Finite Group Databasebeta

← #27145005

Subgroups — #27145005

Order 12416.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#15690775C211yesmaximal, characteristic, complemented
2C2×#15690766C211yesmaximal, characteristic
3#20260673C211yesmaximal, characteristic, complemented
4#20260666C212yesmaximal, complemented
5#20260658C212yesmaximal
6#20260656C211yesmaximal, characteristic, complemented
7#20260647C211yesmaximal, characteristic
8C4⋊C4×Dic97C212yesmaximal, complemented
9#20260586C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic97C212yesmaximal
11#399×C97C211yesmaximal, characteristic
12C22×Dic194C2211yescharacteristic
13C23×Dic97C2211yescharacteristic
14#15690775—24no—
15#15690766—24no—
16C2×C4⋊Dic97C2212yes—
17C2×#13095490C2211yescharacteristic
18C2×#13095490C2212yes—
19C2×#13095490C2212yes—
20C2×#13095490C2211yescharacteristic
21C2×C22⋊Dic97C2212yes—
22C2×C22⋊Dic97C2211yescharacteristic
23C2×C4×Dic97C2211yescharacteristic
24C2×C4×Dic97C2212yes—
25C2×C4×Dic97C2212yes—
26C2×C4×Dic97C2211yescharacteristic
27#15690717C2212yes—
28#15690717C2211yescharacteristic
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32C2×C4⋊C4×C97C2212yes—
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C22⋊C4×C97C2211yescharacteristic
35C2×C22⋊C4×C97C2212yes—
36#95×C97C2211yescharacteristic
37C2×Dic194—22no—
38C2×Dic194—24no—
39C22×Dic97—22no—
40C22×Dic97—22no—
41C22×Dic97Q814yescomplemented
42C22×Dic97C2311yescharacteristic
43C22×Dic97C2312yes—
44C22×Dic97C2311yescharacteristic
45C22×Dic97C2311yescharacteristic
46C22×Dic97C2312yes—
47C22×Dic97C2311yescharacteristic
48C4⋊Dic97—24no—
49#13095490—22no—
50#13095490—24no—
51#13095490—22no—
52#13095490—24no—
53C22⋊Dic97—22no—
54C22⋊Dic97—24no—
55C4×Dic97—22no—
56C4×Dic97—24no—
57C4×Dic97—24no—
58C4×Dic97D414yes—
59C24×C97C2311yescharacteristic
60C22×C4×C97C2311yescharacteristic
61C22×C4×C97C2312yes—
62C22×C4×C97C2311yescharacteristic
63C22×C4×C97C2312yes—
64C4⋊C4×C97—22no—
65C4⋊C4×C97—24no—
66C22⋊C4×C97—22no—
67C22⋊C4×C97—24no—
68C23×C97—21no—
69C23×C97—22no—
70C23×C97—22no—
71C23×C97—21no—
72C23×C97C2×Q812yes—
73C23×C97C2411yescharacteristic
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—21no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97C2×D412yes—
83C2×C4×C97#4114yes—
84C2×Dic97—22no—
85C2×Dic97—21no—
86C2×Dic97—22no—
87C2×Dic97—21no—
88C2×Dic97—22no—
89C2×Dic97—28no—
90C2×Dic97—21no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97C2×D412yes—
96C2×Dic97C2×D412yes—
97C2×Dic97C2×Q812yes—
98C2×Dic97C2×Q812yes—
99C2×Dic97#4114yes—
100C2×Dic97#4112yes—
101Dic194—28no—
102C22×C97—22no—
103C22×C97—24no—
104C22×C97—41no—
105C22×C97—42no—
106C22×C97—41no—
107C22×C97C22×D411yescharacteristic
108C22×C97C22×Q811yescharacteristic
109C22×C97C2×#4111yescharacteristic
110C22×C97C2×#4112yes—
111C22×C97C2×#4112yes—
112C4×C97—22no—
113C4×C97—24no—
114C4×C97—41no—
115C4×C97—42no—
116Dic97—22no—
117Dic97—24no—
118Dic97—22no—
119Dic97—41no—
120Dic97—42no—
121Dic97C22⋊Q814yes—
122C2×C97—42no—
