Finite Group Databasebeta

← #27140727

Subgroups — #27140727

Order 12416.

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

normal subgroups only

This group has 228 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#20261063C212yesmaximal, complemented
2#20260660C211yesmaximal, characteristic
3#20260472C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic
5#543×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2211yescharacteristic
7C2×C22⋊Dic97—21no—
8#15690741—22no—
9#15690734—22no—
10C22⋊#11715895C2212yes—
11#15690717—21no—
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—
23C22⋊Dic97—42no—
24C24×C97D411yescharacteristic
25C2×Q8×C97—21no—
26C2×Q8×C97D412yes—
27C22×C4×C97—21no—
28C22×C4×C97—21no—
29C22×C4×C97D411yescharacteristic
30C22×C4×C97C2311yescharacteristic
31C4⋊C4×C97—21no—
32C4⋊C4×C97—22no—
33C4⋊C4×C97—22no—
34C4⋊C4×C97D412yes—
35C22⋊C4×C97—22no—
36C22⋊C4×C97—22no—
37C22⋊C4×C97—22no—
38C23×C97—21no—
39C23×C97—21no—
40C23×C97—41no—
41C23×C97—41no—
42C23×C97SD1612yes—
43C23×C97C2×D411yescharacteristic
44Q8×C97—42no—
45Q8×C97—81no—
46C2×C4×C97—22no—
47C2×C4×C97—21no—
48C2×C4×C97—22no—
49C2×C4×C97—41no—
50C2×C4×C97—41no—
51C2×C4×C97—81no—
52C2×C4×C97—81no—
53C2×C4×C97C2×D412yes—
54C2×Dic97—41no—
55C2×Dic97—41no—
56C2×Dic97—41no—
57C2×Dic97—41no—
58C2×Dic97—41no—
59C2×Dic97—41no—
60#11715895—42no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97—22no—
64C22×C97—42no—
65C22×C97—41no—
66C22×C97—41no—
67C22×C97—81no—
68C22×C97#12011yescharacteristic
69C22×C97C2×SD1611yescharacteristic
70C22×C97#13711yescharacteristic
71C4×C97—22no—
72C4×C97—41no—
73C4×C97—42no—
74C4×C97—42no—
75Dic97—81no—
76Dic97—81no—
77C2×C97—21no—
78C2×C97—21no—
79C2×C97—42no—
80C2×C97#44811yescharacteristic
81C2×C97#45511yescharacteristic
82C2×C97#45811yescharacteristic
83#1608—971nomaximal, Sylow
84C97#160811yescharacteristic, complemented, Sylow
85#543D9711yescharacteristic
86#483—972no—
87#397—971no—
88#328—972no—
89#323—971no—
90C22⋊Q8—22no—
91C22⋊Q8—21no—
92C22⋊Q8—22no—
93C22⋊Q8C2×D9712yes—
94C2×C4⋊C4—971no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—1941no—
97C2×C22⋊C4C2×D9711yescharacteristic
98#106—1942no—
99#103—1942no—
100C22⋊C8—972no—
101#95—971no—
102#95—1941no—
103C24C22⋊D9711yescharacteristic
104C2×Q8—21no—
105C2×Q8C22⋊D9712yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—1941no—
109C22×C4—1941no—
110C22×C4C22⋊D9711yescharacteristic
111C22×C4C22×D9711yescharacteristic
112C2×C8—1942no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—1942no—
117C4⋊C4C22⋊D9712yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—3882no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1309549812yes—
127C23C2×C22⋊D9711yescharacteristic
128Q8—42no—
129Q8—81no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—3881no—
138C2×C4—3881no—
139C2×C4—3881no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4—3881no—
143C2×C4C2×C22⋊D9712yes—
144C8—3882no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#1569077911yescharacteristic
153C22C2×#1309549811yescharacteristic
154C22#1569084111yescharacteristic
155C4—22no—
156C4—41no—
157C4—42no—
158C4—42no—
159C4—7761no—
160C4—7761no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#2026087411yescharacteristic
165C2#2026091311yescharacteristic
166C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#543×C97C2
socleC22×C97#120
prime radicalC24×C97D4
hypercenter#543D97
hypocenterC97#1608
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. #20261063C2semidirect
3. #20260472C2semidirect
84. C97#1608semidirect

Maximal subgroups

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

subgroupindex
#202610632
#202606602
#202604722
#202604592
#543×C972
#160897