Finite Group Databasebeta

← #41012418

Subgroups — #41012418

Order 27008.

288 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213462C212yesmaximal, complemented
2#27213082C211yesmaximal, characteristic, complemented
3#27212880C212yesmaximal, complemented
4#27212870C211yesmaximal, characteristic, complemented
5#534×C211C211yesmaximal, characteristic
6C2×C4⋊Dic211C2211yescharacteristic, complemented
7C2×Dic211⋊C4—21no—
8C2×C22⋊Dic211—21no—
9#20269908—22no—
10#20269898—22no—
11C22⋊(C211⋊C8)C2212yescomplemented
12C211⋊#95C2211yescharacteristic, complemented
13C22×D4×C211—21no—
14#123×C211—22no—
15#121×C211—21no—
16#121×C211C2212yes—
17#120×C211—22no—
18C2×C22⋊C4×C211C2211yescharacteristic
19C22×Dic211—21no—
20C22×Dic211—21no—
21C2×C211⋊C8—22no—
22C4⋊Dic211—22no—
23Dic211⋊C4—42no—
24C22⋊Dic211—42no—
25C24×C211—21no—
26C24×C211D411yescharacteristic
27C2×D4×C211—22no—
28C2×D4×C211—22no—
29C2×D4×C211—21no—
30C2×D4×C211—42no—
31C2×D4×C211—42no—
32C2×D4×C211D412yes—
33C22×C4×C211—21no—
34C22×C4×C211D411yescharacteristic
35C22×C4×C211C2311yescharacteristic
36C4⋊C4×C211—21no—
37C4⋊C4×C211D412yes—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40C22⋊C4×C211—22no—
41C2×Dic211—22no—
42C2×Dic211—41no—
43C2×Dic211—41no—
44C2×Dic211—41no—
45C2×Dic211—41no—
46C2×Dic211—41no—
47C211⋊C8—42no—
48C23×C211—21no—
49C23×C211—21no—
50C23×C211—22no—
51C23×C211—41no—
52C23×C211—41no—
53C23×C211—42no—
54C23×C211—81no—
55C23×C211—81no—
56C23×C211SD1612yes—
57C23×C211C2×D411yescharacteristic
58D4×C211—42no—
59D4×C211—44no—
60D4×C211—42no—
61D4×C211—42no—
62D4×C211—81no—
63C2×C4×C211—21no—
64C2×C4×C211—22no—
65C2×C4×C211—41no—
66C2×C4×C211—42no—
67C2×C4×C211—41no—
68C2×C4×C211C2×D412yes—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—22no—
72C22×C211—42no—
73C22×C211—41no—
74C22×C211—41no—
75C22×C211—42no—
76C22×C211—42no—
77C22×C211—42no—
78C22×C211—81no—
79C22×C211—84no—
80C22×C211#12011yescharacteristic
81C22×C211C2×SD1611yescharacteristic
82C22×C211AGL(1,ℤ/8)11yescharacteristic
83C4×C211—22no—
84C4×C211—42no—
85C4×C211—41no—
86Dic211—42no—
87Dic211—81no—
88C2×C211—21no—
89C2×C211—21no—
90C2×C211—42no—
91C2×C211—82no—
92C2×C211#45011yescharacteristic
93C2×C211#45311yescharacteristic
94C2×C211#45811yescharacteristic
95C211#161311yescharacteristic, complemented, Sylow
96#1613—2111nomaximal, Sylow
97#534D21111yescharacteristic
98#481—2112no—
99#399—2111no—
100#327—2112no—
101#323—2111no—
102C22×D4—21no—
103#123—22no—
104#121—21no—
105#121C2×D21112yes—
106#120—22no—
107C2×C4⋊C4—4221no—
108C2×C4⋊C4—2111no—
109C2×C22⋊C4—4221no—
110C2×C22⋊C4C2×D21111yescharacteristic
111#106—4222no—
112#102—4222no—
113C22⋊C8—2112no—
114#95—2111no—
115C24—21no—
116C24C22⋊D21111yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C22⋊D21112yes—
123C22×C4—21no—
124C22×C4—4221no—
125C22×C4—4221no—
126C22×C4C22⋊D21111yescharacteristic
127C22×C4C22×D21111yescharacteristic
128C2×C8—4222no—
129C4⋊C4—21no—
130C4⋊C4—8442no—
131C4⋊C4—4222no—
132C4⋊C4C22⋊D21112yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—8442no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#1680995012yes—
146C23C2×C22⋊D21111yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—8441no—
158C2×C4—8441no—
159C2×C4—8441no—
160C2×C4—8441no—
161C2×C4—8441no—
162C2×C4—4222no—
163C2×C4C2×C22⋊D21112yes—
164C8—8442no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#2026994611yescharacteristic
177C22C2×#1680995011yescharacteristic
178C22#2027000311yescharacteristic
179C4—22no—
180C4—41no—
181C4—42no—
182C4—8442no—
183C4—16881no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#2721329711yescharacteristic
189C2#2721331411yescharacteristic
190C2#2721333611yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#534×C211C2
socleC22×C211#120
hypercenter#534D211
hypocenterC211#1613
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #27213462C2semidirect
2. #27213082C2semidirect
3. #27212880C2semidirect
4. #27212870C2semidirect
6. C2×C4⋊Dic211C22semidirect
11. C22⋊(C211⋊C8)C22semidirect
12. C211⋊#95C22semidirect
95. C211#1613semidirect

Maximal subgroups

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

subgroupindex
#272134622
#272130822
#272128802
#272128702
#534×C2112
#1613211