Finite Group Databasebeta

← #41013785

Subgroups — #41013785

Order 27008.

360 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269965C211yesmaximal, characteristic
2#27213620C211yesmaximal, characteristic
3#27213156C211yesmaximal, characteristic, complemented
4#27213154C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic
6#27212906C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269965—22no—
12#20269965C414yescomplemented
13C4×D844—22no—
14#20269939—22no—
15#20269936C2211yescharacteristic
16C2×(C2×D211)⋊C4—22no—
17C22⋊C4×D211—22no—
18#20269902—22no—
19#20269900—22no—
20#20269893—22no—
21C2×M16×C211C2211yescharacteristic
22#117×C211C2211yescharacteristic
23C2×C42×C211C2211yescharacteristic
24C23×D211D412yes—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422C2×C412yescomplemented
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844C2×C412yes—
36C2×D844D412yes—
37C22×Dic211—21no—
38C2×C4×D211—24no—
39#16809946—21no—
40#16809946—21no—
41C2×C211⋊C8—21no—
42C4⋊Dic211D412yes—
43C22⋊Dic211—21no—
44(C2×D211)⋊C4—24no—
45(C2×D211)⋊C4—28no—
46C4×Dic211—21no—
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49M16×C211—21no—
50M16×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211D412yes—
53C22⋊C4×C211—21no—
54C42×C211—21no—
55C42×C211—21no—
56C42×C211C2×C412yescomplemented
57C22×D211—22no—
58C22×D211—22no—
59C22×D211—44no—
60C22×D211C22⋊C414yes—
61Dic422—22no—
62Dic422—22no—
63Dic422—41no—
64Dic422—41no—
65D844—22no—
66D844—22no—
67D844—22no—
68D844—22no—
69C2×Dic211—22no—
70C2×Dic211—22no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C4×D211—84no—
74C211⋊C8—41no—
75C23×C211C2×D411yescharacteristic
76C2×C4×C211—22no—
77C2×C4×C211—22no—
78C2×C4×C211—42no—
79C2×C4×C211C22×C411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—41no—
86C22×C211—22no—
87C22×C211C2×C22⋊C411yescharacteristic
88C22×C211#12011yescharacteristic
89C22×C211#12311yescharacteristic
90C2×D211—24no—
91C2×D211—44no—
92C2×D211—44no—
93C2×D211—82no—
94C4×C211—42no—
95C4×C211—42no—
96C4×C211C4×D411yescharacteristic
97C4×C211C4×D411yescharacteristic
98C4×C211#12111yescharacteristic
99C4×C211#12111yescharacteristic
100Dic211—42no—
101Dic211—42no—
102C2×C211—21no—
103C2×C211—21no—
104C2×C211#38611yescharacteristic
105C2×C211#45512yes—
106D211—44no—
107C211#188511yescharacteristic, complemented, Sylow
108#1885—2111nomaximal, Sylow
109C2×#124—2111no—
110#518—2111no—
111#419—2111no—
112#419—2111no—
113C2×#100—2111no—
114#339—2111no—
115#339D21111yescharacteristic, complemented
116C22×Q8—2111no—
117C22×D4—2111no—
118C2×M16—2111no—
119C2×M16C2×D21111yescharacteristic
120#124—4222no—
121#124—2114no—
122C4×D4—4222no—
123C4×D4—4222no—
124#117—2111no—
125#117C2×D21111yescharacteristic
126C2×C22⋊C4—4222no—
127C2×C22⋊C4—4222no—
128C2×C42C2×D21111yescharacteristic
129#103—4222no—
130#103—4222no—
131#100—4222no—
132C24—2112no—
133C2×Q8—4221no—
134C2×Q8—4221no—
135C2×Q8—4221no—
136C2×Q8—4221no—
137C2×Q8—4221no—
138C2×Q8—2112no—
139C2×D4—4221no—
140C2×D4—4221no—
141C2×D4—4221no—
142C2×D4—4221no—
143C2×D4—2112no—
144C2×D4—2112no—
145C22×C4—21no—
146C22×C4—4224no—
147C22×C4—4221no—
148C22×C4C22×D21111yescharacteristic
149M16—21no—
150M16—21no—
151M16—4221no—
152M16—4221no—
153C2×C8—21no—
154C2×C8—4221no—
155C4⋊C4—2112no—
156C4⋊C4D84412yes—
157C22⋊C4—21no—
158C22⋊C4—4224no—
159C22⋊C4—4228no—
160C22⋊C4—4221no—
161C42—21no—
162C42—21no—
163C42—4221no—
164C42C4×D21112yescomplemented
165C23—8444no—
166C23—4222no—
167C23—4222no—
168C23—2114no—
169C23D4×D21111yescharacteristic
170Q8—8441no—
171Q8—8441no—
172Q8—4222no—
173Q8—4222no—
174D4—4222no—
175D4—4222no—
176D4—4222no—
177D4—4222no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—8441no—
182C2×C4—8441no—
183C2×C4—16884no—
184C2×C4—4222no—
185C2×C4—4222no—
186C2×C4C2×C4×D21111yescharacteristic
187C2×C4D4×D21111yescharacteristic
188C2×C4D4×D21111yescharacteristic
189C2×C4C2×D84411yescharacteristic
190C2×C4#1680996211yescharacteristic
191C2×C4#1680996311yescharacteristic
192C8—41no—
193C8—8441no—
194C22—22no—
195C22—8444no—
196C22—8444no—
197C22—16882no—
198C22—4224no—
199C22C22⋊C4×D21111yescharacteristic
200C22#2026994711yescharacteristic
201C22#2026996211yescharacteristic
202C4—42no—
203C4—42no—
204C4—8442no—
205C4—8442no—
206C4#2026993911yescharacteristic
207C4C4×D84411yescharacteristic
208C4#2026995011yescharacteristic
209C4#2026995211yescharacteristic
210C2—21no—
211C2—21no—
212C2—8444no—
213C2#2721302411yescharacteristic
214C2#2721332512yes—
subgroupG/H
centerC22#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211#120
prime radicalC22×D844C22
cosocleC22×C4×C211C23
hypercenter#339D211
hypocenterC211#1885
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
3. #27213156C2semidirect
4. #27213154C2semidirect
6. #27212906C2semidirect
7. #339×C211C2semidirect
12. #20269965C4semidirect
30. C2×Dic422C2×C4semidirect
56. C42×C211C2×C4semidirect
107. C211#1885semidirect
115. #339D211semidirect
164. C42C4×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699652
#272136202
#272131562
#272131542
C2×#202698932
#272129062
#339×C2112
#1885211