Finite Group Databasebeta

← #27144305

Subgroups — #27144305

Order 12416.

588 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 237 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#20260632C214yesmaximal
6#20260604C211yesmaximal, characteristic
7#20260586C214yesmaximal
8C42×Dic97C211yesmaximal, 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×#13095490C2214yes—
19C2×C4×Dic97C2212yes—
20C2×C4×Dic97C2212yes—
21C2×C4×Dic97C2212yes—
22C2×C4×Dic97C2211yescharacteristic
23C2×C4×Dic97C2211yescharacteristic
24C2×C4×Dic97C2212yes—
25#15690717C2214yes—
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×Dic97C2311yescharacteristic
36C22×Dic97C2312yes—
37C22×Dic97C2311yescharacteristic
38C4⋊Dic97—22no—
39C4⋊Dic97—24no—
40#13095490—24no—
41#13095490—28no—
42#13095490C2×C4116yes—
43C4×Dic97—24no—
44C4×Dic97—24no—
45C4×Dic97—24no—
46C4×Dic97C2×C414yes—
47C4×Dic97D414yes—
48C4×Dic97Q818yescomplemented
49C2×Q8×C97—22no—
50C2×Q8×C97—22no—
51C2×Q8×C97C2×C414yes—
52C22×C4×C97C2312yes—
53C22×C4×C97C2312yes—
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56C4⋊C4×C97—22no—
57C42×C97—22no—
58C23×C97C2411yescharacteristic
59Q8×C97—28no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—24no—
66C2×C4×C97C22×C414yes—
67C2×C4×C97C22×C412yes—
68C2×C4×C97C2×D412yes—
69C2×C4×C97C2×Q814yes—
70C2×C4×C97#4112yes—
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—
80C2×Dic97C2×Q814yes—
81C2×Dic97C2×Q814yes—
82C2×Dic97#4114yes—
83Dic194—28no—
84C22×C97C23×C411yescharacteristic
85C22×C97C22×D411yescharacteristic
86C22×C97C22×Q812yes—
87C22×C97C2×#4112yes—
88C22×C97C2×#4111yescharacteristic
89C4×C97—24no—
90C4×C97—24no—
91C4×C97—42no—
92C4×C97C4×D414yes—
93Dic97—24no—
94Dic97—22no—
95Dic97—42no—
96Dic97C4×Q818yescomplemented
97Dic97C4⋊Q814yescomplemented
98C2×C97C2×C4×D411yescharacteristic
99C2×C97C2×C4×Q812yes—
100C2×C97C2×C4⋊Q811yescharacteristic
101C2×C97#53211yescharacteristic
102C2×C97#53312yes—
103C4×C4⋊Q8—971nomaximal, Sylow
104C97C4×C4⋊Q811yescharacteristic, complemented, Sylow
105C2×C4⋊Q8—971no—
106C2×C4×Q8—971no—
107C2×C4×Q8—971no—
108#391—971no—
109#391D9711yescharacteristic
110#389—974no—
111#384—971no—
112C4×C4⋊C4—974no—
113C43—971no—
114C22×Q8—971no—
115C22×Q8C2×D9711yescharacteristic
116C4⋊Q8—978no—
117C4×Q8—1944no—
118C4×Q8—1944no—
119C2×C4⋊C4—974no—
120C2×C4⋊C4—972no—
121C2×C4⋊C4—971no—
122C2×C4⋊C4—972no—
123C2×C4⋊C4—974no—
124C2×C4⋊C4C2×D9711yescharacteristic
125C2×C42—971no—
126C2×C42—971no—
127C2×C42—972no—
128C2×C42—972no—
129C2×C42—972no—
130C2×C42—972no—
131C2×C42C2×D9711yescharacteristic
132#95—974no—
133#95C2×D9714yes—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—974no—
137C2×Q8—1942no—
138C2×Q8—1942no—
139C2×Q8C4×D9714yes—
140C22×C4—971no—
141C22×C4—972no—
142C22×C4—972no—
143C22×C4—971no—
144C22×C4—972no—
145C22×C4C22×D9712yes—
146C22×C4C22×D9712yes—
147C22×C4C22×D9712yes—
148C22×C4C22×D9711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—9716no—
151C4⋊C4—1948no—
152C4⋊C4—1944no—
153C4⋊C4—1944no—
154C4⋊C4—1942no—
155C42—22no—
156C42—974no—
157C42—974no—
158C42—978no—
159C42—1944no—
160C42—1944no—
161C42—1944no—
162C23C23×D9711yescharacteristic
163Q8—28no—
164Q8—1948no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—974no—
172C2×C4—972no—
173C2×C4—974no—
174C2×C4—974no—
175C2×C4—974no—
176C2×C4—974no—
177C2×C4—972no—
178C2×C4—1942no—
179C2×C4—1944no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4—1942no—
183C2×C4C2×C4×D9714yes—
184C2×C4C2×C4×D9712yes—
185C2×C4D4×D9712yes—
186C2×C4Q8×D9714yes—
187C2×C4#1309551012yes—
188C22C22×C4×D9711yescharacteristic
189C22C2×D4×D9711yescharacteristic
190C22C2×Q8×D9712yes—
191C22#41×D9712yes—
192C22C2×#1309551011yescharacteristic
193C4—24no—
194C4—24no—
195C4—42no—
196C4—978no—
197C4—974no—
198C4—3882no—
199C4—1944no—
200C4—1942no—
201C4#1569077014yes—
202C2C2×#1569077011yescharacteristic
203C2C4×Q8×D9712yes—
204C2C4⋊Q8×D9711yescharacteristic
205C2#2026128411yescharacteristic
206C2#2026128712yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#391×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenter#391D97
hypocenterC97C4×C4⋊Q8
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #15690808C4semidirect
48. C4×Dic97Q8semidirect
96. Dic97C4×Q8semidirect
97. Dic97C4⋊Q8semidirect
104. C97C4×C4⋊Q8semidirect

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
#202606322
#202606042
#202605862
C42×Dic972
#391×C972
C4×C4⋊Q897