Finite Group Databasebeta

← #27151867

Subgroups — #27151867

Order 12416.

380 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 129 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#20261292C211yesmaximal, characteristic
2C2×C4×Dic194C211yesmaximal, characteristic
3#20261066C212yesmaximal
4#20261021C212yesmaximal, complemented
5#20260946C212yesmaximal, complemented
6#20260893C212yesmaximal
7#20260843C214yesmaximal, complemented
8#432×C97C211yesmaximal, characteristic
9C22×Dic194C2211yescharacteristic
10C2×Dic388C2214yes—
11#15690810C2211yescharacteristic
12#15690810C2211yescharacteristic
13#15690802—21no—
14#15690789—21no—
15#15690789C2212yes—
16C4×Dic194—21no—
17C4×Dic194—21no—
18C4×Dic194—21no—
19C4×Dic194C2212yes—
20C4×Dic194C2212yes—
21C4×Dic194C2212yes—
22#15690769—21no—
23C2×C4⋊Dic97C2211yescharacteristic
24C2×#13095490—21no—
25C2×C4×Dic97—21no—
26#15690742C2214yes—
27#15690735C2214yes—
28#15690735C2214yes—
29C2×C42×C97C2211yescharacteristic
30C4⋊C8×C97C2212yes—
31C22⋊C8×C97C2212yes—
32#97×C97C2212yes—
33C2×Dic194—21no—
34C2×Dic194—24no—
35C2×Dic194—21no—
36C2×Dic194C2312yes—
37C2×Dic194C2312yes—
38C22×Dic97—21no—
39C22×Dic97—21no—
40Dic388—28no—
41C4⋊Dic97—21no—
42C4⋊Dic97—21no—
43C4⋊Dic97—21no—
44C4⋊Dic97—21no—
45C4⋊Dic97C2312yes—
46C4⋊Dic97C2312yes—
47#13095490—22no—
48#13095490—22no—
49#13095490—22no—
50#13095490—22no—
51C22⋊Dic97—22no—
52C4×Dic97—22no—
53C4×Dic97—22no—
54C4×Dic97—22no—
55C22×C4×C97D411yescharacteristic
56C22×C4×C97D411yescharacteristic
57C22×C4×C97C2311yescharacteristic
58C2×C8×C97C2314yes—
59C42×C97D411yescharacteristic
60C42×C97D411yescharacteristic
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97C2×D411yescharacteristic
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2411yescharacteristic
73C2×Dic97—24no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—42no—
79Dic194—24no—
80Dic194—22no—
81Dic194—24no—
82Dic194—42no—
83Dic194#4114yes—
84C8×C97—24no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97#13712yes—
88C22×C97C22×D411yescharacteristic
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97#13712yes—
94C4×C97C2×#4112yes—
95Dic97—24no—
96Dic97—42no—
97Dic97—42no—
98Dic97—42no—
99C2×C97—22no—
100C2×C97#52511yescharacteristic
101C2×C97C2×#13711yescharacteristic
102C2×C97C2×#13711yescharacteristic
103#3113—971nomaximal, Sylow
104C97#311311yescharacteristic, complemented, Sylow
105#533—971no—
106C2×C4×Q8—971no—
107#483—972no—
108#475—972no—
109#462—972no—
110#451—972no—
111#441—974no—
112#432D9711yescharacteristic
113C22×Q8—971no—
114C2×Q16—974no—
115C4⋊Q8—971no—
116C4⋊Q8—971no—
117#125—1941no—
118C22⋊Q8—972no—
119C22⋊Q8—1941no—
120C4×Q8—972no—
121C4×Q8—972no—
122C4×Q8—972no—
123C4×Q8—1941no—
124C4×Q8—1941no—
125C4×Q8—1941no—
126#117—1941no—
127C2×C4⋊C4—971no—
128C2×C4⋊C4—1941no—
129C2×C42—1941no—
130C2×C42C2×D9711yescharacteristic
131#106—974no—
132C4⋊C8C2×D9712yes—
133#103—974no—
134#103—974no—
135C22⋊C8C2×D9712yes—
136#97C2×D9712yes—
137C2×Q8—972no—
138C2×Q8—972no—
139C2×Q8—1944no—
140C2×Q8—1941no—
141C2×Q8—1941no—
142C22×C4—1941no—
143C22×C4—1941no—
144C22×C4D38811yescharacteristic
145C22×C4D38811yescharacteristic
146C22×C4C22×D9711yescharacteristic
147Q16—1948no—
148C2×C8C22×D9714yes—
149C4⋊C4—972no—
150C4⋊C4—972no—
151C4⋊C4—1942no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4—1941no—
156C4⋊C4—1941no—
157C4⋊C4—1941no—
158C4⋊C4—1941no—
159C22⋊C4—1942no—
160C42—1942no—
161C42—1942no—
162C42—1942no—
163C42D38811yescharacteristic
164C42D38811yescharacteristic
165C42C22×D9711yescharacteristic
166C42C22×D9711yescharacteristic
167C23C2×D38811yescharacteristic
168Q8—974no—
169Q8—3882no—
170Q8—1944no—
171Q8—1944no—
172Q8—1942no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—3882no—
177C2×C4—1942no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4—1944no—
181C2×C4—1942no—
182C2×C4C2×D38811yescharacteristic
183C2×C4C2×D38811yescharacteristic
184C2×C4C2×D38811yescharacteristic
185C2×C4C2×D38811yescharacteristic
186C2×C4C2×D38811yescharacteristic
187C2×C4C23×D9711yescharacteristic
188C8—24no—
189C22—21no—
190C22—21no—
191C22#1569084312yes—
192C22C22×D38811yescharacteristic
193C4—21no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—3882no—
198C4—3882no—
199C4—3882no—
200C4—1944no—
201C4#1569084312yes—
202C4#41×D9712yes—
203C2—22no—
204C2#2026124711yescharacteristic
205C2C2×#1569084311yescharacteristic
206C2C2×#1569084311yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#432×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenter#432D97
hypocenterC97#3113
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
4. #20261021C2semidirect
5. #20260946C2semidirect
7. #20260843C2semidirect
104. C97#3113semidirect

Maximal subgroups

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

subgroupindex
#202612922
C2×C4×Dic1942
#202610662
#202610212
#202609462
#202608932
#202608432
#432×C972
#311397