Finite Group Databasebeta

← #27152028

Subgroups — #27152028

Order 12416.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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#20261340C211yesmaximal, characteristic
2#20261212C211yesmaximal, characteristic, complemented
3#20261060C212yesmaximal, complemented
4#20261043C212yesmaximal, complemented
5#20260936C212yesmaximal, complemented
6#20260876C212yesmaximal, complemented
7#20260837C212yesmaximal, complemented
8C4×#13095500C212yesmaximal, complemented
9#433×C97C211yesmaximal, characteristic, complemented
10C22×Dic194C2211yescharacteristic
11C2×#13095500C2212yes—
12C2×#13095500C2212yes—
13#15690807C2211yescharacteristic
14#15690802C2211yescharacteristic
15#15690795—22no—
16#15690784C2212yes—
17#15690783—22no—
18C4×Dic194C2212yes—
19C4×Dic194C2212yes—
20#15690775—22no—
21C4×D388C2212yes—
22#15690769C2211yescharacteristic
23#15690766—22no—
24#15690744C2212yes—
25#15690742C2212yes—
26#15690735C2212yes—
27#15690735C2212yes—
28#15690732C2212yes—
29#15690732C2212yes—
30#117×C97C2211yescharacteristic
31C4⋊C8×C97C2212yes—
32C22⋊C8×C97C2212yes—
33#97×C97C2211yescharacteristic
34C4×C8×C97C2211yescharacteristic
35C2×Dic194—21no—
36C2×Dic194—24no—
37C2×Dic194C2312yes—
38C2×D388—22no—
39C2×D388C2312yes—
40C2×C22⋊D97—22no—
41C22×Dic97—21no—
42C2×C4×D97—22no—
43#13095500—24no—
44#13095500—24no—
45C4⋊Dic97C2312yes—
46C4⋊Dic97C2312yes—
47#13095490—22no—
48#13095490—22no—
49#13095490—22no—
50C22⋊Dic97—21no—
51#13095488—22no—
52#13095488—22no—
53C4×Dic97—22no—
54C4×Dic97—22no—
55C4×Dic97—21no—
56C22×C4×C97C2311yescharacteristic
57C2×C8×C97C2312yes—
58C2×C8×C97C2312yes—
59C4⋊C4×C97D412yes—
60C22⋊C4×C97D412yes—
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97C2×D411yescharacteristic
64C22×D97—22no—
65C2×C4×C97—22no—
66C2×C4×C97C2×D412yes—
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2411yescharacteristic
70C22⋊D97—82no—
71C2×Dic97—24no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75D388—22no—
76D388—22no—
77D388—82no—
78C4×D97—82no—
79Dic194—24no—
80Dic194—22no—
81Dic194#4114yes—
82C8×C97—22no—
83C8×C97—22no—
84C22×C97—21no—
85C22×C97—22no—
86C22×C97C22×D411yescharacteristic
87C2×D97—42no—
88C2×D97—42no—
89C2×D97—42no—
90C4×C97—21no—
91C4×C97—22no—
92C4×C97—22no—
93C4×C97C2×#4112yes—
94Dic97—24no—
95Dic97—42no—
96Dic97—42no—
97C2×C97—41no—
98C2×C97#52511yescharacteristic
99C2×C97#57712yes—
100D97—82no—
101#3126—971nomaximal, Sylow
102C97#312611yescharacteristic, complemented, Sylow
103#540—971no—
104#519—971no—
105#482—972no—
106#479—972no—
107#460—972no—
108#448—972no—
109#440—972no—
110C4×SD16—972no—
111#433D9711yescharacteristic, complemented
112C22×Q8—971no—
113C2×SD16—972no—
114C2×SD16—972no—
115#127—971no—
116#125—971no—
117#123—1942no—
118#121—972no—
119#121—1942no—
120C4×Q8—972no—
121C4×Q8—972no—
122C4×Q8—1942no—
123C4×D4—972no—
124#117—971no—
125#117—1942no—
126#117C2×D9711yescharacteristic
127#107—972no—
128#106—972no—
129C4⋊C8C2×D9712yes—
130#103—972no—
131#103—972no—
132#102—972no—
133#102—972no—
134C22⋊C8C2×D9712yes—
135#97C2×D9711yescharacteristic
136C4×C8C2×D9711yescharacteristic
137C2×Q8—972no—
138C2×Q8—1944no—
139C2×Q8—1941no—
140C2×D4—972no—
141C2×D4—1942no—
142C2×D4—1942no—
143C22×C4—1942no—
144C22×C4—1941no—
145C22×C4C22×D9711yescharacteristic
146SD16—1944no—
147SD16—1944no—
148C2×C8C22×D9712yes—
149C2×C8C22×D9712yes—
150C4⋊C4—972no—
151C4⋊C4—972no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4D38812yes—
156C22⋊C4—1941no—
157C22⋊C4—1942no—
158C22⋊C4—1942no—
159C22⋊C4D38812yes—
160C42—1941no—
161C42—1942no—
162C42—1942no—
163C42C22×D9711yescharacteristic
164C42C22×D9711yescharacteristic
165C23—1942no—
166C23C2×D38811yescharacteristic
167Q8—974no—
168Q8—1944no—
169Q8—1942no—
170D4—7762no—
171D4—7762no—
172D4—1942no—
173D4—1942no—
174C2×C4—22no—
175C2×C4—7762no—
176C2×C4—1942no—
177C2×C4—1942no—
178C2×C4—1944no—
179C2×C4—1942no—
180C2×C4C2×D38812yes—
181C2×C4C2×D38812yes—
182C2×C4C2×D38811yescharacteristic
183C2×C4C23×D9711yescharacteristic
184C8—22no—
185C8—22no—
186C22—21no—
187C22—22no—
188C22—3882no—
189C22—3882no—
190C22—3882no—
191C22C22×D38811yescharacteristic
192C4—21no—
193C4—22no—
194C4—22no—
195C4—3882no—
196C4—3882no—
197C4—1944no—
198C4#41×D9712yes—
199C2—41no—
200C2—7762no—
201C2#2026124711yescharacteristic
202C2#2026154712yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#433×C97C2
socleC22×C97C22×D4
prime radical#20261340C2
hypercenter#433D97
hypocenterC97#3126
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #20261212C2semidirect
3. #20261060C2semidirect
4. #20261043C2semidirect
5. #20260936C2semidirect
6. #20260876C2semidirect
7. #20260837C2semidirect
8. C4×#13095500C2semidirect
9. #433×C97C2semidirect
102. C97#3126semidirect
111. #433D97semidirect

Maximal subgroups

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

subgroupindex
#202613402
#202612122
#202610602
#202610432
#202609362
#202608762
#202608372
C4×#130955002
#433×C972
#312697