Finite Group Databasebeta

← #41013933

Subgroups — #41013933

Order 27008.

314 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 85 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#27213720C211yesmaximal, characteristic
2#27213624C211yesmaximal, characteristic, complemented
3#27213162C212yesmaximal
4#27212903C212yesmaximal, complemented
5C2×#101×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2211yescharacteristic
7#20269975C412yescomplemented
8#20269966C412yescomplemented
9#20269960—22no—
10#20269953—22no—
11Q8×Dic211—24no—
12#20269936—22no—
13#20269936C2211yescharacteristic
14#20269936C2212yes—
15#20269903—24no—
16C22×Q8×C211C2211yescharacteristic
17C2×M16×C211C2212yes—
18#101×C211—22no—
19#16809962—22no—
20#16809962—22no—
21C2×Dic422C2×C411yescharacteristic
22C2×D844C2×C411yescharacteristic
23C2×C22⋊D211—21no—
24C2×C4×D211—21no—
25C4⋊Dic211—24no—
26C4⋊Dic211—22no—
27C4⋊Dic211—21no—
28Dic211⋊C4—24no—
29C22⋊Dic211—22no—
30C22⋊Dic211—21no—
31(C2×D211)⋊C4—24no—
32C4×Dic211—24no—
33C4×Dic211C2×C412yes—
34C4×Dic211D414yes—
35C2×Q8×C211—22no—
36C2×Q8×C211—22no—
37C2×Q8×C211C2×C412yes—
38C2×Q8×C211D412yes—
39C22×C4×C211D412yes—
40C22×C4×C211C2311yescharacteristic
41M16×C211—22no—
42M16×C211—22no—
43C2×C8×C211—22no—
44C22×D211—21no—
45Dic422—22no—
46D844—22no—
47C22⋊D211—41no—
48C22⋊D211—41no—
49C2×Dic211—22no—
50C2×Dic211—21no—
51C2×Dic211—24no—
52C4×D211—42no—
53C23×C211C2×D411yescharacteristic
54Q8×C211—24no—
55Q8×C211—24no—
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211C22⋊C414yes—
59C2×C4×C211C22×C411yescharacteristic
60C2×C4×C211C2×D412yes—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211#4112yes—
63C8×C211—42no—
64C22×C211—41no—
65C22×C211C2×C22⋊C411yescharacteristic
66C22×C211#12011yescharacteristic
67C22×C211#12311yescharacteristic
68C2×D211—42no—
69C2×D211—41no—
70C4×C211—24no—
71C4×C211C4×D412yes—
72C4×C211#12112yes—
73Dic211—44no—
74Dic211—42no—
75Dic211—41no—
76C2×C211—21no—
77C2×C211—21no—
78C2×C211—21no—
79C2×C211#38611yescharacteristic
80D211—81no—
81C211#191211yescharacteristic, complemented, Sylow
82#1912—2111nomaximal, Sylow
83#536—2111no—
84#519—2111no—
85#421—2112no—
86C2×#101D21111yescharacteristic, complemented
87H(3,ℤ/4)—2112no—
88C2×#41—2111no—
89C22×Q8C2×D21111yescharacteristic
90C2×M16C2×D21112yes—
91C4⋊Q8—2112no—
92#124—2112no—
93#123—4222no—
94C22⋊Q8—4222no—
95C4×Q8—4224no—
96#117—4222no—
97#117—2112no—
98#117—2111no—
99#104—4224no—
100#101—22no—
101#41—4222no—
102#41—4222no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—2111no—
106C2×Q8C4×D21112yes—
107C2×Q8C22⋊D21112yes—
108C2×D4—4221no—
109C2×D4—2111no—
110C22×C4—4221no—
111C22×C4D84412yes—
112C22×C4C22×D21111yescharacteristic
113M16—22no—
114M16—22no—
115C2×C8—22no—
116C4⋊C4—4224no—
117C4⋊C4—4224no—
118C4⋊C4—4222no—
119C4⋊C4—4221no—
120C22⋊C4—4222no—
121C22⋊C4—4221no—
122C22⋊C4—4224no—
123C42—4224no—
124C42—2112no—
125C42—2114no—
126C23—4221no—
127C23C2×D84411yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—4222no—
131D4—8441no—
132D4—8441no—
133D4—4222no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—8442no—
137C2×C4—4222no—
138C2×C4—4221no—
139C2×C4—4224no—
140C2×C4(C2×D211)⋊C414yes—
141C2×C4C2×C4×D21111yescharacteristic
142C2×C4D4×D21112yes—
143C2×C4C2×C22⋊D21111yescharacteristic
144C2×C4#1680996112yes—
145C8—42no—
146C22—41no—
147C22—8442no—
148C22—8441no—
149C22C2×(C2×D211)⋊C411yescharacteristic
150C22#2026994711yescharacteristic
151C22#2026996111yescharacteristic
152C4—24no—
153C4—8444no—
154C4—8442no—
155C4—8441no—
156C4(C22⋊D211)⋊C412yes—
157C4#2026994912yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—16881no—
162C2#2721302311yescharacteristic
subgroupG/H
centerC2#27213023
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC2×C211#386
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenterC2×#101D211
hypocenterC211#1912
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #27213624C2semidirect
4. #27212903C2semidirect
5. C2×#101×C211C2semidirect
7. #20269975C4semidirect
8. #20269966C4semidirect
81. C211#1912semidirect
86. C2×#101D211semidirect

Maximal subgroups

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

subgroupindex
#272137202
#272136242
#272131622
#272129032
C2×#101×C2112
#1912211