Finite Group Databasebeta

← #27157174

Subgroups — #27157174

Order 12416.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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×#15690867C211yesmaximal, characteristic
2#20261677C211yesmaximal, characteristic
3#20261589C212yesmaximal, complemented
4#20261588C212yesmaximal, complemented
5C22⋊Q8×D97C211yesmaximal, characteristic
6C22×#11715897—21no—
7C2×C4⋊#87604—21no—
8C2×C22⋊#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—
24C2×#11715897—22no—
25C2×#11715897—42no—
26C4⋊#87604—42no—
27C22⋊#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—41no—
55C22×D97—41no—
56C22×D97C22⋊C411yescharacteristic
57C22×D97SD1612yes—
58C22×D97Q1612yes—
59C22×D97C2×D411yescharacteristic
60C2×#87604—42no—
61C2×#87604—42no—
62C2×#87604—42no—
63Q8×C97—41no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—41no—
67C2×C4×C97C22⋊C411yescharacteristic
68C2×C4×C97C22×C411yescharacteristic
69C2×Dic97—22no—
70C2×Dic97—21no—
71C2×Dic97—21no—
72C2×Dic97C22⋊C411yescharacteristic
73C2×Dic97C2×D411yescharacteristic
74C4×D97—22no—
75C4×D97—22no—
76C4×D97—41no—
77C4×D97—41no—
78C4×D97—42no—
79C4×D97—42no—
80Dic194—42no—
81Dic194—41no—
82#11715897—42no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97#10312yes—
86C22×C97C2×C22⋊C411yescharacteristic
87C2×D97—21no—
88C2×D97—21no—
89C2×D97—21no—
90C2×D97—21no—
91C2×D97—42no—
92C2×D97—42no—
93C2×D97#10312yes—
94C2×D97C2×SD1611yescharacteristic
95C2×D97C2×Q1611yescharacteristic
96#87604—82no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—41no—
100C4×C97—41no—
101Dic97—21no—
102Dic97—41no—
103Dic97—41no—
104Dic97#9912yes—
105C2×C97—22no—
106C2×C97C2×#9911yescharacteristic
107C2×C97C2×#10311yescharacteristic
108C2×C97#42111yescharacteristic
109D97—21no—
110D97—22no—
111D97—21no—
112#1520—971nomaximal, Sylow
113C97#152011yescharacteristic, complemented, Sylow
114C2×C22⋊Q8—971no—
115C2×C22⋊C8—971no—
116#380—971no—
117#328—972no—
118#328—972no—
119C22×Q8—971no—
120C23×C4—971no—
121C22×C8—1941no—
122C22⋊Q8—971no—
123C22⋊Q8—971no—
124C22⋊Q8—971no—
125C22⋊Q8—1941no—
126C22⋊Q8—1941no—
127C22⋊Q8#8760411yescharacteristic, complemented
128C2×C4⋊C4—971no—
129C2×C4⋊C4—1941no—
130C2×C4⋊C4—1941no—
131C2×C22⋊C4—1941no—
132C2×C22⋊C4—1941no—
133C22⋊C8—972no—
134C22⋊C8—1941no—
135#95—972no—
136C24—971no—
137C2×Q8—971no—
138C2×Q8—3882no—
139C2×Q8—1941no—
140C2×Q8C2×#8760411yescharacteristic
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—971no—
146C22×C4—3882no—
147C22×C4—1941no—
148C22×C4—1941no—
149C22×C4—1942no—
150C22×C4C2×#8760411yescharacteristic
151C2×C8—3882no—
152C2×C8—1942no—
153C4⋊C4—21no—
154C4⋊C4—971no—
155C4⋊C4—3882no—
156C4⋊C4—1941no—
157C4⋊C4—1941no—
158C4⋊C4—1941no—
159C4⋊C4—1941no—
160C4⋊C4C2×#8760411yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3882no—
163C22⋊C4—1941no—
164C22⋊C4—1941no—
165C22⋊C4—1941no—
166C23—972no—
167C23—972no—
168C23—971no—
169C23—971no—
170C23—3881no—
171C23—3881no—
172C23C22⋊#8760411yescharacteristic
173Q8—41no—
174Q8—3881no—
175Q8—3882no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—3881no—
182C2×C4—3881no—
183C2×C4—3882no—
184C2×C4—3882no—
185C2×C4—3882no—
186C2×C4—3882no—
187C2×C4—3882no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4—1941no—
192C2×C4—1941no—
193C2×C4C22⋊#8760411yescharacteristic
194C2×C4C22×#8760411yescharacteristic
195C8—3882no—
196C22—21no—
197C22—21no—
198C22—971no—
199C22—972no—
200C22—971no—
201C22—3882no—
202C22—3882no—
203C22—1941no—
204C22—1941no—
205C22—1941no—
206C22—1941no—
207C22#1569087312yes—
208C22C2×C22⋊#8760411yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—7762no—
214C4—972no—
215C4—3881no—
216C4—3881no—
217C4—1941no—
218C2—22no—
219C2—1941no—
220C2—1942no—
221C2—1941no—
222C2D97⋊#9911yescharacteristic
223C2C2×#1569087311yescharacteristic
224C2#2026173211yescharacteristic
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#1520
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
113. C97#1520semidirect
127. C22⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908672
#202616772
#202615892
#202615882
C22⋊Q8×D972
#152097