Finite Group Databasebeta

← #41014377

Subgroups — #41014377

Order 27008.

244 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 83 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#27213699C211yesmaximal, characteristic
2#27213169C211yesmaximal, characteristic
3#27212992C211yesmaximal, characteristic
4#27212907C211yesmaximal, characteristic
5#27212906C211yesmaximal, characteristic
6#27212905C211yesmaximal, characteristic
7#339×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9#20269976C412yescomplemented
10#20269968C412yescomplemented
11#20269954—22no—
12C4×Dic422—22no—
13#20269936C2211yescharacteristic
14C2×C4×Dic211C2211yescharacteristic
15C2×C4×Dic211C2211yescharacteristic
16Dic211⋊C8—22no—
17C211⋊#95—22no—
18C2×M16×C211C2211yescharacteristic
19#117×C211C2211yescharacteristic
20C2×C42×C211C2211yescharacteristic
21C2×Dic422—22no—
22C22×Dic211—21no—
23C22×Dic211D412yes—
24#16809946—21no—
25#16809946—21no—
26C2×C211⋊C8—21no—
27C4⋊Dic211—22no—
28C4⋊Dic211C2×C412yes—
29Dic211⋊C4—22no—
30Dic211⋊C4—22no—
31C22⋊Dic211—21no—
32C4×Dic211—21no—
33C4×Dic211—21no—
34C4×Dic211C2×C411yescharacteristic
35C4×Dic211C2×C411yescharacteristic
36C4×Dic211D412yes—
37C4×Dic211Q812yescomplemented
38C22×C4×C211—21no—
39C22×C4×C211C2311yescharacteristic
40M16×C211—21no—
41M16×C211—21no—
42C2×C8×C211—21no—
43C4⋊C4×C211C2×C412yes—
44C22⋊C4×C211—21no—
45C42×C211—21no—
46C42×C211—21no—
47C42×C211Q812yescomplemented
48Dic422—44no—
49C2×Dic211—22no—
50C2×Dic211—22no—
51C2×Dic211—42no—
52C2×Dic211—42no—
53C2×Dic211C22⋊C412yes—
54C2×Dic211C22⋊C412yes—
55C211⋊C8—41no—
56C23×C211C2×D411yescharacteristic
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211—42no—
60C2×C4×C211C22×C411yescharacteristic
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×Q811yescharacteristic
63C2×C4×C211C2×Q811yescharacteristic
64C2×C4×C211#4111yescharacteristic
65C2×C4×C211#4111yescharacteristic
66C8×C211—41no—
67C22×C211—22no—
68C22×C211C2×C22⋊C411yescharacteristic
69C22×C211C22⋊Q811yescharacteristic
70C22×C211C22⋊Q811yescharacteristic
71C4×C211—42no—
72C4×C211—42no—
73C4×C211C4×Q811yescharacteristic
74C4×C211C4×Q811yescharacteristic
75C4×C211#12411yescharacteristic
76C4×C211C4⋊Q811yescharacteristic
77Dic211—22no—
78Dic211—22no—
79Dic211—42no—
80Dic211—42no—
81C2×C211—21no—
82C2×C211—21no—
83C2×C211#39111yescharacteristic
84C2×C211#42112yes—
85C211#199511yescharacteristic, complemented, Sylow
86#1995—2111nomaximal, Sylow
87#533—2111no—
88#423—2111no—
89#376—2111no—
90#339—2111no—
91#339—2111no—
92#339—2111no—
93#339D21111yescharacteristic
94C2×M16—2111no—
95C2×M16C2×D21111yescharacteristic
96C4⋊Q8—2112no—
97#125—2112no—
98C22⋊Q8—4222no—
99C4×Q8—4222no—
100#117—2111no—
101#117C2×D21111yescharacteristic
102C2×C42—2111no—
103C2×C42—2111no—
104C2×C42C2×D21111yescharacteristic
105C4⋊C8—4222no—
106#95—4222no—
107C2×Q8—4222no—
108C22×C4—21no—
109C22×C4—4221no—
110C22×C4—2112no—
111C22×C4C22×D21111yescharacteristic
112M16—21no—
113M16—21no—
114M16—4221no—
115M16—4221no—
116C2×C8—21no—
117C2×C8—4221no—
118C4⋊C4—4222no—
119C4⋊C4—4222no—
120C4⋊C4—4222no—
121C4⋊C4—2112no—
122C4⋊C4C4×D21112yes—
123C22⋊C4—21no—
124C22⋊C4—4221no—
125C42—21no—
126C42—21no—
127C42—4221no—
128C42—4221no—
129C42—2112no—
130C42—2111no—
131C42—2111no—
132C42—2112no—
133C42Dic42212yescomplemented
134C23D4×D21111yescharacteristic
135Q8—8444no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—8442no—
140C2×C4—8442no—
141C2×C4—4222no—
142C2×C4—4222no—
143C2×C4—2112no—
144C2×C4—2112no—
145C2×C4C2×C4×D21111yescharacteristic
146C2×C4D4×D21111yescharacteristic
147C2×C4Q8×D21111yescharacteristic
148C2×C4C2×Dic42211yescharacteristic
149C2×C4#1680996111yescharacteristic
150C2×C4#1680996211yescharacteristic
151C8—41no—
152C8—8441no—
153C22—22no—
154C22C22⋊C4×D21111yescharacteristic
155C22#2026995411yescharacteristic
156C22#2026995511yescharacteristic
157C4—42no—
158C4—42no—
159C4—8442no—
160C4—8442no—
161C4—4222no—
162C4—4222no—
163C4#2026994211yescharacteristic
164C4C4×Dic42211yescharacteristic
165C4#2026996311yescharacteristic
166C4#2026997611yescharacteristic
167C2—21no—
168C2—21no—
169C2#2721305011yescharacteristic
170C2#2721316112yes—
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1995
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #20269976C4semidirect
10. #20269968C4semidirect
37. C4×Dic211Q8semidirect
47. C42×C211Q8semidirect
85. C211#1995semidirect
133. C42Dic422semidirect

Maximal subgroups

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

subgroupindex
#272136992
#272131692
#272129922
#272129072
#272129062
#272129052
#339×C2112
#1995211