Finite Group Databasebeta

← #41013705

Subgroups — #41013705

Order 27008.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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
5C22×C4⋊C4×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×C211—21no—
16C23×C4×C211C2211yescharacteristic
17C2×C4⋊C4×C211—21no—
18C2×C4⋊C4×C211—24no—
19C2×C4⋊C4×C211C2212yes—
20C2×Dic422—21no—
21C2×Dic422—22no—
22C2×Dic422—22no—
23C2×Dic422C2×C412yescomplemented
24C22×Dic211—21no—
25C22×Dic211—21no—
26C2×C211⋊C8—22no—
27C2×C211⋊C8—22no—
28C2×C211⋊C8—22no—
29C4⋊Dic211—21no—
30C4⋊Dic211C2×C412yescomplemented
31Dic211⋊C4—24no—
32C22⋊Dic211—24no—
33C24×C211D411yescharacteristic
34C22×C4×C211—22no—
35C22×C4×C211—22no—
36C22×C4×C211—24no—
37C22×C4×C211—22no—
38C22×C4×C211—42no—
39C22×C4×C211—42no—
40C22×C4×C211C2×C412yes—
41C22×C4×C211D411yescharacteristic
42C22×C4×C211D412yes—
43C22×C4×C211C2311yescharacteristic
44C4⋊C4×C211—24no—
45C4⋊C4×C211—22no—
46C4⋊C4×C211D414yes—
47Dic422—24no—
48Dic422—42no—
49C2×Dic211—22no—
50C2×Dic211—22no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C211⋊C8—42no—
56C23×C211—21no—
57C23×C211—21no—
58C23×C211—21no—
59C23×C211—21no—
60C23×C211C22⋊C412yes—
61C23×C211SD1612yes—
62C23×C211Q1612yes—
63C23×C211C2×D411yescharacteristic
64C2×C4×C211—21no—
65C2×C4×C211—21no—
66C2×C4×C211—24no—
67C2×C4×C211—24no—
68C2×C4×C211—24no—
69C2×C4×C211—48no—
70C2×C4×C211—42no—
71C2×C4×C211—42no—
72C2×C4×C211C22⋊C412yes—
73C2×C4×C211C22×C411yescharacteristic
74C2×C4×C211C2×D412yes—
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211#4112yes—
77C22×C211—22no—
78C22×C211—22no—
79C22×C211—22no—
80C22×C211—22no—
81C22×C211—22no—
82C22×C211—22no—
83C22×C211#10314yescomplemented
84C22×C211C2×C22⋊C411yescharacteristic
85C22×C211#12011yescharacteristic
86C22×C211#12311yescharacteristic
87C22×C211C2×SD1611yescharacteristic
88C22×C211C2×Q1611yescharacteristic
89C22×C211AGL(1,ℤ/8)11yescharacteristic
90C22×C211#13711yescharacteristic
91C4×C211—22no—
92C4×C211—44no—
93C4×C211C4×D412yes—
94C4×C211#12112yes—
95Dic211—42no—
96Dic211—42no—
97C2×C211—24no—
98C2×C211#38611yescharacteristic
99C2×C211C2×#10311yescharacteristic
100C2×C211#41711yescharacteristic
101C2×C211#45011yescharacteristic
102C2×C211#45111yescharacteristic
103C2×C211#48311yescharacteristic
104C2×C211#48411yescharacteristic
105C211#187211yescharacteristic, complemented, Sylow
106#1872—2111nomaximal, Sylow
107C2×C22⋊Q8—2111no—
108C22×C4⋊C4D21111yescharacteristic
109C2×#103—2112no—
110C2×C22⋊C8—2111no—
111#340—2112no—
112C22×Q8—2111no—
113C23×C4—21no—
114C23×C4C2×D21111yescharacteristic
115C22×C8—2112no—
116C22⋊Q8—4222no—
117C22⋊Q8—2114no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—4221no—
121C2×C4⋊C4—2111no—
122C2×C4⋊C4C2×D21112yes—
123C2×C22⋊C4—4221no—
124#103—4224no—
125C22⋊C8—4222no—
126C24C22⋊D21111yescharacteristic
127C2×Q8—4222no—
128C2×Q8—4222no—
129C2×Q8—4221no—
130C2×Q8—2112no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—4221no—
138C22×C4—4221no—
139C22×C4C4×D21112yes—
140C22×C4C22⋊D21111yescharacteristic
141C22×C4D84412yes—
142C22×C4C22×D21111yescharacteristic
143C2×C8—4222no—
144C2×C8—4222no—
145C2×C8—4222no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—4224no—
149C4⋊C4—4221no—
150C4⋊C4—2112no—
151C4⋊C4C22⋊D21114yes—
152C22⋊C4—4224no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23(C2×D211)⋊C412yes—
158C23#1680995012yes—
159C23#1680995212yes—
160C23C2×C22⋊D21111yescharacteristic
161Q8—8442no—
162Q8—4224no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—48no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—8441no—
172C2×C4—8441no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—4222no—
176C2×C4—4222no—
177C2×C4(C2×D211)⋊C412yes—
178C2×C4C2×C4×D21111yescharacteristic
179C2×C4C2×C22⋊D21112yes—
180C2×C4C2×D84411yescharacteristic
181C2×C4#1680996212yes—
182C8—8442no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22#2026990014yescomplemented
190C22C2×(C2×D211)⋊C411yescharacteristic
191C22#2026994611yescharacteristic
192C22#2026996011yescharacteristic
193C22C2×#1680995011yescharacteristic
194C22C2×#1680995211yescharacteristic
195C22#2027000311yescharacteristic
196C22#2027000811yescharacteristic
197C4—22no—
198C4—44no—
199C4—8442no—
200C4—8442no—
201C4C4×C22⋊D21112yes—
202C4#2026995112yes—
203C2—24no—
204C2#2721302511yescharacteristic
205C2C2×#2026990011yescharacteristic
206C2#2721314511yescharacteristic
207C2#2721329711yescharacteristic
208C2#2721330211yescharacteristic
209C2#2721347611yescharacteristic
210C2#2721348111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×C4⋊C4×C211C2
socleC23×C211C2×D4
prime radicalC24×C211D4
cosocleC22×C4×C211C23
hypercenterC22×C4⋊C4D211
hypocenterC211#1872
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#20269900C2semidirect
4. #27212908C2semidirect
9. #20269956C4semidirect
23. C2×Dic422C2×C4semidirect
30. C4⋊Dic211C2×C4semidirect
83. C22×C211#103semidirect
105. C211#1872semidirect
189. C22#20269900semidirect

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
C22×C4⋊C4×C2112
#1872211