Finite Group Databasebeta

← #41014752

Subgroups — #41014752

Order 27008.

314 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 75 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
2#27213021C212yesmaximal
3#27213018C211yesmaximal, characteristic
4#27212912C212yesmaximal
5#342×C211C211yesmaximal, characteristic
6C23×Dic211C2211yescharacteristic
7#20269956—21no—
8#20269956—21no—
9#20269954—22no—
10#20269954—22no—
11C2×C4⋊Dic211—22no—
12C2×Dic211⋊C4—21no—
13C2×C22⋊Dic211C2212yes—
14C2×C22⋊Dic211C2211yescharacteristic
15C211⋊#95—22no—
16C211⋊#95—21no—
17C2×C22⋊C4×C211C2211yescharacteristic
18C2×C22⋊C4×C211C2212yes—
19C2×Dic422—21no—
20C2×Dic422—21no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C22×Dic211—22no—
24C22×Dic211—42no—
25C22×Dic211D412yes—
26C22×Dic211D412yes—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211—44no—
30Dic211⋊C4—22no—
31Dic211⋊C4—22no—
32Dic211⋊C4—42no—
33C22⋊Dic211—22no—
34C22⋊Dic211—22no—
35C22⋊Dic211—21no—
36C22⋊Dic211Q812yescomplemented
37C24×C211C2311yescharacteristic
38C22×C4×C211—22no—
39C22×C4×C211D411yescharacteristic
40C22×C4×C211D411yescharacteristic
41C22⋊C4×C211—22no—
42C22⋊C4×C211—22no—
43C22⋊C4×C211Q814yescomplemented
44Dic422—81no—
45Dic422—81no—
46C2×Dic211—22no—
47C2×Dic211—24no—
48C2×Dic211—22no—
49C2×Dic211—22no—
50C2×Dic211—22no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—42no—
55C2×Dic211—41no—
56C2×Dic211—42no—
57C2×Dic211—42no—
58C23×C211—41no—
59C23×C211—41no—
60C23×C211C2×D412yes—
61C23×C211C2×D411yescharacteristic
62C23×C211C2×Q811yescharacteristic
63C23×C211C2×Q812yes—
64C23×C211#4111yescharacteristic
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—24no—
68C2×C4×C211—42no—
69C2×C4×C211—42no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C22×C211—22no—
73C22×C211—21no—
74C22×C211—22no—
75C22×C211—21no—
76C22×C211—41no—
77C22×C211—42no—
78C22×C211—41no—
79C22×C211#12011yescharacteristic
80C22×C211C22⋊Q812yes—
81C22×C211C22⋊Q811yescharacteristic
82C22×C211C4⋊Q812yes—
83C22×C211C4⋊Q811yescharacteristic
84C4×C211—44no—
85C4×C211—41no—
86C4×C211—41no—
87Dic211—42no—
88Dic211—42no—
89Dic211—42no—
90Dic211—42no—
91C2×C211—21no—
92C2×C211—22no—
93C2×C211—22no—
94C2×C211—21no—
95C2×C211#39511yescharacteristic
96C2×C211B(4,𝔽2)11yescharacteristic
97C2×C211#45811yescharacteristic
98C211#206211yescharacteristic, complemented, Sylow
99#2062—2111nomaximal, Sylow
100#543—2111no—
101#385—2111no—
102#385—2112no—
103#342—2112no—
104#342D21111yescharacteristic
105C23×C4—2111no—
106C22⋊Q8—4222no—
107C22⋊Q8—4222no—
108C22⋊Q8—4221no—
109C22⋊Q8—4221no—
110C2×C4⋊C4—4222no—
111C2×C4⋊C4—4221no—
112C2×C22⋊C4—2112no—
113C2×C22⋊C4—2111no—
114C2×C22⋊C4C2×D21111yescharacteristic
115C2×C22⋊C4C2×D21112yes—
116#95—4222no—
117#95—4221no—
118C24C22×D21111yescharacteristic
119C2×Q8—4221no—
120C2×Q8—4221no—
121C22×C4—22no—
122C22×C4—8442no—
123C22×C4—4222no—
124C22×C4—4221no—
125C22×C4—4221no—
126C22×C4—2112no—
127C22×C4—2112no—
128C22×C4D84411yescharacteristic
129C22×C4D84411yescharacteristic
130C4⋊C4—8442no—
131C4⋊C4—8444no—
132C4⋊C4—4222no—
133C4⋊C4—4222no—
134C4⋊C4—4221no—
135C4⋊C4—4221no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—4222no—
139C22⋊C4—4222no—
140C22⋊C4—4221no—
141C22⋊C4—2112no—
142C22⋊C4Dic42214yescomplemented
143C23—41no—
144C23—41no—
145C23D4×D21112yes—
146C23C2×D84411yescharacteristic
147C23Q8×D21111yescharacteristic
148C23C2×Dic42212yes—
149C23#1680996111yescharacteristic
150Q8—16881no—
151Q8—16881no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—8441no—
160C2×C4—8442no—
161C2×C4—8441no—
162C2×C4—8441no—
163C2×C4—8441no—
164C2×C4—8442no—
165C2×C4—8442no—
166C2×C4—4224no—
167C2×C4—4222no—
168C2×C4—4222no—
169C2×C4—4222no—
170C2×C4—4222no—
171C22—22no—
172C22—21no—
173C22—22no—
174C22—21no—
175C22—41no—
176C22—42no—
177C22—41no—
178C22#2026994711yescharacteristic
179C22#2026995412yes—
180C22#2026995711yescharacteristic
181C22#2026997612yes—
182C22#2026997711yescharacteristic
183C4—44no—
184C4—41no—
185C4—41no—
186C4—8442no—
187C4—8442no—
188C4—8442no—
189C4—8442no—
190C2—22no—
191C2—21no—
192C2—22no—
193C2—21no—
194C2#2721306711yescharacteristic
195C2#2721333411yescharacteristic
196C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, prime radical, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#342×C211C2
socleC22×C211#120
hypercenter#342D211
hypocenterC211#2062
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
36. C22⋊Dic211Q8semidirect
43. C22⋊C4×C211Q8semidirect
98. C211#2062semidirect
142. C22⋊C4Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137722
#272130212
#272130182
#272129122
#342×C2112
#2062211