Finite Group Databasebeta

← #27142205

Subgroups — #27142205

Order 12416.

338 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#13095495C211yesmaximal, characteristic
2C2×#15690729C212yesmaximal, complemented
3#20260699C211yesmaximal, characteristic
4#20260504C212yesmaximal, complemented
5C2×C22⋊Q8×C97C211yesmaximal, characteristic
6C2×#13095495—24no—
7C2×#13095495—21no—
8C2×#13095495C2212yes—
9C2×#13095495C2212yes—
10C22×#11715895—21no—
11#15690729—24no—
12C22⋊#11715895—21no—
13C22⋊#11715895—21no—
14C22×Q8×C97C2211yescharacteristic
15C23×C4×C97C2211yescharacteristic
16C22⋊Q8×C97—21no—
17C22⋊Q8×C97—21no—
18C22⋊Q8×C97—21no—
19C22⋊Q8×C97—21no—
20C2×C4⋊C4×C97C412yes—
21C2×C4⋊C4×C97C2211yescharacteristic
22C2×C22⋊C4×C97C412yes—
23#13095495—24no—
24#13095495—22no—
25#13095495—22no—
26#13095495—22no—
27#13095495D414yes—
28C2×#11715895—24no—
29C2×#11715895—22no—
30C2×#11715895—22no—
31C24×C97C2×C411yescharacteristic
32C2×Q8×C97—21no—
33C2×Q8×C97—21no—
34C2×Q8×C97—22no—
35C2×Q8×C97—22no—
36C22×C4×C97—22no—
37C22×C4×C97—22no—
38C22×C4×C97C2×C412yes—
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2×C412yes—
41C22×C4×C97D411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C22×C4×C97D411yescharacteristic
44C22×C4×C97D411yescharacteristic
45C22×C4×C97C2311yescharacteristic
46C4⋊C4×C97—22no—
47C4⋊C4×C97—22no—
48C4⋊C4×C97—21no—
49C4⋊C4×C97—21no—
50C22⋊C4×C97—22no—
51C22⋊C4×C97—22no—
52C23×C97—22no—
53C23×C97—22no—
54C23×C97—41no—
55C23×C97C22⋊C411yescharacteristic
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58Q8×C97—41no—
59Q8×C97—42no—
60Q8×C97—41no—
61C2×C4×C97—22no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—24no—
71C2×C4×C97—42no—
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C2×D412yes—
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97#4112yes—
78#11715895—24no—
79#11715895—42no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—44no—
88C22×C97#10112yescomplemented
89C22×C97C2×C22⋊C411yescharacteristic
90C22×C97C4×D412yes—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—42no—
94C4×C97—42no—
95C4×C97#12011yescharacteristic
96C4×C97#12112yes—
97C4×C97#12311yescharacteristic
98C2×C97—22no—
99C2×C97—22no—
100C2×C97—41no—
101C2×C97#38611yescharacteristic
102C2×C97C2×#10111yescharacteristic
103C2×C97#41311yescharacteristic
104#1901—971nomaximal, Sylow
105C97#190111yescharacteristic, complemented, Sylow
106C22×M16—971no—
107C2×C22⋊Q8D9711yescharacteristic
108C2×#101—972no—
109#407—971no—
110#343—972no—
111C22×Q8C2×D9711yescharacteristic
112C23×C4C2×D9711yescharacteristic
113C2×M16—972no—
114C2×M16—972no—
115C2×M16—1944no—
116C2×M16—1941no—
117C22×C8—1941no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C2×C4⋊C4Dic9712yes—
123C2×C4⋊C4C2×D9711yescharacteristic
124C2×C22⋊C4Dic9712yes—
125#101—1944no—
126C22⋊C8—1941no—
127C22⋊C8—1941no—
128C24C2×Dic9711yescharacteristic
129C2×Q8—22no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4C2×Dic9712yes—
136C22×C4C2×Dic9711yescharacteristic
137C22×C4C2×Dic9712yes—
138C22×C4C22⋊D9711yescharacteristic
139C22×C4C22⋊D9711yescharacteristic
140C22×C4C22⋊D9711yescharacteristic
141C22×C4C22⋊D9711yescharacteristic
142C22×C4C22×D9711yescharacteristic
143M16—974no—
144M16—1942no—
145M16—1942no—
146M16—1944no—
147M16—1942no—
148C2×C8—1942no—
149C2×C8—1944no—
150C2×C8—1942no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C22⋊C4—22no—
156C22⋊C4—22no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23C22⋊Dic9711yescharacteristic
161C23C22⋊Dic9711yescharacteristic
162C23C22×Dic9711yescharacteristic
163Q8—41no—
164Q8—42no—
165Q8—41no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—42no—
177C2×C4C22⋊Dic9711yescharacteristic
178C2×C4C22⋊Dic9711yescharacteristic
179C2×C4D4×D9712yes—
180C2×C4C2×C22⋊D9711yescharacteristic
181C2×C4C2×C22⋊D9711yescharacteristic
182C2×C4#1309551012yes—
183C8—3882no—
184C8—1944no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#1569072912yescomplemented
194C22C2×C22⋊Dic9711yescharacteristic
195C22D4×Dic9712yes—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—42no—
200C4#1569077811yescharacteristic
201C4#1569078212yes—
202C4#1569079111yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#2026061511yescharacteristic
207C2C2×#1569072911yescharacteristic
208C2#2026071911yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊Q8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊Q8D97
hypocenterC97#1901
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15690729C2semidirect
4. #20260504C2semidirect
88. C22×C97#101semidirect
105. C97#1901semidirect
193. C22#15690729semidirect

Maximal subgroups

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

subgroupindex
C22×#130954952
C2×#156907292
#202606992
#202605042
C2×C22⋊Q8×C972
#190197