Finite Group Databasebeta

← #41014331

Subgroups — #41014331

Order 27008.

344 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 113 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×C4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
2C2×#20269901C212yesmaximal, complemented
3C2×#20269898C212yesmaximal
4C4×C4⋊Dic211C211yesmaximal, characteristic, complemented
5C2×#124×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2212yes—
7C2×C4⋊Dic211C2212yes—
8C2×C4×Dic211—21no—
9C4⋊(C211⋊C8)—22no—
10#20269901—24no—
11#20269898—24no—
12C211⋊#95—21no—
13C22×Q8×C211C2211yescharacteristic
14C22×D4×C211C2211yescharacteristic
15#124×C211—22no—
16#124×C211C414yescomplemented
17C2×C22⋊C4×C211—21no—
18C2×C22⋊C4×C211—21no—
19C2×C42×C211C2211yescharacteristic
20C22×Dic211—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23C2×C211⋊C8—22no—
24C4⋊Dic211—22no—
25C4⋊Dic211D414yes—
26C4×Dic211—42no—
27C24×C211—21no—
28C2×Q8×C211—21no—
29C2×Q8×C211—22no—
30C2×Q8×C211—22no—
31C2×Q8×C211C2×C412yes—
32C2×D4×C211—22no—
33C2×D4×C211—21no—
34C2×D4×C211—22no—
35C2×D4×C211C2×C412yes—
36C22×C4×C211—21no—
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211C2311yescharacteristic
40C22⋊C4×C211—24no—
41C22⋊C4×C211—24no—
42C42×C211C2×C412yes—
43C42×C211D412yes—
44C2×Dic211—24no—
45C2×Dic211—42no—
46C2×Dic211—42no—
47C211⋊C8—42no—
48C23×C211—22no—
49C23×C211—42no—
50C23×C211—42no—
51C23×C211—41no—
52C23×C211—41no—
53C23×C211C2×D411yescharacteristic
54Q8×C211—24no—
55Q8×C211—41no—
56Q8×C211—41no—
57D4×C211—24no—
58D4×C211—41no—
59D4×C211—41no—
60C2×C4×C211—22no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—22no—
64C2×C4×C211—42no—
65C2×C4×C211C22⋊C412yes—
66C2×C4×C211C22⋊C412yes—
67C2×C4×C211C22×C411yescharacteristic
68C2×C4×C211C2×D412yes—
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211#4112yes—
71C22×C211—42no—
72C22×C211—42no—
73C22×C211—42no—
74C22×C211—82no—
75C22×C211—82no—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#12112yes—
78C22×C211#13512yes—
79C22×C211AGL(1,ℤ/8)11yescharacteristic
80C22×C211#13711yescharacteristic
81C4×C211—22no—
82C4×C211—22no—
83C4×C211—42no—
84C4×C211C4×D412yes—
85C4×C211#12411yescharacteristic
86C4×C211#12711yescharacteristic
87Dic211—44no—
88C2×C211—82no—
89C2×C211#39011yescharacteristic
90C2×C211#41611yescharacteristic
91C2×C211#41711yescharacteristic
92C2×C211#46312yes—
93C2×C211#46412yes—
94C211#198311yescharacteristic, complemented, Sylow
95#1983—2111nomaximal, Sylow
96C2×#124D21111yescharacteristic
97C2×C4⋊C8—2111no—
98C2×#103—2112no—
99C2×#102—2112no—
100C4×C4⋊C4—2111no—
101C22×Q8C2×D21111yescharacteristic
102C22×D4C2×D21111yescharacteristic
103C22×C8—2112no—
104#124—22no—
105#124Dic21114yescomplemented
106C2×C4⋊C4—2112no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—4221no—
110C2×C42C2×D21111yescharacteristic
111C4⋊C8—4222no—
112#103—4224no—
113#102—4224no—
114#95—4221no—
115C24—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8C2×Dic21112yes—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4C2×Dic21112yes—
124C22×C4—21no—
125C22×C4—4222no—
126C22×C4C22⋊D21111yescharacteristic
127C22×C4C22⋊D21111yescharacteristic
128C22×C4C22×D21111yescharacteristic
129C2×C8—4222no—
130C2×C8—4222no—
131C2×C8—4222no—
132C4⋊C4—4222no—
133C4⋊C4—2114no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C42—8442no—
137C42C2×Dic21112yes—
138C42C22⋊D21112yes—
139C23—22no—
140C23—42no—
141C23—42no—
142C23—41no—
143C23—41no—
144C23C2×C22⋊D21111yescharacteristic
145Q8—24no—
146Q8—41no—
147Q8—41no—
148D4—24no—
149D4—41no—
150D4—41no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—8442no—
157C2×C4—8442no—
158C2×C4—4224no—
159C2×C4C22⋊Dic21112yes—
160C2×C4C22⋊Dic21112yes—
161C2×C4C22×Dic21111yescharacteristic
162C2×C4D4×D21112yes—
163C2×C4C2×C22⋊D21111yescharacteristic
164C2×C4#1680996112yes—
165C8—8442no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—82no—
170C22—82no—
171C22C2×C22⋊Dic21111yescharacteristic
172C22#2026994812yes—
173C22#2026999712yes—
174C22#2027000211yescharacteristic
175C22#2027000711yescharacteristic
176C4—22no—
177C4—22no—
178C4—42no—
179C4—8444no—
180C4D4×Dic21112yes—
181C4#2026996411yescharacteristic
182C4#2026997311yescharacteristic
183C2—82no—
184C2#2721304911yescharacteristic
185C2#2721314211yescharacteristic
186C2#2721314711yescharacteristic
187C2#2721335812yes—
188C2#2721336512yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#124×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#124D211
hypocenterC211#1983
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C211⋊C8)C2semidirect
2. C2×#20269901C2semidirect
4. C4×C4⋊Dic211C2semidirect
16. #124×C211C4semidirect
94. C211#1983semidirect
105. #124Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C211⋊C8)2
C2×#202699012
C2×#202698982
C4×C4⋊Dic2112
C2×#124×C2112
#1983211