Finite Group Databasebeta

← #27152022

Subgroups — #27152022

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#20261212C211yesmaximal, characteristic, complemented
2#20261056C212yesmaximal, complemented
3#20261039C212yesmaximal, complemented
4#20260932C212yesmaximal, complemented
5#20260873C212yesmaximal, complemented
6#20260835C212yesmaximal, complemented
7C4×#13095499C212yesmaximal, complemented
8#20260796C211yesmaximal, characteristic, complemented
9#540×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C2×#13095499C2212yes—
12C2×#13095499C2212yes—
13C4×Dic194C2212yes—
14C4×Dic194C2212yes—
15#15690775—22no—
16#15690769C2211yescharacteristic
17#15690766—22no—
18#15690743C2212yes—
19#15690741C2212yes—
20C4⋊#11715895C2212yes—
21#15690733C2212yes—
22#15690733C2212yes—
23#15690731C2212yes—
24#15690731C2212yes—
25C22⋊#11715895C2212yes—
26#15690721C2211yescharacteristic
27C4×#11715895C2211yescharacteristic
28#127×C97C2211yescharacteristic
29#125×C97C2211yescharacteristic
30#123×C97—22no—
31#121×C97—22no—
32#121×C97C2212yes—
33C4×D4×C97C2212yes—
34#117×C97C2211yescharacteristic
35C2×Dic194—21no—
36C2×Dic194—24no—
37C2×Dic194C2312yes—
38C22×Dic97—21no—
39#13095499—24no—
40#13095499—24no—
41C2×#11715895C2312yes—
42C2×#11715895C2312yes—
43C4⋊Dic97C2312yes—
44#13095490—22no—
45#13095490—22no—
46C22⋊Dic97—21no—
47C4×Dic97—22no—
48C4×Dic97—22no—
49C4×Dic97—21no—
50C2×D4×C97—22no—
51C2×D4×C97—22no—
52C2×D4×C97C2312yes—
53C22×C4×C97—22no—
54C22×C4×C97C2311yescharacteristic
55C4⋊C4×C97—22no—
56C4⋊C4×C97D412yes—
57C4⋊C4×C97C2312yes—
58C22⋊C4×C97—22no—
59C22⋊C4×C97—22no—
60C22⋊C4×C97D412yes—
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97—22no—
64C23×C97C2×D411yescharacteristic
65D4×C97—22no—
66D4×C97—22no—
67D4×C97—82no—
68D4×C97—82no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—82no—
72C2×C4×C97C2×D412yes—
73C2×C4×C97C2×D412yes—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2411yescharacteristic
76C2×Dic97—24no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79Dic194—24no—
80Dic194—22no—
81Dic194#4114yes—
82#11715895—22no—
83#11715895—22no—
84C22×C97—21no—
85C22×C97—22no—
86C22×C97—42no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97C22×D411yescharacteristic
90C4×C97—21no—
91C4×C97—22no—
92C4×C97—22no—
93C4×C97—42no—
94C4×C97C2×#4112yes—
95Dic97—24no—
96Dic97—42no—
97C2×C97—41no—
98C2×C97—82no—
99C2×C97#52511yescharacteristic
100C2×C97#57712yes—
101#3126—971nomaximal, Sylow
102C97#312611yescharacteristic, complemented, Sylow
103#540D9711yescharacteristic
104#519—971no—
105#482—972no—
106#479—972no—
107#460—972no—
108#448—972no—
109#440—972no—
110C4×SD16—972no—
111#433—971no—
112C22×Q8—971no—
113C2×SD16—972no—
114C2×SD16—972no—
115#127C2×D9711yescharacteristic
116#125C2×D9711yescharacteristic
117#123—22no—
118#121—22no—
119#121C2×D9712yes—
120C4×Q8—972no—
121C4×Q8—972no—
122C4×Q8—1942no—
123C4×D4C2×D9712yes—
124#117—971no—
125#117—1942no—
126#117C2×D9711yescharacteristic
127#107—972no—
128#106—972no—
129C4⋊C8—972no—
130#103—972no—
131#103—972no—
132#102—972no—
133#102—972no—
134C22⋊C8—972no—
135#97—971no—
136C4×C8—971no—
137C2×Q8—972no—
138C2×Q8—1944no—
139C2×Q8—1941no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22×D9712yes—
143C22×C4—22no—
144C22×C4—1941no—
145C22×C4C22×D9711yescharacteristic
146SD16—1944no—
147SD16—1944no—
148C2×C8—972no—
149C2×C8—972no—
150C4⋊C4—22no—
151C4⋊C4—972no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4C22⋊D9712yes—
155C4⋊C4C22×D9712yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—1941no—
159C22⋊C4C22⋊D9712yes—
160C42—1941no—
161C42—1942no—
162C42—1942no—
163C42C22×D9711yescharacteristic
164C42C22×D9711yescharacteristic
165C23—22no—
166C23C2×C22⋊D9711yescharacteristic
167Q8—974no—
168Q8—1944no—
169Q8—1942no—
170D4—22no—
171D4—22no—
172D4—82no—
173D4—82no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—82no—
177C2×C4—1942no—
178C2×C4—1944no—
179C2×C4—1942no—
180C2×C4C2×C22⋊D9712yes—
181C2×C4C2×C22⋊D9712yes—
182C2×C4C2×C22⋊D9711yescharacteristic
183C2×C4C23×D9711yescharacteristic
184C8—1942no—
185C8—1942no—
186C22—21no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C22×C22⋊D9711yescharacteristic
192C4—21no—
193C4—22no—
194C4—22no—
195C4—42no—
196C4—3882no—
197C4—1944no—
198C4#41×D9712yes—
199C2—41no—
200C2—82no—
201C2#2026124211yescharacteristic
202C2#2026154612yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radical#540×C97C2
socleC22×C97C22×D4
hypercenter#540D97
hypocenterC97#3126
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20261212C2semidirect
2. #20261056C2semidirect
3. #20261039C2semidirect
4. #20260932C2semidirect
5. #20260873C2semidirect
6. #20260835C2semidirect
7. C4×#13095499C2semidirect
8. #20260796C2semidirect
102. C97#3126semidirect

Maximal subgroups

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

subgroupindex
#202612122
#202610562
#202610392
#202609322
#202608732
#202608352
C4×#130954992
#202607962
#540×C972
#312697