Finite Group Databasebeta

← #27140189

Subgroups — #27140189

Order 12416.

232 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 75 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#20261289C211yesmaximal, characteristic
2#20260757C211yesmaximal, characteristic
3C4×C8×D97C211yesmaximal, characteristic
4#20260467C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260465C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095494—21no—
9C2×C8×D97—21no—
10#15690809C411yescharacteristic
11#15690808C411yescharacteristic
12#15690800—21no—
13#15690790C411yescharacteristic
14#15690788—21no—
15#15690786C411yescharacteristic
16C4×Dic194—21no—
17C4×Dic194C2211yescharacteristic
18#15690767—21no—
19C42×D97C2211yescharacteristic
20C4⋊#11715895C2211yescharacteristic
21#15690739—21no—
22C4×#11715895C2211yescharacteristic
23C8×Dic97—21no—
24C4×Q8×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32#13095494—42no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C8×D97—24no—
36C4⋊Dic97—21no—
37C4⋊Dic97C2×C411yescharacteristic
38#13095490—21no—
39#13095490—21no—
40#13095490—21no—
41#13095490—21no—
42#13095490—21no—
43#13095488—21no—
44#13095488—21no—
45C4×Dic97—21no—
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×Q8×C97C2×C411yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97C2×C411yescharacteristic
54C42×C97—21no—
55C42×C97C2311yescharacteristic
56C22×D97C22⋊C411yescharacteristic
57Q8×C97—21no—
58Q8×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97C22×C411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69C4×D97—21no—
70C4×D97—21no—
71C4×D97—22no—
72C4×D97SD1612yes—
73C4×D97Q1612yes—
74Dic194—21no—
75Dic194—21no—
76Dic194—42no—
77C8×C97—22no—
78C8×C97—41no—
79#11715895—22no—
80#11715895—41no—
81C22×C97C2×C22⋊C411yescharacteristic
82C2×D97—21no—
83C2×D97—21no—
84C2×D97#10312yes—
85C4×C97—21no—
86C4×C97—41no—
87C4×C97—41no—
88C4×C97#13111yescharacteristic
89C4×C97#13111yescharacteristic
90C4×C97C2×SD1611yescharacteristic
91C4×C97C2×Q1611yescharacteristic
92Dic97—21no—
93Dic97—41no—
94Dic97—41no—
95Dic97#10312yes—
96Dic97#10412yes—
97C2×C97#40811yescharacteristic
98C2×C97C2×#10311yescharacteristic
99C2×C97C2×#10411yescharacteristic
100D97—22no—
101#1493—971nomaximal, Sylow
102C97#149311yescharacteristic, complemented, Sylow
103#533—971no—
104#423—971no—
105C2×C4×C8—971no—
106Q8⋊C8—971no—
107Q8⋊C8—971no—
108Q8⋊C8—971no—
109Q8⋊C8D9711yescharacteristic, complemented
110C2×M16—1941no—
111C22×C8—1941no—
112C4⋊Q8—971no—
113C4⋊Q8—971no—
114#125—1941no—
115C22⋊Q8—971no—
116C22⋊Q8—971no—
117C22⋊Q8—1941no—
118C4×Q8—971no—
119C4×Q8—1941no—
120C4×Q8C2×D9711yescharacteristic
121#117—1941no—
122C2×C42—971no—
123C4⋊C8—971no—
124C4⋊C8—1941no—
125C4⋊C8C2×D9711yescharacteristic
126C4×C8—971no—
127C4×C8—1941no—
128C4×C8C2×D9711yescharacteristic
129C2×Q8—971no—
130C2×Q8—1941no—
131C2×Q8C4×D9711yescharacteristic
132C22×C4—971no—
133C22×C4—971no—
134C22×C4—971no—
135M16—3882no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—1944no—
139C2×C8—1941no—
140C2×C8—1941no—
141C4⋊C4—21no—
142C4⋊C4—971no—
143C4⋊C4—1941no—
144C4⋊C4—1941no—
145C4⋊C4—1941no—
146C4⋊C4—1941no—
147C4⋊C4—1941no—
148C4⋊C4—1941no—
149C4⋊C4C4×D9711yescharacteristic
150C22⋊C4—1941no—
151C22⋊C4—1941no—
152C42—21no—
153C42—971no—
154C42—971no—
155C42—971no—
156C42—1941no—
157C42C22×D9711yescharacteristic
158C23—971no—
159Q8—21no—
160Q8—21no—
161Q8—3882no—
162Q8—1941no—
163Q8—1941no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—971no—
167C2×C4—971no—
168C2×C4—972no—
169C2×C4—972no—
170C2×C4—971no—
171C2×C4—1942no—
172C2×C4—1941no—
173C2×C4—1941no—
174C2×C4—1941no—
175C2×C4—1941no—
176C2×C4C2×C4×D9711yescharacteristic
177C2×C4D4×D9711yescharacteristic
178C2×C4D4×D9711yescharacteristic
179C8—22no—
180C8—41no—
181C8—3881no—
182C8—1942no—
183C22—972no—
184C22—1941no—
185C22—1941no—
186C22C22⋊C4×D9711yescharacteristic
187C4—21no—
188C4—41no—
189C4—41no—
190C4—972no—
191C4—972no—
192C4—3881no—
193C4—3881no—
194C4—1941no—
195C4#1569081611yescharacteristic
196C4#1569081811yescharacteristic
197C4SD16×D9711yescharacteristic
198C4Q16×D9711yescharacteristic
199C2—1942no—
200C2#2026070011yescharacteristic
201C2#103×D9711yescharacteristic
202C2#104×D9711yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenterQ8⋊C8D97
hypocenterC97#1493
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #20260467C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260465C2semidirect
7. Q8⋊C8×C97C2semidirect
102. C97#1493semidirect
109. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612892
#202607572
C4×C8×D972
#202604672
Q8⋊#117158952
#202604652
Q8⋊C8×C972
#149397