Finite Group Databasebeta

← #400×D97 #27144986

Subgroups — #400×D97 #27144986

Order 12416.

604 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic
3#20260677C212yesmaximal, complemented
4#20260676C212yesmaximal, complemented
5#20260675C211yesmaximal, characteristic, complemented
6#20260674C211yesmaximal, characteristic, complemented
7#20260673C211yesmaximal, characteristic, complemented
8#95×D97C212yesmaximal
9#95×D97C212yesmaximal
10#95×D97C211yesmaximal, characteristic
11#400×C97C211yesmaximal, characteristic, complemented
12C22×C4×D97C2211yescharacteristic
13C22×C4×D97C2212yes—
14C22×C4×D97C2212yes—
15C22×C4×D97C2211yescharacteristic
16C22×C4×D97C2211yescharacteristic
17C2×C4⋊Dic97C2211yescharacteristic
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2212yes—
22C4⋊C4×D97—24no—
23C4⋊C4×D97—24no—
24#15690717C2211yescharacteristic
25#15690717C2212yes—
26#15690717C2212yes—
27#15690717C2212yes—
28#15690717C2212yes—
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32C2×C4⋊C4×C97C2211yescharacteristic
33C2×C4⋊C4×C97C2211yescharacteristic
34#95×C97C2211yescharacteristic
35#95×C97C2212yes—
36#95×C97C2212yes—
37C23×D97C2311yescharacteristic
38C22×Dic97C2311yescharacteristic
39C22×Dic97C2312yes—
40C22×Dic97C2312yes—
41C22×Dic97C2311yescharacteristic
42C22×Dic97C2311yescharacteristic
43C2×C4×D97—22no—
44C2×C4×D97—22no—
45C2×C4×D97—24no—
46C2×C4×D97—22no—
47C2×C4×D97—24no—
48C2×C4×D97—24no—
49C2×C4×D97—28no—
50C2×C4×D97—24no—
51C2×C4×D97—24no—
52C2×C4×D97—24no—
53C2×C4×D97D414yes—
54C2×C4×D97Q814yes—
55C4⋊Dic97—22no—
56C4⋊Dic97—22no—
57#13095490—22no—
58#13095490—22no—
59#13095490—24no—
60C22×C4×C97C2311yescharacteristic
61C22×C4×C97C2312yes—
62C22×C4×C97C2312yes—
63C22×C4×C97C2311yescharacteristic
64C22×C4×C97C2311yescharacteristic
65C4⋊C4×C97—22no—
66C4⋊C4×C97—22no—
67C23×C97C2411yescharacteristic
68C22×D97C2×D412yes—
69C22×D97C2×Q812yes—
70C22×D97#4112yes—
71C22×D97#4114yes—
72C22×D97#4114yes—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97—22no—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—24no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97C2×D412yes—
84C2×C4×C97C2×Q812yes—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—22no—
88C2×Dic97—21no—
89C2×Dic97—22no—
90C2×Dic97—22no—
91C2×Dic97—24no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97C2×D412yes—
96C2×Dic97C2×Q812yes—
97C4×D97—28no—
98C4×D97—28no—
99C4×D97—44no—
100C4×D97—48no—
101C4×D97—48no—
102C22×C97C22×D411yescharacteristic
103C22×C97C22×Q811yescharacteristic
104C22×C97C2×#4111yescharacteristic
105C22×C97C2×#4112yes—
106C22×C97C2×#4112yes—
107C2×D97C22⋊Q814yes—
108C2×D97#12314yes—
109C2×D97#12414yes—
110C2×D97#12518yes—
111C2×D97#12618yes—
112C4×C97—22no—
113C4×C97—22no—
114C4×C97—41no—
115C4×C97—42no—
116C4×C97—42no—
117Dic97—22no—
118Dic97—22no—
119Dic97—41no—
120Dic97—42no—
121Dic97—42no—
122C2×C97C2×C22⋊Q811yescharacteristic
123C2×C97C2×#12311yescharacteristic
124C2×C97C2×#12411yescharacteristic
