Finite Group Databasebeta

← C2×#20260928 #27151146

Subgroups — C2×#20260928 #27151146

Order 12416.

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×#13095496C211yesmaximal, characteristic, complemented
2C2×#15690806C211yesmaximal, characteristic, complemented
3C2×D4×Dic97C211yesmaximal, characteristic, complemented
4#20260928C218yesmaximal, direct product factor
5C2×#15690739C211yesmaximal, characteristic, complemented
6C2×#15690732C211yesmaximal, characteristic, complemented
7C2×#15690731C211yesmaximal, characteristic, complemented
8C22×D8×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C22×C22⋊D97—21no—
11C22×C22⋊D97—21no—
12C23×Dic97—21no—
13C2×#13095496—24no—
14C2×#13095496C2214yescomplemented
15C22×#11715895C2211yescharacteristic
16#15690806—22no—
17#15690806C2214yescomplemented
18D4×Dic97—22no—
19D4×Dic97C2214yescomplemented
20C2×C4⋊Dic97C2211yescharacteristic
21C2×C22⋊Dic97—21no—
22C2×C4×Dic97C2211yescharacteristic
23#15690739C2214yescomplemented
24#15690732C2214yescomplemented
25#15690731C2214yescomplemented
26C22×D4×C97C2211yescharacteristic
27C22×D4×C97C2211yescharacteristic
28C2×D8×C97—24no—
29C2×D8×C97C2214yescomplemented
30C22×C8×C97C2211yescharacteristic
31C23×D97—21no—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388—21no—
35C2×D388C2312yes—
36C2×C22⋊D97—24no—
37C2×C22⋊D97—24no—
38C2×C22⋊D97—44no—
39C2×C22⋊D97—44no—
40C22×Dic97—21no—
41C22×Dic97—24no—
42C22×Dic97—44no—
43C22×Dic97D411yescharacteristic
44C22×Dic97D411yescharacteristic
45#13095496—28no—
46C2×#11715895—22no—
47C2×#11715895C2312yes—
48C4⋊Dic97—21no—
49C4⋊Dic97C2312yes—
50C22⋊Dic97—24no—
51C4×Dic97D412yes—
52C4×Dic97C2312yes—
53C24×C97—21no—
54C24×C97—21no—
55C2×D4×C97—21no—
56C2×D4×C97—22no—
57C2×D4×C97—21no—
58C2×D4×C97—22no—
59C2×D4×C97—22no—
60C2×D4×C97D414yescomplemented
61C2×D4×C97C2312yes—
62C2×D4×C97C2312yes—
63C22×C4×C97C2311yescharacteristic
64D8×C97—28no—
65C2×C8×C97—22no—
66C2×C8×C97C2312yes—
67C23×C97—24no—
68C23×C97—22no—
69C23×C97—22no—
70C23×C97—42no—
71C23×C97—42no—
72C23×C97—42no—
73C23×C97—42no—
74C23×C97—42no—
75C23×C97C2×D411yescharacteristic
76C22×D97—22no—
77C22×D97—42no—
78C22×D97—42no—
79C22×D97—42no—
80D4×C97—24no—
81D4×C97—22no—
82D4×C97—22no—
83D4×C97—22no—
84D4×C97—42no—
85D4×C97C2×D414yescomplemented
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2×D412yes—
88C2×C4×C97#4112yes—
89C2×C4×C97C2411yescharacteristic
90C22⋊D97—48no—
91C22⋊D97—48no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—42no—
95C2×Dic97—48no—
96C2×Dic97D814yescomplemented
97C2×Dic97C2×D412yes—
98C2×Dic97C2×D412yes—
99D388—22no—
100D388—22no—
101D388—42no—
102C8×C97—22no—
103#11715895—22no—
104C22×C97—24no—
105C22×C97—42no—
106C22×C97—42no—
107C22×C97—48no—
108C22×C97—42no—
109C22×C97—42no—
110C22×C97—42no—
111C22×C97—84no—
112C22×C97#12112yes—
113C22×C97C2×D812yes—
114C22×C97AGL(1,ℤ/8)12yes—
115C22×C97C22×D411yescharacteristic
116C2×D97—42no—
117C2×D97—42no—
118C2×D97—42no—
119C2×D97—84no—
120C4×C97#12112yes—
121C4×C97C22×D411yescharacteristic
122C4×C97C2×#4111yescharacteristic
123Dic97—22no—
124Dic97—42no—
125Dic97C2×D814yescomplemented
126C2×C97—44no—
127C2×C97—82no—
128C2×C97#45914yescomplemented
129C2×C97C2×#12111yescharacteristic
130C2×C97C22×D811yescharacteristic
131C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
132D97—82no—
133C2×#459—971nomaximal, Sylow
134C97C2×#45911yescharacteristic, complemented, Sylow
