Finite Group Databasebeta

← #41012340

Subgroups — #41012340

Order 27008.

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

normal subgroups only

This group has 266 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#27213777C211yesmaximal, characteristic, complemented
2#27213299C212yesmaximal
3#27213063C211yesmaximal, characteristic
4#27212884C212yesmaximal, complemented
5#324×C211C211yesmaximal, characteristic, complemented
6C22×D844C2211yescharacteristic
7C2×#16809951—22no—
8#20269976—22no—
9#20269968—22no—
10#20269956—21no—
11#20269956C2212yes—
12#20269936—21no—
13C2×(C2×D211)⋊C4—21no—
14C2×(C2×D211)⋊C4—21no—
15#20269899—22no—
16C211⋊#95C2211yescharacteristic
17C2×C4⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19C23×D211—21no—
20C2×Dic422—21no—
21C2×Dic422D412yes—
22C2×D844—22no—
23C2×D844—42no—
24C22×Dic211—21no—
25#16809951—44no—
26C4⋊Dic211—21no—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211D412yes—
30Dic211⋊C4—22no—
31Dic211⋊C4—22no—
32C22⋊Dic211—22no—
33(C2×D211)⋊C4—42no—
34(C2×D211)⋊C4—42no—
35C4×Dic211—22no—
36C22×C4×C211D411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211C2311yescharacteristic
39C2×C8×C211—22no—
40C4⋊C4×C211—22no—
41C22×D211—22no—
42C22×D211—41no—
43C22×D211—41no—
44C22×D211—41no—
45C22×D211—41no—
46C22×D211—41no—
47C22×D211—41no—
48Dic422—42no—
49Dic422—81no—
50D844—24no—
51D844—42no—
52C2×Dic211—22no—
53C2×Dic211—22no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C23×C211C2×D411yescharacteristic
58C2×C4×C211—21no—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211SD1612yescomplemented
64C2×C4×C211C2×D412yes—
65C8×C211—42no—
66C22×C211—21no—
67C22×C211—21no—
68C22×C211#12011yescharacteristic
69C22×C211C2×SD1611yescharacteristic
70C22×C211#13711yescharacteristic
71C2×D211—42no—
72C2×D211—42no—
73C2×D211—42no—
74C2×D211—81no—
75C2×D211—81no—
76C2×D211—81no—
77C2×D211—81no—
78C4×C211—22no—
79C4×C211—22no—
80C4×C211—41no—
81Dic211—42no—
82Dic211—42no—
83Dic211—81no—
84C2×C211—21no—
85C2×C211—21no—
86C2×C211#44811yescharacteristic
87C2×C211#45411yescharacteristic
88C2×C211B(4,𝔽2)11yescharacteristic
89D211—82no—
90C211#159911yescharacteristic, complemented, Sylow
91#1599—2111nomaximal, Sylow
92#544—2111no—
93#450—2112no—
94#394—2111no—
95#328—2112no—
96#324D21111yescharacteristic, complemented
97C22×D4—2111no—
98C2×SD16—4222no—
99C4⋊Q8—4222no—
100#125—4222no—
101C22⋊Q8—4221no—
102C22⋊Q8—2112no—
103#117—4221no—
104C2×C4⋊C4C2×D21111yescharacteristic
105C2×C22⋊C4—4221no—
106C2×C22⋊C4—4221no—
107#102—4222no—
108C22⋊C8C2×D21112yes—
109#95—2111no—
110C24—4221no—
111C2×Q8—4221no—
112C2×Q8—2112no—
113C2×D4—8442no—
114C2×D4—4222no—
115C22×C4—4221no—
116C22×C4D84411yescharacteristic
117C22×C4D84411yescharacteristic
118C22×C4C22×D21111yescharacteristic
119SD16—8444no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—4221no—
123C4⋊C4—4221no—
124C4⋊C4—4221no—
125C4⋊C4—4222no—
126C4⋊C4—4222no—
127C4⋊C4—2112no—
128C22⋊C4—8442no—
129C22⋊C4—8442no—
130C22⋊C4—4222no—
131C42—4222no—
132C23—8441no—
133C23—8441no—
134C23—8441no—
135C23—8441no—
136C23—8441no—
137C23—8441no—
138C23—4222no—
139C23C2×D84411yescharacteristic
140Q8—8442no—
141Q8—16881no—
142D4—8442no—
143D4—4224no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—8441no—
150C2×C4—8441no—
151C2×C4—8441no—
152C2×C4—4222no—
153C2×C4—4222no—
154C2×C4#1680995112yescomplemented
155C2×C4D4×D21112yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—8442no—
160C22—8442no—
161C22—8442no—
162C22—16881no—
163C22—16881no—
164C22—16881no—
165C22—16881no—
166C22#2026994711yescharacteristic
167C22C2×#1680995111yescharacteristic
168C22#2027001011yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—8442no—
173C4—8442no—
174C4—16881no—
175C2—21no—
176C2—21no—
177C2—16882no—
178C2#2721328711yescharacteristic
179C2#2721332111yescharacteristic
180C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211#120
prime radicalC22×D844C22
hypercenter#324D211
hypocenterC211#1599
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #27213777C2semidirect
4. #27212884C2semidirect
5. #324×C211C2semidirect
63. C2×C4×C211SD16semidirect
90. C211#1599semidirect
96. #324D211semidirect
154. C2×C4#16809951semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272132992
#272130632
#272128842
#324×C2112
#1599211