Finite Group Databasebeta

← C2×#20260900 #27150948

Subgroups — C2×#20260900 #27150948

Order 12416.

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

normal subgroups only

This group has 760 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×#13095512C211yesmaximal, characteristic
2C22×#13095501C211yesmaximal, characteristic, complemented
3C22×#13095500C211yesmaximal, characteristic, complemented
4C2×#15690790C211yesmaximal, characteristic, complemented
5#20260900C218yesmaximal, direct product factor
6C2×#15690730C211yesmaximal, characteristic, complemented
7C2×#15690722C211yesmaximal, characteristic, complemented
8C2×#103×C97C211yesmaximal, characteristic, complemented
9C2×#13095512—24no—
10C2×#13095512—22no—
11C2×#13095512—24no—
12C2×#13095512C2214yes—
13C22×Dic194C2211yescharacteristic
14C22×D388—21no—
15C22×D388C2211yescharacteristic
16C22×C4×D97—21no—
17C22×C4×D97C2211yescharacteristic
18C2×#13095501—24no—
19C2×#13095501C2214yescomplemented
20C2×#13095500—24no—
21C2×#13095500C2214yescomplemented
22C22×#11715895C2211yescharacteristic
23#15690790—22no—
24#15690790C2214yescomplemented
25C2×C4⋊Dic97—21no—
26C2×#13095488—21no—
27#15690730C2214yescomplemented
28#15690722C2214yescomplemented
29C22×Q8×C97C2211yescharacteristic
30C22×C8×C97C2211yescharacteristic
31C2×C4⋊C4×C97C2211yescharacteristic
32#103×C97C2214yescomplemented
33C23×D97—21no—
34C23×D97D411yescharacteristic
35#13095512—216no—
36#13095512—28no—
37#13095512—28no—
38C2×Dic194—22no—
39C2×Dic194—22no—
40C2×Dic194—21no—
41C2×Dic194C2312yes—
42C2×D388—28no—
43C2×D388—24no—
44C2×D388—21no—
45C2×D388—22no—
46C2×D388D414yes—
47C2×D388C2312yes—
48C22×Dic97—21no—
49C22×Dic97D411yescharacteristic
50C2×C4×D97—28no—
51C2×C4×D97—22no—
52C2×C4×D97—22no—
53C2×C4×D97—24no—
54C2×C4×D97D412yes—
55C2×C4×D97C2312yes—
56#13095501—28no—
57#13095500—28no—
58C2×#11715895—22no—
59C2×#11715895C2312yes—
60C4⋊Dic97—24no—
61#13095488—24no—
62C2×Q8×C97—22no—
63C2×Q8×C97—21no—
64C2×Q8×C97D414yes—
65C2×Q8×C97C2312yes—
66C22×C4×C97—21no—
67C22×C4×C97—21no—
68C22×C4×C97C2311yescharacteristic
69C2×C8×C97—22no—
70C2×C8×C97C2312yes—
71C4⋊C4×C97—21no—
72C4⋊C4×C97C2312yes—
73C23×C97C2×D411yescharacteristic
74C22×D97—22no—
75C22×D97—24no—
76C22×D97—22no—
77C22×D97—22no—
78C22×D97—22no—
79C22×D97—42no—
80C22×D97—42no—
81C22×D97C2×D412yes—
82Q8×C97—24no—
83Q8×C97—22no—
84Q8×C97C2×D414yes—
85C2×C4×C97—24no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97—42no—
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×D412yes—
91C2×C4×C97C2×D412yes—
92C2×C4×C97C2411yescharacteristic
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—42no—
96C2×Dic97C2×D412yes—
97D388—216no—
98D388—24no—
99D388—22no—
100D388C2×D414yes—
101C4×D97—216no—
102C4×D97—24no—
103C4×D97—22no—
104C4×D97—22no—
105Dic194—22no—
106Dic194—22no—
107Dic194—42no—
108C8×C97—22no—
109#11715895—22no—
110C22×C97#12012yes—
111C22×C97#13512yes—
112C22×C97#13712yes—
113C22×C97C22×D411yescharacteristic
114C2×D97—22no—
115C2×D97—24no—
116C2×D97—22no—
117C2×D97—22no—
118C2×D97—42no—
119C2×D97—42no—
120C2×D97—44no—
121C2×D97—48no—
122C4×C97—24no—
123C4×C97—42no—
124C4×C97#12012yes—
125C4×C97C22×D411yescharacteristic
126C4×C97C22×D411yescharacteristic
127Dic97—22no—
128Dic97—42no—
129C2×C97#45214yescomplemented
130C2×C97C2×#12011yescharacteristic
131C2×C97C2×#13511yescharacteristic
132C2×C97C2×#13711yescharacteristic
133D97—42no—
134D97—44no—
135C2×#452—971nomaximal, Sylow
