Finite Group Databasebeta

← #41012665

Subgroups — #41012665

Order 27008.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213870C211yesmaximal, characteristic
2#27213423C211yesmaximal, characteristic
3#27213353C211yesmaximal, characteristic, complemented
4#27212892C211yesmaximal, characteristic, complemented
5#27212878C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic
7#498×C211C211yesmaximal, characteristic, complemented
8C2×#16809952—21no—
9#20269972—22no—
10#20269966—21no—
11#20269957—21no—
12Q8×Dic211—21no—
13C4×Dic422C2211yescharacteristic
14C4×D844C2211yescharacteristic
15#20269935—21no—
16#20269910—21no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269901—21no—
20#20269897—21no—
21C4⋊Q8×C211C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23#106×C211—22no—
24C4×C8×C211C2211yescharacteristic
25C2×Dic422D411yescharacteristic
26C2×D844D411yescharacteristic
27C2×C4×D211—21no—
28#16809952—42no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C4⋊Dic211—22no—
32C4⋊Dic211D411yescharacteristic
33C4⋊Dic211D411yescharacteristic
34Dic211⋊C4—21no—
35(C2×D211)⋊C4—22no—
36(C2×D211)⋊C4—22no—
37(C2×D211)⋊C4—21no—
38C4×Dic211—22no—
39C4×Dic211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×C8×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211D412yes—
47C42×C211C2311yescharacteristic
48C22×D211—21no—
49Dic422—21no—
50Dic422—21no—
51D844—21no—
52D844—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C4×D211—41no—
57C4×D211—41no—
58C211⋊C8—41no—
59C211⋊C8—41no—
60Q8×C211—41no—
61Q8×C211—41no—
62Q8×C211—41no—
63Q8×C211—41no—
64C2×C4×C211—22no—
65C2×C4×C211—41no—
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C8×C211—22no—
70C22×C211#12011yescharacteristic
71C2×D211—41no—
72C2×D211—41no—
73C2×D211—41no—
74C4×C211—21no—
75C4×C211—42no—
76C4×C211—81no—
77C4×C211#13511yescharacteristic
78C4×C211#13511yescharacteristic
79C4×C211AGL(1,ℤ/8)11yescharacteristic
80C4×C211#13711yescharacteristic
81Dic211—41no—
82Dic211—41no—
83Dic211—41no—
84C2×C211#44911yescharacteristic
85C2×C211#45211yescharacteristic
86C2×C211#45611yescharacteristic
87D211—81no—
88C211#165211yescharacteristic, complemented, Sylow
89#1652—2111nomaximal, Sylow
90#556—2111no—
91#498D21111yescharacteristic, complemented
92#474—2111no—
93#462—2111no—
94#332—2111no—
95Q8⋊C8—2111no—
96D4⋊C8—2111no—
97C2×Q16—4221no—
98C4⋊Q8C2×D21111yescharacteristic
99C4⋊Q8C2×D21111yescharacteristic
100#126—4222no—
101#124—4221no—
102C22⋊Q8—4221no—
103C4×Q8—4221no—
104C4×Q8—2111no—
105C4×D4—2111no—
106#117—4221no—
107#107—4221no—
108#106—22no—
109C4⋊C8—2111no—
110C4⋊C8—2111no—
111#103—4221no—
112#102—4221no—
113C4×C8C2×D21111yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—2111no—
117C2×D4—2111no—
118C22×C4—4221no—
119Q16—8442no—
120C2×C8—21no—
121C2×C8—4221no—
122C2×C8—4221no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—4222no—
127C4⋊C4—4221no—
128C4⋊C4—2111no—
129C4⋊C4—2111no—
130C4⋊C4C22⋊D21112yes—
131C22⋊C4—4222no—
132C22⋊C4—4222no—
133C22⋊C4—4221no—
134C42—4222no—
135C42—4221no—
136C42C22×D21111yescharacteristic
137C23—4221no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—4221no—
143Q8—4221no—
144D4—4221no—
145D4—4221no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—8441no—
149C2×C4—8441no—
150C2×C4—4221no—
151C2×C4—4221no—
152C2×C4—4221no—
153C2×C4D4×D21111yescharacteristic
154C2×C4D4×D21111yescharacteristic
155C2×C4C2×C22⋊D21111yescharacteristic
156C8—22no—
157C8—8441no—
158C8—8441no—
159C22—8441no—
160C22—8441no—
161C22—8441no—
162C22#2026994511yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—8441no—
167C4—8441no—
168C4—8441no—
169C4#2026999811yescharacteristic
170C4#2026999811yescharacteristic
171C4#2027000411yescharacteristic
172C4#2027000611yescharacteristic
173C2—16881no—
174C2#2721329011yescharacteristic
175C2#2721330511yescharacteristic
176C2#2721332811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#498×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#498D211
hypocenterC211#1652
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. #27213353C2semidirect
4. #27212892C2semidirect
5. #27212878C2semidirect
7. #498×C211C2semidirect
88. C211#1652semidirect
91. #498D211semidirect

Maximal subgroups

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

subgroupindex
#272138702
#272134232
#272133532
#272128922
#272128782
#272128752
#498×C2112
#1652211