211yescharacteristic
16#20263322—22no—
17#20263322C2214yescomplemented
18D4×Dic197—22no—
19D4×Dic197C2214yescomplemented
20C2×C4⋊Dic197C2211yescharacteristic
21C2×C22⋊Dic197—21no—
22C2×C4×Dic197C2211yescharacteristic
23#20263255C2214yescomplemented
24#20263248C2214yescomplemented
25#20263247C2214yescomplemented
26C22×D4×C197C2211yescharacteristic
27C22×D4×C197C2211yescharacteristic
28C2×D8×C197—24no—
29C2×D8×C197C2214yescomplemented
30C22×C8×C197C2211yescharacteristic
31C23×D197—21no—
32C2×D788—22no—
33C2×D788—21no—
34C2×D788—22no—
35C2×D788C2312yes—
36C2×C22⋊D197—24no—
37C2×C22⋊D197—24no—
38C2×C22⋊D197—44no—
39C2×C22⋊D197—44no—
40C22×Dic197—24no—
41C22×Dic197—21no—
42C22×Dic197—44no—
43C22×Dic197D411yescharacteristic
44C22×Dic197D411yescharacteristic
45#15691343—28no—
46C2×#13096802—22no—
47C2×#13096802C2312yes—
48C4⋊Dic197—21no—
49C4⋊Dic197C2312yes—
50C22⋊Dic197—24no—
51C4×Dic197D412yes—
52C4×Dic197C2312yes—
53C24×C197—21no—
54C24×C197—21no—
55C2×D4×C197—21no—
56C2×D4×C197—22no—
57C2×D4×C197—21no—
58C2×D4×C197—22no—
59C2×D4×C197—22no—
60C2×D4×C197D414yescomplemented
61C2×D4×C197C2312yes—
62C2×D4×C197C2312yes—
63C22×C4×C197C2311yescharacteristic
64D8×C197—28no—
65C2×C8×C197—22no—
66C2×C8×C197C2312yes—
67C22×D197—22no—
68C22×D197—42no—
69C22×D197—42no—
70C22×D197—42no—
71D788—22no—
72D788—22no—
73D788—42no—
74C22⋊D197—48no—
75C22⋊D197—48no—
76C2×Dic197—22no—
77C2×Dic197—22no—
78C2×Dic197—42no—
79C2×Dic197—48no—
80C2×Dic197D814yescomplemented
81C2×Dic197C2×D412yes—
82C2×Dic197C2×D412yes—
83#13096802—22no—
84C23×C197—24no—
85C23×C197—22no—
86C23×C197—22no—
87C23×C197—42no—
88C23×C197—42no—
89C23×C197—42no—
90C23×C197—42no—
91C23×C197—42no—
92C23×C197C2×D411yescharacteristic
93D4×C197—24no—
94D4×C197—22no—
95D4×C197—22no—
96D4×C197—22no—
97D4×C197—42no—
98D4×C197C2×D414yescomplemented
99C2×C4×C197C2×D411yescharacteristic
100C2×C4×C197C2×D412yes—
101C2×C4×C197#4112yes—
102C2×C4×C197C2411yescharacteristic
103C8×C197—22no—
104C22×C197—24no—
105C22×C197—42no—
106C22×C197—42no—
107C22×C197—48no—
108C22×C197—42no—
109C22×C197—42no—
110C22×C197—42no—
111C22×C197—84no—
112C22×C197#12112yes—
113C22×C197C2×D812yes—
114C22×C197AGL(1,ℤ/8)12yes—
115C22×C197C22×D411yescharacteristic
116C2×D197—42no—
117C2×D197—42no—
118C2×D197—42no—
119C2×D197—84no—
120C4×C197#12112yes—
121C4×C197C22×D411yescharacteristic
122C4×C197C2×#4111yescharacteristic
123Dic197—22no—
124Dic197—42no—
125Dic197C2×D814yescomplemented
126C2×C197—44no—
127C2×C197—82no—
128C2×C197#45914yescomplemented
129C2×C197C2×#12111yescharacteristic
130C2×C197C22×D811yescharacteristic
131C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
132D197—82no—
133C197C2×#45911yescharacteristic, complemented, Sylow
134C2×#459—1971nomaximal, Sylow
135C22×D8—1971no—
136C22×D8D19711yescharacteristic, complemented
137C2×#127—1971no—
138C2×C4×D4—1971no—
139#459—1978no—
140C2×C4⋊C8—1971no—
141C2×#102—1971no—
142C2×#102—1971no—
143C22×D4—3941no—
144C22×D4—3941no—
145C22×D4—1971no—
146C22×D4C2×D19711yescharacteristic
147C22×D4C2×D19711yescharacteristic
148C23×C4—3941no—
149C2×D8—24no—
150C2×D8—3944no—
