Finite Group Databasebeta

← #27144454

Subgroups — #27144454

Order 12416.

516 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 197 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×#15690808C211yesmaximal, characteristic
2C2×Q8×Dic97C211yesmaximal, characteristic
3C2×#15690775C211yesmaximal, characteristic
4#20260641C211yesmaximal, characteristic
5#20260622C214yesmaximal
6#20260617C214yesmaximal
7#20260600C211yesmaximal, characteristic
8#20260581C211yesmaximal, characteristic
9#391×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11#15690808C418yescomplemented
12Q8×Dic97—24no—
13#15690775—24no—
14C2×C4⋊Dic97C2212yes—
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490C2212yes—
17C2×#13095490C2214yes—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2212yes—
20C2×C4×Dic97C2211yescharacteristic
21C2×C4×Dic97C2211yescharacteristic
22#15690717C2212yes—
23#15690717C2214yes—
24#15690717C2214yes—
25#15690717C2212yes—
26C22×Q8×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C42×C97C2211yescharacteristic
29#95×C97C2214yes—
30C2×Dic194—22no—
31C2×Dic194—22no—
32C2×Dic194C2×C414yes—
33C22×Dic97C2312yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2311yescharacteristic
37C22×Dic97C2311yescharacteristic
38C4⋊Dic97—22no—
39C4⋊Dic97—24no—
40#13095490—24no—
41#13095490C2×C4116yes—
42C4×Dic97—24no—
43C4×Dic97—24no—
44C4×Dic97C2×C414yes—
45C4×Dic97D414yes—
46C2×Q8×C97—22no—
47C2×Q8×C97—22no—
48C2×Q8×C97C2×C414yes—
49C22×C4×C97C2312yes—
50C22×C4×C97C2312yes—
51C22×C4×C97C2312yes—
52C22×C4×C97C2311yescharacteristic
53C4⋊C4×C97—22no—
54C42×C97—22no—
55C23×C97C2411yescharacteristic
56Q8×C97—28no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—24no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97C22×C414yes—
65C2×C4×C97C22×C412yes—
66C2×C4×C97C2×D412yes—
67C2×C4×C97#4112yes—
68C2×Dic97—22no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—24no—
76C2×Dic97C22×C414yes—
77C2×Dic97C22×C414yes—
78C2×Dic97C2×D412yes—
79C2×Dic97C2×D412yes—
80Dic194—28no—
81C22×C97C23×C411yescharacteristic
82C22×C97C22×D411yescharacteristic
83C22×C97C2×#4111yescharacteristic
84C22×C972−1+412yes—
85C22×C972−1+412yes—
86C4×C97—24no—
87C4×C97—42no—
88C4×C97—42no—
89C4×C97C4×D414yes—
90Dic97—24no—
91Dic97—24no—
92Dic97—22no—
93Dic97—22no—
94Dic97—42no—
95C2×C97C2×C4×D411yescharacteristic
96C2×C97#51912yes—
97C2×C97#53611yescharacteristic
98C2×C97#53611yescharacteristic
99C2×C97#54112yes—
100#2327—971nomaximal, Sylow
101C97#232711yescharacteristic, complemented, Sylow
102C2×C4⋊Q8—971no—
103C2×C4×Q8—971no—
104C2×C4×Q8—971no—
105#391—971no—
106#391D9711yescharacteristic
107#387—974no—
108#387—974no—
109#382—971no—
110#376—971no—
111C22×Q8—971no—
112C22×Q8C2×D9711yescharacteristic
113C4⋊Q8—978no—
114C4×Q8—1944no—
115C4×Q8—1944no—
116C2×C4⋊C4—974no—
117C2×C4⋊C4—972no—
118C2×C4⋊C4—971no—
119C2×C4⋊C4—972no—
120C2×C4⋊C4C2×D9711yescharacteristic
121C2×C42—971no—
122C2×C42—971no—
123C2×C42—972no—
124C2×C42—972no—
125C2×C42C2×D9711yescharacteristic
126#95—972no—
127#95—972no—
128#95—974no—
129#95—974no—
130#95C2×D9714yes—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—974no—
134C2×Q8—1942no—
135C2×Q8—1942no—
136C2×Q8C4×D9714yes—
137C22×C4—972no—
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—972no—
141C22×C4—972no—
142C22×C4C22×D9712yes—
143C22×C4C22×D9712yes—
144C22×C4C22×D9712yes—
145C22×C4C22×D9711yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—9716no—
148C4⋊C4—1944no—
149C4⋊C4—1944no—
150C4⋊C4—1942no—
151C42—22no—
152C42—974no—
153C42—974no—
154C42—1944no—
155C42—1944no—
156C23C23×D9711yescharacteristic
157Q8—28no—
158Q8—1948no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—974no—
167C2×C4—974no—
168C2×C4—972no—
169C2×C4—972no—
170C2×C4—1942no—
171C2×C4—1944no—
172C2×C4—1942no—
173C2×C4—1942no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4—1942no—
178C2×C4C2×C4×D9714yes—
179C2×C4C2×C4×D9712yes—
180C2×C4D4×D9712yes—
181C2×C4#1309551012yes—
182C22C22×C4×D9711yescharacteristic
183C22C2×D4×D9711yescharacteristic
184C22C2×#1309551011yescharacteristic
185C22#1569085712yes—
186C22#1569085812yes—
187C4—24no—
188C4—42no—
189C4—42no—
190C4—3882no—
191C4—1944no—
192C4—1944no—
193C4—1942no—
194C4—1942no—
195C4#1569077014yes—
196C2C2×#1569077011yescharacteristic
197C2#2026121112yes—
198C2#2026130511yescharacteristic
199C2#2026131011yescharacteristic
200C2#2026134512yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#391×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenter#391D97
hypocenterC97#2327
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #15690808C4semidirect
101. C97#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#156908082
C2×Q8×Dic972
C2×#156907752
#202606412
#202606222
#202606172
#202606002
#202605812
#391×C972
#232797