Finite Group Databasebeta

← #41013846

Subgroups — #41013846

Order 27008.

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

normal subgroups only

This group has 358 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
1#27213657C211yesmaximal, characteristic
2C2×#20269905C212yesmaximal, complemented
3#27213110C211yesmaximal, characteristic, complemented
4#27212906C212yesmaximal, complemented
5C2×#117×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2211yescharacteristic
7C2×#16809946C2212yes—
8#20269960—21no—
9#20269956C412yescomplemented
10#20269951C412yescomplemented
11#20269946—21no—
12C4×C22⋊D211—22no—
13#20269936C2211yescharacteristic
14#20269905—24no—
15C22⋊(C211⋊C8)—22no—
16C23×C4×C211C2211yescharacteristic
17#117×C211—24no—
18C2×C4⋊C4×C211—21no—
19C2×C22⋊C4×C211—21no—
20C2×C42×C211C2212yes—
21#16809962—22no—
22#16809962—22no—
23C2×Dic422C2×C411yescharacteristic
24C2×D844C2×C411yescharacteristic
25C2×C22⋊D211—22no—
26C2×C22⋊D211—21no—
27C2×C4×D211—21no—
28#16809946—22no—
29#16809946—22no—
30C2×C211⋊C8—22no—
31C4⋊Dic211C2×C412yes—
32Dic211⋊C4—22no—
33C22⋊Dic211—22no—
34C22⋊Dic211—21no—
35(C2×D211)⋊C4—22no—
36C4×Dic211—21no—
37C24×C211D411yescharacteristic
38C22×C4×C211—22no—
39C22×C4×C211—22no—
40C22×C4×C211—22no—
41C22×C4×C211C2×C412yes—
42C22×C4×C211D411yescharacteristic
43C22×C4×C211D412yes—
44C22×C4×C211C2311yescharacteristic
45C4⋊C4×C211—24no—
46C22⋊C4×C211—24no—
47C42×C211—22no—
48C42×C211D414yescomplemented
49C22×D211—21no—
50Dic422—22no—
51D844—22no—
52C22⋊D211—44no—
53C22⋊D211—41no—
54C22⋊D211—41no—
55C2×Dic211—22no—
56C2×Dic211—21no—
57C4×D211—42no—
58C211⋊C8—42no—
59C23×C211—21no—
60C23×C211—21no—
61C23×C211—42no—
62C23×C211C22⋊C412yes—
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—44no—
70C2×C4×C211C22⋊C412yes—
71C2×C4×C211C22×C411yescharacteristic
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211#4112yes—
75C22×C211—22no—
76C22×C211—22no—
77C22×C211—22no—
78C22×C211—22no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211C2×C22⋊C411yescharacteristic
82C22×C211#12011yescharacteristic
83C22×C211#12311yescharacteristic
84C2×D211—41no—
85C2×D211—42no—
86C4×C211—22no—
87C4×C211—44no—
88C4×C211C4×D412yes—
89C4×C211#12112yes—
90Dic211—41no—
91Dic211—42no—
92C2×C211—21no—
93C2×C211—21no—
94C2×C211—42no—
95C2×C211#38611yescharacteristic
96C2×C211#42112yes—
97D211—81no—
98C211#189711yescharacteristic, complemented, Sylow
99#1897—2111nomaximal, Sylow
100#525—2111no—
101C2×#117D21111yescharacteristic, complemented
102C2×#104—2112no—
103#407—2111no—
104#339—2112no—
105C2×#41—2111no—
106C23×C4C2×D21111yescharacteristic
107C2×M16—2112no—
108#123—4221no—
109C22⋊Q8—2112no—
110#121—2112no—
111#120—4221no—
112C4×D4—4222no—
113#117—24no—
114#117—2111no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D21112yes—
118#104—4224no—
119C22⋊C8—4222no—
120C24C22⋊D21111yescharacteristic
121#41—4222no—
122#41—4222no—
123C2×Q8—2111no—
124C2×D4—4222no—
125C2×D4—4221no—
126C2×D4—2111no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—4221no—
131C22×C4C4×D21112yes—
132C22×C4C22⋊D21111yescharacteristic
133C22×C4D84412yes—
134C22×C4C22×D21111yescharacteristic
135M16—4222no—
136M16—4222no—
137C2×C8—4222no—
138C4⋊C4—24no—
139C4⋊C4—4222no—
140C4⋊C4—2112no—
141C22⋊C4—24no—
142C22⋊C4—4222no—
143C22⋊C4—4221no—
144C22⋊C4—4222no—
145C42—22no—
146C42—4221no—
147C42C22⋊D21114yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—4221no—
152C23(C2×D211)⋊C412yes—
153C23C2×C22⋊D21111yescharacteristic
154Q8—4222no—
155D4—8444no—
156D4—8441no—
157D4—8441no—
158D4—4222no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—8442no—
166C2×C4—4222no—
167C2×C4—4221no—
168C2×C4(C2×D211)⋊C412yes—
169C2×C4C2×C4×D21111yescharacteristic
170C2×C4C2×C22⋊D21112yes—
171C2×C4C2×D84411yescharacteristic
172C2×C4#1680996212yes—
173C8—8442no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—8442no—
181C22—8441no—
182C22C2×(C2×D211)⋊C411yescharacteristic
183C22#2026994611yescharacteristic
184C22#2026996011yescharacteristic
185C4—22no—
186C4—44no—
187C4—8442no—
188C4—8441no—
189C4C4×C22⋊D21112yes—
190C4#2026995112yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—16881no—
195C2#2721302511yescharacteristic
196C2#2721316412yes—
subgroupG/H
centerC2×C4C2×D844
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#117×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213657C2
cosocleC22×C4×C211C23
hypercenterC2×#117D211
hypocenterC211#1897
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#20269905C2semidirect
3. #27213110C2semidirect
4. #27212906C2semidirect
5. C2×#117×C211C2semidirect
9. #20269956C4semidirect
10. #20269951C4semidirect
48. C42×C211D4semidirect
98. C211#1897semidirect
101. C2×#117D211semidirect
147. C42C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272136572
C2×#202699052
#272131102
#272129062
C2×#117×C2112
#1897211