Finite Group Databasebeta

← #27142181

Subgroups — #27142181

Order 12416.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#20261242C211yesmaximal, characteristic
2C2×#15690767C211yesmaximal, characteristic, complemented
3C2×#15690737C211yesmaximal, characteristic, complemented
4#20260698C211yesmaximal, characteristic, complemented
5#20260496C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7C2×#104×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690791—21no—
11#15690786C412yescomplemented
12#15690781C412yescomplemented
13#15690778—21no—
14C4×C22⋊D97—22no—
15#15690769C2211yescharacteristic
16#15690767—22no—
17#15690767—22no—
18C2×#13095490—21no—
19C2×#13095488—21no—
20C2×C4×Dic97C2211yescharacteristic
21#15690737—22no—
22#15690722—22no—
23C2×#41×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25C2×C42×C97C2211yescharacteristic
26#104×C97—22no—
27C23×D97D411yescharacteristic
28C2×C22⋊D97—22no—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97C2×C412yes—
36C2×C4×D97D412yes—
37#13095495—21no—
38#13095495—21no—
39C2×#11715895—21no—
40C4⋊Dic97C2×C412yes—
41#13095490—24no—
42#13095490—22no—
43C22⋊Dic97—21no—
44C22⋊Dic97—22no—
45#13095488—24no—
46#13095488—22no—
47C4×Dic97—21no—
48C4×Dic97—21no—
49C4×Dic97D412yescomplemented
50#41×C97—21no—
51#41×C97—21no—
52#41×C97—21no—
53#41×C97—21no—
54C2×Q8×C97C2×C411yescharacteristic
55C2×D4×C97—21no—
56C2×D4×C97C2×C411yescharacteristic
57C22×C4×C97—21no—
58C22×C4×C97—21no—
59C22×C4×C97C2311yescharacteristic
60M16×C97—21no—
61M16×C97—21no—
62C2×C8×C97—21no—
63C42×C97—21no—
64C42×C97D412yescomplemented
65C23×C97—21no—
66C23×C97C2×D411yescharacteristic
67C22×D97—21no—
68C22×D97—21no—
69C22×D97—42no—
70C22×D97C22⋊C412yes—
71Q8×C97—21no—
72Q8×C97—21no—
73D4×C97—21no—
74D4×C97—21no—
75D4×C97—41no—
76D4×C97—41no—
77C2×C4×C97—21no—
78C2×C4×C97—22no—
79C2×C4×C97—42no—
80C2×C4×C97—41no—
81C2×C4×C97—41no—
82C2×C4×C97C22×C411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D411yescharacteristic
86C2×C4×C97#4111yescharacteristic
87C2×C4×C97#4111yescharacteristic
88C22⋊D97—44no—
89C2×Dic97—21no—
90C2×Dic97—21no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97—42no—
94C2×Dic97C22⋊C412yes—
95C4×D97—22no—
96C4×D97—22no—
97C4×D97—22no—
98C4×D97—22no—
99C8×C97—41no—
100#11715895—41no—
101C22×C97—22no—
102C22×C97—41no—
103C22×C97—42no—
104C22×C97C2×C22⋊C411yescharacteristic
105C22×C97#12011yescharacteristic
106C22×C97#12311yescharacteristic
107C2×D97—22no—
108C2×D97—22no—
109C2×D97—22no—
110C2×D97—42no—
111C2×D97—42no—
112C4×C97—42no—
113C4×C97—41no—
114C4×C97C4×D411yescharacteristic
115C4×C97C4×D411yescharacteristic
116C4×C97#12111yescharacteristic
117C4×C97#12111yescharacteristic
118Dic97—22no—
119Dic97—42no—
120Dic97—42no—
121C2×C97—21no—
122C2×C97—21no—
123C2×C97—81no—
124C2×C97#38611yescharacteristic
125C2×C97#42112yes—
126D97—42no—
127#1897—971nomaximal, Sylow
128C97#189711yescharacteristic, complemented, Sylow
129#525—971no—
130C2×#117—971no—
131C2×#104—971no—
