Finite Group Databasebeta

← #27140761

Subgroups — #27140761

Order 12416.

236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261068C212yesmaximal, complemented
2#20260671C211yesmaximal, characteristic
3#20260472C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic
5#543×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2211yescharacteristic
7C2×#13095490—21no—
8C2×C22⋊Dic97—21no—
9#15690743—22no—
10#15690734—22no—
11C22⋊#11715895C2212yes—
12#15690717C2211yescharacteristic
13C22⋊Q8×C97—22no—
14C22⋊Q8×C97—21no—
15C22⋊Q8×C97—22no—
16C22⋊Q8×C97C2212yes—
17C2×C22⋊C4×C97—21no—
18C2×C22⋊C4×C97C2211yescharacteristic
19C22×Dic97—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97—22no—
23#13095490—42no—
24C22⋊Dic97—42no—
25C24×C97D411yescharacteristic
26C2×Q8×C97—21no—
27C2×Q8×C97D412yes—
28C22×C4×C97—21no—
29C22×C4×C97—21no—
30C22×C4×C97D411yescharacteristic
31C22×C4×C97C2311yescharacteristic
32C4⋊C4×C97—21no—
33C4⋊C4×C97—22no—
34C4⋊C4×C97—22no—
35C4⋊C4×C97D412yes—
36C22⋊C4×C97—22no—
37C22⋊C4×C97—22no—
38C22⋊C4×C97—22no—
39C23×C97—21no—
40C23×C97—21no—
41C23×C97—41no—
42C23×C97—41no—
43C23×C97Q1612yes—
44C23×C97C2×D411yescharacteristic
45Q8×C97—42no—
46Q8×C97—81no—
47C2×C4×C97—22no—
48C2×C4×C97—21no—
49C2×C4×C97—22no—
50C2×C4×C97—41no—
51C2×C4×C97—41no—
52C2×C4×C97—81no—
53C2×C4×C97—81no—
54C2×C4×C97C2×D412yes—
55C2×Dic97—22no—
56C2×Dic97—41no—
57C2×Dic97—41no—
58C2×Dic97—41no—
59C2×Dic97—41no—
60C2×Dic97—41no—
61#11715895—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—
74C4×C97—42no—
75C4×C97—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#543D9711yescharacteristic
87#484—972no—
88#399—971no—
89#328—972no—
90#323—971no—
91C22⋊Q8—22no—
92C22⋊Q8—21no—
93C22⋊Q8—22no—
94C22⋊Q8C2×D9712yes—
95C2×C4⋊C4—971no—
96C2×C4⋊C4—1941no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—1941no—
99C2×C22⋊C4C2×D9711yescharacteristic
100#107—1942no—
101#103—1942no—
102C22⋊C8—972no—
103#95—971no—
104C24C22⋊D9711yescharacteristic
105C2×Q8—21no—
106C2×Q8C22⋊D9712yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—1941no—
110C22×C4—1941no—
111C22×C4C22⋊D9711yescharacteristic
112C22×C4C22×D9711yescharacteristic
113C2×C8—1942no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—3882no—
118C4⋊C4—1942no—
119C4⋊C4C22⋊D9712yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—3882no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23#1309550112yes—
129C23C2×C22⋊D9711yescharacteristic
130Q8—42no—
131Q8—81no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—3881no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4—3881no—
143C2×C4—3881no—
144C2×C4—1942no—
145C2×C4C2×C22⋊D9712yes—
146C8—3882no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1569077911yescharacteristic
155C22C2×#1309550111yescharacteristic
156C22#1569084111yescharacteristic
157C4—22no—
158C4—41no—
159C4—42no—
160C4—42no—
161C4—7761no—
162C4—3882no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2026089111yescharacteristic
167C2#2026090311yescharacteristic
168C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#543×C97C2
socleC22×C97#120
prime radicalC24×C97D4
hypercenter#543D97
hypocenterC97#1614
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20261068C2semidirect
3. #20260472C2semidirect
85. C97#1614semidirect

Maximal subgroups

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

subgroupindex
#202610682
#202606712
#202604722
#202604592
#543×C972
#161497