Finite Group Databasebeta

← #41014555

Subgroups — #41014555

Order 27008.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213777C211yesmaximal, characteristic, complemented
2#27213152C212yesmaximal
3#27213047C211yesmaximal, characteristic
4#27212907C212yesmaximal, complemented
5#343×C211C211yesmaximal, characteristic, complemented
6C22×D844C2211yescharacteristic
7#20269976—22no—
8#20269968—22no—
9#20269956—22no—
10#20269936C2212yes—
11C2×(C2×D211)⋊C4—22no—
12C2×C4×Dic211C2211yescharacteristic
13#20269899—22no—
14#20269899—22no—
15C2×M16×C211C2212yes—
16C2×C4⋊C4×C211C2211yescharacteristic
17C23×D211—21no—
18C2×Dic422—22no—
19C2×D844—22no—
20C2×D844—22no—
21C2×D844—21no—
22C2×D844Q812yescomplemented
23C22×Dic211—21no—
24C4⋊Dic211—22no—
25C4⋊Dic211D412yes—
26C4⋊Dic211D412yes—
27Dic211⋊C4—24no—
28C22⋊Dic211—22no—
29(C2×D211)⋊C4—44no—
30C4×Dic211—22no—
31C4×Dic211—22no—
32C22×C4×C211D412yes—
33C22×C4×C211C2311yescharacteristic
34M16×C211—22no—
35M16×C211—22no—
36C2×C8×C211—22no—
37C4⋊C4×C211—22no—
38C22×D211—22no—
39C22×D211—42no—
40C22×D211—42no—
41C22×D211—41no—
42C22×D211—41no—
43Dic422—82no—
44D844—24no—
45D844—42no—
46C2×Dic211—22no—
47C2×Dic211—22no—
48C2×Dic211—22no—
49C2×Dic211—41no—
50C2×Dic211—41no—
51C23×C211C2×D411yescharacteristic
52C2×C4×C211—22no—
53C2×C4×C211—42no—
54C2×C4×C211C2×D412yes—
55C2×C4×C211C2×Q811yescharacteristic
56C2×C4×C211#4111yescharacteristic
57C2×C4×C211#4112yes—
58C8×C211—42no—
59C22×C211—21no—
60C22×C211—21no—
61C22×C211#12011yescharacteristic
62C22×C211C22⋊Q811yescharacteristic
63C22×C211#12411yescharacteristic
64C2×D211—44no—
65C2×D211—42no—
66C2×D211—82no—
67C2×D211—81no—
68C2×D211—81no—
69C4×C211—42no—
70C4×C211C22⋊Q812yes—
71C4×C211#12412yes—
72Dic211—42no—
73Dic211—42no—
74Dic211—42no—
75C2×C211—21no—
76C2×C211—21no—
77C2×C211#39311yescharacteristic
78C2×C211#45311yescharacteristic
79C2×C211#45411yescharacteristic
80D211—82no—
81C211#202711yescharacteristic, complemented, Sylow
82#2027—2111nomaximal, Sylow
83#544—2111no—
84#418—2112no—
85#390—2111no—
86#343D21111yescharacteristic, complemented
87#339—2112no—
88C22×D4—2111no—
89C2×M16C2×D21112yes—
90C4⋊Q8—4222no—
91#125—4222no—
92C22⋊Q8—4222no—
93#117—2112no—
94C2×C4⋊C4C2×D21111yescharacteristic
95C2×C22⋊C4—4222no—
96C2×C42—2111no—
97#102—4222no—
98#102—4222no—
99C24—4221no—
100C2×Q8—4222no—
101C2×D4—4222no—
102C2×D4—4221no—
103C2×D4—4222no—
104C2×D4—2112no—
105C22×C4—4221no—
106C22×C4D84412yes—
107C22×C4C22×D21111yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—4222no—
113C4⋊C4—4224no—
114C4⋊C4—2112no—
115C4⋊C4—2112no—
116C22⋊C4—8444no—
117C22⋊C4—4222no—
118C42—4222no—
119C42—4222no—
120C23—8442no—
121C23—8442no—
122C23—8441no—
123C23—8441no—
124C23—4222no—
125C23C2×D84411yescharacteristic
126Q8—16882no—
127D4—8442no—
128D4—4224no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—8441no—
132C2×C4—8441no—
133C2×C4—4222no—
134C2×C4—4222no—
135C2×C4—4222no—
136C2×C4D4×D21112yes—
137C2×C4Q8×D21111yescharacteristic
138C2×C4#1680996111yescharacteristic
139C2×C4#1680996212yes—
140C8—42no—
141C22—21no—
142C22—21no—
143C22—8444no—
144C22—8442no—
145C22—16882no—
146C22—16881no—
147C22—16881no—
148C22#2026994711yescharacteristic
149C22#2026995711yescharacteristic
150C22#2026996511yescharacteristic
151C4—42no—
152C4—8442no—
153C4—8442no—
154C4—8442no—
155C4#2026995512yes—
156C4#2026996312yes—
157C2—21no—
158C2—21no—
159C2—16882no—
160C2#2721305911yescharacteristic
161C2#2721331611yescharacteristic
162C2#2721332111yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#120
prime radicalC22×D844C22
hypercenter#343D211
hypocenterC211#2027
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. #27213777C2semidirect
4. #27212907C2semidirect
5. #343×C211C2semidirect
22. C2×D844Q8semidirect
81. C211#2027semidirect
86. #343D211semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272131522
#272130472
#272129072
#343×C2112
#2027211