Finite Group Databasebeta

← #41014566

Subgroups — #41014566

Order 27008.

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

normal subgroups only

This group has 290 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#27213772C211yesmaximal, characteristic
2#27213007C212yesmaximal
3#27213007C211yesmaximal, characteristic
4#27212912C212yesmaximal
5#342×C211C211yesmaximal, characteristic
6C23×Dic211C2211yescharacteristic
7#20269956—22no—
8#20269954—24no—
9C2×C22⋊Dic211C2211yescharacteristic
10C2×C22⋊Dic211C2212yes—
11C211⋊#95—24no—
12C211⋊#95—22no—
13C2×C22⋊C4×C211C2212yes—
14C2×C22⋊C4×C211C2211yescharacteristic
15C2×Dic422—22no—
16C22×Dic211—22no—
17C22×Dic211—21no—
18C22×Dic211—21no—
19C22×Dic211—21no—
20C22×Dic211—22no—
21C22×Dic211D414yes—
22C22×Dic211Q812yescomplemented
23C4⋊Dic211—22no—
24Dic211⋊C4—24no—
25C22⋊Dic211—22no—
26C22⋊Dic211—22no—
27C22⋊Dic211—22no—
28C24×C211C2311yescharacteristic
29C22×C4×C211—22no—
30C22×C4×C211D412yes—
31C22⋊C4×C211—22no—
32C22⋊C4×C211—22no—
33C22⋊C4×C211—22no—
34Dic422—82no—
35C2×Dic211—24no—
36C2×Dic211—24no—
37C2×Dic211—22no—
38C2×Dic211—41no—
39C2×Dic211—42no—
40C2×Dic211—41no—
41C2×Dic211—44no—
42C2×Dic211—42no—
43C2×Dic211—41no—
44C23×C211—42no—
45C23×C211C2×D412yes—
46C23×C211C2×D411yescharacteristic
47C23×C211C2×Q811yescharacteristic
48C23×C211#4111yescharacteristic
49C23×C211#4112yes—
50C2×C4×C211—22no—
51C2×C4×C211—42no—
52C2×C4×C211—44no—
53C2×C4×C211—42no—
54C22×C211—24no—
55C22×C211—22no—
56C22×C211—41no—
57C22×C211—41no—
58C22×C211—42no—
59C22×C211#12011yescharacteristic
60C22×C211C22⋊Q812yes—
61C22×C211C22⋊Q811yescharacteristic
62C22×C211#12412yes—
63C22×C211#12411yescharacteristic
64C4×C211—42no—
65C4×C211—82no—
66Dic211—42no—
67Dic211—44no—
68Dic211—81no—
69C2×C211—21no—
70C2×C211—22no—
71C2×C211—22no—
72C2×C211—21no—
73C2×C211#39311yescharacteristic
74C2×C211#45812yes—
75C211#203011yescharacteristic, complemented, Sylow
76#2030—2111nomaximal, Sylow
77#543—2111no—
78#381—2112no—
79#381—2111no—
80#342—2112no—
81#342D21111yescharacteristic
82C23×C4—2111no—
83C22⋊Q8—4224no—
84C22⋊Q8—4222no—
85C2×C22⋊C4—2111no—
86C2×C22⋊C4—2112no—
87C2×C22⋊C4C2×D21112yes—
88C2×C22⋊C4C2×D21111yescharacteristic
89#95—4224no—
90#95—4222no—
91C24C22×D21111yescharacteristic
92C2×Q8—4222no—
93C22×C4—22no—
94C22×C4—4221no—
95C22×C4—4222no—
96C22×C4—4221no—
97C22×C4—4221no—
98C22×C4—4222no—
99C22×C4—2114no—
100C22×C4—2112no—
101C22×C4D84412yes—
102C4⋊C4—4224no—
103C4⋊C4—4222no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—4222no—
108C22⋊C4—4222no—
109C22⋊C4—4222no—
110C23—42no—
111C23D4×D21112yes—
112C23C2×D84411yescharacteristic
113C23Q8×D21111yescharacteristic
114C23#1680996111yescharacteristic
115C23#1680996212yes—
116Q8—16882no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—8441no—
122C2×C4—8442no—
123C2×C4—8441no—
124C2×C4—8444no—
125C2×C4—8442no—
126C2×C4—8441no—
127C2×C4—4224no—
128C2×C4—4224no—
129C2×C4—4222no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#2026994711yescharacteristic
136C22#2026995512yes—
137C22#2026995711yescharacteristic
138C22#2026996312yes—
139C22#2026996511yescharacteristic
140C4—42no—
141C4—82no—
142C4—8442no—
143C4—8444no—
144C4—16881no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2721305911yescharacteristic
150C2#2721333712yes—
subgroupG/H
centerC22#20269947
commutator, prime radical, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#342×C211C2
socleC22×C211#120
hypercenter#342D211
hypocenterC211#2030
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic211Q8semidirect
75. C211#2030semidirect

Maximal subgroups

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

subgroupindex
#272137722
#272130072
#272129122
#342×C2112
#2030211