151C2×D8—1974no—
152C2×D8C2×D19714yescomplemented
153C22×C8—1971no—
154C22×C8C2×D19711yescharacteristic
155#127—3942no—
156#127—1974no—
157C4×D4—3942no—
158C4×D4—1974no—
159C2×C4⋊C4—1971no—
160C2×C22⋊C4—3941no—
161C2×C42—1971no—
162C4⋊C8—1974no—
163#102—1974no—
164#102—1974no—
165C24—21no—
166C24—21no—
167C24—3941no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—7884no—
174C2×D4—7884no—
175C2×D4—3944no—
176C2×D4—3944no—
177C2×D4—3941no—
178C2×D4—3942no—
179C2×D4—3942no—
180C2×D4—1972no—
181C2×D4C22⋊D19714yescomplemented
182C2×D4C22×D19712yes—
183C2×D4C22×D19712yes—
184C22×C4—7884no—
185C22×C4—3941no—
186C22×C4—3944no—
187C22×C4—1971no—
188C22×C4—1971no—
189C22×C4C22×D19711yescharacteristic
190D8—28no—
191D8—3948no—
192C2×C8—22no—
193C2×C8—3942no—
194C2×C8—1972no—
195C2×C8C22×D19712yes—
196C4⋊C4—3941no—
197C4⋊C4—1972no—
198C22⋊C4—3944no—
199C42—1972no—
200C42—1972no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—7882no—
210C23—7882no—
211C23—7882no—
212C23—3942no—
213C23D4×D19711yescharacteristic
214D4—24no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—42no—
219D4—7888no—
220D4—7888no—
221D4—7882no—
222D4—3942no—
223D4—3942no—
224D4C2×C22⋊D19714yescomplemented
225C2×C4—7882no—
226C2×C4—7888no—
227C2×C4—3942no—
228C2×C4—3942no—
229C2×C4—1972no—
230C2×C4—1974no—
231C2×C4—1972no—
232C2×C4D4×D19711yescharacteristic
233C2×C4C2×C22⋊D19712yes—
234C2×C4#1569135712yes—
235C2×C4C23×D19711yescharacteristic
236C8—22no—
237C8—3942no—
238C22—24no—
239C22—42no—
240C22—48no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—84no—
246C22—7882no—
247C22—7882no—
248C22—7882no—
249C22—15764no—
250C22#2026329812yes—
251C22D8×D19712yes—
252C22#2026335012yes—
253C22C2×D4×D19711yescharacteristic
254C4—7882no—
255C4—3942no—
256C4—1974no—
257C4#2026329812yes—
258C4C22×C22⋊D19711yescharacteristic
259C4C2×#1569135711yescharacteristic
260C2—44no—
261C2—82no—
262C2—15762no—
263C2#2716127214yesdirect product factor
264C2C2×#2026329811yescharacteristic
265C2C2×D8×D19711yescharacteristic
266C2C2×#2026335011yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×D8×C197C2
socleC23×C197C2×D4
hypercenterC22×D8D197
hypocenterC197C2×#459
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#15691343C2semidirect
2. C2×#20263322C2semidirect
3. C2×D4×Dic197C2semidirect
4. #27161272C2direct (N × H)
5. C2×#20263255C2semidirect
6. C2×#20263248C2semidirect
7. C2×#20263247C2semidirect
8. C22×D8×C197C2semidirect
14. C2×#15691343C22semidirect
17. #20263322C22semidirect
19. D4×Dic197C22semidirect
23. #20263255C22semidirect
24. #20263248C22semidirect
25. #20263247C22semidirect
29. C2×D8×C197C22semidirect
60. C2×D4×C197D4semidirect
80. C2×Dic197D8semidirect
98. D4×C197C2×D4semidirect
125. Dic197C2×D8semidirect
128. C2×C197#459semidirect
133. C197C2×#459semidirect
136. C22×D8D197semidirect
152. C2×D8C2×D197semidirect
181. C2×D4C22⋊D197semidirect
224. D4C2×C22⋊D197semidirect
263. C2#27161272direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913432
C2×#202633222
C2×D4×Dic1972
#271612722
C2×#202632552
C2×#202632482
C2×#202632472
C22×D8×C1972
C2×#459197