Finite Group Databasebeta

← #27142145

Subgroups — #27142145

Order 12416.

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

normal subgroups only

This group has 578 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, complemented
2C2×#15690725C211yesmaximal, characteristic, complemented
3#20260591C212yesmaximal, complemented
4#20260501C212yesmaximal, complemented
5C2×#120×C97C211yesmaximal, characteristic
6D4×Dic97—24no—
7#15690769—21no—
8C2×C22⋊Dic97—24no—
9C2×C22⋊Dic97—21no—
10C2×C22⋊Dic97C2212yes—
11C2×C22⋊Dic97C2212yes—
12#15690725—22no—
13C25×C97C2211yescharacteristic
14C22×D4×C97—21no—
15C22×D4×C97C2211yescharacteristic
16#120×C97—22no—
17#120×C97C414yescomplemented
18C2×C22⋊C4×C97—21no—
19C2×C22⋊C4×C97C2211yescharacteristic
20C22×Dic97—22no—
21C22×Dic97—22no—
22C22×Dic97D414yescomplemented
23C4⋊Dic97—22no—
24C22⋊Dic97—22no—
25C22⋊Dic97—22no—
26C22⋊Dic97—22no—
27C22⋊Dic97—22no—
28C22⋊Dic97—22no—
29C22⋊Dic97—48no—
30C4×Dic97—22no—
31C24×C97—22no—
32C24×C97—21no—
33C24×C97—21no—
34C24×C97—22no—
35C24×C97—44no—
36C24×C97C2×C412yescomplemented
37C24×C97D411yescharacteristic
38C24×C97C2311yescharacteristic
39C2×D4×C97—22no—
40C2×D4×C97—22no—
41C2×D4×C97—24no—
42C2×D4×C97—44no—
43C2×D4×C97C2×C412yes—
44C2×D4×C97D412yes—
45C22×C4×C97—21no—
46C22×C4×C97D411yescharacteristic
47C22⋊C4×C97—21no—
48C22⋊C4×C97—24no—
49C22⋊C4×C97C2×C412yescomplemented
50C23×C97—24no—
51C23×C97—28no—
52C23×C97—22no—
53C23×C97—24no—
54C23×C97—21no—
55C23×C97—21no—
56C23×C97—44no—
57C23×C97—42no—
58C23×C97—44no—
59C23×C97—42no—
60C23×C97—44no—
61C23×C97—48no—
62C23×C97—42no—
63C23×C97—41no—
64C23×C97—42no—
65C23×C97—41no—
66C23×C97C22⋊C412yescomplemented
67C23×C97C22×C411yescharacteristic
68C23×C97C2×D412yes—
69C23×C97C2×D411yescharacteristic
70C23×C97C2×D411yescharacteristic
71C23×C97#4112yes—
72D4×C97—24no—
73D4×C97—24no—
74D4×C97—48no—
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—42no—
79C2×C4×C97C22⋊C412yescomplemented
80C2×Dic97—24no—
81C2×Dic97—44no—
82C2×Dic97—44no—
83C2×Dic97—42no—
84C2×Dic97—82no—
85C22×C97—24no—
86C22×C97—22no—
87C22×C97—24no—
88C22×C97—28no—
89C22×C97—22no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97—42no—
93C22×C97—42no—
94C22×C97—48no—
95C22×C97—42no—
96C22×C97—44no—
97C22×C97—44no—
98C22×C97—42no—
99C22×C97—42no—
100C22×C97—44no—
101C22×C97—81no—
102C22×C97C2×C22⋊C411yescharacteristic
103C22×C97C4×D412yes—
104C22×C97#12011yescharacteristic
105C22×C97#12112yes—
106C22×C97#12311yescharacteristic
107C4×C97—22no—
108C4×C97—42no—
109Dic97—44no—
110Dic97—82no—
111C2×C97—22no—
112C2×C97—21no—
113C2×C97—22no—
114C2×C97—21no—
115C2×C97—44no—
116C2×C97—42no—
117C2×C97#38611yescharacteristic
118C2×C97B(4,𝔽2)12yes—
119#1889—971nomaximal, Sylow
120C97#188911yescharacteristic, complemented, Sylow
121C2×#120D9711yescharacteristic
122#518—971no—
123C2×#99—971no—
124#379—972no—
