Finite Group Databasebeta

← #41012820

Subgroups — #41012820

Order 27008.

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

normal subgroups only

This group has 166 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#27213423C211yesmaximal, characteristic
3#27213352C211yesmaximal, characteristic
4#27213211C211yesmaximal, characteristic
5#27212896C211yesmaximal, characteristic, complemented
6#27212891C211yesmaximal, characteristic, complemented
7#332×C211C211yesmaximal, characteristic, complemented
8C2×#16809951—21no—
9C2×C8×D211—21no—
10#20269977C2211yescharacteristic
11C4×D844C2211yescharacteristic
12#20269910—21no—
13C4⋊(C211⋊C8)C2211yescharacteristic
14#20269902—21no—
15#20269901—21no—
16#20269901—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—
26#16809951—42no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8D411yescharacteristic
29C2×C211⋊C8D411yescharacteristic
30C8×D211—42no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211D411yescharacteristic
34(C2×D211)⋊C4—21no—
35C2×Q8×C211—21no—
36C2×C8×C211—21no—
37C2×C8×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—21no—
40C42×C211C2311yescharacteristic
41C22×D211—21no—
42Dic422—41no—
43Dic422—41no—
44D844—21no—
45D844SD1612yes—
46C2×Dic211—21no—
47C2×Dic211—41no—
48C4×D211—41no—
49C4×D211—41no—
50C211⋊C8—21no—
51C211⋊C8—21no—
52C211⋊C8—41no—
53Q8×C211—41no—
54Q8×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211C2×D411yescharacteristic
57C2×C4×C211C2×D411yescharacteristic
58C2×C4×C211#4111yescharacteristic
59C8×C211—41no—
60C8×C211—41no—
61C22×C211#12111yescharacteristic
62C2×D211—22no—
63C2×D211—41no—
64C2×D211—41no—
65C4×C211—21no—
66C4×C211—81no—
67C4×C211C2×SD1611yescharacteristic
68C4×C211#13511yescharacteristic
69C4×C211#13511yescharacteristic
70C4×C211AGL(1,ℤ/8)11yescharacteristic
71Dic211—41no—
72Dic211—81no—
73C2×C211#46311yescharacteristic
74C2×C211#46511yescharacteristic
75C2×C211#47311yescharacteristic
76D211—42no—
77C211#167911yescharacteristic, complemented, Sylow
78#1679—2111nomaximal, Sylow
79#487—2111no—
80#474—2111no—
81#461—2111no—
82C8×D4—2111no—
83#334—2111no—
84#332—2111no—
85#332D21111yescharacteristic, complemented
86C2×SD16—4221no—
87C22×C8—4221no—
88C4⋊Q8—2111no—
89C4⋊Q8C2×D21111yescharacteristic
90C4×D4—2111no—
91#107—4221no—
92C4⋊C8—2111no—
93C4⋊C8C2×D21111yescharacteristic
94C4⋊C8C2×D21111yescharacteristic
95#103—4221no—
96#103—4221no—
97#103—4221no—
98#102—4221no—
99C22⋊C8—4221no—
100C4×C8—2111no—
101C2×Q8—21no—
102C2×Q8—4221no—
103C2×D4—2111no—
104C22×C4—4221no—
105SD16—8442no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—8442no—
109C2×C8—4221no—
110C2×C8—2111no—
111C2×C8—2111no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—4221no—
115C4⋊C4—4221no—
116C4⋊C4—2111no—
117C22⋊C4—4221no—
118C42C22×D21111yescharacteristic
119C23—4221no—
120Q8—41no—
121Q8—41no—
122Q8—8441no—
123Q8—8441no—
124D4—4221no—
125D4—2112no—
126C2×C4—41no—
127C2×C4—8441no—
128C2×C4—8441no—
129C2×C4—8441no—
130C2×C4—4221no—
131C2×C4D4×D21111yescharacteristic
132C2×C4D4×D21111yescharacteristic
133C2×C4#1680996211yescharacteristic
134C8—41no—
135C8—41no—
136C8—8441no—
137C8—4221no—
138C8—4221no—
139C22—8441no—
140C22—8441no—
141C22—4222no—
142C22#2026995011yescharacteristic
143C4—21no—
144C4—81no—
145C4—8441no—
146C4—16881no—
147C4SD16×D21111yescharacteristic
148C4#2026999811yescharacteristic
149C4#2027000011yescharacteristic
150C4#2027000111yescharacteristic
151C2—8442no—
152C2#2721336011yescharacteristic
153C2#2721337811yescharacteristic
154C2#2721342011yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#332×C211C2
socleC22×C211#121
prime radicalC2×D844D4
hypercenter#332D211
hypocenterC211#1679
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. #27212896C2semidirect
6. #27212891C2semidirect
7. #332×C211C2semidirect
77. C211#1679semidirect
85. #332D211semidirect

Maximal subgroups

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

subgroupindex
#272134962
#272134232
#272133522
#272132112
#272128962
#272128912
#332×C2112
#1679211