Finite Group Databasebeta

← #41014393

Subgroups — #41014393

Order 27008.

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

normal subgroups only

This group has 496 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×#20269948C212yesmaximal, complemented
2C2×#20269898C212yesmaximal, complemented
3C2×(C2×D211)⋊C8C211yesmaximal, characteristic, complemented
4#27212909C211yesmaximal, characteristic, complemented
5C22×D8×C211C211yesmaximal, characteristic, complemented
6C22×C22⋊D211—22no—
7C22×C4×D211C2211yescharacteristic
8C22×C211⋊C8C2211yescharacteristic
9#20269948—24no—
10#20269948—24no—
11C2×C4⋊Dic211C2212yes—
12C2×C22⋊Dic211—22no—
13#20269898—24no—
14(C2×D211)⋊C8—22no—
15C22×D4×C211C2212yes—
16C2×D8×C211—24no—
17C2×D8×C211—22no—
18C22×C8×C211C2211yescharacteristic
19C23×D211D411yescharacteristic
20C2×C22⋊D211—28no—
21C2×C22⋊D211—44no—
22C2×C22⋊D211—44no—
23C22×Dic211—22no—
24C22×Dic211D411yescharacteristic
25C2×C4×D211—21no—
26C2×C4×D211—22no—
27C2×C4×D211—22no—
28C2×C4×D211—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—22no—
31C4⋊Dic211—22no—
32C4⋊Dic211D414yescomplemented
33C22⋊Dic211—28no—
34C24×C211—22no—
35C2×D4×C211—22no—
36C2×D4×C211—22no—
37C2×D4×C211—22no—
38C2×D4×C211—22no—
39C2×D4×C211—22no—
40C2×D4×C211D414yescomplemented
41C22×C4×C211C2311yescharacteristic
42D8×C211—28no—
43C2×C8×C211—22no—
44C2×C8×C211D412yescomplemented
45C22×D211—22no—
46C22×D211—21no—
47C22×D211—21no—
48C22×D211—22no—
49C22×D211D812yescomplemented
50C22⋊D211—416no—
51C2×Dic211—22no—
52C2×Dic211—24no—
53C2×Dic211—21no—
54C2×Dic211—42no—
55C2×Dic211—42no—
56C4×D211—41no—
57C4×D211—42no—
58C4×D211—41no—
59C211⋊C8—41no—
60C23×C211—24no—
61C23×C211—42no—
62C23×C211—42no—
63C23×C211—42no—
64C23×C211—42no—
65C23×C211—42no—
66C23×C211—42no—
67C23×C211C2×D411yescharacteristic
68D4×C211—24no—
69D4×C211—24no—
70D4×C211—42no—
71D4×C211—42no—
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D412yes—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211#4111yescharacteristic
76C8×C211—22no—
77C22×C211—44no—
78C22×C211—44no—
79C22×C211—44no—
80C22×C211—82no—
81C22×C211—82no—
82C22×C211—82no—
83C22×C211—82no—
84C22×C211#12012yes—
85C22×C211#12111yescharacteristic
86C22×C211C2×D811yescharacteristic
87C22×C211#13511yescharacteristic
88C22×C211AGL(1,ℤ/8)12yes—
89C2×D211—24no—
90C2×D211—22no—
91C2×D211—41no—
92C2×D211—42no—
93C2×D211—41no—
94C4×C211#12011yescharacteristic
95C4×C211#12112yes—
96C4×C211#12711yescharacteristic
97Dic211—41no—
98Dic211—44no—
99C2×C211—84no—
100C2×C211#39211yescharacteristic
101C2×C211#44712yes—
102C2×C211#44912yes—
103C2×C211#46611yescharacteristic
104C2×C211#46911yescharacteristic
105D211—42no—
106C211#199911yescharacteristic, complemented, Sylow
107#1999—2111nomaximal, Sylow
108C22×D8D21111yescharacteristic, complemented
109C2×#121—2112no—
110C2×#102—2112no—
111C2×C22⋊C8—2111no—
112#340—2111no—
113C22×D4—4222no—
114C22×D4C2×D21112yes—
115C23×C4—2111no—
116C2×D8—24no—
117C2×D8—22no—
118C22×C8—2111no—
119C22×C8C2×D21111yescharacteristic
120#121—4224no—
121#121—4224no—
122C2×C4⋊C4—2112no—
123C2×C22⋊C4—4222no—
124#102—4224no—
125C22⋊C8—4222no—
126C24—22no—
127C24—2111no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—8444no—
134C2×D4—8444no—
135C2×D4—4228no—
136C2×D4C22⋊D21114yescomplemented
137C22×C4—4222no—
138C22×C4—4222no—
139C22×C4—4221no—
140C22×C4—4222no—
141C22×C4—4221no—
142C22×C4—2111no—
143C22×C4C22×D21111yescharacteristic
144D8—28no—
145C2×C8—22no—
146C2×C8—4222no—
147C2×C8—4221no—
148C2×C8C22⋊D21112yescomplemented
149C4⋊C4—4222no—
150C4⋊C4—2114no—
151C22⋊C4—4228no—
152C23—24no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—4222no—
160C23—4221no—
161C23—4221no—
162C23—4222no—
163C23—2112no—
164C23D4×D21111yescharacteristic
165D4—24no—
166D4—24no—
167D4—42no—
168D4—42no—
169D4—84416no—
170C2×C4—8442no—
171C2×C4—8442no—
172C2×C4—8441no—
173C2×C4—8442no—
174C2×C4—8441no—
175C2×C4—4222no—
176C2×C4—4224no—
177C2×C4—4221no—
178C2×C4D4×D21112yes—
179C2×C4C2×C22⋊D21112yes—
180C2×C4C2×C22⋊D21111yescharacteristic
181C2×C4#1680996111yescharacteristic
182C8—22no—
183C8—8441no—
184C22—44no—
185C22—44no—
186C22—44no—
187C22—82no—
188C22—82no—
189C22—82no—
190C22—82no—
191C22—8441no—
192C22—8442no—
193C22—8441no—
194C22—4222no—
195C22—4224no—
196C22#2026994512yes—
197C22#2026994811yescharacteristic
198C22D8×D21111yescharacteristic
199C22#2026999611yescharacteristic
200C22#2027000112yes—
201C4—8444no—
202C4—8441no—
203C4#2026994611yescharacteristic
204C4#2026994912yes—
205C4#2026997311yescharacteristic
206C2—84no—
207C2—8442no—
208C2#2721305411yescharacteristic
209C2#2721327812yes—
210C2#2721328812yes—
211C2#2721337911yescharacteristic
212C2#2721339611yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×D8×C211C2
socleC23×C211C2×D4
hypercenterC22×D8D211
hypocenterC211#1999
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20269948C2semidirect
2. C2×#20269898C2semidirect
3. C2×(C2×D211)⋊C8C2semidirect
4. #27212909C2semidirect
5. C22×D8×C211C2semidirect
32. C4⋊Dic211D4semidirect
40. C2×D4×C211D4semidirect
44. C2×C8×C211D4semidirect
49. C22×D211D8semidirect
106. C211#1999semidirect
108. C22×D8D211semidirect
136. C2×D4C22⋊D211semidirect
148. C2×C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699482
C2×#202698982
C2×(C2×D211)⋊C82
#272129092
C22×D8×C2112
#1999211