Finite Group Databasebeta

← #41012323

Subgroups — #41012323

Order 27008.

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

normal subgroups only

This group has 316 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#27213772C211yesmaximal, characteristic, complemented
2#27213299C212yesmaximal, complemented
3#27213055C211yesmaximal, characteristic
4#27212884C212yesmaximal, complemented
5#323×C211C211yesmaximal, characteristic, complemented
6C22×D844C2211yescharacteristic
7C22×C22⋊D211—21no—
8C2×#16809951—22no—
9#20269956—21no—
10#20269956C2212yes—
11#20269954—22no—
12#20269954—22no—
13C2×C22⋊Dic211—21no—
14C2×(C2×D211)⋊C4—21no—
15#20269899—22no—
16C211⋊#95C2211yescharacteristic
17C2×C22⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19C23×D211—21no—
20C2×Dic422—21no—
21C2×Dic422D412yescomplemented
22C2×D844—22no—
23C2×D844—42no—
24C2×C22⋊D211—42no—
25C2×C22⋊D211—42no—
26C2×C22⋊D211—42no—
27C22×Dic211—21no—
28C22×Dic211—21no—
29C22×Dic211—21no—
30#16809951—44no—
31C4⋊Dic211—21no—
32C4⋊Dic211D412yescomplemented
33Dic211⋊C4—22no—
34Dic211⋊C4—22no—
35C22⋊Dic211—22no—
36C22⋊Dic211—22no—
37(C2×D211)⋊C4—42no—
38C24×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211C2311yescharacteristic
41C2×C8×C211—22no—
42C22⋊C4×C211—22no—
43C22×D211—22no—
44C22×D211—41no—
45C22×D211—41no—
46C22×D211—41no—
47C22×D211—41no—
48C22×D211—41no—
49C22×D211—41no—
50Dic422—42no—
51Dic422—81no—
52D844—24no—
53D844—42no—
54C22⋊D211—48no—
55C2×Dic211—22no—
56C2×Dic211—22no—
57C2×Dic211—22no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—81no—
61C2×Dic211—81no—
62C23×C211—21no—
63C23×C211—21no—
64C23×C211—41no—
65C23×C211—41no—
66C23×C211SD1612yescomplemented
67C23×C211C2×D411yescharacteristic
68C2×C4×C211—21no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211C2×D412yes—
72C8×C211—42no—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211—22no—
76C22×C211—42no—
77C22×C211—41no—
78C22×C211—41no—
79C22×C211—81no—
80C22×C211#12011yescharacteristic
81C22×C211C2×SD1611yescharacteristic
82C22×C211#13711yescharacteristic
83C2×D211—42no—
84C2×D211—42no—
85C2×D211—42no—
86C2×D211—81no—
87C2×D211—81no—
88C2×D211—81no—
89C2×D211—81no—
90C4×C211—22no—
91C4×C211—41no—
92Dic211—42no—
93Dic211—42no—
94Dic211—42no—
95C2×C211—21no—
96C2×C211—21no—
97C2×C211—42no—
98C2×C211#44811yescharacteristic
99C2×C211#45311yescharacteristic
100C2×C211B(4,𝔽2)11yescharacteristic
101D211—82no—
102C211#159611yescharacteristic, complemented, Sylow
103#1596—2111nomaximal, Sylow
104#543—2111no—
105#450—2112no—
106#392—2111no—
107#328—2112no—
108#323D21111yescharacteristic, complemented
109C22×D4—4221no—
110C22×D4—2111no—
111C2×SD16—4222no—
112C22⋊Q8—4222no—
113C22⋊Q8—4222no—
114C22⋊Q8—4221no—
115C22⋊Q8—2112no—
116C2×C22⋊C4—4221no—
117C2×C22⋊C4—4221no—
118C2×C22⋊C4C2×D21111yescharacteristic
119#102—4222no—
120C22⋊C8C2×D21112yes—
121#95—2111no—
122C24—4221no—
123C24D84411yescharacteristic
124C2×Q8—4221no—
125C2×Q8—2112no—
126C2×D4—8442no—
127C2×D4—8442no—
128C2×D4—8442no—
129C2×D4—8442no—
130C2×D4—4222no—
131C22×C4—4221no—
132C22×C4—4221no—
133C22×C4—4221no—
134C22×C4D84411yescharacteristic
135C22×C4C22×D21111yescharacteristic
136SD16—8444no—
137C2×C8—22no—
138C4⋊C4—4222no—
139C4⋊C4—4221no—
140C4⋊C4—4222no—
141C4⋊C4—2112no—
142C22⋊C4—22no—
143C22⋊C4—8442no—
144C22⋊C4—4222no—
145C22⋊C4—4222no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—8441no—
151C23—8441no—
152C23—8441no—
153C23—8441no—
154C23—8441no—
155C23—8441no—
156C23—4222no—
157C23#1680995112yescomplemented
158C23C2×D84411yescharacteristic
159Q8—8442no—
160Q8—16881no—
161D4—8442no—
162D4—8448no—
163D4—4224no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—8441no—
168C2×C4—8441no—
169C2×C4—16881no—
170C2×C4—16881no—
171C2×C4—4222no—
172C2×C4—4222no—
173C2×C4—4222no—
174C2×C4D4×D21112yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—8442no—
184C22—8442no—
185C22—8442no—
186C22—16881no—
187C22—16881no—
188C22—16881no—
189C22—16881no—
190C22#2026994711yescharacteristic
191C22C2×#1680995111yescharacteristic
192C22#2027001011yescharacteristic
193C4—22no—
194C4—41no—
195C4—8442no—
196C4—8442no—
197C4—8442no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—16882no—
202C2#2721328711yescharacteristic
203C2#2721331611yescharacteristic
204C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#323×C211C2
socleC22×C211#120
prime radical#27213055C2
hypercenter#323D211
hypocenterC211#1596
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #27213772C2semidirect
2. #27213299C2semidirect
4. #27212884C2semidirect
5. #323×C211C2semidirect
21. C2×Dic422D4semidirect
32. C4⋊Dic211D4semidirect
66. C23×C211SD16semidirect
102. C211#1596semidirect
108. #323D211semidirect
157. C23#16809951semidirect

Maximal subgroups

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

subgroupindex
#272137722
#272132992
#272130552
#272128842
#323×C2112
#1596211