Finite Group Databasebeta

← #27145025

Subgroups — #27145025

Order 12416.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#15690772C211yesmaximal, characteristic
3#20260673C211yesmaximal, characteristic, complemented
4#20260664C212yesmaximal, complemented
5#20260653C211yesmaximal, characteristic
6#20260652C211yesmaximal, characteristic
7#20260648C212yesmaximal, complemented
8#20260582C212yesmaximal, complemented
9#20260582C212yesmaximal, complemented
10#20260580C211yesmaximal, characteristic, complemented
11#400×C97C211yesmaximal, characteristic, complemented
12C22×Dic194C2211yescharacteristic
13C22×D388C2211yescharacteristic
14C22×C4×D97C2212yes—
15#15690775—24no—
16#15690772—24no—
17C2×#13095490C2211yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×#13095488C2212yes—
20C2×#13095488C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2212yes—
23C2×#13095488C2212yes—
24C2×C4×Dic97C2211yescharacteristic
25C2×C4×Dic97C2211yescharacteristic
26C2×C4×Dic97C2212yes—
27C2×C4×Dic97C2211yescharacteristic
28#15690717C2212yes—
29#15690717C2212yes—
30#15690717C2211yescharacteristic
31#15690717C2212yes—
32C2×C4⋊C4×C97C2211yescharacteristic
33C2×C4⋊C4×C97C2211yescharacteristic
34#95×C97C2212yes—
35#95×C97C2211yescharacteristic
36#95×C97C2212yes—
37C23×D97C2312yes—
38C2×Dic194—22no—
39C2×Dic194—22no—
40C2×Dic194—22no—
41C2×D388—24no—
42C2×D388—22no—
43C22×Dic97C2311yescharacteristic
44C22×Dic97C2312yes—
45C22×Dic97C2311yescharacteristic
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2311yescharacteristic
48C2×C4×D97—24no—
49C2×C4×D97—24no—
50C2×C4×D97—24no—
51#13095490—22no—
52#13095490—22no—
53#13095488—24no—
54#13095488—24no—
55#13095488—24no—
56#13095488—24no—
57#13095488—24no—
58C4×Dic97—22no—
59C4×Dic97—22no—
60C4×Dic97—24no—
61C4×Dic97D414yes—
62C22×C4×C97C2311yescharacteristic
63C22×C4×C97C2312yes—
64C22×C4×C97C2312yes—
65C22×C4×C97C2311yescharacteristic
66C22×C4×C97C2311yescharacteristic
67C4⋊C4×C97—22no—
68C4⋊C4×C97—22no—
69C23×C97C2411yescharacteristic
70C22×D97—22no—
71C22×D97—22no—
72C22×D97—22no—
73C22×D97—22no—
74C22×D97—22no—
75C22×D97—22no—
76C22×D97#4114yes—
77C2×C4×C97—21no—
78C2×C4×C97—22no—
79C2×C4×C97—21no—
80C2×C4×C97—21no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97C2×D412yes—
89C2×C4×C97#4112yes—
90C2×Dic97—21no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—21no—
94C2×Dic97—21no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97C2×D412yes—
100C2×Dic97C2×D412yes—
101C2×Dic97#4112yes—
102C2×Dic97#4114yes—
103D388—28no—
104C4×D97—48no—
105Dic194—28no—
106C22×C97C22×D411yescharacteristic
107C22×C97C2×#4112yes—
108C22×C97C2×#4111yescharacteristic
109C22×C97C2×#4112yes—
110C22×C97C2×#4111yescharacteristic
111C2×D97—24no—
112C2×D97—24no—
113C2×D97—24no—
114C2×D97—42no—
115C2×D97—42no—
116C2×D97—42no—
117C2×D97—42no—
118C4×C97—22no—
119C4×C97—22no—
120C4×C97—41no—
121C4×C97—42no—
122C4×C97—42no—
123Dic97—22no—
124Dic97—22no—
125Dic97—41no—
126Dic97—42no—