135C22×D8—971no—
136C22×D8D9711yescharacteristic, complemented
137C2×#127—971no—
138C2×C4×D4—971no—
139#459—978no—
140C2×C4⋊C8—971no—
141C2×#102—971no—
142C2×#102—971no—
143C22×D4—971no—
144C22×D4—1941no—
145C22×D4—1941no—
146C22×D4C2×D9711yescharacteristic
147C22×D4C2×D9711yescharacteristic
148C23×C4—1941no—
149C2×D8—24no—
150C2×D8—974no—
151C2×D8—1944no—
152C2×D8C2×D9714yescomplemented
153C22×C8—971no—
154C22×C8C2×D9711yescharacteristic
155#127—974no—
156#127—1942no—
157C4×D4—974no—
158C4×D4—1942no—
159C2×C4⋊C4—971no—
160C2×C22⋊C4—1941no—
161C2×C42—971no—
162C4⋊C8—974no—
163#102—974no—
164#102—974no—
165C24—21no—
166C24—21no—
167C24—1941no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—972no—
174C2×D4—3884no—
175C2×D4—3884no—
176C2×D4—1944no—
177C2×D4—1944no—
178C2×D4—1941no—
179C2×D4—1942no—
180C2×D4—1942no—
181C2×D4C22⋊D9714yescomplemented
182C2×D4C22×D9712yes—
183C2×D4C22×D9712yes—
184C22×C4—971no—
185C22×C4—971no—
186C22×C4—3884no—
187C22×C4—1944no—
188C22×C4—1941no—
189C22×C4C22×D9711yescharacteristic
190D8—28no—
191D8—1948no—
192C2×C8—22no—
193C2×C8—972no—
194C2×C8—1942no—
195C2×C8C22×D9712yes—
196C4⋊C4—972no—
197C4⋊C4—1941no—
198C22⋊C4—1944no—
199C42—972no—
200C42—972no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—3882no—
210C23—3882no—
211C23—3882no—
212C23—1942no—
213C23D4×D9711yescharacteristic
214D4—24no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—42no—
219D4—3888no—
220D4—3888no—
221D4—3882no—
222D4—1942no—
223D4—1942no—
224D4C2×C22⋊D9714yescomplemented
225C2×C4—974no—
226C2×C4—972no—
227C2×C4—972no—
228C2×C4—3888no—
229C2×C4—3882no—
230C2×C4—1942no—
231C2×C4—1942no—
232C2×C4D4×D9711yescharacteristic
233C2×C4C2×C22⋊D9712yes—
234C2×C4#1309551012yes—
235C2×C4C23×D9711yescharacteristic
236C8—22no—
237C8—1942no—
238C22—24no—
239C22—48no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—84no—
246C22—7764no—
247C22—3882no—
248C22—3882no—
249C22—3882no—
250C22#1569078212yes—
251C22D8×D9712yes—
252C22#1569083412yes—
253C22C2×D4×D9711yescharacteristic
254C4—974no—
255C4—3882no—
256C4—1942no—
257C4#1569078212yes—
258C4C22×C22⋊D9711yescharacteristic
259C4C2×#1309551011yescharacteristic
260C2—44no—
261C2—82no—
262C2—7762no—
263C2#2026092814yesdirect product factor
264C2C2×#1569078211yescharacteristic
265C2C2×D8×D9711yescharacteristic
266C2C2×#1569083411yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×D8×C97C2
socleC23×C97C2×D4
hypercenterC22×D8D97
hypocenterC97C2×#459
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#13095496C2semidirect
2. C2×#15690806C2semidirect
3. C2×D4×Dic97C2semidirect
4. #20260928C2direct (N × H)
5. C2×#15690739C2semidirect
6. C2×#15690732C2semidirect
7. C2×#15690731C2semidirect
8. C22×D8×C97C2semidirect
14. C2×#13095496C22semidirect
17. #15690806C22semidirect
19. D4×Dic97C22semidirect
23. #15690739C22semidirect
24. #15690732C22semidirect
25. #15690731C22semidirect
29. C2×D8×C97C22semidirect
60. C2×D4×C97D4semidirect
85. D4×C97C2×D4semidirect
96. C2×Dic97D8semidirect
125. Dic97C2×D8semidirect
128. C2×C97#459semidirect
134. C97C2×#459semidirect
136. C22×D8D97semidirect
152. C2×D8C2×D97semidirect
181. C2×D4C22⋊D97semidirect
224. D4C2×C22⋊D97semidirect
263. C2#20260928direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130954962
C2×#156908062
C2×D4×Dic972
#202609282
C2×#156907392
C2×#156907322
C2×#156907312
C22×D8×C972
C2×#45997