Finite Group Databasebeta

← #27150039

Subgroups — #27150039

Order 12416.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×#15690799C211yesmaximal, characteristic, complemented
2C2×#15690784C211yesmaximal, characteristic
3C22×C4×Dic97C211yesmaximal, characteristic, complemented
4#20260642C211yesmaximal, characteristic, complemented
5#20260636C211yesmaximal, characteristic, complemented
6#20260625C214yesmaximal, complemented
7#20260612C214yesmaximal, complemented
8#20260600C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C22×C22⋊D97C2212yes—
12C23×Dic97C2212yes—
13#15690799—24no—
14#15690784—24no—
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095488C2214yes—
17C2×#13095488C2214yes—
18C2×#13095488C2212yes—
19C2×C4×Dic97—24no—
20C2×C4×Dic97C2212yes—
21C2×C4×Dic97C2211yescharacteristic
22C2×C4×Dic97C2211yescharacteristic
23#15690717C2214yes—
24#15690717C2214yes—
25C22×Q8×C97C2211yescharacteristic
26C23×C4×C97C2211yescharacteristic
27C22⋊Q8×C97—24no—
28C2×C4⋊C4×C97C2212yes—
29C2×C4⋊C4×C97C2211yescharacteristic
30C2×C22⋊C4×C97C2212yes—
31C23×D97C2312yes—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388—22no—
35C2×C22⋊D97—24no—
36C2×C22⋊D97—28no—
37C22×Dic97—24no—
38C22×Dic97—24no—
39C22×Dic97—24no—
40C22×Dic97C2312yes—
41C22×Dic97C2312yes—
42C22×Dic97C2312yes—
43C4⋊Dic97—22no—
44#13095488—24no—
45#13095488—28no—
46#13095488—28no—
47C4×Dic97—22no—
48C4×Dic97—24no—
49C4×Dic97D414yescomplemented
50C24×C97C2311yescharacteristic
51C2×Q8×C97—22no—
52C2×Q8×C97—22no—
53C2×Q8×C97—22no—
54C22×C4×C97—22no—
55C22×C4×C97—22no—
56C22×C4×C97D414yes—
57C22×C4×C97C2312yes—
58C22×C4×C97C2312yes—
59C22×C4×C97C2311yescharacteristic
60C22×C4×C97C2311yescharacteristic
61C4⋊C4×C97—22no—
62C4⋊C4×C97—24no—
63C22⋊C4×C97—24no—
64C23×C97—21no—
65C23×C97—21no—
66C23×C97—22no—
67C23×C97C2×D412yes—
68C23×C97#4114yes—
69C23×C97C2411yescharacteristic
70C22×D97—22no—
71C22×D97—22no—
72C22×D97—24no—
73C22×D97—24no—
74C22×D97—22no—
75Q8×C97—44no—
76C2×C4×C97—21no—
77C2×C4×C97—21no—
78C2×C4×C97—28no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—24no—
84C2×C4×C97C2×D412yes—
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D412yes—
87C2×C4×C97#4112yes—
88C22⋊D97—48no—
89C2×Dic97—216no—
90C2×Dic97—22no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—24no—
96C2×Dic97C2×D414yes—
97C2×Dic97#4114yes—
98D388—44no—
99C22×C97—22no—
100C22×C97—24no—
101C22×C97—24no—
102C22×C97—22no—
103C22×C97#12414yescomplemented
104C22×C97C22×D411yescharacteristic
105C22×C97C22×D411yescharacteristic
106C22×C97C2×#4112yes—
107C22×C97C2×#4111yescharacteristic
108C22×C97C2×#4112yes—
109C2×D97—42no—
110C2×D97—44no—
111C2×D97—42no—
112C2×D97—44no—
113C2×D97—42no—
114C4×C97—22no—
115C4×C97—42no—
116C4×C97—42no—
117C4×C97#12114yes—
118Dic97—24no—
119Dic97—24no—
120Dic97—42no—
121C2×C97—24no—
122C2×C97C2×#12111yescharacteristic
123C2×C97#52512yes—
124C2×C97C2×#12411yescharacteristic
125C2×C97#52912yes—
126C2×C97#53211yescharacteristic
127D97—82no—
128#2850—971nomaximal, Sylow
129C97#285011yescharacteristic, complemented, Sylow
130C2×#124—971no—
