Finite Group Databasebeta

← #342×D97 #27141464

Subgroups — #342×D97 #27141464

Order 12416.

680 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 145 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×C22⋊C4×D97C211yesmaximal, characteristic
2C2×C22⋊C4×D97C212yesmaximal
3#20260502C211yesmaximal, characteristic, complemented
4#20260501C212yesmaximal, complemented
5#342×C97C211yesmaximal, characteristic, complemented
6C24×D97C2211yescharacteristic
7C22×C4×D97—22no—
8C22×C4×D97C412yescomplemented
9C2×C22⋊Dic97C2211yescharacteristic
10C2×C22⋊Dic97C2212yes—
11C2×#13095488—22no—
12C2×#13095488C412yescomplemented
13C22⋊C4×D97—22no—
14C22⋊C4×D97—22no—
15C22⋊C4×D97—24no—
16C22⋊C4×D97C418yescomplemented
17C2×C22⋊C4×C97C2211yescharacteristic
18C2×C22⋊C4×C97C2212yes—
19C23×D97—44no—
20C23×D97C2×C411yescharacteristic
21C23×D97C2×C414yes—
22C23×D97C2×C411yescharacteristic
23C23×D97D412yes—
24C23×D97D412yes—
25C23×D97D412yes—
26C23×D97Q812yes—
27C22×Dic97—22no—
28C22×Dic97C2×C412yescomplemented
29C2×C4×D97—22no—
30C2×C4×D97—22no—
31C2×C4×D97—28no—
32C2×C4×D97—48no—
33C2×C4×D97—44no—
34C22⋊Dic97—22no—
35C22⋊Dic97—22no—
36C22⋊Dic97C2×C414yescomplemented
37#13095488—24no—
38#13095488—28no—
39C24×C97C2311yescharacteristic
40C22×C4×C97—22no—
41C22×C4×C97C2×C412yescomplemented
42C22⋊C4×C97—22no—
43C22⋊C4×C97—22no—
44C22⋊C4×C97C2×C414yescomplemented
45C23×C97—42no—
46C23×C97C22×C411yescharacteristic
47C23×C97C22×C412yes—
48C23×C97C2×D412yes—
49C23×C97C2×D411yescharacteristic
50C23×C97C2×Q811yescharacteristic
51C22×D97—24no—
52C22×D97—28no—
53C22×D97—48no—
54C22×D97—44no—
55C22×D97—44no—
56C22×D97—42no—
57C22×D97—44no—
58C22×D97C4212yes—
59C22×D97C4212yes—
60C22×D97C22⋊C412yes—
61C22×D97C22⋊C414yes—
62C22×D97C22⋊C411yescharacteristic
63C22×D97C22⋊C414yes—
64C22×D97C22⋊C411yescharacteristic
65C22×D97C4⋊C412yes—
66C22×D97C4⋊C414yes—
67C22×D97C4⋊C414yes—
68C22×D97C4⋊C412yes—
69C2×C4×C97—22no—
70C2×C4×C97—24no—
71C2×C4×C97—44no—
72C2×C4×C97—42no—
73C2×Dic97—22no—
74C2×Dic97—24no—
75C2×Dic97—44no—
76C2×Dic97—42no—
77C4×D97—416no—
78C4×D97—44no—
79C4×D97—44no—
80C22×C97—22no—
81C22×C97—24no—
82C22×C97—42no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97C2×C4211yescharacteristic
86C22×C97C2×C22⋊C412yes—
87C22×C97C2×C22⋊C411yescharacteristic
88C22×C97C2×C4⋊C411yescharacteristic
89C22×C97C2×C4⋊C412yes—
90C2×D97—22no—
91C2×D97—24no—
92C2×D97—28no—
93C2×D97—24no—
94C2×D97—24no—
95C2×D97—28no—
96C2×D97—42no—
97C2×D97—44no—
98C2×D97—44no—
99C2×D97—42no—
100C2×D97#9512yes—
101C2×D97#9514yes—
102C2×D97#9512yes—
103C2×D97#9914yes—
104C4×C97—44no—
105C4×C97—42no—
106Dic97—44no—
107Dic97—42no—
108C2×C97—22no—
109C2×C97—21no—
110C2×C97—21no—
111C2×C97—22no—
112C2×C97C2×#9511yescharacteristic
113C2×C97C2×#9912yes—
114D97—22no—
115D97—24no—
116D97#34214yesdirect product factor
117C2×#342—971nomaximal, Sylow
