Finite Group Databasebeta

← #41013710

Subgroups — #41013710

Order 27008.

360 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 109 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×#20269956C211yesmaximal, characteristic
2C2×#20269900C212yesmaximal, complemented
3C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
4#27212908C212yesmaximal, complemented
5C2×#117×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C22×C211⋊C8C2212yes—
8#20269956—22no—
9#20269956C414yescomplemented
10C2×C4⋊Dic211C2211yescharacteristic
11C2×Dic211⋊C4—21no—
12C2×C22⋊Dic211—21no—
13#20269900—24no—
14C22⋊(C211⋊C8)—22no—
15C23×C4×C211C2211yescharacteristic
16#117×C211—24no—
17C2×C4⋊C4×C211C2212yes—
18C2×C22⋊C4×C211—21no—
19C2×C42×C211—21no—
20C2×Dic422—21no—
21C2×Dic422—22no—
22C2×Dic422—22no—
23C2×Dic422C2×C412yes—
24C22×Dic211—21no—
25C22×Dic211—21no—
26C2×C211⋊C8—22no—
27C2×C211⋊C8—22no—
28C2×C211⋊C8—22no—
29C4⋊Dic211—21no—
30C4⋊Dic211C2×C412yes—
31Dic211⋊C4—24no—
32C22⋊Dic211—24no—
33C24×C211D411yescharacteristic
34C22×C4×C211—22no—
35C22×C4×C211—22no—
36C22×C4×C211—22no—
37C22×C4×C211C2×C412yes—
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D412yes—
40C22×C4×C211C2311yescharacteristic
41C4⋊C4×C211—22no—
42C4⋊C4×C211D414yes—
43C22⋊C4×C211—24no—
44C42×C211—24no—
45Dic422—24no—
46Dic422—42no—
47C2×Dic211—22no—
48C2×Dic211—22no—
49C2×Dic211—41no—
50C2×Dic211—41no—
51C2×Dic211—42no—
52C211⋊C8—42no—
53C23×C211—21no—
54C23×C211—21no—
55C23×C211—22no—
56C23×C211—22no—
57C23×C211C22⋊C412yes—
58C23×C211C2×D411yescharacteristic
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—24no—
62C2×C4×C211—24no—
63C2×C4×C211—24no—
64C2×C4×C211—42no—
65C2×C4×C211—42no—
66C2×C4×C211C22⋊C412yes—
67C2×C4×C211C22×C411yescharacteristic
68C2×C4×C211C2×D412yes—
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211#4112yes—
71C22×C211—22no—
72C22×C211—22no—
73C22×C211—22no—
74C22×C211—42no—
75C22×C211—42no—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#12011yescharacteristic
78C22×C211#12311yescharacteristic
79C22×C211#13512yes—
80C22×C211#13712yes—
81C4×C211—22no—
82C4×C211—44no—
83C4×C211C4×D412yes—
84C4×C211#12112yes—
85Dic211—42no—
86Dic211—42no—
87C2×C211—42no—
88C2×C211#38611yescharacteristic
89C2×C211#41611yescharacteristic
90C2×C211#41911yescharacteristic
91C2×C211#45212yes—
92C2×C211#48512yes—
93C211#187311yescharacteristic, complemented, Sylow
94#1873—2111nomaximal, Sylow
95C2×C22⋊Q8—2111no—
96C2×#117D21111yescharacteristic
97C2×#103—2112no—
98C2×C22⋊C8—2111no—
99#340—2112no—
100C22×Q8—2111no—
101C23×C4C2×D21111yescharacteristic
102C22×C8—2112no—
103C22⋊Q8—4222no—
104C22⋊Q8—2114no—
105#117—24no—
106C2×C4⋊C4—4221no—
107C2×C4⋊C4—2111no—
108C2×C4⋊C4C2×D21112yes—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—4221no—
111C2×C42—21no—
112#103—4224no—
113C22⋊C8—4222no—
114C24C22⋊D21111yescharacteristic
115C2×Q8—4222no—
116C2×Q8—4222no—
117C2×Q8—4221no—
118C2×Q8—2112no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—4221no—
123C22×C4—4221no—
124C22×C4C4×D21112yes—
125C22×C4C22⋊D21111yescharacteristic
126C22×C4D84412yes—
127C22×C4C22×D21111yescharacteristic
128C2×C8—4222no—
129C2×C8—4222no—
130C2×C8—4222no—
131C4⋊C4—22no—
132C4⋊C4—4224no—
133C4⋊C4—4221no—
134C4⋊C4—2112no—
135C4⋊C4C22⋊D21114yes—
136C22⋊C4—24no—
137C22⋊C4—4224no—
138C42—24no—
139C23—21no—
140C23—21no—
141C23—22no—
142C23—22no—
143C23(C2×D211)⋊C412yes—
144C23C2×C22⋊D21111yescharacteristic
145Q8—8442no—
146Q8—4224no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—8441no—
155C2×C4—8441no—
156C2×C4—8442no—
157C2×C4—4222no—
158C2×C4—4222no—
159C2×C4(C2×D211)⋊C412yes—
160C2×C4C2×C4×D21111yescharacteristic
161C2×C4C2×C22⋊D21112yes—
162C2×C4C2×D84411yescharacteristic
163C2×C4#1680996212yes—
164C8—8442no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22C2×(C2×D211)⋊C411yescharacteristic
171C22#2026994611yescharacteristic
172C22#2026996011yescharacteristic
173C22#2026999712yes—
174C22#2027000712yes—
175C4—22no—
176C4—44no—
177C4—8442no—
178C4—8442no—
179C4C4×C22⋊D21112yes—
180C4#2026995112yes—
181C2—42no—
182C2#2721302511yescharacteristic
183C2#2721314011yescharacteristic
184C2#2721315411yescharacteristic
185C2#2721330812yes—
186C2#2721348712yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#117×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC2×#117D211
hypocenterC211#1873
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#20269900C2semidirect
4. #27212908C2semidirect
9. #20269956C4semidirect
93. C211#1873semidirect

Maximal subgroups

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

subgroupindex
C2×#202699562
C2×#202699002
C2×C22⋊(C211⋊C8)2
#272129082
C2×#117×C2112
#1873211