Finite Group Databasebeta

← #41014560

Subgroups — #41014560

Order 27008.

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

normal subgroups only

This group has 246 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
2#27213156C212yesmaximal
3#27213051C211yesmaximal, characteristic
4#27212907C212yesmaximal
5#343×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7#20269976—22no—
8#20269968—22no—
9#20269956—22no—
10#20269936C2212yes—
11C2×C4×Dic211C2211yescharacteristic
12#20269902—22no—
13#20269902—22no—
14C211⋊#95—22no—
15C2×M16×C211C2212yes—
16C2×C4⋊C4×C211C2211yescharacteristic
17C2×Dic422—22no—
18C2×Dic422—22no—
19C2×Dic422—21no—
20C2×Dic422—22no—
21C2×Dic422Q812yescomplemented
22C22×Dic211—21no—
23C22×Dic211—21no—
24C4⋊Dic211—22no—
25C4⋊Dic211D412yes—
26C4⋊Dic211D412yes—
27Dic211⋊C4—24no—
28C22⋊Dic211—22no—
29C4×Dic211—22no—
30C4×Dic211—22no—
31C22×C4×C211D412yes—
32C22×C4×C211C2311yescharacteristic
33M16×C211—22no—
34M16×C211—22no—
35C2×C8×C211—22no—
36C4⋊C4×C211—22no—
37Dic422—24no—
38Dic422—42no—
39Dic422—82no—
40C2×Dic211—22no—
41C2×Dic211—22no—
42C2×Dic211—22no—
43C2×Dic211—22no—
44C2×Dic211—41no—
45C2×Dic211—41no—
46C2×Dic211—41no—
47C2×Dic211—41no—
48C23×C211C2×D411yescharacteristic
49C2×C4×C211—22no—
50C2×C4×C211—42no—
51C2×C4×C211C2×D412yes—
52C2×C4×C211C2×Q811yescharacteristic
53C2×C4×C211#4111yescharacteristic
54C2×C4×C211#4112yes—
55C8×C211—42no—
56C22×C211—21no—
57C22×C211—21no—
58C22×C211#12011yescharacteristic
59C22×C211C22⋊Q811yescharacteristic
60C22×C211#12411yescharacteristic
61C4×C211—42no—
62C4×C211C22⋊Q812yes—
63C4×C211#12412yes—
64Dic211—42no—
65Dic211—42no—
66Dic211—42no—
67Dic211—42no—
68C2×C211—21no—
69C2×C211—21no—
70C2×C211#39311yescharacteristic
71C2×C211#45511yescharacteristic
72C2×C211#45611yescharacteristic
73C211#202811yescharacteristic, complemented, Sylow
74#2028—2111nomaximal, Sylow
75#544—2111no—
76#419—2112no—
77#391—2111no—
78#343D21111yescharacteristic
79#339—2112no—
80C22×Q8—2111no—
81C2×M16C2×D21112yes—
82C4⋊Q8—4222no—
83#125—4222no—
84C22⋊Q8—4222no—
85#117—2112no—
86C2×C4⋊C4C2×D21111yescharacteristic
87C2×C42—2111no—
88#103—4222no—
89#103—4222no—
90#95—4222no—
91C2×Q8—4222no—
92C2×Q8—4222no—
93C2×Q8—4222no—
94C2×Q8—4221no—
95C2×Q8—2112no—
96C22×C4—4221no—
97C22×C4—4221no—
98C22×C4D84412yes—
99C22×C4C22×D21111yescharacteristic
100M16—22no—
101M16—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—4222no—
105C4⋊C4—4224no—
106C4⋊C4—2112no—
107C4⋊C4—2112no—
108C22⋊C4—4222no—
109C42—4222no—
110C42—4222no—
111C23C2×D84411yescharacteristic
112Q8—8442no—
113Q8—16882no—
114Q8—4224no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—8441no—
118C2×C4—8441no—
119C2×C4—8441no—
120C2×C4—8441no—
121C2×C4—4222no—
122C2×C4—4222no—
123C2×C4—4222no—
124C2×C4—4222no—
125C2×C4D4×D21112yes—
126C2×C4Q8×D21111yescharacteristic
127C2×C4#1680996111yescharacteristic
128C2×C4#1680996212yes—
129C8—42no—
130C22—21no—
131C22—21no—
132C22#2026994711yescharacteristic
133C22#2026995711yescharacteristic
134C22#2026996511yescharacteristic
135C4—42no—
136C4—8442no—
137C4—8442no—
138C4—8442no—
139C4—8442no—
140C4#2026995512yes—
141C4#2026996312yes—
142C2—21no—
143C2—21no—
144C2#2721305911yescharacteristic
145C2#2721332611yescharacteristic
146C2#2721333111yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#343D211
hypocenterC211#2028
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
21. C2×Dic422Q8semidirect
73. C211#2028semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272131562
#272130512
#272129072
#343×C2112
#2028211