Finite Group Databasebeta

← #27150036

Subgroups — #27150036

Order 12416.

696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15690797C211yesmaximal, characteristic, complemented
2C2×#15690790C211yesmaximal, characteristic, complemented
3C2×#15690781C211yesmaximal, characteristic
4C2×C42×D97C211yesmaximal, characteristic, complemented
5#20260641C211yesmaximal, characteristic, complemented
6#20260628C214yesmaximal, complemented
7#20260611C214yesmaximal, complemented
8#20260600C211yesmaximal, characteristic, complemented
9#390×C97C211yesmaximal, characteristic, complemented
10C22×Dic194C2211yescharacteristic
11C22×C22⋊D97C2212yes—
12C22×C4×D97C2212yes—
13C22×C4×D97C2211yescharacteristic
14#15690797—24no—
15#15690790—24no—
16#15690781—24no—
17C2×C4⋊Dic97C2212yes—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×C22⋊Dic97C2214yes—
20C2×C22⋊Dic97C2212yes—
21C2×#13095488C2212yes—
22C2×C4×Dic97C2212yes—
23C2×C4×Dic97C2211yescharacteristic
24C42×D97—24no—
25#15690717C2214yes—
26#15690717C2214yes—
27C22×D4×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2211yescharacteristic
29C2×C22⋊C4×C97C2214yes—
30C2×C42×C97C2211yescharacteristic
31C23×D97C2311yescharacteristic
32C2×Dic194—22no—
33C2×Dic194—22no—
34C2×Dic194—22no—
35C2×C22⋊D97—24no—
36C2×C22⋊D97—28no—
37C22×Dic97C2312yes—
38C22×Dic97C2312yes—
39C22×Dic97C2311yescharacteristic
40C22×Dic97C2312yes—
41C2×C4×D97—24no—
42C2×C4×D97—22no—
43C2×C4×D97—24no—
44C2×C4×D97—22no—
45C2×C4×D97—24no—
46C2×C4×D97D414yes—
47C4⋊Dic97—22no—
48C4⋊Dic97—24no—
49C22⋊Dic97—24no—
50C22⋊Dic97—28no—
51#13095488—24no—
52C4×Dic97—24no—
53C4×Dic97D414yescomplemented
54C24×C97C2312yes—
55C2×D4×C97—22no—
56C2×D4×C97—22no—
57C2×D4×C97—22no—
58C22×C4×C97C2312yes—
59C22×C4×C97C2312yes—
60C22×C4×C97C2311yescharacteristic
61C4⋊C4×C97—22no—
62C22⋊C4×C97—28no—
63C42×C97—22no—
64C23×C97—22no—
65C23×C97—22no—
66C23×C97—24no—
67C23×C97—22no—
68C23×C97—24no—
69C23×C97C2411yescharacteristic
70C22×D97—21no—
71C22×D97—22no—
72C22×D97—21no—
73C22×D97C2×D412yes—
74C22×D97#4114yes—
75D4×C97—44no—
76C2×C4×C97—22no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—24no—
80C2×C4×C97C2×D412yes—
81C2×C4×C97C2×D412yes—
82C2×C4×C97#4114yes—
83C2×C4×C97#4112yes—
84C22⋊D97—48no—
85C2×Dic97—21no—
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—21no—
89C2×Dic97—22no—
90C2×Dic97—22no—
91C2×Dic97—22no—
92C2×Dic97—24no—
93C2×Dic97—22no—
94C2×Dic97C2×D414yes—
95C2×Dic97C2×D412yes—
96C4×D97—216no—
97C4×D97—28no—
98Dic194—44no—
99C22×C97—42no—
100C22×C97—44no—
101C22×C97—42no—
102C22×C97—44no—
103C22×C97—42no—
104C22×C97C22×D411yescharacteristic
105C22×C97C22×D411yescharacteristic
106C22×C97C2×#4112yes—
107C22×C97C2×#4112yes—
108C22×C97C2×#4111yescharacteristic
109C2×D97—22no—
110C2×D97—24no—
111C2×D97—24no—
112C2×D97—22no—
113C2×D97#12414yescomplemented
114C4×C97—24no—
115C4×C97—42no—
116C4×C97#12114yes—
117Dic97—22no—
118Dic97—24no—
119Dic97—42no—
120Dic97—42no—
121C2×C97—82no—
122C2×C97C2×#12111yescharacteristic
123C2×C97#52512yes—
124C2×C97C2×#12411yescharacteristic
125C2×C97#52912yes—
