Finite Group Databasebeta

← #41014539

Subgroups — #41014539

Order 27008.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#20269977C211yesmaximal, characteristic
2C2×#20269901C212yesmaximal
3#27212908C212yesmaximal
4#27212901C211yesmaximal, characteristic
5#391×C211C211yesmaximal, characteristic
6C22×Dic422—21no—
7C22×C211⋊C8C2212yes—
8#20269977—24no—
9C2×C4⋊Dic211—21no—
10C2×C4⋊Dic211C2212yes—
11#20269901—24no—
12C22×Q8×C211C2211yescharacteristic
13C2×C4⋊C4×C211C2211yescharacteristic
14C2×C42×C211C2211yescharacteristic
15#95×C211—21no—
16#95×C211—21no—
17C2×Dic422—24no—
18C2×Dic422—42no—
19C2×Dic422—42no—
20C22×Dic211—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23C2×C211⋊C8—22no—
24C4⋊Dic211—24no—
25C4⋊Dic211—22no—
26C4⋊Dic211D414yes—
27C2×Q8×C211—21no—
28C2×Q8×C211—22no—
29C2×Q8×C211—22no—
30C2×Q8×C211Q812yescomplemented
31C22×C4×C211—21no—
32C22×C4×C211—21no—
33C22×C4×C211D411yescharacteristic
34C22×C4×C211D411yescharacteristic
35C22×C4×C211C2311yescharacteristic
36C4⋊C4×C211—22no—
37C42×C211—22no—
38Dic422—48no—
39C2×Dic211—24no—
40C2×Dic211—42no—
41C2×Dic211—42no—
42C211⋊C8—42no—
43C23×C211C2×D411yescharacteristic
44Q8×C211—24no—
45Q8×C211—42no—
46C2×C4×C211—21no—
47C2×C4×C211—21no—
48C2×C4×C211—21no—
49C2×C4×C211—21no—
50C2×C4×C211—22no—
51C2×C4×C211—41no—
52C2×C4×C211—41no—
53C2×C4×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211SD1612yes—
57C2×C4×C211Q1612yes—
58C2×C4×C211C2×D412yes—
59C2×C4×C211C2×Q811yescharacteristic
60C2×C4×C211#4112yes—
61C2×C4×C211#4111yescharacteristic
62C22×C211#12011yescharacteristic
63C22×C211C22⋊Q811yescharacteristic
64C22×C211#12411yescharacteristic
65C22×C211C2×SD1611yescharacteristic
66C22×C211C2×Q1611yescharacteristic
67C22×C211AGL(1,ℤ/8)11yescharacteristic
68C22×C211#13711yescharacteristic
69C4×C211—22no—
70C4×C211—22no—
71C4×C211—42no—
72C4×C211—81no—
73C4×C211C22⋊Q812yes—
74C4×C211#12412yes—
75Dic211—44no—
76C2×C211#39311yescharacteristic
77C2×C211#45011yescharacteristic
78C2×C211#45111yescharacteristic
79C2×C211#47511yescharacteristic
80C2×C211#47711yescharacteristic
81C2×C211#48711yescharacteristic
82C2×C211#48911yescharacteristic
83C211#202411yescharacteristic, complemented, Sylow
84#2024—2111nomaximal, Sylow
85C2×C4⋊Q8—2111no—
86C2×#103—2112no—
87#391D21111yescharacteristic
88#340—2112no—
89#336—2111no—
90C22×Q8—4221no—
91C22×Q8C2×D21111yescharacteristic
92C22×C8—2112no—
93C4⋊Q8—4224no—
94C2×C4⋊C4—4221no—
95C2×C4⋊C4—2112no—
96C2×C4⋊C4C2×D21111yescharacteristic
97C2×C42C2×D21111yescharacteristic
98#103—4224no—
99#95—21no—
100#95—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—8442no—
105C2×Q8—8442no—
106C2×Q8—4224no—
107C2×Q8Dic42212yescomplemented
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—4222no—
111C22×C4C22⋊D21111yescharacteristic
112C22×C4C22⋊D21111yescharacteristic
113C22×C4C22×D21111yescharacteristic
114C2×C8—4222no—
115C2×C8—4222no—
116C2×C8—4222no—
117C4⋊C4—22no—
118C4⋊C4—4224no—
119C4⋊C4—4222no—
120C4⋊C4—2114no—
121C42—22no—
122C23C2×C22⋊D21111yescharacteristic
123Q8—24no—
124Q8—42no—
125Q8—8448no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—8442no—
137C2×C4—8442no—
138C2×C4—4224no—
139C2×C4#1680994912yes—
140C2×C4#1680995212yes—
141C2×C4D4×D21112yes—
142C2×C4C2×Dic42211yescharacteristic
143C2×C4#1680996112yes—
144C2×C4#1680996211yescharacteristic
145C8—8442no—
146C22#2026994511yescharacteristic
147C22#2026995611yescharacteristic
148C22#2026996411yescharacteristic
149C22C2×#1680994911yescharacteristic
150C22C2×#1680995211yescharacteristic
151C22#2027000211yescharacteristic
152C22#2027000711yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—81no—
157C4—8444no—
158C4#2026995412yes—
159C4#2026996312yes—
160C2#2721305811yescharacteristic
161C2#2721329611yescharacteristic
162C2#2721330111yescharacteristic
163C2#2721343111yescharacteristic
164C2#2721344111yescharacteristic
165C2#2721349611yescharacteristic
166C2#2721350511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#391×C211C2
socle, prime radicalC23×C211C2×D4
hypercenter#391D211
hypocenterC211#2024
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C211Q8semidirect
83. C211#2024semidirect
107. C2×Q8Dic422semidirect

Maximal subgroups

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

subgroupindex
C2×#202699772
C2×#202699012
#272129082
#272129012
#391×C2112
#2024211