Finite Group Databasebeta

← #27145024

Subgroups — #27145024

Order 12416.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic97C211yesmaximal, characteristic
3#20260675C211yesmaximal, characteristic, complemented
4#20260674C211yesmaximal, characteristic, complemented
5#20260663C212yesmaximal, complemented
6#20260649C211yesmaximal, characteristic
7#20260647C212yesmaximal, complemented
8C22⋊C4×Dic97C212yesmaximal, complemented
9C22⋊C4×Dic97C212yesmaximal, complemented
10#20260580C211yesmaximal, characteristic, complemented
11#394×C97C211yesmaximal, characteristic
12C22×Dic194C2211yescharacteristic
13C23×Dic97C2212yes—
14#15690775—24no—
15D4×Dic97—24no—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490C2211yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×C22⋊Dic97C2212yes—
20C2×C22⋊Dic97C2212yes—
21C2×C22⋊Dic97C2212yes—
22C2×C4×Dic97C2211yescharacteristic
23C2×C4×Dic97C2211yescharacteristic
24C2×C4×Dic97C2212yes—
25C2×C4×Dic97C2211yescharacteristic
26#15690717C2212yes—
27#15690717C2212yes—
28#15690717C2212yes—
29#15690717C2211yescharacteristic
30#15690717C2212yes—
31#15690717C2212yes—
32C22×D4×C97C2211yescharacteristic
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C22⋊C4×C97C2212yes—
35C2×C22⋊C4×C97C2212yes—
36#95×C97C2211yescharacteristic
37C2×Dic194—22no—
38C2×Dic194—22no—
39C2×Dic194—22no—
40C22×Dic97—24no—
41C22×Dic97—24no—
42C22×Dic97—24no—
43C22×Dic97C2311yescharacteristic
44C22×Dic97C2312yes—
45C22×Dic97C2312yes—
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2311yescharacteristic
49C4⋊Dic97—22no—
50#13095490—22no—
51#13095490—22no—
52C22⋊Dic97—24no—
53C22⋊Dic97—24no—
54C22⋊Dic97—24no—
55C4×Dic97—22no—
56C4×Dic97—22no—
57C4×Dic97—24no—
58C4×Dic97D414yes—
59C24×C97C2312yes—
60C2×D4×C97—24no—
61C2×D4×C97—22no—
62C22×C4×C97C2311yescharacteristic
63C22×C4×C97C2312yes—
64C22×C4×C97C2311yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—22no—
67C22⋊C4×C97—24no—
68C22⋊C4×C97—24no—
69C23×C97—22no—
70C23×C97—22no—
71C23×C97—22no—
72C23×C97—22no—
73C23×C97—22no—
74C23×C97—22no—
75C23×C97#4114yes—
76C23×C97C2411yescharacteristic
77D4×C97—28no—
78C2×C4×C97—21no—
79C2×C4×C97—22no—
80C2×C4×C97—21no—
81C2×C4×C97—21no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97C2×D412yes—
87C2×C4×C97#4112yes—
88C2×Dic97—21no—
89C2×Dic97—21no—
90C2×Dic97—21no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97—48no—
101C2×Dic97C2×D412yes—
102C2×Dic97C2×D412yes—
103C2×Dic97#4112yes—
104C2×Dic97#4114yes—
105Dic194—28no—
106C22×C97—24no—
107C22×C97—24no—
108C22×C97—24no—
109C22×C97—42no—
110C22×C97—42no—
111C22×C97—42no—
112C22×C97—42no—
113C22×C97C22×D411yescharacteristic
114C22×C97C2×#4111yescharacteristic
115C22×C97C2×#4112yes—
116C22×C97C2×#4112yes—
117C22×C97C2×#4111yescharacteristic
118C4×C97—22no—
119C4×C97—22no—
120C4×C97—41no—
121C4×C97—42no—
122Dic97—22no—
123Dic97—22no—
124Dic97—41no—
125Dic97—42no—
126Dic97—42no—
127Dic97#12414yes—
128C2×C97—44no—
129C2×C97#52511yescharacteristic
130C2×C97#52511yescharacteristic