126C2×C97#53211yescharacteristic
127D97—24no—
128#2850—971nomaximal, Sylow
129C97#285011yescharacteristic, complemented, Sylow
130C2×#124—971no—
131C2×C22⋊Q8—971no—
132C2×#121—971no—
133C22×C42—971no—
134#391—971no—
135#390D9711yescharacteristic, complemented
136#388—974no—
137#386—974no—
138#382—971no—
139C22×Q8—971no—
140C22×D4—972no—
141C22×D4C2×D9711yescharacteristic
142C23×C4—971no—
143C23×C4—972no—
144#124—1944no—
145C22⋊Q8—1944no—
146#121—1944no—
147C2×C4⋊C4—971no—
148C2×C4⋊C4—972no—
149C2×C4⋊C4C2×D9711yescharacteristic
150C2×C22⋊C4—972no—
151C2×C22⋊C4—974no—
152C2×C22⋊C4—972no—
153C2×C22⋊C4C2×D9714yes—
154C2×C42—971no—
155C2×C42—972no—
156C2×C42—1944no—
157C2×C42C2×D9711yescharacteristic
158#95—974no—
159#95—974no—
160C24—971no—
161C24C22×D9712yes—
162C2×Q8—1942no—
163C2×Q8—1942no—
164C2×Q8—1942no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—1944no—
169C2×D4—1948no—
170C22×C4—972no—
171C22×C4—971no—
172C22×C4—974no—
173C22×C4—972no—
174C22×C4—972no—
175C22×C4—1944no—
176C22×C4—1944no—
177C22×C4—1942no—
178C22×C4—1944no—
179C22×C4—1942no—
180C22×C4C22×D9712yes—
181C22×C4C22×D9712yes—
182C22×C4C22×D9711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—1944no—
185C4⋊C4—1942no—
186C22⋊C4—28no—
187C22⋊C4—1944no—
188C22⋊C4—1944no—
189C22⋊C4—1948no—
190C42—22no—
191C42—974no—
192C42—1944no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—22no—
198C23—972no—
199C23—974no—
200C23—1941no—
201C23—1942no—
202C23—1941no—
203C23C23×D9711yescharacteristic
204Q8—3884no—
205D4—44no—
206D4—3888no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—974no—
212C2×C4—972no—
213C2×C4—19416no—
214C2×C4—1942no—
215C2×C4—1942no—
216C2×C4—1942no—
217C2×C4—1944no—
218C2×C4—1942no—
219C2×C4—1941no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1941no—
223C2×C4—1948no—
224C2×C4D4×D9712yes—
225C2×C4D4×D9712yes—
226C2×C4#1309551014yes—
227C2×C4#1309551112yes—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—974no—
234C22—1942no—
235C22—1944no—
236C22—1944no—
237C22—1942no—
238C22C2×D4×D9711yescharacteristic
239C22C2×D4×D9711yescharacteristic
240C22#41×D9712yes—
241C22C2×#1309551012yes—
242C22C2×#1309551111yescharacteristic
243C4—24no—
244C4—42no—
245C4—3882no—
246C4—3882no—
247C4—1942no—
248C4—1944no—
249C4#1569078314yes—
250C2—82no—
251C2—1944no—
252C2C2×#1569078311yescharacteristic
253C2#2026124512yes—
254C2#124×D9711yescharacteristic
255C2#2026126712yes—
256C2#2026128311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#390×C97C2
prime radicalC2×#15690781C2
hypercenter#390D97
hypocenterC97#2850
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#15690797C2semidirect
2. C2×#15690790C2semidirect
4. C2×C42×D97C2semidirect
5. #20260641C2semidirect
6. #20260628C2semidirect
7. #20260611C2semidirect
8. #20260600C2semidirect
9. #390×C97C2semidirect
53. C4×Dic97D4semidirect
113. C2×D97#124semidirect
129. C97#2850semidirect
135. #390D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907972
C2×#156907902
C2×#156907812
C2×C42×D972
#202606412
#202606282
#202606112
#202606002
#390×C972
#285097