Finite Group Databasebeta

← #27140763

Subgroups — #27140763

Order 12416.

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261361C211yesmaximal, characteristic
2#20261071C212yesmaximal, complemented
3#20260671C211yesmaximal, characteristic
4#20260473C212yesmaximal, complemented
5#323×C97C211yesmaximal, characteristic
6#15690789—21no—
7#15690789C2212yes—
8#15690787—22no—
9#15690787—22no—
10C2×C4⋊Dic97C2211yescharacteristic
11C2×#13095490—21no—
12C2×C22⋊Dic97—21no—
13C2×C22⋊Dic97—21no—
14#15690744—22no—
15#15690735—22no—
16#15690717C2211yescharacteristic
17C2×C22⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C2×Dic194—21no—
20C2×Dic194D412yes—
21C22×Dic97—21no—
22C22×Dic97—21no—
23C22×Dic97—21no—
24C22×Dic97—21no—
25C4⋊Dic97—21no—
26C4⋊Dic97—22no—
27C4⋊Dic97D412yes—
28#13095490—22no—
29#13095490—22no—
30#13095490—42no—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33C22⋊Dic97—42no—
34C24×C97D411yescharacteristic
35C22×C4×C97D411yescharacteristic
36C22×C4×C97C2311yescharacteristic
37C2×C8×C97—22no—
38C22⋊C4×C97—22no—
39C23×C97—21no—
40C23×C97—21no—
41C23×C97—41no—
42C23×C97—41no—
43C23×C97Q1612yes—
44C23×C97C2×D411yescharacteristic
45C2×C4×C97—21no—
46C2×C4×C97—41no—
47C2×C4×C97—41no—
48C2×C4×C97C2×D412yes—
49C2×Dic97—22no—
50C2×Dic97—22no—
51C2×Dic97—22no—
52C2×Dic97—41no—
53C2×Dic97—41no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56C2×Dic97—41no—
57C2×Dic97—81no—
58C2×Dic97—81no—
59Dic194—42no—
60Dic194—81no—
61C8×C97—42no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97—22no—
65C22×C97—42no—
66C22×C97—41no—
67C22×C97—41no—
68C22×C97—81no—
69C22×C97#12011yescharacteristic
70C22×C97C2×Q1611yescharacteristic
71C22×C97#13711yescharacteristic
72C4×C97—22no—
73C4×C97—41no—
74Dic97—42no—
75Dic97—42no—
76Dic97—42no—
77Dic97—81no—
78C2×C97—21no—
79C2×C97—21no—
80C2×C97—42no—
81C2×C97#45111yescharacteristic
82C2×C97#45311yescharacteristic
83C2×C97#45811yescharacteristic
84#1614—971nomaximal, Sylow
85C97#161411yescharacteristic, complemented, Sylow
86#543—971no—
87#484—972no—
88#399—971no—
89#328—972no—
90#323D9711yescharacteristic
91C22⋊Q8—972no—
92C22⋊Q8—1942no—
93C22⋊Q8—1942no—
94C22⋊Q8—1941no—
95C2×C4⋊C4—971no—
96C2×C4⋊C4—1941no—
97C2×C22⋊C4—1941no—
98C2×C22⋊C4—1941no—
99C2×C22⋊C4C2×D9711yescharacteristic
100#107—1942no—
101#103—1942no—
102C22⋊C8C2×D9712yes—
103#95—971no—
104C24D38811yescharacteristic
105C2×Q8—972no—
106C2×Q8—1941no—
107C22×C4—1941no—
108C22×C4—1941no—
109C22×C4—1941no—
110C22×C4—1941no—
111C22×C4D38811yescharacteristic
112C22×C4C22×D9711yescharacteristic
113C2×C8—22no—
114C4⋊C4—972no—
115C4⋊C4—3882no—
116C4⋊C4—1942no—
117C4⋊C4—1941no—
118C4⋊C4—1942no—
119C4⋊C4—1942no—
120C22⋊C4—22no—
121C22⋊C4—3882no—
122C22⋊C4—1942no—
123C22⋊C4—1942no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic38812yes—
129C23C2×D38811yescharacteristic
130Q8—7761no—
131Q8—3882no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—7761no—
136C2×C4—7761no—
137C2×C4—3881no—
138C2×C4—3881no—
139C2×C4—3881no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4—1942no—
143C2×C4—1942no—
144C2×C4—1942no—
145C2×C4D4×D9712yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1569078011yescharacteristic
155C22C2×Dic38811yescharacteristic
156C22#1569084311yescharacteristic
157C4—22no—
158C4—41no—
159C4—7761no—
160C4—3882no—
161C4—3882no—
162C4—3882no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2026089311yescharacteristic
167C2#2026090511yescharacteristic
168C2#2026092611yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#323×C97C2
socleC22×C97#120
prime radicalC24×C97D4
hypercenter#323D97
hypocenterC97#1614
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20261071C2semidirect
4. #20260473C2semidirect
85. C97#1614semidirect

Maximal subgroups

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

subgroupindex
#202613612
#202610712
#202606712
#202604732
#323×C972
#161497