Finite Group Databasebeta

← #41014641

Subgroups — #41014641

Order 27008.

298 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 95 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×#20269901C212yesmaximal
2C2×#20269901C212yesmaximal
3#27213044C211yesmaximal, characteristic
4#27212901C211yesmaximal, characteristic
5C2×C4⋊Q8×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2212yes—
7C2×C4⋊Dic211C2211yescharacteristic
8C2×C4⋊Dic211C2211yescharacteristic
9C2×Dic211⋊C4—21no—
10#20269901—24no—
11#20269901—24no—
12C211⋊#95—21no—
13C22×Q8×C211C2212yes—
14C4⋊Q8×C211—24no—
15C2×C4⋊C4×C211—21no—
16C2×C4⋊C4×C211—21no—
17C2×C42×C211C2211yescharacteristic
18C22×Dic211—21no—
19C22×Dic211—21no—
20C2×C211⋊C8—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23C4⋊Dic211—22no—
24C4⋊Dic211—21no—
25C4⋊Dic211D412yes—
26Dic211⋊C4—42no—
27C2×Q8×C211—22no—
28C2×Q8×C211—22no—
29C2×Q8×C211—22no—
30C2×Q8×C211—22no—
31C2×Q8×C211—22no—
32C2×Q8×C211D414yes—
33C22×C4×C211—22no—
34C22×C4×C211D411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37C4⋊C4×C211—24no—
38C4⋊C4×C211—24no—
39C42×C211—22no—
40C2×Dic211—22no—
41C2×Dic211—41no—
42C2×Dic211—41no—
43C2×Dic211—41no—
44C2×Dic211—41no—
45C2×Dic211—41no—
46C211⋊C8—42no—
47C23×C211C2×D411yescharacteristic
48Q8×C211—24no—
49Q8×C211—24no—
50Q8×C211—42no—
51Q8×C211—42no—
52C2×C4×C211—21no—
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—21no—
56C2×C4×C211—24no—
57C2×C4×C211—42no—
58C2×C4×C211—42no—
59C2×C4×C211SD1612yes—
60C2×C4×C211Q1612yes—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D412yes—
63C2×C4×C211#4112yes—
64C2×C4×C211#4111yescharacteristic
65C22×C211#12011yescharacteristic
66C22×C211#12111yescharacteristic
67C22×C211#12411yescharacteristic
68C22×C211C2×SD1611yescharacteristic
69C22×C211C2×Q1611yescharacteristic
70C22×C211#13711yescharacteristic
71C22×C211#13711yescharacteristic
72C4×C211—22no—
73C4×C211—22no—
74C4×C211—44no—
75C4×C211#12112yes—
76C4×C211#12312yes—
77Dic211—42no—
78Dic211—81no—
79C2×C211#39411yescharacteristic
80C2×C211#44811yescharacteristic
81C2×C211#45111yescharacteristic
82C2×C211#46111yescharacteristic
83C2×C211#46211yescharacteristic
84C2×C211#48711yescharacteristic
85C2×C211#49111yescharacteristic
86C211#204311yescharacteristic, complemented, Sylow
87#2043—2111nomaximal, Sylow
88C2×C4⋊Q8D21111yescharacteristic
89C2×#103—2112no—
90C2×#103—2112no—
91#389—2111no—
92#336—2111no—
93C22×Q8C2×D21112yes—
94C22×C8—2112no—
95C4⋊Q8—24no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—4221no—
99C2×C4⋊C4—2111no—
100C2×C4⋊C4—2111no—
101C2×C42C2×D21111yescharacteristic
102#103—4224no—
103#103—4224no—
104#95—4221no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8C22⋊D21114yes—
111C22×C4—22no—
112C22×C4—4221no—
113C22×C4—4221no—
114C22×C4C22⋊D21111yescharacteristic
115C22×C4C22⋊D21111yescharacteristic
116C22×C4C22×D21111yescharacteristic
117C2×C8—4222no—
118C2×C8—4222no—
119C2×C8—4222no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—8442no—
123C4⋊C4—4222no—
124C4⋊C4—4221no—
125C4⋊C4—2112no—
126C42—22no—
127C23C2×C22⋊D21111yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—42no—
131Q8—42no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—24no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—8441no—
140C2×C4—8441no—
141C2×C4—8441no—
142C2×C4—8441no—
143C2×C4—8441no—
144C2×C4—4222no—
145C2×C4#1680994912yes—
146C2×C4#1680995212yes—
147C2×C4D4×D21111yescharacteristic
148C2×C4C2×C22⋊D21112yes—
149C2×C4#1680996112yes—
150C2×C4#1680996111yescharacteristic
151C8—8442no—
152C22#2026994611yescharacteristic
153C22#2026994811yescharacteristic
154C22#2026996411yescharacteristic
155C22C2×#1680994911yescharacteristic
156C22C2×#1680995211yescharacteristic
157C22#2027000811yescharacteristic
158C22#2027000811yescharacteristic
159C4—22no—
160C4—22no—
161C4—44no—
162C4—8442no—
163C4—16881no—
164C4#2026994912yes—
165C4#2026995812yes—
166C2#2721306011yescharacteristic
167C2#2721328511yescharacteristic
168C2#2721330211yescharacteristic
169C2#2721334811yescharacteristic
170C2#2721335311yescharacteristic
171C2#2721349611yescharacteristic
172C2#2721351411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×C4⋊Q8×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC2×C4⋊Q8D211
hypocenterC211#2043
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
86. C211#2043semidirect

Maximal subgroups

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

subgroupindex
C2×#202699012
#272130442
#272129012
C2×C4⋊Q8×C2112
#2043211