Finite Group Databasebeta

← #328×D97 #27140215

Subgroups — #328×D97 #27140215

Order 12416.

336 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 77 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⋊Q8×D97C211yesmaximal, characteristic
2C22⋊C8×D97C211yesmaximal, characteristic
3#95×D97C211yesmaximal, characteristic
4#20260473C211yesmaximal, characteristic, complemented
5#20260472C211yesmaximal, characteristic, complemented
6#20260471C211yesmaximal, characteristic, complemented
7#328×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97C411yescharacteristic, complemented
9C22×C4×D97—21no—
10C22×C4×D97C2211yescharacteristic
11C2×C8×D97—21no—
12#15690790C411yescharacteristic, complemented
13#15690789C2211yescharacteristic
14#15690788—21no—
15#15690787—21no—
16#15690786C411yescharacteristic, complemented
17C4⋊C4×D97—21no—
18C4⋊C4×D97C411yescharacteristic, complemented
19C22⋊C4×D97—21no—
20C22⋊#11715895C2211yescharacteristic
21#15690722—21no—
22#15690717—21no—
23#15690717C2211yescharacteristic
24C22⋊Q8×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C23×D97D411yescharacteristic
28C2×Dic194—21no—
29C2×Dic194C2×C411yescharacteristic, complemented
30Q8×D97—42no—
31C22×Dic97—21no—
32C22×Dic97D411yescharacteristic
33C2×C4×D97—24no—
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—42no—
37C2×C4×D97—42no—
38C2×C4×D97—42no—
39C2×C4×D97C2×C411yescharacteristic
40C2×C4×D97C2×C411yescharacteristic
41C2×C4×D97D411yescharacteristic
42C2×C4×D97D411yescharacteristic
43C2×#11715895—21no—
44C8×D97—42no—
45C4⋊Dic97—21no—
46C4⋊Dic97C2×C411yescharacteristic, complemented
47#13095490—21no—
48#13095490—21no—
49#13095490—21no—
50C22⋊Dic97—21no—
51#13095488—21no—
52#13095488—21no—
53C2×Q8×C97C2×C411yescharacteristic, complemented
54C22×C4×C97—21no—
55C22×C4×C97C2311yescharacteristic
56C2×C8×C97—21no—
57C4⋊C4×C97—21no—
58C4⋊C4×C97C2×C411yescharacteristic, complemented
59C22⋊C4×C97—21no—
60C23×C97C2×D411yescharacteristic
61C22×D97—22no—
62C22×D97—22no—
63C22×D97C22⋊C411yescharacteristic
64C22×D97C22⋊C411yescharacteristic
65C22×D97SD1612yes—
66C22×D97Q1612yes—
67Q8×C97—41no—
68C2×C4×C97—22no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97C22×C411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×Dic97—22no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97C22⋊C411yescharacteristic
83C2×Dic97C22⋊C411yescharacteristic
84C4×D97—22no—
85C4×D97—22no—
86C4×D97—41no—
87C4×D97—41no—
88C4×D97—48no—
89C4×D97—42no—
90C4×D97—42no—
91Dic194—42no—
92Dic194—41no—
93C8×C97—41no—
94#11715895—41no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97C2×C22⋊C411yescharacteristic
98C22×C97C2×SD1611yescharacteristic
99C22×C97C2×Q1611yescharacteristic
100C2×D97—21no—
101C2×D97—21no—
102C2×D97—24no—
103C2×D97—24no—
104C2×D97#9912yes—
105C2×D97#10312yes—
106C2×D97#10312yes—
107C2×D97#10412yes—
108C4×C97—21no—
109C4×C97—21no—
110C4×C97—42no—
111C4×C97—41no—
112C4×C97—41no—
113Dic97—21no—
114Dic97—21no—
115Dic97—42no—
116Dic97—41no—
117Dic97—41no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97C2×#9911yescharacteristic
121C2×C97C2×#10311yescharacteristic
122C2×C97C2×#10411yescharacteristic