118C97C2×#34211yescharacteristic, complemented, Sylow
119C22×C22⋊C4—972no—
120C22×C22⋊C4—971no—
121#342—972no—
122#342—971no—
123#342D9711yescharacteristic, direct product factor
124C25—971no—
125C23×C4—972no—
126C23×C4—1942no—
127C2×C22⋊C4—972no—
128C2×C22⋊C4—978no—
129C2×C22⋊C4—971no—
130C2×C22⋊C4—972no—
131C2×C22⋊C4—1944no—
132C2×C22⋊C4—1942no—
133C2×C22⋊C4—1942no—
134C2×C22⋊C4—1942no—
135C2×C22⋊C4C2×D9711yescharacteristic
136C2×C22⋊C4C2×D9712yes—
137C24—971no—
138C24—974no—
139C24—972no—
140C24—972no—
141C24—972no—
142C24—971no—
143C24—972no—
144C24—3884no—
145C24C22×D9711yescharacteristic
146C22×C4—22no—
147C22×C4—972no—
148C22×C4—3888no—
149C22×C4—3884no—
150C22×C4—1948no—
151C22×C4—1942no—
152C22×C4—1942no—
153C22×C4—1942no—
154C22×C4C4×D9712yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—974no—
158C22⋊C4—1948no—
159C22⋊C4—1944no—
160C22⋊C4—1942no—
161C22⋊C4—1942no—
162C22⋊C4C4×D9714yescomplemented
163C23—42no—
164C23—971no—
165C23—972no—
166C23—974no—
167C23—972no—
168C23—974no—
169C23—972no—
170C23—974no—
171C23—972no—
172C23—971no—
173C23—974no—
174C23—972no—
175C23—3882no—
176C23—3884no—
177C23—3888no—
178C23—3884no—
179C23—3884no—
180C23—1944no—
181C23—1948no—
182C23C2×C4×D9711yescharacteristic
183C23C2×C4×D9712yes—
184C23D4×D9712yes—
185C23D4×D9711yescharacteristic
186C23Q8×D9711yescharacteristic
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—3884no—
192C2×C4—38816no—
193C2×C4—3884no—
194C2×C4—3884no—
195C2×C4—3882no—
196C2×C4—1944no—
197C2×C4—1942no—
198C22—22no—
199C22—24no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—974no—
204C22—972no—
205C22—974no—
206C22—972no—
207C22—3884no—
208C22—3882no—
209C22—3882no—
210C22—3884no—
211C22—1948no—
212C22—1944no—
213C22—1944no—
214C22—1948no—
215C22—1942no—
216C22—1944no—
217C22C42×D9711yescharacteristic
218C22C22⋊C4×D9712yes—
219C22C22⋊C4×D9711yescharacteristic
220C22C4⋊C4×D9711yescharacteristic
221C22C4⋊C4×D9712yes—
222C4—44no—
223C4—42no—
224C4—3884no—
225C4—3882no—
226C2—22no—
227C2—21no—
228C2—21no—
229C2—22no—
230C2—974no—
231C2—1944no—
232C2—1942no—
233C2#95×D9711yescharacteristic
234C2#99×D9712yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC22×C97C2×C42
FrattiniC24C22×D97
Fitting#342×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×D97C22
cosocleC24×C97C23
hypercenter#342D97
hypocenterC97C2×#342
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #20260502C2semidirect
4. #20260501C2semidirect
5. #342×C97C2semidirect
8. C22×C4×D97C4semidirect
12. C2×#13095488C4semidirect
16. C22⋊C4×D97C4semidirect
28. C22×Dic97C2×C4semidirect
36. C22⋊Dic97C2×C4semidirect
41. C22×C4×C97C2×C4semidirect
44. C22⋊C4×C97C2×C4semidirect
116. D97#342direct (N × H)
118. C97C2×#342semidirect
123. #342D97direct (N × H)
154. C22×C4C4×D97semidirect
162. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D972
#202605022
#202605012
#342×C972
C2×#34297