131C2×C22⋊Q8D9711yescharacteristic, complemented
132C2×#121—971no—
133C22×C42—971no—
134#391—971no—
135#390—971no—
136#388—974no—
137#386—974no—
138#382—971no—
139C22×Q8C2×D9711yescharacteristic
140C22×D4—972no—
141C22×D4—971no—
142C23×C4—972no—
143C23×C4C2×D9711yescharacteristic
144#124—1944no—
145C22⋊Q8—24no—
146#121—1944no—
147C2×C4⋊C4—971no—
148C2×C4⋊C4C2×D9711yescharacteristic
149C2×C4⋊C4C2×D9712yes—
150C2×C22⋊C4—972no—
151C2×C22⋊C4—974no—
152C2×C22⋊C4—974no—
153C2×C22⋊C4C2×D9712yes—
154C2×C42—971no—
155C2×C42—972no—
156C2×C42—971no—
157C2×C42—1944no—
158#95—974no—
159#95—974no—
160C24—972no—
161C24C22×D9711yescharacteristic
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—1944no—
166C2×D4—1948no—
167C2×D4—1942no—
168C2×D4—1942no—
169C2×D4—1942no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—972no—
173C22×C4—972no—
174C22×C4—972no—
175C22×C4—1944no—
176C22×C4—1944no—
177C22×C4—1944no—
178C22×C4C22⋊D9714yes—
179C22×C4C22×D9712yes—
180C22×C4C22×D9712yes—
181C22×C4C22×D9711yescharacteristic
182C22×C4C22×D9711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—1942no—
186C22⋊C4—24no—
187C22⋊C4—1948no—
188C22⋊C4—1948no—
189C22⋊C4—1944no—
190C42—974no—
191C42—1944no—
192C42—1942no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—1942no—
197C23—1942no—
198C23—1944no—
199C23—1942no—
200C23—1944no—
201C23C2×C22⋊D9712yes—
202C23#1309551214yes—
203C23C23×D9711yescharacteristic
204Q8—44no—
205D4—3888no—
206D4—3884no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—974no—
216C2×C4—974no—
217C2×C4—1942no—
218C2×C4—1944no—
219C2×C4—19416no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1942no—
223C2×C4—1942no—
224C2×C4D4×D9712yes—
225C2×C4C2×C22⋊D9712yes—
226C2×C4C2×C22⋊D9712yes—
227C2×C4#1309551012yes—
228C22—22no—
229C22—24no—
230C22—24no—
231C22—22no—
232C22—3882no—
233C22—3884no—
234C22—3882no—
235C22—3884no—
236C22—3882no—
237C22#1569079914yescomplemented
238C22C2×D4×D9711yescharacteristic
239C22C22×C22⋊D9711yescharacteristic
240C22#41×D9712yes—
241C22C2×#1309551011yescharacteristic
242C22C2×#1309551212yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—3882no—
247C4—1944no—
248C4—1944no—
249C4#1569078214yes—
250C2—24no—
251C2—7762no—
252C2C2×#1569078211yescharacteristic
253C2#2026124812yes—
254C2C2×#1569079911yescharacteristic
255C2#2026127112yes—
256C2#2026128111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×C22⋊Q8×C97C2
prime radicalC2×#15690784C2
hypercenterC2×C22⋊Q8D97
hypocenterC97#2850
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15690799C2semidirect
3. C22×C4×Dic97C2semidirect
4. #20260642C2semidirect
5. #20260636C2semidirect
6. #20260625C2semidirect
7. #20260612C2semidirect
8. #20260600C2semidirect
9. C2×C22⋊Q8×C97C2semidirect
49. C4×Dic97D4semidirect
103. C22×C97#124semidirect
129. C97#2850semidirect
131. C2×C22⋊Q8D97semidirect
237. C22#15690799semidirect

Maximal subgroups

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

subgroupindex
C2×#156907992
C2×#156907842
C22×C4×Dic972
#202606422
#202606362
#202606252
#202606122
#202606002
C2×C22⋊Q8×C972
#285097