Finite Group Databasebeta

← #41014554

Subgroups — #41014554

Order 27008.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213151C212yesmaximal
3#27212915C211yesmaximal, characteristic, complemented
4#27212906C212yesmaximal, complemented
5#390×C211C211yesmaximal, characteristic
6C2×#16809946C2212yes—
7#20269976—22no—
8#20269968—22no—
9#20269956—22no—
10#20269936C2212yes—
11#20269898—22no—
12#20269898—22no—
13C22×D4×C211C2211yescharacteristic
14C2×C4⋊C4×C211C2211yescharacteristic
15C2×C22⋊C4×C211—22no—
16C2×C42×C211C2211yescharacteristic
17C2×Dic422—22no—
18#16809946—22no—
19#16809946—22no—
20C2×C211⋊C8—22no—
21C4⋊Dic211—22no—
22C4⋊Dic211D412yes—
23C4⋊Dic211D412yes—
24Dic211⋊C4—24no—
25C22⋊Dic211—22no—
26C4×Dic211—22no—
27C24×C211—21no—
28C2×D4×C211—22no—
29C2×D4×C211—22no—
30C2×D4×C211—21no—
31C2×D4×C211Q812yescomplemented
32C22×C4×C211—21no—
33C22×C4×C211D412yes—
34C22×C4×C211C2311yescharacteristic
35C4⋊C4×C211—22no—
36C22⋊C4×C211—44no—
37C42×C211—22no—
38Dic422—82no—
39C2×Dic211—22no—
40C2×Dic211—22no—
41C211⋊C8—42no—
42C23×C211—22no—
43C23×C211—42no—
44C23×C211—42no—
45C23×C211—41no—
46C23×C211—41no—
47C23×C211C2×D411yescharacteristic
48D4×C211—24no—
49D4×C211—42no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—41no—
53C2×C4×C211—41no—
54C2×C4×C211—42no—
55C2×C4×C211C2×D412yes—
56C2×C4×C211C2×Q811yescharacteristic
57C2×C4×C211#4112yes—
58C2×C4×C211#4111yescharacteristic
59C22×C211—21no—
60C22×C211—21no—
61C22×C211—44no—
62C22×C211—42no—
63C22×C211—82no—
64C22×C211—81no—
65C22×C211—81no—
66C22×C211#12011yescharacteristic
67C22×C211C22⋊Q811yescharacteristic
68C22×C211#12411yescharacteristic
69C4×C211—42no—
70C4×C211—42no—
71C4×C211C22⋊Q812yes—
72C4×C211#12412yes—
73Dic211—42no—
74Dic211—42no—
75C2×C211—21no—
76C2×C211—21no—
77C2×C211—82no—
78C2×C211#39311yescharacteristic
79C2×C211#45311yescharacteristic
80C2×C211#45411yescharacteristic
81C211#202711yescharacteristic, complemented, Sylow
82#2027—2111nomaximal, Sylow
83#544—2111no—
84#418—2112no—
85#390D21111yescharacteristic
86#343—2111no—
87#339—2112no—
88C22×D4C2×D21111yescharacteristic
89C2×M16—2112no—
90C4⋊Q8—4222no—
91#125—4222no—
92C22⋊Q8—4222no—
93#117—2112no—
94C2×C4⋊C4C2×D21111yescharacteristic
95C2×C22⋊C4—22no—
96C2×C42C2×D21111yescharacteristic
97#102—4222no—
98#102—4222no—
99C24—21no—
100C2×Q8—4222no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4Dic42212yescomplemented
105C22×C4—21no—
106C22×C4C22⋊D21112yes—
107C22×C4C22×D21111yescharacteristic
108M16—4222no—
109M16—4222no—
110C2×C8—4222no—
111C4⋊C4—22no—
112C4⋊C4—4222no—
113C4⋊C4—4224no—
114C4⋊C4—2112no—
115C4⋊C4—2112no—
116C22⋊C4—44no—
117C22⋊C4—4222no—
118C42—22no—
119C42—4222no—
120C23—22no—
121C23—42no—
122C23—42no—
123C23—41no—
124C23—41no—
125C23C2×C22⋊D21111yescharacteristic
126Q8—16882no—
127D4—24no—
128D4—42no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—4222no—
135C2×C4—4222no—
136C2×C4D4×D21112yes—
137C2×C4C2×Dic42211yescharacteristic
138C2×C4#1680996112yes—
139C2×C4#1680996211yescharacteristic
140C8—8442no—
141C22—21no—
142C22—21no—
143C22—44no—
144C22—42no—
145C22—82no—
146C22—81no—
147C22—81no—
148C22#2026994511yescharacteristic
149C22#2026995611yescharacteristic
150C22#2026996411yescharacteristic
151C4—42no—
152C4—42no—
153C4—8442no—
154C4—8442no—
155C4#2026995412yes—
156C4#2026996312yes—
157C2—21no—
158C2—21no—
159C2—82no—
160C2#2721305811yescharacteristic
161C2#2721331311yescharacteristic
162C2#2721331811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#390×C211C2
socleC22×C211#120
prime radicalC22×D4×C211C22
hypercenter#390D211
hypocenterC211#2027
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #27213777C2semidirect
3. #27212915C2semidirect
4. #27212906C2semidirect
31. C2×D4×C211Q8semidirect
81. C211#2027semidirect
104. C2×D4Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272131512
#272129152
#272129062
#390×C2112
#2027211