Finite Group Databasebeta

← #398×D97 #27144965

Subgroups — #398×D97 #27144965

Order 12416.

692 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C212yesmaximal
2C2×C4⋊C4×D97C211yesmaximal, characteristic
3C2×C4⋊C4×D97C211yesmaximal, characteristic
4#20260669C212yesmaximal, complemented
5#20260668C212yesmaximal, complemented
6#20260667C211yesmaximal, characteristic, complemented
7#20260666C211yesmaximal, characteristic, complemented
8#20260665C211yesmaximal, characteristic, complemented
9#95×D97C212yesmaximal
10#95×D97C211yesmaximal, characteristic
11#398×C97C211yesmaximal, characteristic, complemented
12C22×C4×D97C2212yes—
13C22×C4×D97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15C22×C4×D97C2211yescharacteristic
16C22×C4×D97C2212yes—
17C2×C4⋊Dic97C2211yescharacteristic
18C2×C4⋊Dic97C2212yes—
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095490C2212yes—
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2212yes—
23C2×#13095490C2212yes—
24C2×#13095490C2211yescharacteristic
25C4⋊C4×D97—28no—
26C4⋊C4×D97—24no—
27C4⋊C4×D97—24no—
28#15690717C2211yescharacteristic
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2211yescharacteristic
35C2×C4⋊C4×C97C2212yes—
36#95×C97C2211yescharacteristic
37#95×C97C2212yes—
38C23×D97C2311yescharacteristic
39C22×Dic97C2312yes—
40C22×Dic97C2311yescharacteristic
41C22×Dic97C2311yescharacteristic
42C22×Dic97C2311yescharacteristic
43C22×Dic97C2312yes—
44C2×C4×D97—22no—
45C2×C4×D97—24no—
46C2×C4×D97—22no—
47C2×C4×D97—24no—
48C2×C4×D97—24no—
49C2×C4×D97—24no—
50C2×C4×D97—24no—
51C2×C4×D97—22no—
52C2×C4×D97—24no—
53C2×C4×D97—24no—
54C2×C4×D97D414yes—
55C2×C4×D97D414yes—
56C2×C4×D97Q818yes—
57C4⋊Dic97—22no—
58C4⋊Dic97—22no—
59C4⋊Dic97—24no—
60#13095490—24no—
61#13095490—22no—
62#13095490—22no—
63#13095490—24no—
64#13095490—24no—
65C22×C4×C97C2312yes—
66C22×C4×C97C2311yescharacteristic
67C22×C4×C97C2311yescharacteristic
68C22×C4×C97C2311yescharacteristic
69C22×C4×C97C2312yes—
70C4⋊C4×C97—24no—
71C4⋊C4×C97—22no—
72C4⋊C4×C97—22no—
73C23×C97C2411yescharacteristic
74C22×D97C2×D412yes—
75C22×D97C2×D412yes—
76C22×D97C2×Q814yes—
77C22×D97#4112yes—
78C22×D97#4114yes—
79C2×C4×C97—21no—
80C2×C4×C97—22no—
81C2×C4×C97—21no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—21no—
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2×D412yes—
91C2×C4×C97C2×Q814yes—
92C2×Dic97—21no—
93C2×Dic97—22no—
94C2×Dic97—21no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—21no—
100C2×Dic97—22no—
101C2×Dic97—22no—
102C2×Dic97C2×D412yes—
103C2×Dic97C2×D412yes—
104C2×Dic97C2×Q814yes—
105C4×D97—28no—
106C4×D97—216no—
107C4×D97—28no—
108C4×D97—48no—
109C4×D97—44no—
110C22×C97C22×D411yescharacteristic
111C22×C97C22×D411yescharacteristic
112C22×C97C22×Q812yes—
113C22×C97C2×#4111yescharacteristic
114C22×C97C2×#4112yes—
115C2×D97#12114yes—
116C2×D97C22⋊Q818yes—
117C2×D97#12314yes—
118C2×D97#12518yes—
119C2×D97C4⋊Q814yes—
120C4×C97—22no—
121C4×C97—24no—
122C4×C97—22no—
123C4×C97—42no—
124C4×C97—41no—
125Dic97—22no—
126Dic97—24no—
127Dic97—22no—
128Dic97—42no—
129Dic97—41no—
130C2×C97C2×#12111yescharacteristic
131C2×C97C2×C22⋊Q812yes—
132C2×C97C2×#12311yescharacteristic