123D97—22no—
124D97#32814yesdirect product factor
125C2×#328—971nomaximal, Sylow
126C97C2×#32811yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—971no—
128C2×C22⋊C8—971no—
129C2×#95—971no—
130#328—971no—
131#328—971no—
132#328—971no—
133#328D9711yescharacteristic, direct product factor
134C22×Q8—971no—
135C23×C4—971no—
136C23×C4—1941no—
137C22×C8—1941no—
138C22⋊Q8—971no—
139C22⋊Q8—971no—
140C22⋊Q8—971no—
141C22⋊Q8—1941no—
142C22⋊Q8—1941no—
143C22⋊Q8C2×D9711yescharacteristic
144C2×C4⋊C4—971no—
145C2×C4⋊C4—1941no—
146C2×C22⋊C4—1941no—
147C22⋊C8—971no—
148C22⋊C8—1941no—
149C22⋊C8C2×D9711yescharacteristic
150#95—971no—
151#95—1941no—
152#95C2×D9711yescharacteristic
153C24—971no—
154C2×Q8—971no—
155C2×Q8—3882no—
156C2×Q8—1941no—
157C2×Q8C4×D9711yescharacteristic, complemented
158C22×C4—21no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—971no—
163C22×C4—971no—
164C22×C4—3882no—
165C22×C4—3882no—
166C22×C4—3882no—
167C22×C4—1941no—
168C22×C4—1941no—
169C22×C4—1944no—
170C22×C4—1941no—
171C22×C4C22×D9711yescharacteristic
172C2×C8—21no—
173C2×C8—3882no—
174C2×C8—1941no—
175C4⋊C4—21no—
176C4⋊C4—971no—
177C4⋊C4—1941no—
178C4⋊C4—1941no—
179C4⋊C4—1941no—
180C4⋊C4—1941no—
181C4⋊C4C4×D9711yescharacteristic, complemented
182C22⋊C4—21no—
183C22⋊C4—1941no—
184C22⋊C4—1941no—
185C22⋊C4—1941no—
186C23—971no—
187C23—971no—
188C23—972no—
189C23—972no—
190C23—1942no—
191C23—1942no—
192C23D4×D9711yescharacteristic
193Q8—41no—
194Q8—3881no—
195Q8—3882no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—971no—
203C2×C4—971no—
204C2×C4—3881no—
205C2×C4—3881no—
206C2×C4—3882no—
207C2×C4—3882no—
208C2×C4—3888no—
209C2×C4—3881no—
210C2×C4—3881no—
211C2×C4—3881no—
212C2×C4—1942no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4—1941no—
216C2×C4—1941no—
217C2×C4C2×C4×D9711yescharacteristic
218C2×C4D4×D9711yescharacteristic
219C8—41no—
220C8—3881no—
221C22—21no—
222C22—21no—
223C22—972no—
224C22—972no—
225C22—972no—
226C22—972no—
227C22—1944no—
228C22—1944no—
229C22—1941no—
230C22—1941no—
231C22C22⋊C4×D9711yescharacteristic
232C22SD16×D9711yescharacteristic
233C22Q16×D9711yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—3882no—
240C4—3881no—
241C4—3881no—
242C4—1941no—
243C4—1941no—
244C2—21no—
245C2—21no—
246C2—974no—
247C2—1942no—
248C2#99×D9711yescharacteristic
249C2#103×D9711yescharacteristic
250C2#104×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#328×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC22×C4×C97C23
hypercenter#328D97
hypocenterC97C2×#328
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #20260473C2semidirect
5. #20260472C2semidirect
6. #20260471C2semidirect
7. #328×C97C2semidirect
8. C2×Q8×D97C4semidirect
12. #15690790C4semidirect
16. #15690786C4semidirect
18. C4⋊C4×D97C4semidirect
29. C2×Dic194C2×C4semidirect
46. C4⋊Dic97C2×C4semidirect
53. C2×Q8×C97C2×C4semidirect
58. C4⋊C4×C97C2×C4semidirect
124. D97#328direct (N × H)
126. C97C2×#328semidirect
133. #328D97direct (N × H)
157. C2×Q8C4×D97semidirect
181. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D972
C22⋊C8×D972
#95×D972
#202604732
#202604722
#202604712
#328×C972
C2×#32897