Finite Group Databasebeta

← #41012848

Subgroups — #41012848

Order 27008.

160 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213496C211yesmaximal, characteristic, complemented
2#27213437C211yesmaximal, characteristic
3#27213423C211yesmaximal, characteristic
4#27213211C211yesmaximal, characteristic
5#27212898C211yesmaximal, characteristic, complemented
6#27212894C211yesmaximal, characteristic, complemented
7#332×C211C211yesmaximal, characteristic, complemented
8C2×C8×D211—21no—
9#20269977C2211yescharacteristic
10C4×D844C2211yescharacteristic
11#20269910—21no—
12#20269909—21no—
13C4⋊(C211⋊C8)C2211yescharacteristic
14#20269901—21no—
15#20269901—21no—
16#20269899—21no—
17#20269897—21no—
18(C2×D211)⋊C8—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22C4⋊C8×C211C2211yescharacteristic
23C2×Dic422—21no—
24C2×D844D411yescharacteristic
25C2×C4×D211—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8D411yescharacteristic
28C2×C211⋊C8D411yescharacteristic
29C8×D211—42no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211D411yescharacteristic
33(C2×D211)⋊C4—21no—
34C2×Q8×C211—21no—
35C2×C8×C211—21no—
36C2×C8×C211—21no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—21no—
39C42×C211C2311yescharacteristic
40C22×D211—21no—
41Dic422—81no—
42D844—21no—
43D844Q1612yes—
44C2×Dic211—21no—
45C2×Dic211—41no—
46C4×D211—41no—
47C4×D211—41no—
48C211⋊C8—21no—
49C211⋊C8—21no—
50C211⋊C8—41no—
51Q8×C211—41no—
52Q8×C211—41no—
53C2×C4×C211—41no—
54C2×C4×C211C2×D411yescharacteristic
55C2×C4×C211C2×D411yescharacteristic
56C2×C4×C211#4111yescharacteristic
57C8×C211—41no—
58C8×C211—41no—
59C22×C211#12111yescharacteristic
60C2×D211—22no—
61C2×D211—41no—
62C2×D211—41no—
63C4×C211—21no—
64C4×C211—81no—
65C4×C211C2×Q1611yescharacteristic
66C4×C211#13511yescharacteristic
67C4×C211#13511yescharacteristic
68C4×C211AGL(1,ℤ/8)11yescharacteristic
69Dic211—41no—
70Dic211—81no—
71C2×C211#46311yescharacteristic
72C2×C211#46711yescharacteristic
73C2×C211#47111yescharacteristic
74D211—42no—
75C211#168311yescharacteristic, complemented, Sylow
76#1683—2111nomaximal, Sylow
77#487—2111no—
78#476—2111no—
79#474—2111no—
80C8×D4—2111no—
81#335—2111no—
82#333—2111no—
83#332D21111yescharacteristic, complemented
84C22×C8—4221no—
85C4⋊Q8—2111no—
86C4⋊Q8C2×D21111yescharacteristic
87C4×D4—2111no—
88#107—4221no—
89#106—4221no—
90C4⋊C8—2111no—
91C4⋊C8C2×D21111yescharacteristic
92C4⋊C8C2×D21111yescharacteristic
93#103—4221no—
94#103—4221no—
95#102—4221no—
96#102—4221no—
97C22⋊C8—4221no—
98C4×C8—2111no—
99C2×Q8—21no—
100C2×Q8—4221no—
101C2×D4—2111no—
102C22×C4—4221no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—8442no—
106C2×C8—4221no—
107C2×C8—2111no—
108C2×C8—2111no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—4221no—
112C4⋊C4—4221no—
113C4⋊C4—2111no—
114C22⋊C4—4221no—
115C42C22×D21111yescharacteristic
116C23—4221no—
117Q8—41no—
118Q8—41no—
119Q8—16881no—
120D4—4221no—
121D4—2112no—
122C2×C4—41no—
123C2×C4—8441no—
124C2×C4—8441no—
125C2×C4—8441no—
126C2×C4—4221no—
127C2×C4D4×D21111yescharacteristic
128C2×C4D4×D21111yescharacteristic
129C2×C4#1680996211yescharacteristic
130C8—41no—
131C8—41no—
132C8—8441no—
133C8—4221no—
134C8—4221no—
135C22—8441no—
136C22—8441no—
137C22—4222no—
138C22#2026995011yescharacteristic
139C4—21no—
140C4—81no—
141C4—8441no—
142C4—16881no—
143C4Q16×D21111yescharacteristic
144C4#2026999811yescharacteristic
145C4#2026999811yescharacteristic
146C4#2027000111yescharacteristic
147C2—8442no—
148C2#2721336011yescharacteristic
149C2#2721338611yescharacteristic
150C2#2721341211yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#332×C211C2
socleC22×C211#121
prime radicalC2×D844D4
hypercenter#332D211
hypocenterC211#1683
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #27213496C2semidirect
5. #27212898C2semidirect
6. #27212894C2semidirect
7. #332×C211C2semidirect
75. C211#1683semidirect
83. #332D211semidirect

Maximal subgroups

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

subgroupindex
#272134962
#272134372
#272134232
#272132112
#272128982
#272128942
#332×C2112
#1683211