Finite Group Databasebeta

← #324×D97 #27140044

Subgroups — #324×D97 #27140044

Order 12416.

328 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 87 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×C4⋊C4×D97C211yesmaximal, characteristic
2C22⋊C8×D97C212yesmaximal
3#20260461C211yesmaximal, characteristic, complemented
4#20260460C212yesmaximal, complemented
5#324×C97C211yesmaximal, characteristic, complemented
6C22×C4×D97C411yescharacteristic
7C22×C4×D97C411yescharacteristic
8C22×C4×D97C2211yescharacteristic
9C2×C8×D97—22no—
10C2×C4⋊Dic97C411yescharacteristic
11C2×#13095490C411yescharacteristic
12C2×#13095490C2211yescharacteristic
13C4⋊C4×D97—22no—
14C4⋊C4×D97—22no—
15C22⋊#11715895C2212yes—
16#15690722—22no—
17C2×C4⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C23×D97C2×C411yescharacteristic
20C22×Dic97C2×C411yescharacteristic
21C22×Dic97C2×C411yescharacteristic
22C22×Dic97C2×C411yescharacteristic
23C2×C4×D97—21no—
24C2×C4×D97—21no—
25C2×C4×D97—22no—
26C2×C4×D97—21no—
27C2×C4×D97—21no—
28C2×C4×D97—42no—
29C2×C4×D97—42no—
30C2×C4×D97C814yescomplemented
31C2×C4×D97D412yes—
32C2×C4×D97D412yes—
33C2×#11715895—22no—
34C8×D97—44no—
35C4⋊Dic97—22no—
36#13095490—22no—
37#13095490—22no—
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2311yescharacteristic
41C2×C8×C97—22no—
42C4⋊C4×C97—22no—
43C23×C97C22×C411yescharacteristic
44C22×D97—22no—
45C22×D97—22no—
46C22×D97C22⋊C411yescharacteristic
47C22×D97C22⋊C411yescharacteristic
48C22×D97C2×C812yes—
49C22×D97M1612yes—
50C2×C4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97C2×C812yescomplemented
56C2×C4×C97C2×D412yes—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—41no—
61C2×Dic97—41no—
62C2×Dic97C22⋊C412yes—
63C2×Dic97C2×C812yescomplemented
64C4×D97—24no—
65C4×D97—24no—
66C4×D97—24no—
67C4×D97—42no—
68C4×D97—42no—
69C4×D97—42no—
70C8×C97—42no—
71#11715895—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97C2×C22⋊C411yescharacteristic
75C22×C97C22×C811yescharacteristic
76C22×C97C2×M1611yescharacteristic
77C2×D97—21no—
78C2×D97—21no—
79C2×D97—24no—
80C2×D97—24no—
81C2×D97C22⋊C812yes—
82C2×D97C22⋊C812yes—
83C2×D97#9912yes—
84C2×D97#10112yes—
85C4×C97—22no—
86C4×C97—22no—
87C4×C97—41no—
88Dic97—22no—
89Dic97—22no—
90Dic97—41no—
91C2×C97—21no—
92C2×C97—21no—
93C2×C97C2×C22⋊C811yescharacteristic
94C2×C97C2×#9911yescharacteristic
95C2×C97C2×#10111yescharacteristic
96D97—22no—
97D97#32414yesdirect product factor
98C2×#324—971nomaximal, Sylow
99C97C2×#32411yescharacteristic, complemented, Sylow
100C22×C4⋊C4—971no—
101C2×C22⋊C8—972no—
102#324—971no—
103#324—972no—
104#324D9711yescharacteristic, direct product factor
105C23×C4—971no—
106C23×C4—971no—
107C23×C4—971no—
108C22×C8—1942no—
109C2×C4⋊C4—971no—
110C2×C4⋊C4—971no—
111C2×C4⋊C4—971no—
112C2×C4⋊C4—1942no—
113C2×C4⋊C4—1942no—
114C2×C4⋊C4C2×D9711yescharacteristic
115C22⋊C8—972no—
116C22⋊C8—1942no—
117C22⋊C8C2×D9712yes—
118C24—971no—
119C22×C4—971no—
120C22×C4—971no—
121C22×C4—972no—
122C22×C4—974no—
123C22×C4—972no—
124C22×C4—971no—
125C22×C4—3882no—
126C22×C4—3882no—
127C22×C4—1941no—
128C22×C4—1941no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4—1942no—
132C22×C4C4×D9711yescharacteristic
133C22×C4C4×D9711yescharacteristic
134C22×C4C22×D9711yescharacteristic
135C2×C8—22no—
136C2×C8—3884no—
137C2×C8—1942no—
138C4⋊C4—22no—
139C4⋊C4—1942no—
140C4⋊C4—1942no—
141C4⋊C4—1942no—
142C23—971no—
143C23—971no—
144C23—972no—
145C23—972no—
146C23—1942no—
147C23—1942no—
148C23C2×C4×D9711yescharacteristic
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—972no—
155C2×C4—972no—
156C2×C4—3882no—
157C2×C4—3882no—
158C2×C4—3882no—
159C2×C4—3881no—
160C2×C4—3881no—
161C2×C4—1944no—
162C2×C4—1944no—
163C2×C4—1944no—
164C2×C4—1941no—
165C2×C4—1941no—
166C2×C4—1941no—
167C2×C4C8×D9712yescomplemented
168C2×C4D4×D9712yes—
169C8—42no—
170C8—3882no—
171C22—21no—
172C22—21no—
173C22—972no—
174C22—972no—
175C22—972no—
176C22—972no—
177C22—1944no—
178C22—1944no—
179C22—1941no—
180C22—1941no—
181C22C22⋊C4×D9711yescharacteristic
182C22C2×C8×D9711yescharacteristic
183C22M16×D9711yescharacteristic
184C4—22no—
185C4—22no—
186C4—41no—
187C4—3881no—
188C4—1942no—
189C4—1942no—
190C2—21no—
191C2—21no—
192C2—974no—
193C2—1942no—
194C2C22⋊C8×D9711yescharacteristic
195C2#99×D9711yescharacteristic
196C2#101×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC22×C97C22×C8
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×C4×C97C23
hypercenter#324D97
hypocenterC97C2×#324
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. #20260461C2semidirect
4. #20260460C2semidirect
5. #324×C97C2semidirect
30. C2×C4×D97C8semidirect
55. C2×C4×C97C2×C8semidirect
63. C2×Dic97C2×C8semidirect
97. D97#324direct (N × H)
99. C97C2×#324semidirect
104. #324D97direct (N × H)
167. C2×C4C8×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
C22⋊C8×D972
#202604612
#202604602
#324×C972
C2×#32497