Finite Group Databasebeta

← #27151672

Subgroups — #27151672

Order 12416.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15690769C211yesmaximal, characteristic, complemented
2#20261056C214yesmaximal, complemented
3#20261051C212yesmaximal, complemented
4#20261046C212yesmaximal, complemented
5#20260775C212yesmaximal, complemented
6#20260740C212yesmaximal, complemented
7#20260699C211yesmaximal, characteristic, complemented
8C2×#121×C97C211yesmaximal, characteristic
9C2×#13095495C2212yes—
10#15690769—21no—
11#15690769—21no—
12#15690769C2212yes—
13#15690769C2212yescomplemented
14C2×C4⋊Dic97C2211yescharacteristic, complemented
15C2×C4⋊Dic97C2211yescharacteristic, complemented
16C2×C22⋊Dic97—21no—
17C2×C4×Dic97—21no—
18#15690743C2214yescomplemented
19#15690741C2214yescomplemented
20#15690731C2214yescomplemented
21#15690731C2214yescomplemented
22C22⋊#11715895C2212yescomplemented
23C22⋊#11715895C2212yescomplemented
24C22×D4×C97—21no—
25C22×D4×C97C2211yescharacteristic
26C23×C4×C97C2211yescharacteristic
27#121×C97—21no—
28#121×C97—21no—
29#121×C97C2212yes—
30#121×C97C2212yes—
31C2×C4⋊C4×C97C2211yescharacteristic
32C2×C22⋊C4×C97—21no—
33C22×Dic97—21no—
34C22×Dic97—21no—
35#13095495—24no—
36C2×#11715895C2314yes—
37C4⋊Dic97—21no—
38C4⋊Dic97—21no—
39C4⋊Dic97C2312yes—
40C4⋊Dic97C2312yes—
41C22⋊Dic97—22no—
42C22⋊Dic97—22no—
43C4×Dic97—22no—
44C4×Dic97—22no—
45C24×C97—21no—
46C24×C97D411yescharacteristic
47C2×D4×C97—22no—
48C2×D4×C97—22no—
49C2×D4×C97—21no—
50C2×D4×C97—24no—
51C2×D4×C97—44no—
52C2×D4×C97C2312yes—
53C22×C4×C97—24no—
54C22×C4×C97—21no—
55C22×C4×C97D411yescharacteristic
56C22×C4×C97D411yescharacteristic
57C22×C4×C97D411yescharacteristic
58C22×C4×C97C2311yescharacteristic
59C22×C4×C97C2311yescharacteristic
60C22×C4×C97C2311yescharacteristic
61C4⋊C4×C97—21no—
62C4⋊C4×C97C2312yes—
63C22⋊C4×C97—22no—
64C22⋊C4×C97—22no—
65C23×C97—22no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97—41no—
69C23×C97—42no—
70C23×C97—42no—
71C23×C97—42no—
72C23×C97C2×D411yescharacteristic
73C23×C97C2×D411yescharacteristic
74C23×C97C2×D411yescharacteristic
75D4×C97—24no—
76D4×C97—44no—
77D4×C97—44no—
78D4×C97—42no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—42no—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97#4114yes—
90C2×C4×C97C2411yescharacteristic
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97—42no—
94C2×Dic97—42no—
95#11715895—24no—
96C22×C97—21no—
97C22×C97—21no—
98C22×C97—21no—
99C22×C97—21no—
100C22×C97—21no—
101C22×C97—21no—
102C22×C97—21no—
103C22×C97—44no—
104C22×C97—42no—
105C22×C97—42no—
106C22×C97—42no—
107C22×C97—84no—
108C22×C97#12312yes—
109C22×C97AGL(1,ℤ/8)12yes—
110C22×C97C22×D411yescharacteristic
111C4×C97—21no—
112C4×C97—21no—
113C4×C97—42no—
114C4×C97#12312yes—
115C4×C97C2×#4112yes—
116Dic97—42no—
117Dic97—42no—
118C2×C97—22no—
119C2×C97—22no—
120C2×C97—41no—
121C2×C97—82no—
122C2×C97C2×#12311yescharacteristic
123C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C97#57511yescharacteristic
125#3094—971nomaximal, Sylow
