Finite Group Databasebeta

← #41014629

Subgroups — #41014629

Order 27008.

402 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 95 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×#20269898C212yesmaximal, complemented
2C2×#20269898C212yesmaximal, complemented
3#27213044C211yesmaximal, characteristic, complemented
4#27212901C211yesmaximal, characteristic, complemented
5C2×#127×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2212yes—
7C2×C4⋊Dic211C2211yescharacteristic
8C2×C4⋊Dic211C2211yescharacteristic
9C2×Dic211⋊C4—21no—
10#20269898—24no—
11#20269898—24no—
12C211⋊#95—21no—
13C22×D4×C211—21no—
14C22×D4×C211—21no—
15C22×D4×C211C2212yes—
16#127×C211—24no—
17C2×C42×C211C2211yescharacteristic
18C22×Dic211—21no—
19C22×Dic211—21no—
20C2×C211⋊C8—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23C4⋊Dic211—22no—
24C4⋊Dic211—21no—
25C4⋊Dic211D412yescomplemented
26Dic211⋊C4—42no—
27C24×C211—22no—
28C2×D4×C211—24no—
29C2×D4×C211—24no—
30C2×D4×C211—22no—
31C2×D4×C211—22no—
32C2×D4×C211—22no—
33C2×D4×C211—22no—
34C2×D4×C211—22no—
35C2×D4×C211—42no—
36C2×D4×C211—42no—
37C2×D4×C211—42no—
38C2×D4×C211—42no—
39C2×D4×C211D414yes—
40C22×C4×C211D411yescharacteristic
41C22×C4×C211D411yescharacteristic
42C22×C4×C211C2311yescharacteristic
43C42×C211—22no—
44C2×Dic211—22no—
45C2×Dic211—41no—
46C2×Dic211—41no—
47C2×Dic211—41no—
48C2×Dic211—41no—
49C2×Dic211—41no—
50C211⋊C8—42no—
51C23×C211—24no—
52C23×C211—42no—
53C23×C211—42no—
54C23×C211—42no—
55C23×C211—42no—
56C23×C211—42no—
57C23×C211—42no—
58C23×C211C2×D411yescharacteristic
59D4×C211—24no—
60D4×C211—24no—
61D4×C211—48no—
62D4×C211—48no—
63D4×C211—42no—
64D4×C211—42no—
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211D812yes—
70C2×C4×C211SD1612yes—
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D412yes—
73C2×C4×C211#4112yes—
74C2×C4×C211#4111yescharacteristic
75C22×C211—44no—
76C22×C211—44no—
77C22×C211—44no—
78C22×C211—82no—
79C22×C211—82no—
80C22×C211—82no—
81C22×C211—82no—
82C22×C211#12011yescharacteristic
83C22×C211#12111yescharacteristic
84C22×C211#12411yescharacteristic
85C22×C211C2×D811yescharacteristic
86C22×C211C2×SD1611yescharacteristic
87C22×C211AGL(1,ℤ/8)11yescharacteristic
88C22×C211AGL(1,ℤ/8)11yescharacteristic
89C4×C211—22no—
90C4×C211—22no—
91C4×C211#12112yes—
92C4×C211#12312yes—
93Dic211—42no—
94Dic211—81no—
95C2×C211—84no—
96C2×C211#39411yescharacteristic
97C2×C211#44711yescharacteristic
98C2×C211#45011yescharacteristic
99C2×C211#45911yescharacteristic
100C2×C211#46011yescharacteristic
101C2×C211#48611yescharacteristic
102C2×C211#49011yescharacteristic
103C211#204111yescharacteristic, complemented, Sylow
104#2041—2111nomaximal, Sylow
105C2×#127D21111yescharacteristic
106C2×#102—2112no—
107C2×#102—2112no—
108#389—2111no—
109#336—2111no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4C2×D21112yes—
113C22×C8—2112no—
114#127—24no—
115C2×C4⋊C4—4221no—
116C2×C4⋊C4—2111no—
117C2×C4⋊C4—2111no—
118C2×C42C2×D21111yescharacteristic
119#102—4224no—
120#102—4224no—
121#95—4221no—
122C24—22no—
123C2×D4—24no—
124C2×D4—24no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4C22⋊D21114yes—
135C22×C4—4221no—
136C22×C4—4221no—
137C22×C4C22⋊D21111yescharacteristic
138C22×C4C22⋊D21111yescharacteristic
139C22×C4C22×D21111yescharacteristic
140C2×C8—4222no—
141C2×C8—4222no—
142C2×C8—4222no—
143C4⋊C4—8442no—
144C4⋊C4—4222no—
145C4⋊C4—4221no—
146C4⋊C4—2112no—
147C42—22no—
148C23—24no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23C2×C22⋊D21111yescharacteristic
156D4—24no—
157D4—24no—
158D4—48no—
159D4—48no—
160D4—42no—
161D4—42no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—8441no—
167C2×C4—8441no—
168C2×C4—8441no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—4222no—
172C2×C4#1680994712yes—
173C2×C4#1680995012yes—
174C2×C4D4×D21111yescharacteristic
175C2×C4C2×C22⋊D21112yes—
176C2×C4#1680996112yes—
177C2×C4#1680996111yescharacteristic
178C8—8442no—
179C22—44no—
180C22—44no—
181C22—44no—
182C22—82no—
183C22—82no—
184C22—82no—
185C22—82no—
186C22#2026994611yescharacteristic
187C22#2026994811yescharacteristic
188C22#2026996411yescharacteristic
189C22C2×#1680994711yescharacteristic
190C22C2×#1680995011yescharacteristic
191C22#2027000311yescharacteristic
192C22#2027000311yescharacteristic
193C4—22no—
194C4—22no—
195C4—8442no—
196C4—16881no—
197C4#2026994912yes—
198C4#2026995812yes—
199C2—84no—
200C2#2721306011yescharacteristic
201C2#2721328011yescharacteristic
202C2#2721329711yescharacteristic
203C2#2721333811yescharacteristic
204C2#2721334311yescharacteristic
205C2#2721349211yescharacteristic
206C2#2721351011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×#127×C211C2
socleC23×C211C2×D4
hypercenterC2×#127D211
hypocenterC211#2041
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#20269898C2semidirect
2. C2×#20269898C2semidirect
3. #27213044C2semidirect
4. #27212901C2semidirect
25. C4⋊Dic211D4semidirect
103. C211#2041semidirect

Maximal subgroups

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

subgroupindex
C2×#202698982
#272130442
#272129012
C2×#127×C2112
#2041211