Finite Group Databasebeta

← #41012404

Subgroups — #41012404

Order 27008.

210 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213775C211yesmaximal, characteristic
2#27213476C211yesmaximal, characteristic
3#27213077C211yesmaximal, characteristic
4#27212884C211yesmaximal, characteristic
5#27212883C211yesmaximal, characteristic
6#27212871C211yesmaximal, characteristic
7#483×C211C211yesmaximal, characteristic
8#20269976—21no—
9#20269975—21no—
10#20269967—21no—
11#20269967—21no—
12#20269956C2211yescharacteristic
13#20269954—21no—
14#20269933—21no—
15C2×Dic211⋊C4—21no—
16C2×Dic211⋊C4C2211yescharacteristic
17#20269908—21no—
18#20269900—21no—
19C22⋊(C211⋊C8)C2211yescharacteristic
20C211⋊#95—21no—
21C211⋊#95C2211yescharacteristic
22C22⋊Q8×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#106×C211—21no—
25#103×C211—21no—
26C22⋊C8×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422D411yescharacteristic
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35Dic211⋊C4—21no—
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—42no—
43C22⋊Dic211—21no—
44C4×Dic211—21no—
45C4×Dic211—21no—
46C2×Q8×C211D411yescharacteristic
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49C2×C8×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D411yescharacteristic
54C22⋊C4×C211—21no—
55Dic422—41no—
56Dic422—81no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—22no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211SD1612yes—
67C211⋊C8—41no—
68C23×C211C2×D411yescharacteristic
69Q8×C211—41no—
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C8×C211—41no—
79C22×C211—21no—
80C22×C211—21no—
81C22×C211#12011yescharacteristic
82C22×C211C2×SD1611yescharacteristic
83C22×C211#13711yescharacteristic
84C4×C211—21no—
85C4×C211—21no—
86C4×C211—41no—
87C4×C211—41no—
88C4×C211—81no—
89Dic211—22no—
90Dic211—42no—
91Dic211—41no—
92Dic211—41no—
93Dic211—41no—
94C2×C211—21no—
95C2×C211—21no—
96C2×C211#44811yescharacteristic
97C2×C211#45611yescharacteristic
98C2×C211#45811yescharacteristic
99C211#161111yescharacteristic, complemented, Sylow
100#1611—2111nomaximal, Sylow
101#544—2111no—
102#483—2111no—
103#483D21111yescharacteristic
104#398—2111no—
105#328—2111no—
106#328—2111no—
107#324—2111no—
108C4⋊Q8—4221no—
109C4⋊Q8—4221no—
110#125—4221no—
111#125—4221no—
112C22⋊Q8—4221no—
113C22⋊Q8—2111no—
114C22⋊Q8C2×D21111yescharacteristic
115#117—4221no—
116C2×C4⋊C4—4221no—
117C2×C4⋊C4—2111no—
118C2×C4⋊C4C2×D21111yescharacteristic
119#106—21no—
120#106—4221no—
121#103—21no—
122#103—4221no—
123C22⋊C8—2111no—
124C22⋊C8C2×D21111yescharacteristic
125#95—4221no—
126#95—2111no—
127C2×Q8—4221no—
128C2×Q8—2111no—
129C2×Q8C22⋊D21111yescharacteristic
130C22×C4—21no—
131C22×C4—4221no—
132C22×C4—2111no—
133C22×C4—2111no—
134C22×C4C22×D21111yescharacteristic
135C2×C8—21no—
136C2×C8—4221no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—8442no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C4⋊C4—2111no—
150C4⋊C4C22⋊D21111yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—4221no—
153C42—4221no—
154C42—4221no—
155C23D4×D21111yescharacteristic
156Q8—41no—
157Q8—8441no—
158Q8—16881no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—8441no—
166C2×C4—8441no—
167C2×C4—8441no—
168C2×C4—4222no—
169C2×C4—4221no—
170C2×C4—4221no—
171C2×C4—4221no—
172C2×C4—4221no—
173C2×C4—4221no—
174C2×C4—2112no—
175C2×C4D4×D21111yescharacteristic
176C2×C4C2×C22⋊D21111yescharacteristic
177C8—41no—
178C8—8441no—
179C22—21no—
180C22—21no—
181C22#2026994511yescharacteristic
182C22SD16×D21111yescharacteristic
183C22#2027000611yescharacteristic
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—81no—
189C4—8442no—
190C4—8441no—
191C4—8441no—
192C4—8441no—
193C4—4222no—
194C2—21no—
195C2—21no—
196C2#2721328311yescharacteristic
197C2#2721332711yescharacteristic
198C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#483×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#483D211
hypocenterC211#1611
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
99. C211#1611semidirect

Maximal subgroups

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

subgroupindex
#272137752
#272134762
#272130772
#272128842
#272128832
#272128712
#483×C2112
#1611211