123C2×C97C2×C22⋊Q811yescharacteristic
124C2×C97#52511yescharacteristic
125C2×C97#52912yes—
126C2×C97#53211yescharacteristic
127C2×C97#53312yes—
128#2396—971nomaximal, Sylow
129C97#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—971no—
131C2×#117—971no—
132#400—971no—
133#399D9711yescharacteristic
134#398—972no—
135#396—972no—
136#395—971no—
137#393—971no—
138C4×C4⋊C4—972no—
139C4×C4⋊C4—971no—
140C4×C22⋊C4—972no—
141C22×Q8—971no—
142C23×C4—971no—
143C4×Q8—1944no—
144#117—1944no—
145C2×C4⋊C4—971no—
146C2×C4⋊C4—971no—
147C2×C4⋊C4—972no—
148C2×C4⋊C4—972no—
149C2×C4⋊C4—972no—
150C2×C4⋊C4C2×D9711yescharacteristic
151C2×C4⋊C4C2×D9712yes—
152C2×C22⋊C4—971no—
153C2×C22⋊C4—972no—
154C2×C22⋊C4C2×D9711yescharacteristic
155C2×C22⋊C4C2×D9712yes—
156C2×C42—971no—
157C2×C42—971no—
158C2×C42—972no—
159C2×C42—972no—
160#95—972no—
161#95—972no—
162#95—972no—
163#95—971no—
164#95—972no—
165#95C2×D9711yescharacteristic
166C24C22×D9711yescharacteristic
167C2×Q8—1944no—
168C2×Q8—1942no—
169C22×C4—971no—
170C22×C4—974no—
171C22×C4—971no—
172C22×C4—972no—
173C22×C4—971no—
174C22×C4—972no—
175C22×C4—971no—
176C22×C4—1942no—
177C22×C4—1942no—
178C22×C4C22×D9711yescharacteristic
179C22×C4C22×D9712yes—
180C22×C4C22×D9711yescharacteristic
181C22×C4C22×D9712yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—1942no—
185C4⋊C4—1944no—
186C4⋊C4—1944no—
187C4⋊C4—1942no—
188C4⋊C4—1944no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—1944no—
192C22⋊C4—1942no—
193C42—974no—
194C42—1944no—
195C42—1942no—
196C42—1944no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D9712yes—
202C23C23×D9711yescharacteristic
203Q8—1948no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—972no—
213C2×C4—972no—
214C2×C4—972no—
215C2×C4—972no—
216C2×C4—972no—
217C2×C4—974no—
218C2×C4—1948no—
219C2×C4—1941no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1942no—
223C2×C4—1942no—
224C2×C4—1942no—
225C2×C4—1941no—
226C2×C4—1941no—
227C2×C4—1942no—
228C2×C4—1942no—
229C2×C4D4×D9712yes—
230C2×C4#1309551014yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D9711yescharacteristic
237C22C2×Q8×D9711yescharacteristic
238C22#41×D9711yescharacteristic
239C22#41×D9712yes—
240C22C2×#1309551012yes—
241C4—22no—
242C4—24no—
243C4—41no—
244C4—42no—
245C4—974no—
246C4—3881no—
247C4—3882no—
248C4—1942no—
249C4—1944no—
250C4—1942no—
251C2—42no—
252C2C22⋊Q8×D9711yescharacteristic
253C2#2026124311yescharacteristic
254C2#2026126212yes—
255C2#2026128311yescharacteristic
256C2#2026128712yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#399×C97C2
prime radicalC24×C97C23
hypercenter#399D97
hypocenterC97#2396
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15690775C2semidirect
3. #20260673C2semidirect
4. #20260666C2semidirect
6. #20260656C2semidirect
8. C4⋊C4×Dic97C2semidirect
9. #20260586C2semidirect
41. C22×Dic97Q8semidirect
129. C97#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#156907752
C2×#156907662
#202606732
#202606662
#202606582
#202606562
#202606472
C4⋊C4×Dic972
#202605862
C22⋊C4×Dic972
#399×C972
#239697