131C2×C97C2×#12411yescharacteristic
132C2×C97#52912yes—
133C2×C97#52912yes—
134#2398—971nomaximal, Sylow
135C97#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—971no—
137C2×C4×D4—971no—
138#400—971no—
139#400—971no—
140#397—972no—
141#394—971no—
142#394D9711yescharacteristic
143#393—972no—
144C4×C22⋊C4—972no—
145C4×C22⋊C4—972no—
146#376—971no—
147C22×Q8—971no—
148C22×D4C2×D9711yescharacteristic
149C23×C4—972no—
150C4×Q8—1944no—
151C4×D4—1944no—
152C2×C4⋊C4—971no—
153C2×C4⋊C4—971no—
154C2×C4⋊C4—971no—
155C2×C4⋊C4C2×D9711yescharacteristic
156C2×C22⋊C4—972no—
157C2×C22⋊C4—972no—
158C2×C22⋊C4—972no—
159C2×C22⋊C4C2×D9712yes—
160C2×C22⋊C4C2×D9712yes—
161C2×C42—971no—
162C2×C42—971no—
163C2×C42—971no—
164C2×C42—972no—
165#95—972no—
166#95—971no—
167#95—972no—
168#95—972no—
169#95—972no—
170#95—972no—
171#95C2×D9711yescharacteristic
172C24C22×D9712yes—
173C2×Q8—1942no—
174C2×Q8—1942no—
175C2×Q8—1942no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—971no—
179C22×C4—971no—
180C22×C4—971no—
181C22×C4—972no—
182C22×C4—972no—
183C22×C4—971no—
184C22×C4—1944no—
185C22×C4—1944no—
186C22×C4—1944no—
187C22×C4C22×D9712yes—
188C22×C4C22×D9711yescharacteristic
189C22×C4C22×D9711yescharacteristic
190C22×C4C22×D9711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—1942no—
193C4⋊C4—1942no—
194C4⋊C4—1942no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—1944no—
198C22⋊C4—1944no—
199C22⋊C4—1944no—
200C42—974no—
201C42—1944no—
202C42—1942no—
203C42—1942no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#1309551014yes—
211C23C23×D9711yescharacteristic
212Q8—1948no—
213D4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—972no—
223C2×C4—974no—
224C2×C4—972no—
225C2×C4—972no—
226C2×C4—3888no—
227C2×C4—1941no—
228C2×C4—1942no—
229C2×C4—1942no—
230C2×C4—1942no—
231C2×C4—1942no—
232C2×C4—1942no—
233C2×C4—1942no—
234C2×C4—1942no—
235C2×C4—1941no—
236C2×C4—1941no—
237C2×C4—1941no—
238C2×C4—1941no—
239C2×C4D4×D9712yes—
240C2×C4#1309551012yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D9711yescharacteristic
249C22#41×D9711yescharacteristic
250C22#41×D9712yes—
251C22C2×#1309551012yes—
252C22C2×#1309551011yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—974no—
258C4—3881no—
259C4—3882no—
260C4—3882no—
261C4—1942no—
262C4—1942no—
263C2—44no—
264C2#2026124111yescharacteristic
265C2#2026124511yescharacteristic
266C2#124×D9711yescharacteristic
267C2#2026126212yes—
268C2#2026126612yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#394×C97C2
prime radicalC22×D4×C97C22
hypercenter#394D97
hypocenterC97#2398
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#15690775C2semidirect
3. #20260675C2semidirect
4. #20260674C2semidirect
5. #20260663C2semidirect
7. #20260647C2semidirect
8. C22⋊C4×Dic97C2semidirect
9. C22⋊C4×Dic97C2semidirect
10. #20260580C2semidirect
135. C97#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#156907752
C2×D4×Dic972
#202606752
#202606742
#202606632
#202606492
#202606472
C22⋊C4×Dic972
#202605802
#394×C972
#239897