Finite Group Databasebeta

← #27157134

Subgroups — #27157134

Order 12416.

294 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261707C211yesmaximal, characteristic
2C4×C22⋊#87604C211yesmaximal, characteristic
3#20261589C212yesmaximal, complemented
4#20261588C212yesmaximal, complemented
5C22⋊Q8×D97C211yesmaximal, characteristic
6C2×#13095520—21no—
7C2×C22⋊#87604—21no—
8C2×C4×#87604—21no—
9#15690867—21no—
10#15690867C2212yes—
11(C2×#87604)⋊C4C2212yes—
12C2×Q8×D97C411yescharacteristic, complemented
13C22×C4×D97C2211yescharacteristic
14#15690790C2211yescharacteristic
15#15690789C411yescharacteristic, complemented
16#15690788—21no—
17#15690787—21no—
18#15690786C2211yescharacteristic
19C4⋊C4×D97—21no—
20C4⋊C4×D97C411yescharacteristic, complemented
21C22⋊C4×D97—21no—
22C22⋊Q8×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095520—42no—
25C2×#11715897—22no—
26C22⋊#87604—24no—
27C4×#87604—42no—
28C23×D97D411yescharacteristic
29C2×Dic194—21no—
30C2×Dic194C2×C411yescharacteristic
31Q8×D97—42no—
32C22×Dic97D411yescharacteristic
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97—42no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97C2311yescharacteristic
40C4⋊Dic97—21no—
41C4⋊Dic97C2×C411yescharacteristic
42#13095490—21no—
43#13095490—21no—
44#13095490—21no—
45C22⋊Dic97—21no—
46#13095488—21no—
47#13095488—21no—
48C2×Q8×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C4⋊C4×C97—21no—
51C4⋊C4×C97C2×C411yescharacteristic
52C22⋊C4×C97—21no—
53C23×C97C22⋊C411yescharacteristic
54C22×D97—22no—
55C22×D97—22no—
56C22×D97—41no—
57C22×D97C22⋊C411yescharacteristic
58C22×D97C2×D411yescharacteristic
59C2×#87604—24no—
60C2×#87604—42no—
61C2×#87604—42no—
62Q8×C97—41no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97—41no—
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22×C411yescharacteristic
68C2×Dic97—22no—
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73C4×D97—22no—
74C4×D97—22no—
75C4×D97—41no—
76C4×D97—41no—
77C4×D97—42no—
78C4×D97—42no—
79Dic194—42no—
80Dic194—41no—
81#11715897—42no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97C2×C22⋊C411yescharacteristic
86C2×D97—21no—
87C2×D97—21no—
88C2×D97—21no—
89C2×D97—21no—
90C2×D97—44no—
91C2×D97#10412yes—
92C2×D97#13712yes—
93#87604—44no—
94C4×C97—21no—
95C4×C97—21no—
96C4×C97—41no—
97C4×C97—41no—
98Dic97—21no—
99Dic97—41no—
100Dic97—41no—
101Dic97#9912yes—
102C2×C97—41no—
103C2×C97C2×#9911yescharacteristic
104C2×C97#41911yescharacteristic
105C2×C97C2×#10411yescharacteristic
106D97—22no—
107D97—22no—
108#1508—971nomaximal, Sylow
109C97#150811yescharacteristic, complemented, Sylow
110C2×C22⋊Q8—971no—
111#407—971no—
112C4×C22⋊C4—971no—
113#328—972no—
114#328—972no—
115C22×Q8—971no—
116C23×C4—971no—
117C2×M16—1941no—
118C22⋊Q8—971no—
119C22⋊Q8—971no—
120C22⋊Q8—971no—
121C22⋊Q8—1941no—
122C22⋊Q8—1941no—
123C22⋊Q8#8760411yescharacteristic, complemented
124C2×C4⋊C4—971no—
125C2×C4⋊C4—1941no—
126C2×C22⋊C4—1941no—
127C2×C22⋊C4—1941no—
128C2×C42—1941no—
129C22⋊C8—972no—
130C22⋊C8—1941no—
131#95—972no—
132C24—971no—
133C2×Q8—971no—
134C2×Q8—3882no—
135C2×Q8—1941no—
136C2×Q8C2×#8760411yescharacteristic
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—3882no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4—1942no—
146C22×C4C2×#8760411yescharacteristic
147M16—3882no—
148C2×C8—1942no—
149C4⋊C4—21no—
150C4⋊C4—971no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4—1941no—
154C4⋊C4—1941no—
155C4⋊C4C2×#8760411yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—1941no—
158C22⋊C4—1941no—
159C22⋊C4—1944no—
160C22⋊C4—1941no—
161C42—3882no—
162C23—971no—
163C23—971no—
164C23—3881no—
165C23—1942no—
166C23—1942no—
167C23C22⋊#8760411yescharacteristic
168Q8—41no—
169Q8—3881no—
170Q8—3882no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—971no—
175C2×C4—971no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—3882no—
179C2×C4—3882no—
180C2×C4—3882no—
181C2×C4—3882no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4—1944no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4C22⋊#8760411yescharacteristic
189C2×C4C22×#8760411yescharacteristic
190C8—3882no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—972no—
195C22—972no—
196C22—3884no—
197C22—1941no—
198C22—1941no—
199C22—1941no—
200C22—1941no—
201C22C2×C22⋊#8760411yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—972no—
207C4—3884no—
208C4—3881no—
209C4—3881no—
210C4—1941no—
211C2—41no—
212C2—1942no—
213C2—1942no—
214C2D97⋊#9911yescharacteristic
215C2#2026172811yescharacteristic
216C2D97⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC22⋊Q8#87604
hypocenterC97#1508
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20261589C2semidirect
4. #20261588C2semidirect
12. C2×Q8×D97C4semidirect
15. #15690789C4semidirect
20. C4⋊C4×D97C4semidirect
22. C22⋊Q8×C97C4semidirect
109. C97#1508semidirect
123. C22⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617072
C4×C22⋊#876042
#202615892
#202615882
C22⋊Q8×D972
#150897