Finite Group Databasebeta

← #41012441

Subgroups — #41012441

Order 27008.

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

normal subgroups only

This group has 202 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#27213482C211yesmaximal, characteristic
3#27213479C211yesmaximal, characteristic
4#27213086C211yesmaximal, characteristic
5#27212882C211yesmaximal, characteristic
6#27212871C211yesmaximal, characteristic
7#328×C211C211yesmaximal, characteristic
8#20269976—21no—
9#20269975—21no—
10#20269967—21no—
11#20269967—21no—
12#20269956C2211yescharacteristic
13#20269954—21no—
14#20269933—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4C2211yescharacteristic
17#20269911—21no—
18#20269910—21no—
19#20269902—21no—
20#20269901—21no—
21C22⋊(C211⋊C8)C2211yescharacteristic
22C211⋊#95—21no—
23C211⋊#95—21no—
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422D411yescharacteristic
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D411yescharacteristic
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43Dic211⋊C4—21no—
44C22⋊Dic211—21no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C2×Q8×C211D411yescharacteristic
48C22×C4×C211—21no—
49C22×C4×C211C2311yescharacteristic
50C2×C8×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211D411yescharacteristic
53C22⋊C4×C211—21no—
54Dic422—41no—
55Dic422—81no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—41no—
62C2×Dic211—42no—
63C2×Dic211—41no—
64C2×Dic211Q1612yes—
65C211⋊C8—41no—
66C23×C211C2×D411yescharacteristic
67Q8×C211—41no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C8×C211—41no—
77C22×C211—21no—
78C22×C211—21no—
79C22×C211#12011yescharacteristic
80C22×C211C2×Q1611yescharacteristic
81C22×C211#13711yescharacteristic
82C4×C211—21no—
83C4×C211—21no—
84C4×C211—41no—
85C4×C211—41no—
86C4×C211—81no—
87Dic211—22no—
88Dic211—41no—
89Dic211—41no—
90Dic211—41no—
91Dic211—81no—
92C2×C211—21no—
93C2×C211—21no—
94C2×C211#45111yescharacteristic
95C2×C211#45411yescharacteristic
96C2×C211#45811yescharacteristic
97C211#161711yescharacteristic, complemented, Sylow
98#1617—2111nomaximal, Sylow
99#544—2111no—
100#484—2111no—
101#484—2111no—
102#400—2111no—
103#328—2111no—
104#328D21111yescharacteristic
105#324—2111no—
106C4⋊Q8—4221no—
107C4⋊Q8—4221no—
108#125—4221no—
109#125—4221no—
110C22⋊Q8—4221no—
111C22⋊Q8—2111no—
112C22⋊Q8C2×D21111yescharacteristic
113#117—4221no—
114C2×C4⋊C4—2111no—
115C2×C4⋊C4—2111no—
116#107—4221no—
117#107—4221no—
118#103—4221no—
119#103—4221no—
120C22⋊C8—2111no—
121C22⋊C8C2×D21111yescharacteristic
122#95—4221no—
123#95—4221no—
124#95C2×D21111yescharacteristic
125C2×Q8—4221no—
126C2×Q8—2111no—
127C2×Q8D84411yescharacteristic
128C22×C4—21no—
129C22×C4—4221no—
130C22×C4—2111no—
131C22×C4—2111no—
132C22×C4C22×D21111yescharacteristic
133C2×C8—21no—
134C2×C8—4221no—
135C4⋊C4—21no—
136C4⋊C4—4221no—
137C4⋊C4—4221no—
138C4⋊C4—4221no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C4⋊C4—2111no—
147C4⋊C4D84411yescharacteristic
148C22⋊C4—21no—
149C22⋊C4—4221no—
150C42—4221no—
151C42—4221no—
152C23D4×D21111yescharacteristic
153Q8—41no—
154Q8—8441no—
155Q8—16881no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—8441no—
163C2×C4—8442no—
164C2×C4—8441no—
165C2×C4—4221no—
166C2×C4—4221no—
167C2×C4—4221no—
168C2×C4—4221no—
169C2×C4—4221no—
170C2×C4—2112no—
171C2×C4D4×D21111yescharacteristic
172C2×C4C2×D84411yescharacteristic
173C8—41no—
174C8—8441no—
175C22—21no—
176C22—21no—
177C22#2026994711yescharacteristic
178C22Q16×D21111yescharacteristic
179C22#2027000611yescharacteristic
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—81no—
185C4—8441no—
186C4—8441no—
187C4—8441no—
188C4—16881no—
189C4—4222no—
190C2—21no—
191C2—21no—
192C2#2721330311yescharacteristic
193C2#2721332011yescharacteristic
194C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#328D211
hypocenterC211#1617
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
97. C211#1617semidirect

Maximal subgroups

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

subgroupindex
#272137752
#272134822
#272134792
#272130862
#272128822
#272128712
#328×C2112
#1617211