132C2×#104D9711yescharacteristic, complemented
133#407—971no—
134#339—971no—
135#339—971no—
136C2×#41C2×D9711yescharacteristic
137C23×C4—971no—
138C2×M16—971no—
139C2×M16C2×D9711yescharacteristic
140#123—1941no—
141C22⋊Q8—972no—
142#121—972no—
143#120—1941no—
144C4×D4—1942no—
145#117—971no—
146#117—1942no—
147#117—1942no—
148C2×C4⋊C4—1941no—
149C2×C22⋊C4—1941no—
150C2×C42—971no—
151C2×C42C2×D9711yescharacteristic
152#104—22no—
153#104—1942no—
154C22⋊C8—1942no—
155C24—971no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160C2×Q8C4×D9711yescharacteristic
161C2×D4—21no—
162C2×D4—1942no—
163C2×D4C4×D9711yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—971no—
167C22×C4—972no—
168C22×C4—972no—
169C22×C4—1941no—
170C22×C4—1941no—
171C22×C4—1941no—
172C22×C4—1941no—
173C22×C4—1941no—
174C22×C4C22×D9711yescharacteristic
175M16—21no—
176M16—21no—
177M16—1941no—
178M16—1941no—
179C2×C8—21no—
180C2×C8—1941no—
181C4⋊C4—972no—
182C4⋊C4—1942no—
183C4⋊C4—1944no—
184C22⋊C4—1942no—
185C22⋊C4—1944no—
186C22⋊C4—1942no—
187C22⋊C4—1941no—
188C42—21no—
189C42—972no—
190C42—1941no—
191C42—1941no—
192C42C22⋊D9712yescomplemented
193C23—21no—
194C23—972no—
195C23—3882no—
196C23—1941no—
197C23—1941no—
198C23D4×D9711yescharacteristic
199Q8—21no—
200Q8—21no—
201D4—21no—
202D4—21no—
203D4—41no—
204D4—41no—
205D4—3884no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—972no—
212C2×C4—3882no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4—1941no—
216C2×C4—1941no—
217C2×C4—1942no—
218C2×C4—1942no—
219C2×C4—1942no—
220C2×C4—1942no—
221C2×C4C2×C4×D9711yescharacteristic
222C2×C4D4×D9711yescharacteristic
223C2×C4D4×D9711yescharacteristic
224C2×C4C2×C22⋊D9711yescharacteristic
225C2×C4#1309551011yescharacteristic
226C2×C4#1309551111yescharacteristic
227C8—41no—
228C8—3881no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—3882no—
233C22—3882no—
234C22—1942no—
235C22—1942no—
236C22—1942no—
237C22C22⋊C4×D9711yescharacteristic
238C22#1569077811yescharacteristic
239C22#1569079211yescharacteristic
240C4—42no—
241C4—41no—
242C4—3882no—
243C4—3882no—
244C4—1942no—
245C4#1569077011yescharacteristic
246C4C4×C22⋊D9711yescharacteristic
247C4#1569078111yescharacteristic
248C4#1569078311yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—3882no—
253C2#2026061111yescharacteristic
254C2#2026075012yes—
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#104×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261242C2
cosocleC22×C4×C97C23
hypercenterC2×#104D97
hypocenterC97#1897
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#15690767C2semidirect
3. C2×#15690737C2semidirect
4. #20260698C2semidirect
5. #20260496C2semidirect
6. #20260495C2semidirect
7. C2×#104×C97C2semidirect
11. #15690786C4semidirect
12. #15690781C4semidirect
49. C4×Dic97D4semidirect
64. C42×C97D4semidirect
128. C97#1897semidirect
132. C2×#104D97semidirect
192. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612422
C2×#156907672
C2×#156907372
#202606982
#202604962
#202604952
C2×#104×C972
#189797