126C97#309411yescharacteristic, complemented, Sylow
127C2×#121D9711yescharacteristic
128C2×#117—971no—
129#482—974no—
130#481—972no—
131#480—972no—
132#428—972no—
133#418—972no—
134#407—971no—
135C22×D4—21no—
136C22×D4C2×D9711yescharacteristic
137C23×C4C2×D9711yescharacteristic
138C2×M16—972no—
139#121—21no—
140#121—21no—
141#121C2×D9712yes—
142#121C2×D9712yes—
143#117—972no—
144#117—972no—
145#117—1941no—
146#117—1941no—
147C2×C4⋊C4—971no—
148C2×C4⋊C4—971no—
149C2×C4⋊C4C2×D9711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—1941no—
152C2×C42—1941no—
153#107—974no—
154#106—974no—
155#102—974no—
156#102—974no—
157C22⋊C8—972no—
158C22⋊C8—972no—
159C24—21no—
160C24C22⋊D9711yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—24no—
165C2×D4—44no—
166C2×D4C22×D9712yes—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—1941no—
170C22×C4—1941no—
171C22×C4C22⋊D9711yescharacteristic
172C22×C4C22⋊D9711yescharacteristic
173C22×C4C22⋊D9711yescharacteristic
174C22×C4C22×D9711yescharacteristic
175C22×C4C22×D9711yescharacteristic
176C22×C4C22×D9711yescharacteristic
177M16—1944no—
178C2×C8—974no—
179C4⋊C4—21no—
180C4⋊C4—972no—
181C4⋊C4—972no—
182C4⋊C4—1941no—
183C4⋊C4—1941no—
184C4⋊C4C22×D9712yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—1942no—
188C22⋊C4—1942no—
189C42—1942no—
190C42—1942no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C2×C22⋊D9711yescharacteristic
199C23C2×C22⋊D9711yescharacteristic
200C23C2×C22⋊D9711yescharacteristic
201D4—24no—
202D4—44no—
203D4—44no—
204D4—42no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—3882no—
213C2×C4—3882no—
214C2×C4—1942no—
215C2×C4—1942no—
216C2×C4C2×C22⋊D9711yescharacteristic
217C2×C4C2×C22⋊D9711yescharacteristic
218C2×C4C2×C22⋊D9711yescharacteristic
219C2×C4#1309551014yes—
220C2×C4C23×D9711yescharacteristic
221C8—1944no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—84no—
234C22#1569079112yes—
235C22#1569083612yes—
236C22C22×C22⋊D9711yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—3882no—
241C4—3882no—
242C4#1569079112yes—
243C4C2×#1309551012yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—82no—
248C2C2×#1569079111yescharacteristic
249C2C2×#1569083611yescharacteristic
250C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radicalC2×#121×C97C2
socleC22×C97C22×D4
hypercenterC2×#121D97
hypocenterC97#3094
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#15690769C2semidirect
2. #20261056C2semidirect
3. #20261051C2semidirect
4. #20261046C2semidirect
5. #20260775C2semidirect
6. #20260740C2semidirect
7. #20260699C2semidirect
13. #15690769C22semidirect
14. C2×C4⋊Dic97C22semidirect
15. C2×C4⋊Dic97C22semidirect
18. #15690743C22semidirect
19. #15690741C22semidirect
20. #15690731C22semidirect
21. #15690731C22semidirect
22. C22⋊#11715895C22semidirect
23. C22⋊#11715895C22semidirect
126. C97#3094semidirect

Maximal subgroups

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

subgroupindex
C2×#156907692
#202610562
#202610512
#202610462
#202607752
#202607402
#202606992
C2×#121×C972
#309497