Finite Group Databasebeta

← #27150062

Subgroups — #27150062

Order 12416.

688 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 177 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×#15690791C212yesmaximal, complemented
2C22×C4×Dic97C211yesmaximal, characteristic, complemented
3#20260638C212yesmaximal, complemented
4#20260631C212yesmaximal, complemented
5#20260623C214yesmaximal, complemented
6#20260615C212yesmaximal, complemented
7#20260595C211yesmaximal, characteristic, complemented
8C2×#121×C97C211yesmaximal, characteristic
9C23×Dic97C2211yescharacteristic
10C23×Dic97C2211yescharacteristic
11#15690791—28no—
12C2×C4⋊Dic97C2212yes—
13C2×#13095490C2214yes—
14C2×C22⋊Dic97C2214yes—
15C2×C22⋊Dic97C2212yes—
16C2×C22⋊Dic97C2214yes—
17C2×C4×Dic97—24no—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2212yes—
20#15690717C2214yes—
21#15690717C2212yes—
22C22×D4×C97C2212yes—
23C22×D4×C97C2211yescharacteristic
24C23×C4×C97C2211yescharacteristic
25#121×C97—24no—
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C22⋊C4×C97C2212yes—
28C22×Dic97—22no—
29C22×Dic97—22no—
30C22×Dic97—22no—
31C22×Dic97—24no—
32C22×Dic97D414yescomplemented
33C22×Dic97C2314yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C4⋊Dic97—24no—
37#13095490—28no—
38C22⋊Dic97—24no—
39C22⋊Dic97—28no—
40C22⋊Dic97—28no—
41C4×Dic97—24no—
42C4×Dic97—24no—
43C24×C97C2312yes—
44C24×C97C2311yescharacteristic
45C2×D4×C97—24no—
46C2×D4×C97—24no—
47C2×D4×C97—24no—
48C2×D4×C97—22no—
49C2×D4×C97—24no—
50C22×C4×C97—24no—
51C22×C4×C97D414yes—
52C22×C4×C97C2312yes—
53C22×C4×C97C2311yescharacteristic
54C22×C4×C97C2311yescharacteristic
55C4⋊C4×C97—22no—
56C22⋊C4×C97—24no—
57C23×C97—22no—
58C23×C97—22no—
59C23×C97—22no—
60C23×C97—22no—
61C23×C97—22no—
62C23×C97—24no—
63C23×C97—24no—
64C23×C97C2×D412yes—
65C23×C97C2×D412yes—
66C23×C97#4112yes—
67C23×C97C2411yescharacteristic
68D4×C97—48no—
69D4×C97—44no—
70C2×C4×C97—22no—
71C2×C4×C97—28no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97C2×D412yes—
76C2×C4×C97C2×D412yes—
77C2×C4×C97#4114yes—
78C2×Dic97—28no—
79C2×Dic97—28no—
80C2×Dic97—22no—
81C2×Dic97—22no—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97—24no—
85C2×Dic97—24no—
86C2×Dic97—24no—
87C2×Dic97C2×D414yes—
88C2×Dic97#4114yes—
89C22×C97—22no—
90C22×C97—22no—
91C22×C97—24no—
92C22×C97—24no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97—42no—
96C22×C97—44no—
97C22×C97—44no—
98C22×C97#12114yes—
99C22×C97C22×D411yescharacteristic
100C22×C97C22×D411yescharacteristic
101C22×C97C2×#4112yes—
102C22×C97C2×#4111yescharacteristic
103C22×C97C2×#4112yes—
104C4×C97—22no—
105C4×C97—42no—
106C4×C97#12314yescomplemented
107Dic97—24no—
108Dic97—24no—
109Dic97—44no—
110C2×C97—24no—
111C2×C97—82no—
112C2×C97C2×#12111yescharacteristic
113C2×C97C2×#12311yescharacteristic
114C2×C97#52511yescharacteristic
115C2×C97#52912yes—
116C2×C97#53212yes—
