Finite Group Databasebeta

← C4×#15690872 #27157502

Subgroups — C4×#15690872 #27157502

Order 12416.

396 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 110 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×#15690872C212yesmaximal
2C4×#13095519C211yesmaximal, characteristic, complemented
3C4×C4⋊#87604C211yesmaximal, characteristic, complemented
4#20261608C212yesmaximal, complemented
5C4×D4×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604C2212yes—
8C2×C4×#87604—21no—
9#15690872C418yesdirect product factor
10C4×#11715897—21no—
11C4×#11715897—21no—
12(C2×#87604)⋊C4—21no—
13C2×D4×D97C2211yescharacteristic
14C22×C4×D97—21no—
15C4×D388C411yescharacteristic, complemented
16D4×Dic97C411yescharacteristic, complemented
17#15690772C411yescharacteristic, complemented
18C4×C22⋊D97—21no—
19#15690770—21no—
20C4⋊C4×D97C2211yescharacteristic
21C22⋊C4×D97—21no—
22C42×D97C2211yescharacteristic
23C4×D4×C97C411yescharacteristic, complemented
24C22×#87604—22no—
25#13095519—22no—
26#13095519C2×C414yescomplemented
27C2×#11715897—22no—
28C4⋊#87604—22no—
29C4⋊#87604C2×C414yescomplemented
30C4×#87604—42no—
31C23×D97—21no—
32C2×D388C2×C411yescharacteristic
33C2×C22⋊D97—21no—
34D4×D97—22no—
35D4×D97C2×C414yes—
36C22×Dic97—21no—
37C2×C4×D97—21no—
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97—42no—
41C2×C4×D97—42no—
42C2×C4×D97D411yescharacteristic
43C2×C4×D97D411yescharacteristic
44C2×C4×D97C2311yescharacteristic
45C4⋊Dic97C2×C411yescharacteristic
46#13095490—21no—
47C22⋊Dic97—21no—
48#13095488—21no—
49#13095488—21no—
50C4×Dic97C2×C411yescharacteristic
51C4×Dic97D411yescharacteristic
52C4×Dic97D411yescharacteristic
53C2×D4×C97C2×C411yescharacteristic
54C22×C4×C97—21no—
55C4⋊C4×C97C2×C411yescharacteristic
56C22⋊C4×C97—21no—
57C42×C97C2×C411yescharacteristic
58C23×C97—21no—
59C22×D97—21no—
60C22×D97—24no—
61C22×D97—41no—
62C22×D97—41no—
63C22×D97—42no—
64C22×D97C2×D411yescharacteristic
65C2×#87604—24no—
66C2×#87604—42no—
67C2×#87604—42no—
68D4×C97—21no—
69D4×C97C4212yescomplemented
70C2×C4×C97—21no—
71C2×C4×C97—42no—
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C22×C411yescharacteristic
75C22⋊D97—24no—
76C2×Dic97—21no—
77C2×Dic97—42no—
78C2×Dic97C22⋊C411yescharacteristic
79C2×Dic97C22⋊C411yescharacteristic
80C2×Dic97C2×D411yescharacteristic
81D388—21no—
82D388C4212yescomplemented
83C4×D97—22no—
84C4×D97—41no—
85C4×D97—41no—
86C4×D97—42no—
87C4×D97—42no—
88C4×D97D812yes—
89C4×D97SD1612yes—
90C4×D97C22×C412yes—
91C4×D97#4112yes—
92#11715897—24no—
93C22×C97—22no—
94C22×C97—41no—
95C22×C97—41no—
96C22×C97C2×C22⋊C411yescharacteristic
97C2×D97—22no—
98C2×D97—41no—
99C2×D97—41no—
100C2×D97—44no—
101C2×D97—44no—
102C2×D97C4×D412yes—
103C2×D97C2×D811yescharacteristic
104C2×D97C2×SD1611yescharacteristic
105C2×D97#13512yes—
106#87604—44no—
107C4×C97—21no—
108C4×C97—41no—
109C4×C97#10212yescomplemented
110C4×C97C2×C4211yescharacteristic
111C4×C97#11711yescharacteristic
112Dic97—21no—
113Dic97—41no—
114Dic97#10212yescomplemented
115Dic97C4×D412yes—
116C2×C97—42no—
117C2×C97C4×C22⋊C411yescharacteristic
118C2×C97C2×#10211yescharacteristic
119C2×C97#41611yescharacteristic
120D97—42no—
121D97C4×D812yes—
122D97C4×SD1612yes—
