Finite Group Databasebeta

← #27140098

Subgroups — #27140098

Order 12416.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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×#15690788C211yesmaximal, characteristic
2#20260697C211yesmaximal, characteristic
3C22⋊C8×D97C211yesmaximal, characteristic
4#20260461C211yesmaximal, characteristic, complemented
5#20260460C211yesmaximal, characteristic, complemented
6#20260460C211yesmaximal, characteristic, complemented
7#324×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C411yescharacteristic
9C22×C4×D97C2211yescharacteristic
10C2×#13095494—21no—
11C2×C8×D97—21no—
12#15690788—21no—
13#15690788—21no—
14#15690788—21no—
15#15690788—21no—
16C2×#13095490C411yescharacteristic
17C2×#13095490C2211yescharacteristic
18C2×#13095488C411yescharacteristic
19C2×#13095488C411yescharacteristic
20C22⋊#11715895C2211yescharacteristic
21C22⋊#11715895C2211yescharacteristic
22#15690722—21no—
23#15690722—21no—
24C2×C4⋊C4×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26C22⋊C8×C97C2211yescharacteristic
27C23×D97C2×C411yescharacteristic
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—42no—
31C22×Dic97C2×C411yescharacteristic
32C22×Dic97C2×C411yescharacteristic
33C22×Dic97C2×C411yescharacteristic
34C2×C4×D97—42no—
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39#13095494—42no—
40C2×#11715895—21no—
41C2×#11715895—21no—
42C8×D97—42no—
43#13095490—21no—
44#13095490—21no—
45#13095490—21no—
46#13095490—21no—
47#13095488—21no—
48#13095488—21no—
49#13095488—21no—
50#13095488—21no—
51C22×C4×C97C2×C411yescharacteristic
52C22×C4×C97C2×C411yescharacteristic
53C22×C4×C97C2311yescharacteristic
54C2×C8×C97—21no—
55C2×C8×C97—21no—
56C4⋊C4×C97—21no—
57C4⋊C4×C97—21no—
58C23×C97C22×C411yescharacteristic
59C22×D97—21no—
60C22×D97—21no—
61C22×D97—22no—
62C22×D97—41no—
63C22×D97C22⋊C411yescharacteristic
64C22×D97C22⋊C411yescharacteristic
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—41no—
70C2×C4×C97—41no—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×Dic97—22no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—41no—
79C2×Dic97C22⋊C411yescharacteristic
80C2×Dic97C22⋊C411yescharacteristic
81C4×D97—22no—
82C4×D97—22no—
83C4×D97—41no—
84C4×D97—41no—
85Dic194—42no—
86Dic194—42no—
87C8×C97—41no—
88C8×C97—41no—
89#11715895—41no—
90#11715895—41no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97C2×C22⋊C411yescharacteristic
94C22×C97#13111yescharacteristic
95C22×C97#13111yescharacteristic
96C2×D97—21no—
97C2×D97—21no—
98C2×D97—21no—
99C2×D97—21no—
100C2×D97—21no—
101C2×D97—42no—
102C2×D97—42no—
103C2×D97#10112yes—
104C4×C97—21no—
105C4×C97—21no—
106C4×C97—41no—
107C4×C97—41no—
108Dic97—21no—
109Dic97—41no—
110Dic97—41no—
111Dic97#9912yes—
112C2×C97—21no—
113C2×C97—21no—
114C2×C97#40811yescharacteristic
115C2×C97C2×#9911yescharacteristic
116C2×C97C2×#10111yescharacteristic
117D97—22no—
118D97—41no—
119#1472—971nomaximal, Sylow
120C97#147211yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—971no—
122#407—971no—
123C2×C22⋊C8—971no—
124#324—971no—
125#324—971no—
126#324—971no—
127#324D9711yescharacteristic, complemented
128C22×Q8—971no—
129C23×C4—971no—
130C2×M16—1941no—
131C22×C8—1941no—
132C22⋊Q8—1941no—
133C22⋊Q8—1941no—
134C22⋊Q8—1941no—
135C22⋊Q8—1941no—
136C2×C4⋊C4—971no—
137C2×C4⋊C4—971no—
138C2×C4⋊C4C2×D9711yescharacteristic
139C2×C22⋊C4—971no—
140C2×C22⋊C4—971no—
141C22⋊C8—971no—
142C22⋊C8—971no—
143C22⋊C8—1941no—
144C22⋊C8—1941no—
145C22⋊C8C2×D9711yescharacteristic
146C22⋊C8C2×D9711yescharacteristic
147C24—971no—
148C2×Q8—3882no—
149C2×Q8—1941no—
150C2×Q8—1941no—
151C22×C4—971no—
152C22×C4—971no—
153C22×C4—971no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—3882no—
159C22×C4C4×D9711yescharacteristic
160C22×C4C4×D9711yescharacteristic
161C22×C4C22×D9711yescharacteristic
162M16—3882no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—3882no—
166C2×C8—1941no—
167C2×C8—1941no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—1941no—
171C4⋊C4—1941no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C22⋊C4—1941no—
175C22⋊C4—1941no—
176C22⋊C4—1941no—
177C22⋊C4—1941no—
178C23—971no—
179C23—971no—
180C23—3881no—
181C23—1942no—
182C23—1941no—
183C23—1941no—
184C23C2×C4×D9711yescharacteristic
185Q8—3882no—
186Q8—3882no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—971no—
194C2×C4—971no—
195C2×C4—3881no—
196C2×C4—3881no—
197C2×C4—3881no—
198C2×C4—1942no—
199C2×C4—1942no—
200C2×C4—1942no—
201C2×C4—1941no—
202C2×C4—1941no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4D4×D9711yescharacteristic
206C2×C4D4×D9711yescharacteristic
207C8—41no—
208C8—41no—
209C8—3881no—
210C8—3881no—
211C22—21no—
212C22—21no—
213C22—972no—
214C22—3882no—
215C22—3882no—
216C22—1941no—
217C22—1941no—
218C22—1941no—
219C22—1941no—
220C22—1941no—
221C22C22⋊C4×D9711yescharacteristic
222C22#1569081611yescharacteristic
223C22#1569081811yescharacteristic
224C4—21no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—972no—
229C4—3881no—
230C4—3881no—
231C4—1941no—
232C2—21no—
233C2—21no—
234C2—3881no—
235C2—1942no—
236C2#2026070011yescharacteristic
237C2#99×D9711yescharacteristic
238C2#101×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×C4×C97C23
hypercenter#324D97
hypocenterC97#1472
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
4. #20260461C2semidirect
5. #20260460C2semidirect
6. #20260460C2semidirect
7. #324×C97C2semidirect
120. C97#1472semidirect
127. #324D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907882
#202606972
C22⋊C8×D972
#202604612
#202604602
#324×C972
#147297