125C2×C97C2×#12512yes—
126C2×C97C2×#12612yes—
127D97#40018yesdirect product factor
128C2×#400—971nomaximal, Sylow
129C97C2×#40011yescharacteristic, complemented, Sylow
130C22×C4⋊C4—971no—
131C22×C4⋊C4—971no—
132#400—971no—
133#400—971no—
134#400—971no—
135#400—972no—
136#400—972no—
137#400D9711yescharacteristic, direct product factor
138C2×#95—971no—
139C2×#95—972no—
140C2×#95—972no—
141C23×C4—971no—
142C23×C4—971no—
143C23×C4—971no—
144C23×C4—972no—
145C23×C4—972no—
146C2×C4⋊C4—971no—
147C2×C4⋊C4—972no—
148C2×C4⋊C4—971no—
149C2×C4⋊C4—971no—
150C2×C4⋊C4—971no—
151C2×C4⋊C4—1944no—
152C2×C4⋊C4—1944no—
153C2×C4⋊C4C2×D9711yescharacteristic
154C2×C4⋊C4C2×D9711yescharacteristic
155#95—972no—
156#95—972no—
157#95—972no—
158#95—971no—
159#95—972no—
160#95—972no—
161#95—972no—
162#95—972no—
163#95C2×D9711yescharacteristic
164#95C2×D9712yes—
165#95C2×D9712yes—
166C24—971no—
167C22×C4—971no—
168C22×C4—971no—
169C22×C4—974no—
170C22×C4—974no—
171C22×C4—972no—
172C22×C4—971no—
173C22×C4—972no—
174C22×C4—1942no—
175C22×C4—1944no—
176C22×C4—1944no—
177C22×C4—1948no—
178C22×C4—1944no—
179C22×C4—1944no—
180C22×C4—1944no—
181C22×C4—1942no—
182C22×C4—1944no—
183C22×C4—1942no—
184C22×C4C22×D9712yes—
185C22×C4C22×D9711yescharacteristic
186C22×C4C22×D9712yes—
187C22×C4C22×D9711yescharacteristic
188C22×C4C22×D9711yescharacteristic
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—1944no—
192C4⋊C4—1942no—
193C4⋊C4—1942no—
194C4⋊C4—1942no—
195C4⋊C4—1942no—
196C23—972no—
197C23—972no—
198C23—972no—
199C23—974no—
200C23—974no—
201C23C23×D9711yescharacteristic
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—972no—
213C2×C4—972no—
214C2×C4—3884no—
215C2×C4—3888no—
216C2×C4—3888no—
217C2×C4—1948no—
218C2×C4—1948no—
219C2×C4—1941no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1944no—
223C2×C4—1942no—
224C2×C4—1942no—
225C2×C4—1942no—
226C2×C4—1941no—
227C2×C4—1941no—
228C2×C4—1942no—
229C2×C4D4×D9712yes—
230C2×C4Q8×D9712yes—
231C22—974no—
232C22—974no—
233C22—974no—
234C22—978no—
235C22—978no—
236C22C2×D4×D9711yescharacteristic
237C22C2×Q8×D9711yescharacteristic
238C22#41×D9711yescharacteristic
239C22#41×D9712yes—
240C22#41×D9712yes—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—3881no—
247C4—3882no—
248C4—3882no—
249C4—1942no—
250C4—1942no—
251C2—978no—
252C2C22⋊Q8×D9711yescharacteristic
253C2#123×D9711yescharacteristic
254C2#124×D9711yescharacteristic
255C2#125×D9712yes—
256C2#126×D9712yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#400×C97C2
prime radicalC23×D97C23
hypercenter#400D97
hypocenterC97C2×#400
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #20260677C2semidirect
4. #20260676C2semidirect
5. #20260675C2semidirect
6. #20260674C2semidirect
7. #20260673C2semidirect
11. #400×C97C2semidirect
127. D97#400direct (N × H)
129. C97C2×#400semidirect
137. #400D97direct (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
#202606772
#202606762
#202606752
#202606742
#202606732
#95×D972
#400×C972
C2×#40097