127Dic97#12414yes—
128C2×C97#52511yescharacteristic
129C2×C97#52511yescharacteristic
130C2×C97C2×#12411yescharacteristic
131C2×C97#52912yes—
132C2×C97#52912yes—
133D97—44no—
134#2398—971nomaximal, Sylow
135C97#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—971no—
137C2×C4×D4—971no—
138#400—971no—
139#400D9711yescharacteristic, complemented
140#397—972no—
141#394—971no—
142#394—971no—
143#393—972no—
144C4×C22⋊C4—972no—
145C4×C22⋊C4—972no—
146#376—971no—
147C22×Q8—971no—
148C22×D4—971no—
149C23×C4—972no—
150C4×Q8—1944no—
151C4×D4—1944no—
152C2×C4⋊C4—971no—
153C2×C4⋊C4—971no—
154C2×C4⋊C4C2×D9711yescharacteristic
155C2×C4⋊C4C2×D9711yescharacteristic
156C2×C22⋊C4—972no—
157C2×C22⋊C4—972no—
158C2×C22⋊C4—972no—
159C2×C22⋊C4—972no—
160C2×C22⋊C4—972no—
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#95C2×D9711yescharacteristic
170#95C2×D9712yes—
171#95C2×D9712yes—
172C24—972no—
173C2×Q8—1942no—
174C2×Q8—1942no—
175C2×Q8—1942no—
176C2×D4—1944no—
177C2×D4—1942no—
178C22×C4—971no—
179C22×C4—971no—
180C22×C4—971no—
181C22×C4—972no—
182C22×C4—971no—
183C22×C4—1944no—
184C22×C4—1944no—
185C22×C4—1944no—
186C22×C4C22×D9712yes—
187C22×C4C22×D9712yes—
188C22×C4C22×D9711yescharacteristic
189C22×C4C22×D9711yescharacteristic
190C22×C4C22×D9711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—1942no—
194C4⋊C4—1942no—
195C22⋊C4—1944no—
196C22⋊C4—1944no—
197C22⋊C4—1944no—
198C22⋊C4—1944no—
199C22⋊C4—1944no—
200C42—974no—
201C42—1944no—
202C42—1942no—
203C42—1942no—
204C23—974no—
205C23—1942no—
206C23—1942no—
207C23—1942no—
208C23—1942no—
209C23—1942no—
210C23—1942no—
211C23C23×D9711yescharacteristic
212Q8—1948no—
213D4—1948no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—972no—
226C2×C4—974no—
227C2×C4—972no—
228C2×C4—972no—
229C2×C4—3888no—
230C2×C4—1941no—
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#1309551212yes—
241C22—3882no—
242C22—3882no—
243C22—3882no—
244C22—3882no—
245C22—1944no—
246C22—1944no—
247C22—1944no—
248C22C2×D4×D9711yescharacteristic
249C22#41×D9712yes—
250C22#41×D9711yescharacteristic
251C22#41×D9712yes—
252C22C2×#1309551211yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—974no—
259C4—3881no—
260C4—3882no—
261C4—1942no—
262C4—1942no—
263C2—3884no—
264C2#2026124311yescharacteristic
265C2#2026124811yescharacteristic
266C2#124×D9711yescharacteristic
267C2#2026126812yes—
268C2#2026127012yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#400×C97C2
prime radicalC22×D388C22
hypercenter#400D97
hypocenterC97#2398
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#15690775C2semidirect
3. #20260673C2semidirect
4. #20260664C2semidirect
7. #20260648C2semidirect
8. #20260582C2semidirect
9. #20260582C2semidirect
10. #20260580C2semidirect
11. #400×C97C2semidirect
135. C97#2398semidirect
139. #400D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907752
C2×#156907722
#202606732
#202606642
#202606532
#202606522
#202606482
#202605822
#202605802
#400×C972
#239897