Finite Group Databasebeta

← #41014405

Subgroups — #41014405

Order 27008.

392 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#20269953C212yesmaximal
2C2×#20269901C212yesmaximal, complemented
3C2×(C2×D211)⋊C8C211yesmaximal, characteristic
4#27212909C211yesmaximal, characteristic, complemented
5C22×Q16×C211C211yesmaximal, characteristic, complemented
6C22×C4×D211C2211yescharacteristic
7C22×C211⋊C8C2211yescharacteristic
8#20269953—24no—
9#20269953—24no—
10C2×C4⋊Dic211C2212yes—
11C2×Dic211⋊C4—22no—
12C2×(C2×D211)⋊C4—22no—
13#20269901—24no—
14(C2×D211)⋊C8—22no—
15C22×Q8×C211C2212yes—
16C2×Q16×C211—24no—
17C2×Q16×C211—22no—
18C22×C8×C211C2211yescharacteristic
19C23×D211D411yescharacteristic
20C22×Dic211—22no—
21C22×Dic211D411yescharacteristic
22C2×C4×D211—21no—
23C2×C4×D211—22no—
24C2×C4×D211—22no—
25C2×C4×D211—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—22no—
28C4⋊Dic211—22no—
29C4⋊Dic211D414yes—
30Dic211⋊C4—28no—
31(C2×D211)⋊C4—28no—
32C2×Q8×C211—22no—
33C2×Q8×C211—22no—
34C2×Q8×C211—22no—
35C2×Q8×C211—22no—
36C2×Q8×C211—22no—
37C2×Q8×C211D414yes—
38C22×C4×C211—22no—
39C22×C4×C211C2311yescharacteristic
40Q16×C211—28no—
41C2×C8×C211—22no—
42C2×C8×C211D412yes—
43C22×D211—22no—
44C22×D211—21no—
45C22×D211—21no—
46C22×D211—22no—
47C22×D211Q1612yescomplemented
48C2×Dic211—24no—
49C2×Dic211—22no—
50C2×Dic211—21no—
51C2×Dic211—42no—
52C2×Dic211—42no—
53C4×D211—41no—
54C4×D211—42no—
55C4×D211—41no—
56C211⋊C8—41no—
57C23×C211C2×D411yescharacteristic
58Q8×C211—24no—
59Q8×C211—24no—
60Q8×C211—42no—
61Q8×C211—42no—
62C2×C4×C211—24no—
63C2×C4×C211—42no—
64C2×C4×C211—42no—
65C2×C4×C211C2×D412yes—
66C2×C4×C211C2×D412yes—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C8×C211—22no—
70C22×C211#12012yes—
71C22×C211#12111yescharacteristic
72C22×C211C2×Q1611yescharacteristic
73C22×C211#13511yescharacteristic
74C22×C211#13712yes—
75C2×D211—24no—
76C2×D211—22no—
77C2×D211—41no—
78C2×D211—42no—
79C2×D211—41no—
80C4×C211—44no—
81C4×C211#12011yescharacteristic
82C4×C211#12112yes—
83C4×C211#12711yescharacteristic
84Dic211—41no—
85Dic211—44no—
86C2×C211#39211yescharacteristic
87C2×C211#45112yes—
88C2×C211#45212yes—
89C2×C211#46711yescharacteristic
90C2×C211#47011yescharacteristic
91D211—42no—
92C211#200111yescharacteristic, complemented, Sylow
93#2001—2111nomaximal, Sylow
94C22×Q16D21111yescharacteristic, complemented
95C2×C22⋊Q8—2112no—
96C2×#103—2112no—
97C2×C22⋊C8—2111no—
98#340—2111no—
99C22×Q8C2×D21112yes—
100C23×C4—2111no—
101C2×Q16—24no—
102C2×Q16—22no—
103C22×C8—2111no—
104C22×C8C2×D21111yescharacteristic
105C22⋊Q8—4224no—
106C22⋊Q8—4224no—
107C2×C4⋊C4—4222no—
108C2×C4⋊C4—2112no—
109C2×C22⋊C4—4222no—
110#103—4224no—
111C22⋊C8—4222no—
112C24—2111no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8C22⋊D21114yes—
119C22×C4—22no—
120C22×C4—4222no—
121C22×C4—4222no—
122C22×C4—4221no—
123C22×C4—4222no—
124C22×C4—4221no—
125C22×C4—2111no—
126C22×C4C22×D21111yescharacteristic
127Q16—28no—
128C2×C8—22no—
129C2×C8—4222no—
130C2×C8—4221no—
131C2×C8C22⋊D21112yes—
132C4⋊C4—4228no—
133C4⋊C4—4222no—
134C4⋊C4—2114no—
135C22⋊C4—4228no—
136C23—4222no—
137C23—4221no—
138C23—4221no—
139C23—4222no—
140C23—2112no—
141C23D4×D21111yescharacteristic
142Q8—24no—
143Q8—24no—
144Q8—42no—
145Q8—42no—
146C2×C4—24no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—8442no—
150C2×C4—8442no—
151C2×C4—8441no—
152C2×C4—8442no—
153C2×C4—8441no—
154C2×C4—4224no—
155C2×C4—4222no—
156C2×C4—4221no—
157C2×C4D4×D21112yes—
158C2×C4C2×C22⋊D21112yes—
159C2×C4C2×C22⋊D21111yescharacteristic
160C2×C4#1680996111yescharacteristic
161C8—22no—
162C8—8441no—
163C22—8441no—
164C22—8442no—
165C22—8441no—
166C22—4222no—
167C22—4224no—
168C22#2026994512yes—
169C22#2026994811yescharacteristic
170C22Q16×D21111yescharacteristic
171C22#2027000011yescharacteristic
172C22#2027000912yes—
173C4—44no—
174C4—8444no—
175C4—8441no—
176C4#2026994611yescharacteristic
177C4#2026994912yes—
178C4#2026997311yescharacteristic
179C2—8442no—
180C2#2721305411yescharacteristic
181C2#2721330012yes—
182C2#2721330712yes—
183C2#2721338411yescharacteristic
184C2#2721340111yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×Q16×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
hypercenterC22×Q16D211
hypocenterC211#2001
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#20269901C2semidirect
4. #27212909C2semidirect
5. C22×Q16×C211C2semidirect
47. C22×D211Q16semidirect
92. C211#2001semidirect
94. C22×Q16D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699532
C2×#202699012
C2×(C2×D211)⋊C82
#272129092
C22×Q16×C2112
#2001211