Finite Group Databasebeta

← #27141645

Subgroups — #27141645

Order 12416.

852 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C23×C4×D97C211yesmaximal, characteristic
2C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
3C2×#15690722C214yesmaximal, complemented
4C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
5C24×D97C411yescharacteristic
6C23×Dic97C411yescharacteristic
7C22×C4×D97—28no—
8C22×C4×D97—22no—
9C22×C4×D97—22no—
10C22×C4×D97C412yes—
11C22×C4×D97C2212yes—
12C22×#11715895C2212yescomplemented
13C22⋊#11715895—22no—
14#15690722—28no—
15C23×C4×C97C2211yescharacteristic
16C22×C8×C97C2212yescomplemented
17C22⋊C8×C97—22no—
18C23×D97—22no—
19C23×D97—22no—
20C23×D97—24no—
21C23×D97—24no—
22C23×D97C814yescomplemented
23C23×D97C2×C412yes—
24C22×Dic97—22no—
25C22×Dic97—24no—
26C22×Dic97C2×C412yes—
27C2×C4×D97—216no—
28C2×C4×D97—216no—
29C2×C4×D97—24no—
30C2×C4×D97—24no—
31C2×C4×D97—24no—
32C2×C4×D97—22no—
33C2×C4×D97—22no—
34C2×C4×D97—24no—
35C2×C4×D97D418yes—
36C2×#11715895—24no—
37C2×#11715895—22no—
38C24×C97C2×C411yescharacteristic
39C22×C4×C97—24no—
40C22×C4×C97C2×C411yescharacteristic
41C22×C4×C97D412yes—
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44C2×C8×C97—22no—
45C2×C8×C97—24no—
46C23×C97—22no—
47C23×C97—22no—
48C23×C97C22⋊C412yes—
49C23×C97C2×C812yes—
50C23×C97M1612yes—
51C23×C97C22×C411yescharacteristic
52C22×D97—28no—
53C22×D97—216no—
54C22×D97—28no—
55C22×D97—28no—
56C22×D97—22no—
57C22×D97—22no—
58C22×D97—22no—
59C22×D97—22no—
60C22×D97—216no—
61C22×D97C22⋊C414yes—
62C22×D97C2×C814yes—
63C22×D97M1614yes—
64C2×C4×C97—22no—
65C2×C4×C97—24no—
66C2×C4×C97—24no—
67C2×C4×C97C22⋊C412yes—
68C2×C4×C97C2×D414yes—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×Dic97—28no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97C22⋊C414yes—
75C4×D97—216no—
76C4×D97—28no—
77C4×D97—28no—
78C8×C97—42no—
79#11715895—42no—
80C22×C97—22no—
81C22×C97—24no—
82C22×C97—22no—
83C22×C97—24no—
84C22×C97C22⋊C814yescomplemented
85C22×C97C2×C22⋊C412yes—
86C22×C97C2×C22⋊C411yescharacteristic
87C22×C97C22×C811yescharacteristic
88C22×C97C2×M1612yes—
89C22×C97C2×M1611yescharacteristic
90C2×D97—232no—
91C2×D97—24no—
92C2×D97—24no—
93C2×D97—24no—
94C2×D97—24no—
95C2×D97—24no—
96C2×D97—24no—
97C2×D97C22⋊C818yescomplemented
98C4×C97—22no—
99C4×C97#12012yes—
100C4×C97#12012yes—
101Dic97—24no—
102C2×C97—24no—
103C2×C97#37911yescharacteristic
104C2×C97C2×C22⋊C812yes—
105C2×C97C2×C22⋊C811yescharacteristic
106C2×C97#40712yes—
107C2×C97#40711yescharacteristic
108D97—28no—
109#1779—971nomaximal, Sylow
110C97#177911yescharacteristic, complemented, Sylow
111C24×C4—971no—
112C2×C22⋊C8—971no—
113C2×C22⋊C8—974no—
114C2×C22⋊C8D9711yescharacteristic, complemented
115C25—971no—
116C23×C4—972no—
117C23×C4—971no—
118C23×C4—972no—
119C23×C4—1948no—
120C23×C4—1942no—
121C23×C4—1942no—
122C23×C4C2×D9711yescharacteristic
123C22×C8—972no—
124C22×C8C2×D9712yescomplemented
125C22⋊C8—22no—
126C22⋊C8—1948no—
127C22⋊C8—1942no—
128C24—972no—
129C24—974no—
130C24—1944no—
131C24—1944no—
132C24—1942no—
133C24—1942no—
134C24C4×D9711yescharacteristic
135C22×C4—24no—
136C22×C4—972no—
137C22×C4—978no—
138C22×C4—1944no—
139C22×C4—1944no—
140C22×C4—19416no—
141C22×C4—19416no—
142C22×C4—1944no—
143C22×C4—1942no—
144C22×C4—1944no—
145C22×C4—1944no—
146C22×C4—1942no—
147C22×C4—1942no—
148C22×C4C4×D9711yescharacteristic
149C22×C4D38812yes—
150C22×C4C22⋊D9712yes—
151C22×C4C22×D9711yescharacteristic
152C2×C8—24no—
153C2×C8—22no—
154C2×C8—1944no—
155C2×C8—1942no—
156C23—22no—
157C23—22no—
158C23—974no—
159C23—974no—
160C23—974no—
161C23—19416no—
162C23—1948no—
163C23—19416no—
164C23—1948no—
165C23—1942no—
166C23—1942no—
167C23—1942no—
168C23—1942no—
169C23—1948no—
170C23#1309548812yes—
171C23C8×D9712yes—
172C23#1309549412yes—
173C23C2×C4×D9711yescharacteristic
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—974no—
178C2×C4—19416no—
179C2×C4—1948no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4—1948no—
183C2×C4—1948no—
184C2×C4#1309548812yes—
185C2×C4D4×D9714yes—
186C2×C4C2×C22⋊D9711yescharacteristic
187C2×C4C2×D38811yescharacteristic
188C8—42no—
189C8—3882no—
190C22—24no—
191C22—24no—
192C22—22no—
193C22—22no—
194C22—978no—
195C22—19432no—
196C22—1944no—
197C22—1944no—
198C22—1944no—
199C22—1944no—
200C22—1944no—
201C22—1944no—
202C22#1569072214yescomplemented
203C22C22⋊C4×D9712yes—
204C22C2×#1309548811yescharacteristic
205C22C2×C8×D9711yescharacteristic
206C22M16×D9712yes—
207C22C2×#1309549411yescharacteristic
208C4—22no—
209C4—1944no—
210C4#1569077812yes—
211C4#1569078012yes—
212C2—24no—
213C2—1948no—
214C2#2026059011yescharacteristic
215C2C22⋊C8×D9712yes—
216C2C2×#1569072211yescharacteristic
217C2#2026069712yes—
218C2#2026069811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×D97
commutatorC22×C97C22×C8
FittingC2×C22⋊C8×C97C2
socleC23×C97C22×C4
prime radicalC24×D97C4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊C8D97
hypocenterC97#1779
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#11715895C2semidirect
3. C2×#15690722C2semidirect
4. C2×C22⋊C8×C97C2semidirect
12. C22×#11715895C22semidirect
16. C22×C8×C97C22semidirect
22. C23×D97C8semidirect
84. C22×C97C22⋊C8semidirect
97. C2×D97C22⋊C8semidirect
110. C97#1779semidirect
114. C2×C22⋊C8D97semidirect
124. C22×C8C2×D97semidirect
202. C22#15690722semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D972
C2×C22⋊#117158952
C2×#156907222
C2×C22⋊C8×C972
#177997