123C4×#102—971nomaximal, Sylow
124C97C4×#10211yescharacteristic, complemented, Sylow
125C2×C4×D4—971no—
126C2×#102—972no—
127C2×C4×C8—971no—
128C4×C4⋊C4—971no—
129#340—972no—
130C22×D4—971no—
131C23×C4—1941no—
132C22×C8—972no—
133C4×D4—971no—
134C4×D4—971no—
135C4×D4—971no—
136C4×D4—1941no—
137C4×D4—1941no—
138C4×D4#8760411yescharacteristic, complemented
139C2×C4⋊C4—972no—
140C2×C4⋊C4—971no—
141C2×C22⋊C4—1941no—
142C2×C42—971no—
143C2×C42—1941no—
144#102—978no—
145C4×C8—1941no—
146C4×C8—1941no—
147#95—1941no—
148C24—1941no—
149C2×D4—974no—
150C2×D4—971no—
151C2×D4—1942no—
152C2×D4—1941no—
153C2×D4C2×#8760411yescharacteristic
154C22×C4—21no—
155C22×C4—971no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—3882no—
159C22×C4—3882no—
160C22×C4—1941no—
161C22×C4—1941no—
162C22×C4—1941no—
163C22×C4—1942no—
164C22×C4—1941no—
165C2×C8—974no—
166C2×C8—1942no—
167C2×C8—1942no—
168C4⋊C4—974no—
169C4⋊C4—971no—
170C4⋊C4—1941no—
171C4⋊C4—1942no—
172C4⋊C4C2×#8760411yescharacteristic
173C22⋊C4—21no—
174C22⋊C4—1941no—
175C22⋊C4—1941no—
176C22⋊C4—1941no—
177C42—971no—
178C42—971no—
179C42—971no—
180C42—3882no—
181C42C2×#8760411yescharacteristic
182C23—21no—
183C23—971no—
184C23—3881no—
185C23—3881no—
186C23—3882no—
187C23—1944no—
188C23—1941no—
189D4—21no—
190D4—972no—
191D4—1944no—
192D4—1941no—
193D4C4×#8760412yescomplemented
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—971no—
197C2×C4—971no—
198C2×C4—972no—
199C2×C4—972no—
200C2×C4—972no—
201C2×C4—972no—
202C2×C4—971no—
203C2×C4—3882no—
204C2×C4—3881no—
205C2×C4—3881no—
206C2×C4—3882no—
207C2×C4—3882no—
208C2×C4—3882no—
209C2×C4—3882no—
210C2×C4—1942no—
211C2×C4—1944no—
212C2×C4—1941no—
213C2×C4C22⋊#8760411yescharacteristic
214C2×C4C22⋊#8760411yescharacteristic
215C2×C4C22×#8760411yescharacteristic
216C8—1944no—
217C22—22no—
218C22—41no—
219C22—41no—
220C22—972no—
221C22—971no—
222C22—971no—
223C22—972no—
224C22—3884no—
225C22—3884no—
226C22—3881no—
227C22—3881no—
228C22—1942no—
229C22C2×C22⋊#8760411yescharacteristic
230C4—21no—
231C4—41no—
232C4—972no—
233C4—972no—
234C4—3884no—
235C4—3881no—
236C4—1941no—
237C4#1569087212yesdirect product factor
238C4C2×C4×#8760411yescharacteristic
239C4#1569088911yescharacteristic
240C2—42no—
241C2—972no—
242C2—972no—
243C2—3882no—
244C2C4×C22⋊#8760411yescharacteristic
245C2C2×#1569087211yescharacteristic
246C2#2026172511yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC4×C97C2×C42
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C22
cosocleC2×C4×D97C23
hypercenterC4×D4#87604
hypocenterC97C4×#102
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×#13095519C2semidirect
3. C4×C4⋊#87604C2semidirect
4. #20261608C2semidirect
9. #15690872C4direct (N × H)
15. C4×D388C4semidirect
16. D4×Dic97C4semidirect
17. #15690772C4semidirect
23. C4×D4×C97C4semidirect
26. #13095519C2×C4semidirect
29. C4⋊#87604C2×C4semidirect
69. D4×C97C42semidirect
82. D388C42semidirect
109. C4×C97#102semidirect
114. Dic97#102semidirect
124. C97C4×#102semidirect
138. C4×D4#87604semidirect
193. D4C4×#87604semidirect
237. C4#15690872direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908722
C4×#130955192
C4×C4⋊#876042
#202616082
C4×D4×D972
C4×#10297