Finite Group Databasebeta

← #27150084

Subgroups — #27150084

Order 12416.

644 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 201 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×#15690808C211yesmaximal, characteristic, complemented
2C2×#15690789C211yesmaximal, characteristic
3C22×C4×Dic97C211yesmaximal, characteristic
4#20260642C211yesmaximal, characteristic, complemented
5#20260640C211yesmaximal, characteristic, complemented
6#20260630C214yesmaximal, complemented
7#20260604C211yesmaximal, characteristic, complemented
8#20260593C214yesmaximal
9C2×C22⋊Q8×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C23×Dic97C2212yes—
12#15690808—24no—
13#15690789—24no—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095490C2212yes—
16C2×#13095490C2214yes—
17C2×#13095490C2214yes—
18C2×C22⋊Dic97C2212yes—
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—
31C2×Dic194—22no—
32C2×Dic194—22no—
33C2×Dic194—22no—
34C22×Dic97—24no—
35C22×Dic97—24no—
36C22×Dic97Q818yescomplemented
37C22×Dic97C2312yes—
38C22×Dic97C2312yes—
39C22×Dic97C2312yes—
40C22×Dic97C2312yes—
41C4⋊Dic97—22no—
42#13095490—24no—
43#13095490—28no—
44#13095490—28no—
45C22⋊Dic97—24no—
46C4×Dic97—22no—
47C4×Dic97—24no—
48C4×Dic97D414yes—
49C24×C97C2311yescharacteristic
50C2×Q8×C97—22no—
51C2×Q8×C97—22no—
52C2×Q8×C97—22no—
53C22×C4×C97—22no—
54C22×C4×C97—22no—
55C22×C4×C97D414yes—
56C22×C4×C97C2312yes—
57C22×C4×C97C2312yes—
58C22×C4×C97C2311yescharacteristic
59C22×C4×C97C2311yescharacteristic
60C4⋊C4×C97—22no—
61C4⋊C4×C97—24no—
62C22⋊C4×C97—24no—
63C23×C97—21no—
64C23×C97—21no—
65C23×C97—22no—
66C23×C97C2×D412yes—
67C23×C97C2×Q814yes—
68C23×C97C2411yescharacteristic
69Q8×C97—44no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—28no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—24no—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D412yes—
80C2×C4×C97C2×D412yes—
81C2×C4×C97#4112yes—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97—216no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—24no—
89C2×Dic97—22no—
90C2×Dic97C2×D414yes—
91C2×Dic97C2×Q814yes—
92C2×Dic97C2×Q814yes—
93C2×Dic97#4114yes—
94Dic194—44no—
95C22×C97—22no—
96C22×C97—24no—
97C22×C97—22no—
98C22×C97—24no—
99C22×C97C4⋊Q814yescomplemented
100C22×C97C22×D411yescharacteristic
101C22×C97C22×D411yescharacteristic
102C22×C97C22×Q812yes—
103C22×C97C2×#4112yes—
104C22×C97C2×#4111yescharacteristic
105C4×C97—22no—
106C4×C97—42no—
107C4×C97—42no—
108C4×C97#12114yes—
109Dic97—24no—
110Dic97—42no—
111Dic97—42no—
112Dic97C22⋊Q818yescomplemented
113C2×C97—24no—
114C2×C97C2×#12111yescharacteristic
115C2×C97C2×C22⋊Q812yes—
116C2×C97C2×C4⋊Q811yescharacteristic
117C2×C97#53211yescharacteristic
118C2×C97#53312yes—
119#2857—971nomaximal, Sylow
120C97#285711yescharacteristic, complemented, Sylow
121C2×C4⋊Q8—971no—
122C2×C22⋊Q8—971no—
123C2×C22⋊Q8D9711yescharacteristic
124C22×C42—971no—
125#391—971no—
126#391—971no—
127#389—974no—
128#384—971no—
129#380—974no—
130C22×Q8—971no—
131C22×Q8C2×D9711yescharacteristic
132C23×C4—972no—
133C23×C4C2×D9711yescharacteristic
134C4⋊Q8—1944no—
135C22⋊Q8—24no—
136C22⋊Q8—1944no—
137C2×C4⋊C4—972no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—974no—
140C2×C4⋊C4—974no—
141C2×C4⋊C4C2×D9711yescharacteristic
142C2×C4⋊C4C2×D9712yes—
143C2×C22⋊C4—972no—
144C2×C22⋊C4C2×D9712yes—
145C2×C42—971no—
146C2×C42—972no—
147C2×C42—971no—
148C2×C42—1944no—
149#95—974no—
150#95—974no—
151C24C22×D9711yescharacteristic
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—1942no—
156C2×Q8—1942no—
157C2×Q8—1942no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—972no—
161C22×C4—978no—
162C22×C4—972no—
163C22×C4—972no—
164C22×C4—972no—
165C22×C4—1944no—
166C22×C4—1944no—
167C22×C4C22⋊D9714yes—
168C22×C4C22×D9712yes—
169C22×C4C22×D9712yes—
170C22×C4C22×D9711yescharacteristic
171C22×C4C22×D9711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—1948no—
175C4⋊C4—1948no—
176C4⋊C4—1944no—
177C4⋊C4—1942no—
178C22⋊C4—24no—
179C22⋊C4—1944no—
180C42—974no—
181C42—1944no—
182C42—1942no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23C2×C22⋊D9712yes—
187C23Q8×D9714yes—
188C23C23×D9711yescharacteristic
189Q8—44no—
190Q8—3884no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—974no—
200C2×C4—974no—
201C2×C4—974no—
202C2×C4—974no—
203C2×C4—19416no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4—1942no—
207C2×C4—1944no—
208C2×C4—1942no—
209C2×C4—1942no—
210C2×C4—1942no—
211C2×C4D4×D9712yes—
212C2×C4C2×C22⋊D9712yes—
213C2×C4C2×C22⋊D9712yes—
214C2×C4#1309551012yes—
215C22—22no—
216C22—24no—
217C22—22no—
218C22—24no—
219C22#1569080814yescomplemented
220C22C2×D4×D9711yescharacteristic
221C22C22×C22⋊D9711yescharacteristic
222C22C2×Q8×D9712yes—
223C22#41×D9712yes—
224C22C2×#1309551011yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—978no—
229C4—3882no—
230C4—3882no—
231C4—1944no—
232C4#1569078214yes—
233C2—24no—
234C2C2×#1569078211yescharacteristic
235C2C22⋊Q8×D9712yes—
236C2C2×#1569080811yescharacteristic
237C2#2026128111yescharacteristic
238C2#2026128712yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×C22⋊Q8×C97C2
prime radicalC24×C97C23
hypercenterC2×C22⋊Q8D97
hypocenterC97#2857
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×#15690808C2semidirect
4. #20260642C2semidirect
5. #20260640C2semidirect
6. #20260630C2semidirect
7. #20260604C2semidirect
36. C22×Dic97Q8semidirect
99. C22×C97C4⋊Q8semidirect
112. Dic97C22⋊Q8semidirect
120. C97#2857semidirect
219. C22#15690808semidirect

Maximal subgroups

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

subgroupindex
C2×#156908082
C2×#156907892
C22×C4×Dic972
#202606422
#202606402
#202606302
#202606042
#202605932
C2×C22⋊Q8×C972
#285797