Finite Group Databasebeta

← #41014556

Subgroups — #41014556

Order 27008.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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#27213154C212yesmaximal
2#27213051C211yesmaximal, characteristic
3#27212915C211yesmaximal, characteristic
4#27212905C212yesmaximal
5#544×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C2×#16809946C2212yes—
8C2×C4×Dic211C2211yescharacteristic
9#20269900—22no—
10#20269900—22no—
11C211⋊#95—22no—
12C4⋊Q8×C211—22no—
13#125×C211—22no—
14C22⋊Q8×C211—22no—
15#117×C211C2212yes—
16C2×C4⋊C4×C211C2211yescharacteristic
17C2×Dic422—22no—
18C2×Dic422—21no—
19C2×Dic422—22no—
20C2×Dic422Q812yescomplemented
21C22×Dic211—21no—
22C22×Dic211—21no—
23#16809946—22no—
24#16809946—22no—
25C2×C211⋊C8—22no—
26C4×Dic211—22no—
27C2×Q8×C211—22no—
28C22×C4×C211D412yes—
29C22×C4×C211C2311yescharacteristic
30C4⋊C4×C211—22no—
31C4⋊C4×C211—24no—
32C4⋊C4×C211—22no—
33C4⋊C4×C211D412yes—
34C4⋊C4×C211D412yes—
35C22⋊C4×C211—22no—
36C42×C211—22no—
37Dic422—24no—
38Dic422—42no—
39C2×Dic211—22no—
40C2×Dic211—22no—
41C2×Dic211—41no—
42C2×Dic211—41no—
43C2×Dic211—41no—
44C2×Dic211—41no—
45C211⋊C8—42no—
46C23×C211C2×D411yescharacteristic
47Q8×C211—82no—
48C2×C4×C211—22no—
49C2×C4×C211—22no—
50C2×C4×C211—22no—
51C2×C4×C211—42no—
52C2×C4×C211C2×D412yes—
53C2×C4×C211C2×Q811yescharacteristic
54C2×C4×C211#4112yes—
55C2×C4×C211#4111yescharacteristic
56C22×C211—21no—
57C22×C211—21no—
58C22×C211#12011yescharacteristic
59C22×C211C22⋊Q811yescharacteristic
60C22×C211#12411yescharacteristic
61C4×C211—42no—
62C4×C211—42no—
63C4×C211—42no—
64C4×C211C22⋊Q812yes—
65C4×C211#12412yes—
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#544D21111yescharacteristic
76#419—2112no—
77#391—2111no—
78#343—2111no—
79#339—2112no—
80C22×Q8—2111no—
81C2×M16—2112no—
82C4⋊Q8—22no—
83#125—22no—
84C22⋊Q8—22no—
85#117C2×D21112yes—
86C2×C4⋊C4C2×D21111yescharacteristic
87C2×C42—2111no—
88#103—4222no—
89#103—4222no—
90#95—4222no—
91C2×Q8—22no—
92C2×Q8—4222no—
93C2×Q8—4222no—
94C2×Q8—4221no—
95C2×Q8—2112no—
96C22×C4—4221no—
97C22×C4—4221no—
98C22×C4C22⋊D21112yes—
99C22×C4C22×D21111yescharacteristic
100M16—4222no—
101M16—4222no—
102C2×C8—4222no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4C22⋊D21112yes—
107C4⋊C4C22⋊D21112yes—
108C22⋊C4—22no—
109C42—22no—
110C42—4222no—
111C23C2×C22⋊D21111yescharacteristic
112Q8—82no—
113Q8—8442no—
114Q8—4224no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—8441no—
120C2×C4—8441no—
121C2×C4—8441no—
122C2×C4—8441no—
123C2×C4—4222no—
124C2×C4—4222no—
125C2×C4C2×C22⋊D21112yes—
126C2×C4Q8×D21111yescharacteristic
127C2×C4#1680996312yes—
128C2×C4#1680996311yescharacteristic
129C8—8442no—
130C22—21no—
131C22—21no—
132C22#2026994611yescharacteristic
133C22#2026995311yescharacteristic
134C22#2026996611yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—8442no—
139C4—8442no—
140C4#2026995312yes—
141C4#2026996612yes—
142C2—21no—
143C2—21no—
144C2#2721305711yescharacteristic
145C2#2721332411yescharacteristic
146C2#2721332911yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#544×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#544D211
hypocenterC211#2028
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. 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
#272131542
#272130512
#272129152
#272129052
#544×C2112
#2028211