125#342—972no—
126C25C2×D9711yescharacteristic
127C22×D4—21no—
128C22×D4C2×D9711yescharacteristic
129#120—22no—
130#120Dic9714yescomplemented
131C4×D4—1944no—
132#117—1941no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—972no—
135C2×C22⋊C4—972no—
136C2×C22⋊C4—1944no—
137C2×C22⋊C4—1941no—
138C2×C22⋊C4C2×D9711yescharacteristic
139#99—1942no—
140C24—22no—
141C24—21no—
142C24—21no—
143C24—22no—
144C24—44no—
145C24C2×Dic9712yescomplemented
146C24C22⋊D9711yescharacteristic
147C24C22×D9711yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—24no—
151C2×D4—44no—
152C2×D4C2×Dic9712yes—
153C2×D4C22⋊D9712yes—
154C22×C4—21no—
155C22×C4—974no—
156C22×C4—1942no—
157C22×C4—1942no—
158C22×C4C22⋊D9711yescharacteristic
159C4⋊C4—1942no—
160C22⋊C4—21no—
161C22⋊C4—24no—
162C22⋊C4—3888no—
163C22⋊C4—1942no—
164C22⋊C4—1942no—
165C22⋊C4—1942no—
166C22⋊C4—1942no—
167C22⋊C4—1942no—
168C22⋊C4C2×Dic9712yescomplemented
169C42—1942no—
170C23—28no—
171C23—24no—
172C23—22no—
173C23—24no—
174C23—21no—
175C23—21no—
176C23—44no—
177C23—42no—
178C23—42no—
179C23—44no—
180C23—44no—
181C23—48no—
182C23—42no—
183C23—41no—
184C23—42no—
185C23—41no—
186C23C22⋊Dic9712yescomplemented
187C23C22×Dic9711yescharacteristic
188C23D4×D9712yes—
189C23C2×C22⋊D9711yescharacteristic
190C23C2×C22⋊D9711yescharacteristic
191C23#1309551012yes—
192D4—24no—
193D4—24no—
194D4—48no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—7762no—
200C2×C4—3884no—
201C2×C4—3882no—
202C2×C4—3884no—
203C2×C4—1944no—
204C2×C4C22⋊Dic9712yescomplemented
205C22—22no—
206C22—24no—
207C22—24no—
208C22—28no—
209C22—22no—
210C22—41no—
211C22—41no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—48no—
216C22—44no—
217C22—44no—
218C22—42no—
219C22—42no—
220C22—44no—
221C22—81no—
222C22C2×C22⋊Dic9711yescharacteristic
223C22D4×Dic9712yes—
224C22#1569077811yescharacteristic
225C22#1569078212yes—
226C22#1569079111yescharacteristic
227C4—22no—
228C4—42no—
229C4—7762no—
230C4—3884no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2—21no—
235C2—44no—
236C2—42no—
237C2#2026061511yescharacteristic
238C2#2026092112yes—
subgroupG/H
centerC22#15690778
commutatorC23×C97C22×C4
FrattiniC24C22×D97
Fitting, prime radicalC2×#120×C97C2
socleC22×C97#120
cosocleC24×C97C23
hypercenterC2×#120D97
hypocenterC97#1889
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #20261210C2semidirect
2. C2×#15690725C2semidirect
3. #20260591C2semidirect
4. #20260501C2semidirect
17. #120×C97C4semidirect
22. C22×Dic97D4semidirect
36. C24×C97C2×C4semidirect
49. C22⋊C4×C97C2×C4semidirect
66. C23×C97C22⋊C4semidirect
79. C2×C4×C97C22⋊C4semidirect
120. C97#1889semidirect
130. #120Dic97semidirect
145. C24C2×Dic97semidirect
168. C22⋊C4C2×Dic97semidirect
186. C23C22⋊Dic97semidirect
204. C2×C4C22⋊Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612102
C2×#156907252
#202605912
#202605012
C2×#120×C972
#188997