117#2854—971nomaximal, Sylow
118C97#285411yescharacteristic, complemented, Sylow
119C2×#123—972no—
120C2×#121D9711yescharacteristic
121C22×C42—971no—
122#390—972no—
123#389—972no—
124#388—974no—
125#386—972no—
126#380—971no—
127C22×D4C2×D9712yes—
128C22×D4C2×D9711yescharacteristic
129C23×C4—971no—
130C23×C4—971no—
131C23×C4C2×D9711yescharacteristic
132#123—1948no—
133#121—24no—
134C2×C4⋊C4—974no—
135C2×C4⋊C4—972no—
136C2×C4⋊C4C2×D9711yescharacteristic
137C2×C22⋊C4—972no—
138C2×C22⋊C4—974no—
139C2×C22⋊C4—974no—
140C2×C22⋊C4C2×D9712yes—
141C2×C42—972no—
142C2×C42—972no—
143C2×C42—1944no—
144#95—972no—
145#95—974no—
146C24C22×D9712yes—
147C24C22×D9711yescharacteristic
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4—24no—
153C22×C4—24no—
154C22×C4—974no—
155C22×C4—972no—
156C22×C4—972no—
157C22×C4—974no—
158C22×C4—1944no—
159C22×C4—1942no—
160C22×C4—1942no—
161C22×C4—1942no—
162C22×C4C22⋊D9714yes—
163C22×C4C22×D9712yes—
164C22×C4C22×D9711yescharacteristic
165C22×C4C22×D9711yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—1948no—
168C4⋊C4—1944no—
169C22⋊C4—24no—
170C22⋊C4—1948no—
171C22⋊C4—1948no—
172C22⋊C4—1944no—
173C42—1944no—
174C42—1944no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—22no—
181C23—24no—
182C23D4×D9712yes—
183C23C2×C22⋊D9712yes—
184C23#1309551012yes—
185C23C23×D9711yescharacteristic
186D4—48no—
187D4—44no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—974no—
194C2×C4—974no—
195C2×C4—1944no—
196C2×C4—1944no—
197C2×C4—1944no—
198C2×C4—1948no—
199C2×C4—1948no—
200C2×C4—1942no—
201C2×C4—1942no—
202C2×C4—1942no—
203C2×C4—1942no—
204C2×C4C2×C22⋊D9712yes—
205C2×C4C2×C22⋊D9712yes—
206C2×C4#1309551014yes—
207C22—22no—
208C22—22no—
209C22—24no—
210C22—24no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—44no—
215C22—44no—
216C22#1569078114yes—
217C22C2×D4×D9711yescharacteristic
218C22C22×C22⋊D9711yescharacteristic
219C22#41×D9712yes—
220C22C2×#1309551011yescharacteristic
221C22C2×#1309551012yes—
222C4—22no—
223C4—42no—
224C4—3884no—
225C4—1944no—
226C4—1944no—
227C4#1569079114yescomplemented
228C2—24no—
229C2—82no—
230C2C2×#1569078111yescharacteristic
231C2C2×#1569079111yescharacteristic
232C2#2026124211yescharacteristic
233C2#2026126612yes—
234C2#2026128012yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting, prime radicalC2×#121×C97C2
hypercenterC2×#121D97
hypocenterC97#2854
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#15690791C2semidirect
2. C22×C4×Dic97C2semidirect
3. #20260638C2semidirect
4. #20260631C2semidirect
5. #20260623C2semidirect
6. #20260615C2semidirect
7. #20260595C2semidirect
32. C22×Dic97D4semidirect
106. C4×C97#123semidirect
118. C97#2854semidirect
227. C4#15690791semidirect

Maximal subgroups

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

subgroupindex
C2×#156907912
C22×C4×Dic972
#202606382
#202606312
#202606232
#202606152
#202605952
C2×#121×C972
#285497