Finite Group Databasebeta

← #41014551

Subgroups — #41014551

Order 27008.

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

normal subgroups only

This group has 282 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#27213150C212yesmaximal
2#27213047C211yesmaximal, characteristic
3#27212915C211yesmaximal, characteristic, complemented
4#27212905C212yesmaximal, complemented
5#544×C211C211yesmaximal, characteristic, complemented
6C22×D844C2211yescharacteristic
7C2×#16809946C2212yes—
8C2×(C2×D211)⋊C4—22no—
9C2×C4×Dic211C2211yescharacteristic
10#20269897—22no—
11#20269897—22no—
12C4⋊Q8×C211—22no—
13#125×C211—22no—
14C22⋊Q8×C211—22no—
15#117×C211C2212yes—
16C2×C4⋊C4×C211C2211yescharacteristic
17C23×D211—21no—
18C2×D844—22no—
19C2×D844—22no—
20C2×D844—21no—
21C2×D844Q812yescomplemented
22C22×Dic211—21no—
23#16809946—22no—
24#16809946—22no—
25C2×C211⋊C8—22no—
26(C2×D211)⋊C4—44no—
27C4×Dic211—22no—
28C2×Q8×C211—22no—
29C22×C4×C211D412yes—
30C22×C4×C211C2311yescharacteristic
31C4⋊C4×C211—22no—
32C4⋊C4×C211—24no—
33C4⋊C4×C211—22no—
34C4⋊C4×C211D412yes—
35C4⋊C4×C211D412yes—
36C22⋊C4×C211—22no—
37C42×C211—22no—
38C22×D211—22no—
39C22×D211—42no—
40C22×D211—42no—
41C22×D211—41no—
42C22×D211—41no—
43D844—24no—
44D844—42no—
45C2×Dic211—22no—
46C2×Dic211—41no—
47C2×Dic211—41no—
48C211⋊C8—42no—
49C23×C211C2×D411yescharacteristic
50Q8×C211—82no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
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#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×C211—42no—
71C4×C211—42no—
72C4×C211C22⋊Q812yes—
73C4×C211#12412yes—
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#544D21111yescharacteristic, complemented
84#418—2112no—
85#390—2111no—
86#343—2111no—
87#339—2112no—
88C22×D4—2111no—
89C2×M16—2112no—
90C4⋊Q8—22no—
91#125—22no—
92C22⋊Q8—22no—
93#117C2×D21112yes—
94C2×C4⋊C4C2×D21111yescharacteristic
95C2×C22⋊C4—4222no—
96C2×C42—2111no—
97#102—4222no—
98#102—4222no—
99C24—4221no—
100C2×Q8—22no—
101C2×D4—4222no—
102C2×D4—4221no—
103C2×D4—4222no—
104C2×D4—2112no—
105C22×C4—4221no—
106C22×C4C22⋊D21112yes—
107C22×C4C22×D21111yescharacteristic
108M16—4222no—
109M16—4222no—
110C2×C8—4222no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4C22⋊D21112yes—
115C4⋊C4C22⋊D21112yes—
116C22⋊C4—22no—
117C22⋊C4—8444no—
118C42—22no—
119C42—4222no—
120C23—8442no—
121C23—8442no—
122C23—8441no—
123C23—8441no—
124C23—4222no—
125C23C2×C22⋊D21111yescharacteristic
126Q8—82no—
127D4—8442no—
128D4—4224no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—42no—
133C2×C4—8441no—
134C2×C4—8441no—
135C2×C4—4222no—
136C2×C4C2×C22⋊D21112yes—
137C2×C4Q8×D21111yescharacteristic
138C2×C4#1680996312yes—
139C2×C4#1680996311yescharacteristic
140C8—8442no—
141C22—21no—
142C22—21no—
143C22—8444no—
144C22—8442no—
145C22—16882no—
146C22—16881no—
147C22—16881no—
148C22#2026994611yescharacteristic
149C22#2026995311yescharacteristic
150C22#2026996611yescharacteristic
151C4—42no—
152C4—42no—
153C4—42no—
154C4—8442no—
155C4#2026995312yes—
156C4#2026996612yes—
157C2—21no—
158C2—21no—
159C2—16882no—
160C2#2721305711yescharacteristic
161C2#2721331411yescharacteristic
162C2#2721331911yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#544×C211C2
socleC22×C211#120
prime radicalC22×D844C22
hypercenter#544D211
hypocenterC211#2027
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
3. #27212915C2semidirect
4. #27212905C2semidirect
5. #544×C211C2semidirect
21. C2×D844Q8semidirect
81. C211#2027semidirect
83. #544D211semidirect

Maximal subgroups

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

subgroupindex
#272131502
#272130472
#272129152
#272129052
#544×C2112
#2027211