133C2×C97C2×#12512yes—
134C2×C97C2×C4⋊Q811yescharacteristic
135D97#39818yesdirect product factor
136C2×#398—971nomaximal, Sylow
137C97C2×#39811yescharacteristic, complemented, Sylow
138C22×C4⋊C4—971no—
139C22×C4⋊C4—972no—
140C22×C4⋊C4—971no—
141#398—971no—
142#398—971no—
143#398—972no—
144#398—971no—
145#398—972no—
146#398D9711yescharacteristic, direct product factor
147C2×#95—971no—
148C2×#95—972no—
149C23×C4—971no—
150C23×C4—971no—
151C23×C4—972no—
152C23×C4—972no—
153C23×C4—971no—
154C2×C4⋊C4—972no—
155C2×C4⋊C4—971no—
156C2×C4⋊C4—972no—
157C2×C4⋊C4—972no—
158C2×C4⋊C4—971no—
159C2×C4⋊C4—971no—
160C2×C4⋊C4—972no—
161C2×C4⋊C4—971no—
162C2×C4⋊C4—1944no—
163C2×C4⋊C4—1948no—
164C2×C4⋊C4—1944no—
165C2×C4⋊C4C2×D9711yescharacteristic
166C2×C4⋊C4C2×D9711yescharacteristic
167C2×C4⋊C4C2×D9712yes—
168#95—971no—
169#95—972no—
170#95—972no—
171#95—972no—
172#95—972no—
173#95C2×D9712yes—
174#95C2×D9711yescharacteristic
175C24—971no—
176C22×C4—972no—
177C22×C4—971no—
178C22×C4—971no—
179C22×C4—974no—
180C22×C4—974no—
181C22×C4—978no—
182C22×C4—971no—
183C22×C4—972no—
184C22×C4—1944no—
185C22×C4—1944no—
186C22×C4—1942no—
187C22×C4—1944no—
188C22×C4—1944no—
189C22×C4—1942no—
190C22×C4—1944no—
191C22×C4—1942no—
192C22×C4—1944no—
193C22×C4—1944no—
194C22×C4C22×D9711yescharacteristic
195C22×C4C22×D9712yes—
196C22×C4C22×D9711yescharacteristic
197C22×C4C22×D9711yescharacteristic
198C22×C4C22×D9712yes—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—1944no—
203C4⋊C4—1944no—
204C4⋊C4—1942no—
205C4⋊C4—1942no—
206C4⋊C4—1944no—
207C4⋊C4—1944no—
208C4⋊C4—1942no—
209C4⋊C4—1942no—
210C23—972no—
211C23—972no—
212C23—974no—
213C23—972no—
214C23—974no—
215C23C23×D9711yescharacteristic
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—972no—
227C2×C4—972no—
228C2×C4—974no—
229C2×C4—3888no—
230C2×C4—3884no—
231C2×C4—1948no—
232C2×C4—19416no—
233C2×C4—1948no—
234C2×C4—1942no—
235C2×C4—1942no—
236C2×C4—1941no—
237C2×C4—1942no—
238C2×C4—1942no—
239C2×C4—1941no—
240C2×C4—1942no—
241C2×C4—1941no—
242C2×C4—1942no—
243C2×C4—1942no—
244C2×C4D4×D9712yes—
245C2×C4D4×D9712yes—
246C2×C4Q8×D9714yes—
247C22—974no—
248C22—978no—
249C22—974no—
250C22—978no—
251C22—974no—
252C22C2×D4×D9711yescharacteristic
253C22C2×D4×D9711yescharacteristic
254C22C2×Q8×D9712yes—
255C22#41×D9711yescharacteristic
256C22#41×D9712yes—
257C4—22no—
258C4—24no—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—3882no—
263C4—3881no—
264C4—1942no—
265C4—1944no—
266C4—1942no—
267C2—978no—
268C2#121×D9711yescharacteristic
269C2C22⋊Q8×D9712yes—
270C2#123×D9711yescharacteristic
271C2#125×D9712yes—
272C2C4⋊Q8×D9711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#398×C97C2
prime radicalC23×D97C23
hypercenter#398D97
hypocenterC97C2×#398
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
4. #20260669C2semidirect
5. #20260668C2semidirect
6. #20260667C2semidirect
7. #20260666C2semidirect
8. #20260665C2semidirect
11. #398×C97C2semidirect
135. D97#398direct (N × H)
137. C97C2×#398semidirect
146. #398D97direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
#202606692
#202606682
#202606672
#202606662
#202606652
#95×D972
#398×C972
C2×#39897