Finite Group Databasebeta

← #27142019

Subgroups — #27142019

Order 12416.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#20261210C211yesmaximal, characteristic
2#20260615C211yesmaximal, characteristic
3#20260610C212yesmaximal, complemented
4#20260501C212yesmaximal, complemented
5C2×#99×C97C211yesmaximal, characteristic
6C23×Dic97C2211yescharacteristic
7D4×Dic97—22no—
8D4×Dic97—22no—
9#15690769—21no—
10C2×C4⋊Dic97—22no—
11C2×C22⋊Dic97—21no—
12C2×C22⋊Dic97—21no—
13C2×C22⋊Dic97C2211yescharacteristic
14C2×C22⋊Dic97C2212yes—
15#15690717—22no—
16#15690717—21no—
17C22×D4×C97C2211yescharacteristic
18C2×C22⋊C4×C97C2212yes—
19#99×C97C414yescomplemented
20C22×Dic97—21no—
21C22×Dic97—22no—
22C22×Dic97—21no—
23C22×Dic97—21no—
24C22×Dic97—21no—
25C22×Dic97—42no—
26C22×Dic97D412yes—
27C22×Dic97D412yes—
28C4⋊Dic97—21no—
29C4⋊Dic97—21no—
30C4⋊Dic97—44no—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33C22⋊Dic97—22no—
34C22⋊Dic97—22no—
35C22⋊Dic97—42no—
36C4×Dic97—22no—
37C24×C97D411yescharacteristic
38C24×C97C2311yescharacteristic
39C2×D4×C97—22no—
40C2×D4×C97—22no—
41C2×D4×C97C2×C412yes—
42C2×D4×C97Q812yes—
43C22×C4×C97—22no—
44C22×C4×C97D411yescharacteristic
45C22⋊C4×C97—22no—
46C22⋊C4×C97C2×C414yescomplemented
47C23×C97—21no—
48C23×C97—21no—
49C23×C97—41no—
50C23×C97—41no—
51C23×C97—41no—
52C23×C97—41no—
53C23×C97C4⋊C412yescomplemented
54C23×C97C22×C411yescharacteristic
55C23×C97C2×D412yes—
56C23×C97C2×D411yescharacteristic
57C23×C97C2×Q811yescharacteristic
58C23×C97#4112yes—
59D4×C97—24no—
60D4×C97—24no—
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—24no—
64C2×C4×C97—42no—
65C2×C4×C97—42no—
66C2×C4×C97C4⋊C412yescomplemented
67C2×Dic97—22no—
68C2×Dic97—24no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—42no—
72C2×Dic97—41no—
73C2×Dic97—41no—
74C2×Dic97—42no—
75C2×Dic97—42no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79C2×Dic97—81no—
80C2×Dic97—81no—
81C22×C97—22no—
82C22×C97—21no—
83C22×C97—22no—
84C22×C97—21no—
85C22×C97—22no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97—81no—
93C22×C97C2×C4⋊C411yescharacteristic
94C22×C97C4×D412yes—
95C22×C97#12011yescharacteristic
96C22×C97C22⋊Q812yes—
97C22×C97#12311yescharacteristic
98C4×C97—22no—
99C4×C97—44no—
100Dic97—42no—
101Dic97—42no—
102Dic97—42no—
103Dic97—81no—
104C2×C97—21no—
105C2×C97—22no—
106C2×C97—21no—
107C2×C97—22no—
108C2×C97—42no—
109C2×C97#38511yescharacteristic
110C2×C97B(4,𝔽2)11yescharacteristic
111C2×C97#45811yescharacteristic
112#1867—971nomaximal, Sylow
113C97#186711yescharacteristic, complemented, Sylow
114#518—971no—
115C2×#99D9711yescharacteristic
116#386—971no—
117#385—972no—
118#342—972no—
119C22×D4C2×D9711yescharacteristic
120C23×C4—971no—
121C4×D4—1942no—
122C4×D4—1942no—
123#117—1941no—
124C2×C4⋊C4—1942no—
125C2×C22⋊C4—972no—
126C2×C22⋊C4—971no—
127C2×C22⋊C4—1941no—
128C2×C22⋊C4—1941no—
129C2×C22⋊C4C2×D9712yes—
130#99Dic9714yescomplemented
131#95—1942no—
132#95—1941no—
133C24D38811yescharacteristic
134C24C22×D9711yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4Dic19412yes—
138C2×D4C2×Dic9712yes—
139C22×C4—22no—
140C22×C4—972no—
141C22×C4—972no—
142C22×C4—3882no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4—1941no—
146C22×C4—1942no—
147C22×C4—1941no—
148C22×C4D38811yescharacteristic
149C4⋊C4—3884no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C22⋊C4—22no—
153C22⋊C4—3882no—
154C22⋊C4—1942no—
155C22⋊C4—1942no—
156C22⋊C4—1942no—
157C22⋊C4—1942no—
158C22⋊C4C2×Dic9714yescomplemented
159C42—1942no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23C4⋊Dic9712yescomplemented
167C23C22×Dic9711yescharacteristic
168C23D4×D9712yes—
169C23C2×D38811yescharacteristic
170C23C2×Dic19411yescharacteristic
171C23#1309551012yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—24no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—7761no—
180C2×C4—7761no—
181C2×C4—3881no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—3882no—
185C2×C4—3881no—
186C2×C4—3881no—
187C2×C4—3882no—
188C2×C4—3882no—
189C2×C4—1942no—
190C2×C4—1944no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4C4⋊Dic9712yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×C4⋊Dic9711yescharacteristic
207C22D4×Dic9712yes—
208C22#1569078011yescharacteristic
209C22#1569078712yes—
210C22#1569079411yescharacteristic
211C4—22no—
212C4—44no—
213C4—7761no—
214C4—3882no—
215C4—3882no—
216C4—3882no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2026061011yescharacteristic
223C2#2026092311yescharacteristic
224C2#2026092611yescharacteristic
subgroupG/H
centerC22#15690780
commutatorC23×C97C22×C4
FrattiniC24C22×D97
FittingC2×#99×C97C2
socleC22×C97#120
prime radicalC22×D4×C97C22
cosocleC24×C97C23
hypercenterC2×#99D97
hypocenterC97#1867
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #20260610C2semidirect
4. #20260501C2semidirect
19. #99×C97C4semidirect
46. C22⋊C4×C97C2×C4semidirect
53. C23×C97C4⋊C4semidirect
66. C2×C4×C97C4⋊C4semidirect
113. C97#1867semidirect
130. #99Dic97semidirect
158. C22⋊C4C2×Dic97semidirect
166. C23C4⋊Dic97semidirect
193. C2×C4C4⋊Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612102
#202606152
#202606102
#202605012
C2×#99×C972
#186797