136C97C2×#45211yescharacteristic, complemented, Sylow
137C22×#41—971no—
138C22×Q16—971no—
139C22×SD16—971no—
140C2×C22⋊Q8—971no—
141#452—978no—
142C2×#103D9711yescharacteristic, complemented
143C2×#102—971no—
144C2×C22⋊C8—971no—
145C2×#41—974no—
146C2×#41—1942no—
147C2×#41—1944no—
148C2×#41—1944no—
149C22×Q8—971no—
150C22×Q8C2×D9711yescharacteristic
151C22×D4—971no—
152C22×D4—1941no—
153C23×C4—971no—
154C23×C4—1941no—
155C2×Q16—974no—
156C2×Q16—1944no—
157C2×SD16—974no—
158C2×SD16—1944no—
159C22×C8—971no—
160C22×C8C2×D9711yescharacteristic
161C22⋊Q8—974no—
162C22⋊Q8—1942no—
163C2×C4⋊C4—1941no—
164C2×C4⋊C4C2×D9711yescharacteristic
165C2×C22⋊C4—1941no—
166#103C2×D9714yescomplemented
167#102—974no—
168C22⋊C8—974no—
169C24—971no—
170C24—1941no—
171#41—1948no—
172#41—19416no—
173#41—1948no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—972no—
177C2×Q8—1942no—
178C2×Q8—1942no—
179C2×Q8—1941no—
180C2×Q8D38814yes—
181C2×Q8C22×D9712yes—
182C2×D4—974no—
183C2×D4—972no—
184C2×D4—1944no—
185C2×D4—1948no—
186C2×D4—1941no—
187C2×D4—1942no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—971no—
191C22×C4—972no—
192C22×C4—972no—
193C22×C4—1944no—
194C22×C4—1948no—
195C22×C4—1941no—
196C22×C4—1942no—
197C22×C4—1942no—
198C22×C4C22×D9711yescharacteristic
199Q16—1948no—
200SD16—1948no—
201C2×C8—22no—
202C2×C8—972no—
203C2×C8—1942no—
204C2×C8C22×D9712yes—
205C4⋊C4—21no—
206C4⋊C4—1944no—
207C4⋊C4C22×D9712yes—
208C22⋊C4—1944no—
209C23—972no—
210C23—3882no—
211C23—3882no—
212C23—1944no—
213C23—1942no—
214C23—1942no—
215C23—1942no—
216C23—1942no—
217C23D4×D9711yescharacteristic
218Q8—24no—
219Q8—22no—
220Q8—3882no—
221Q8—1942no—
222Q8—1942no—
223Q8C2×D38814yes—
224D4—974no—
225D4—19416no—
226D4—1944no—
227D4—1942no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—972no—
233C2×C4—3882no—
234C2×C4—19416no—
235C2×C4—1942no—
236C2×C4—1942no—
237C2×C4—1944no—
238C2×C4—1942no—
239C2×C4—1942no—
240C2×C4D4×D9711yescharacteristic
241C2×C4D4×D9712yes—
242C2×C4C2×D38812yes—
243C2×C4C23×D9711yescharacteristic
244C8—22no—
245C8—1942no—
246C22—3888no—
247C22—3884no—
248C22—3882no—
249C22—3882no—
250C22—1944no—
251C22—1942no—
252C22—1942no—
253C22—1942no—
254C22#1569078012yes—
255C22#1569083112yes—
256C22#1569084212yes—
257C22C2×D4×D9711yescharacteristic
258C4—24no—
259C4—42no—
260C4—3882no—
261C4—1942no—
262C4#1569078012yes—
263C4C2×D4×D9711yescharacteristic
264C4C22×D38811yescharacteristic
265C2—3884no—
266C2—3882no—
267C2#2026090014yesdirect product factor
268C2C2×#1569078011yescharacteristic
269C2C2×#1569083111yescharacteristic
270C2C2×#1569084211yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#103×C97C2
socleC23×C97C2×D4
prime radicalC22×#13095512C2
hypercenterC2×#103D97
hypocenterC97C2×#452
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C22×#13095501C2semidirect
3. C22×#13095500C2semidirect
4. C2×#15690790C2semidirect
5. #20260900C2direct (N × H)
6. C2×#15690730C2semidirect
7. C2×#15690722C2semidirect
8. C2×#103×C97C2semidirect
19. C2×#13095501C22semidirect
21. C2×#13095500C22semidirect
24. #15690790C22semidirect
27. #15690730C22semidirect
28. #15690722C22semidirect
32. #103×C97C22semidirect
129. C2×C97#452semidirect
136. C97C2×#452semidirect
142. C2×#103D97semidirect
166. #103C2×D97semidirect
267. C2#20260900direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955122
C22×#130955012
C22×#130955002
C2×#156907902
#202609002
C2×#156907302
C2×#156907222
C2×#103×C972
C2×#45297