Finite Group Databasebeta

← #41012355

Subgroups — #41012355

Order 27008.

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

normal subgroups only

This group has 256 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#27213301C212yesmaximal, complemented
2#27213058C211yesmaximal, characteristic
3#27212883C212yesmaximal, complemented
4#27212870C211yesmaximal, characteristic
5#543×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C2×#16809952—22no—
8C2×C22⋊Dic211—21no—
9#20269900—22no—
10C22⋊(C211⋊C8)C2212yes—
11C211⋊#95—21no—
12C211⋊#95C2211yescharacteristic
13C22⋊Q8×C211—22no—
14C22⋊Q8×C211—21no—
15C22⋊Q8×C211—22no—
16C22⋊Q8×C211C2212yes—
17C2×C22⋊C4×C211—21no—
18C2×C22⋊C4×C211C2211yescharacteristic
19C2×Dic422—22no—
20C2×Dic422—42no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23#16809952—44no—
24C2×C211⋊C8—22no—
25C22⋊Dic211—42no—
26C24×C211D411yescharacteristic
27C2×Q8×C211—21no—
28C2×Q8×C211D412yes—
29C22×C4×C211—21no—
30C22×C4×C211—21no—
31C22×C4×C211D411yescharacteristic
32C22×C4×C211C2311yescharacteristic
33C4⋊C4×C211—21no—
34C4⋊C4×C211—22no—
35C4⋊C4×C211—22no—
36C4⋊C4×C211D412yes—
37C22⋊C4×C211—22no—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40Dic422—24no—
41Dic422—42no—
42C2×Dic211—22no—
43C2×Dic211—41no—
44C2×Dic211—41no—
45C2×Dic211—41no—
46C2×Dic211—41no—
47C2×Dic211—41no—
48C211⋊C8—42no—
49C23×C211—21no—
50C23×C211—21no—
51C23×C211—41no—
52C23×C211—41no—
53C23×C211Q1612yescomplemented
54C23×C211C2×D411yescharacteristic
55Q8×C211—42no—
56Q8×C211—81no—
57C2×C4×C211—22no—
58C2×C4×C211—21no—
59C2×C4×C211—22no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—81no—
63C2×C4×C211—81no—
64C2×C4×C211C2×D412yes—
65C22×C211—21no—
66C22×C211—21no—
67C22×C211—22no—
68C22×C211—42no—
69C22×C211—41no—
70C22×C211—41no—
71C22×C211—81no—
72C22×C211#12011yescharacteristic
73C22×C211C2×Q1611yescharacteristic
74C22×C211#13711yescharacteristic
75C4×C211—22no—
76C4×C211—41no—
77C4×C211—42no—
78C4×C211—42no—
79Dic211—42no—
80Dic211—81no—
81C2×C211—21no—
82C2×C211—21no—
83C2×C211—42no—
84C2×C211#45111yescharacteristic
85C2×C211#45511yescharacteristic
86C2×C211B(4,𝔽2)11yescharacteristic
87C211#160211yescharacteristic, complemented, Sylow
88#1602—2111nomaximal, Sylow
89#543D21111yescharacteristic
90#451—2112no—
91#393—2111no—
92#328—2112no—
93#323—2111no—
94C22×Q8—2111no—
95C2×Q16—4222no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D21112yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—4221no—
102C2×C22⋊C4C2×D21111yescharacteristic
103#103—4222no—
104C22⋊C8—2112no—
105#95—4221no—
106#95—2111no—
107C24C22⋊D21111yescharacteristic
108C2×Q8—21no—
109C2×Q8—8442no—
110C2×Q8—4222no—
111C2×Q8C22⋊D21112yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—4221no—
115C22×C4—4221no—
116C22×C4C22⋊D21111yescharacteristic
117C22×C4C22×D21111yescharacteristic
118Q16—8444no—
119C2×C8—4222no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D21112yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—8442no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1680995212yescomplemented
133C23C2×C22⋊D21111yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—8442no—
137Q8—4224no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—8441no—
146C2×C4—8441no—
147C2×C4—8441no—
148C2×C4—8441no—
149C2×C4—8441no—
150C2×C4—4222no—
151C2×C4C2×C22⋊D21112yes—
152C8—8442no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2026994611yescharacteristic
161C22C2×#1680995211yescharacteristic
162C22#2027000811yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—8442no—
168C4—16881no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2721330211yescharacteristic
173C2#2721332411yescharacteristic
174C2#2721333311yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#543×C211C2
socleC22×C211#120
prime radicalC24×C211D4
hypercenter#543D211
hypocenterC211#1602
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #27213301C2semidirect
3. #27212883C2semidirect
53. C23×C211Q16semidirect
87. C211#1602semidirect
132. C23#16809952semidirect

Maximal subgroups

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

subgroupindex
#272133012
#272130582
#272128832
#272128702
#543×C2112
#1602211