Finite Group Databasebeta

← #27157927

Subgroups — #27157927

Order 12416.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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
1#20261677C211yesmaximal, characteristic
2C4×C22⋊#87604C211yesmaximal, characteristic
3#20261613C212yesmaximal
4#20261613C212yesmaximal
5C2×#15690788C211yesmaximal, characteristic
6C2×C4⋊#87604—21no—
7C2×C22⋊#87604C2212yes—
8C2×C22⋊#87604C2212yes—
9C2×C4×#87604—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4—21no—
12C22×Dic194C411yescharacteristic, complemented
13C22×C4×D97C2211yescharacteristic
14#15690788—21no—
15#15690788—21no—
16#15690788—21no—
17#15690788—21no—
18C2×#13095490C411yescharacteristic, complemented
19C2×#13095490C411yescharacteristic, complemented
20C2×#13095488C2211yescharacteristic
21C2×#13095488C2211yescharacteristic
22C2×C4⋊C4×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C22×#87604—22no—
25C4⋊#87604—42no—
26C22⋊#87604—22no—
27C22⋊#87604—24no—
28C22⋊#87604Q814yescomplemented
29C4×#87604—42no—
30C23×D97C2311yescharacteristic
31C2×Dic194—21no—
32C2×Dic194—21no—
33C2×Dic194—42no—
34C22×Dic97C2×C411yescharacteristic
35C22×Dic97C2×C411yescharacteristic
36C22×Dic97C2×C411yescharacteristic
37C2×C4×D97—42no—
38C2×C4×D97D411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97D411yescharacteristic
41C2×C4×D97D411yescharacteristic
42#13095490—21no—
43#13095490—21no—
44#13095490—21no—
45#13095490—21no—
46#13095488—21no—
47#13095488—21no—
48#13095488—21no—
49#13095488—21no—
50C22×C4×C97C2×C411yescharacteristic
51C22×C4×C97C2×C411yescharacteristic
52C22×C4×C97C2×C411yescharacteristic
53C4⋊C4×C97—21no—
54C4⋊C4×C97—21no—
55C23×C97C22×C411yescharacteristic
56C22×D97—41no—
57C22×D97—41no—
58C22×D97C2×D411yescharacteristic
59C22×D97C2×D411yescharacteristic
60C22×D97C2×Q812yes—
61C22×D97#4112yes—
62C2×#87604—24no—
63C2×#87604—42no—
64C2×#87604—42no—
65C2×#87604—42no—
66C2×#87604—42no—
67C2×#87604—42no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C22⋊C411yescharacteristic
75C2×Dic97—22no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97C22⋊C411yescharacteristic
81C2×Dic97C22⋊C411yescharacteristic
82C4×D97—22no—
83C4×D97—22no—
84C4×D97—41no—
85C4×D97—41no—
86Dic194—42no—
87Dic194—42no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97C2×C22⋊C411yescharacteristic
91C22×C97C4×Q812yes—
92C2×D97—21no—
93C2×D97—21no—
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—42no—
97C2×D97—42no—
98C2×D97C22⋊Q812yes—
99C2×D97#12411yescharacteristic
100C2×D97C4⋊Q811yescharacteristic
101#87604—44no—
102#87604—82no—
103C4×C97—21no—
104C4×C97—21no—
105C4×C97—41no—
106C4×C97—41no—
107Dic97—21no—
108Dic97—41no—
109Dic97—41no—
110Dic97#9912yes—
111C2×C97—22no—
112C2×C97#39111yescharacteristic
113C2×C97C2×#9911yescharacteristic
114C2×C97#41011yescharacteristic
115D97—22no—
116D97—21no—
117D97—21no—
118#1996—971nomaximal, Sylow
119C97#199611yescharacteristic, complemented, Sylow
120C2×C22⋊Q8—971no—
121#380—971no—
122C4×C22⋊C4—971no—
123#342—972no—
124#342—972no—
125C22×Q8—971no—
126C23×C4—971no—
127C22⋊Q8—1941no—
128C22⋊Q8—1941no—
129C22⋊Q8—1941no—
130C22⋊Q8—1941no—
131C2×C4⋊C4—971no—
132C2×C4⋊C4—971no—
133C2×C4⋊C4—1941no—
134C2×C4⋊C4#8760411yescharacteristic, complemented
135C2×C22⋊C4—972no—
136C2×C22⋊C4—971no—
137C2×C22⋊C4—971no—
138C2×C22⋊C4—972no—
139C2×C42—1941no—
140#95—1941no—
141#95—1941no—
142C24—971no—
143C2×Q8—3882no—
144C2×Q8—1941no—
145C2×Q8—1941no—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—971no—
149C22×C4—971no—
150C22×C4—971no—
151C22×C4—971no—
152C22×C4—971no—
153C22×C4—3882no—
154C22×C4—1942no—
155C22×C4—1942no—
156C22×C4C2×#8760411yescharacteristic
157C22×C4C2×#8760411yescharacteristic
158C22×C4C2×#8760411yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—3882no—
162C4⋊C4—1941no—
163C4⋊C4—1941no—
164C4⋊C4—1941no—
165C4⋊C4—1941no—
166C22⋊C4—974no—
167C22⋊C4—1941no—
168C22⋊C4—1941no—
169C22⋊C4—1941no—
170C22⋊C4—1941no—
171C22⋊C4—1944no—
172C22⋊C4—1942no—
173C42—3882no—
174C23—972no—
175C23—972no—
176C23—971no—
177C23—971no—
178C23—3881no—
179C23—3881no—
180C23C22×#8760411yescharacteristic
181Q8—3882no—
182Q8—3882no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—971no—
189C2×C4—971no—
190C2×C4—3881no—
191C2×C4—3881no—
192C2×C4—3882no—
193C2×C4—3882no—
194C2×C4—3882no—
195C2×C4—3882no—
196C2×C4—3882no—
197C2×C4—3881no—
198C2×C4—3881no—
199C2×C4—1942no—
200C2×C4—1942no—
201C2×C4—1942no—
202C2×C4—1944no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4C22⋊#8760411yescharacteristic
206C2×C4C22⋊#8760411yescharacteristic
207C22—21no—
208C22—21no—
209C22—971no—
210C22—972no—
211C22—971no—
212C22—3882no—
213C22—3882no—
214C22—1941no—
215C22—1941no—
216C22—1941no—
217C22—1941no—
218C22C2×C22⋊#8760411yescharacteristic
219C22Q8×#8760412yes—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—7762no—
225C4—972no—
226C4—3884no—
227C4—3881no—
228C4—3881no—
229C4—1941no—
230C2—22no—
231C2—1941no—
232C2—1942no—
233C2—1941no—
234C2(C2×#87604)⋊Q811yescharacteristic
235C2D97⋊#9911yescharacteristic
236C2#2026171411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radical, cosocleC23×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1996
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. C22×Dic194C4semidirect
18. C2×#13095490C4semidirect
19. C2×#13095490C4semidirect
22. C2×C4⋊C4×C97C4semidirect
28. C22⋊#87604Q8semidirect
119. C97#1996semidirect
134. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616772
C4×C22⋊#876042
#202616132
C2×#156907882
#199697