Finite Group Databasebeta

← #41014553

Subgroups — #41014553

Order 27008.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213774C211yesmaximal, characteristic, complemented
2#27213152C211yesmaximal, characteristic
3#27213150C211yesmaximal, characteristic
4#27213048C211yesmaximal, characteristic
5#27212914C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269977—21no—
11#20269976—21no—
12#20269969—21no—
13#20269968—21no—
14#20269954—22no—
15#20269936C2211yescharacteristic
16C2×C4⋊Dic211C2211yescharacteristic
17C2×(C2×D211)⋊C4—22no—
18#20269899—21no—
19#20269899—21no—
20#20269897—21no—
21#20269897—21no—
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24C2×C42×C211C2211yescharacteristic
25C23×D211—21no—
26C2×Dic422—22no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844Q812yescomplemented
33C22×Dic211D412yes—
34#16809946—21no—
35#16809946—21no—
36C2×C211⋊C8—21no—
37C4⋊Dic211—22no—
38C4⋊Dic211—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211D411yescharacteristic
41C4⋊Dic211D411yescharacteristic
42Dic211⋊C4—22no—
43Dic211⋊C4—22no—
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—44no—
46C4×Dic211—21no—
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49M16×C211—21no—
50M16×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211D411yescharacteristic
53C4⋊C4×C211D411yescharacteristic
54C22⋊C4×C211—21no—
55C42×C211—21no—
56C42×C211—21no—
57C42×C211—21no—
58C22×D211—22no—
59C22×D211—42no—
60C22×D211—42no—
61C22×D211—41no—
62C22×D211—41no—
63Dic422—82no—
64D844—22no—
65D844—22no—
66D844—41no—
67D844—41no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—22no—
71C2×Dic211—42no—
72C211⋊C8—41no—
73C23×C211C2×D411yescharacteristic
74C2×C4×C211—22no—
75C2×C4×C211—21no—
76C2×C4×C211—21no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×Q811yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—41no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211#12011yescharacteristic
89C22×C211C22⋊Q811yescharacteristic
90C22×C211#12411yescharacteristic
91C2×D211—44no—
92C2×D211—42no—
93C2×D211—82no—
94C2×D211—81no—
95C2×D211—81no—
96C4×C211—42no—
97C4×C211—41no—
98C4×C211—41no—
99C4×C211C22⋊Q811yescharacteristic
100C4×C211C22⋊Q811yescharacteristic
101C4×C211#12411yescharacteristic
102C4×C211#12411yescharacteristic
103Dic211—42no—
104Dic211—41no—
105Dic211—41no—
106C2×C211—21no—
107C2×C211—21no—
108C2×C211#39311yescharacteristic
109C2×C211#45311yescharacteristic
110C2×C211#45411yescharacteristic
111D211—82no—
112C211#202711yescharacteristic, complemented, Sylow
113#2027—2111nomaximal, Sylow
114#544—2111no—
115#418—2111no—
116#418—2111no—
117#390—2111no—
118#343—2111no—
119#339—2111no—
120#339D21111yescharacteristic, complemented
121C22×D4—2111no—
122C2×M16—2111no—
123C2×M16C2×D21111yescharacteristic
124C4⋊Q8—4221no—
125C4⋊Q8—4221no—
126#125—4221no—
127#125—4221no—
128C22⋊Q8—4222no—
129#117—2111no—
130#117C2×D21111yescharacteristic
131C2×C4⋊C4—2111no—
132C2×C22⋊C4—4222no—
133C2×C42C2×D21111yescharacteristic
134#102—4221no—
135#102—4221no—
136#102—4221no—
137#102—4221no—
138C24—4221no—
139C2×Q8—4222no—
140C2×D4—4221no—
141C2×D4—4221no—
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—2112no—
146C22×C4—21no—
147C22×C4—2112no—
148C22×C4C22×D21111yescharacteristic
149M16—21no—
150M16—21no—
151M16—4221no—
152M16—4221no—
153C2×C8—21no—
154C2×C8—4221no—
155C4⋊C4—4222no—
156C4⋊C4—4222no—
157C4⋊C4—4222no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C4⋊C4—2111no—
161C4⋊C4—2111no—
162C4⋊C4D84411yescharacteristic
163C4⋊C4D84411yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—8444no—
166C22⋊C4—4221no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—4221no—
171C23—8442no—
172C23—8442no—
173C23—8441no—
174C23—8441no—
175C23—4222no—
176C23D4×D21111yescharacteristic
177Q8—16882no—
178D4—8441no—
179D4—8441no—
180D4—4222no—
181D4—4222no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—8442no—
188C2×C4—4221no—
189C2×C4—4221no—
190C2×C4—4222no—
191C2×C4D4×D21111yescharacteristic
192C2×C4C2×D84411yescharacteristic
193C2×C4Q8×D21111yescharacteristic
194C2×C4#1680996111yescharacteristic
195C2×C4#1680996211yescharacteristic
196C2×C4#1680996211yescharacteristic
197C8—41no—
198C8—8441no—
199C22—21no—
200C22—21no—
201C22—8444no—
202C22—8442no—
203C22—16882no—
204C22—16881no—
205C22—16881no—
206C22#2026994711yescharacteristic
207C22#2026995511yescharacteristic
208C22#2026996311yescharacteristic
209C4—42no—
210C4—41no—
211C4—41no—
212C4—8442no—
213C4—8441no—
214C4—8441no—
215C4#2026995511yescharacteristic
216C4#2026995711yescharacteristic
217C4#2026996311yescharacteristic
218C4#2026996511yescharacteristic
219C2—21no—
220C2—21no—
221C2—16882no—
222C2#2721305911yescharacteristic
223C2#2721331511yescharacteristic
224C2#2721332011yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211#120
prime radicalC22×D844C22
hypercenter#339D211
hypocenterC211#2027
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #27213774C2semidirect
5. #27212914C2semidirect
6. #27212906C2semidirect
7. #339×C211C2semidirect
32. C2×D844Q8semidirect
112. C211#2027semidirect
120. #339D211semidirect

Maximal subgroups

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

subgroupindex
#272137742
#272131522
#272131502
#272130482
#272129142
#272129062
#339×C2112
#2027211