Finite Group Databasebeta

← #41012329

Subgroups — #41012329

Order 27008.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#27213743C211yesmaximal, characteristic, complemented
2#27213299C211yesmaximal, characteristic, complemented
3#27213282C211yesmaximal, characteristic
4#27213063C211yesmaximal, characteristic, complemented
5#27212884C211yesmaximal, characteristic, complemented
6#27212881C211yesmaximal, characteristic, complemented
7#324×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809951—21no—
10C2×D1688—21no—
11#20269972—21no—
12#20269967—21no—
13#20269962—21no—
14#20269960—21no—
15#20269956C2211yescharacteristic
16#20269955—21no—
17#20269951C2211yescharacteristic
18C4×C22⋊D211—21no—
19C2×(C2×D211)⋊C4—21no—
20C2×(C2×D211)⋊C4—21no—
21#20269899—21no—
22#20269899—21no—
23C211⋊#95C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26C22⋊C8×C211C2211yescharacteristic
27C23×D211—21no—
28C2×Dic422D411yescharacteristic
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—42no—
32C2×D844D411yescharacteristic
33C2×C22⋊D211—21no—
34C22×Dic211—21no—
35C2×C4×D211—21no—
36#16809951—42no—
37D1688—42no—
38C4⋊Dic211D411yescharacteristic
39C4⋊Dic211D411yescharacteristic
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43Dic211⋊C4—21no—
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—42no—
50(C2×D211)⋊C4—42no—
51C4×Dic211—21no—
52C22×C4×C211D411yescharacteristic
53C22×C4×C211D411yescharacteristic
54C22×C4×C211C2311yescharacteristic
55C2×C8×C211—21no—
56C2×C8×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59C22×D211—22no—
60C22×D211—21no—
61C22×D211—41no—
62C22×D211—41no—
63C22×D211—41no—
64C22×D211—41no—
65C22×D211—41no—
66C22×D211—41no—
67Dic422—41no—
68D844—22no—
69D844—22no—
70D844—41no—
71D844—41no—
72D844—41no—
73C22⋊D211—41no—
74C22⋊D211—41no—
75C2×Dic211—21no—
76C2×Dic211—21no—
77C2×Dic211—21no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C2×Dic211—41no—
81C4×D211—81no—
82C23×C211C2×D411yescharacteristic
83C2×C4×C211—21no—
84C2×C4×C211—21no—
85C2×C4×C211—21no—
86C2×C4×C211—21no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C8×C211—41no—
92C8×C211—41no—
93C22×C211—21no—
94C22×C211—21no—
95C22×C211#12011yescharacteristic
96C22×C211#13511yescharacteristic
97C22×C211AGL(1,ℤ/8)11yescharacteristic
98C2×D211—42no—
99C2×D211—42no—
100C2×D211—41no—
101C2×D211—41no—
102C2×D211—41no—
103C2×D211—42no—
104C2×D211—81no—
105C2×D211—81no—
106C2×D211—81no—
107C2×D211—81no—
108C4×C211—21no—
109C4×C211—21no—
110C4×C211—41no—
111C4×C211—41no—
112Dic211—41no—
113Dic211—41no—
114Dic211—41no—
115Dic211—81no—
116C2×C211—21no—
117C2×C211—21no—
118C2×C211#44911yescharacteristic
119C2×C211#45411yescharacteristic
120C2×C211B(4,𝔽2)11yescharacteristic
121D211—81no—
122D211—82no—
123C211#159711yescharacteristic, complemented, Sylow
124#1597—2111nomaximal, Sylow
125#539—2111no—
126#450—2111no—
127#447—2111no—
128#394—2111no—
129#328—2111no—
130#327—2111no—
131#324D21111yescharacteristic, complemented
132C22×D4—2111no—
133C2×SD16—4221no—
134C2×D8—4221no—
135#126—4221no—
136#125—4221no—
137#123—4221no—
138#123—4221no—
139C22⋊Q8—4221no—
140C22⋊Q8—2111no—
141#121—2111no—
142C4×D4—4221no—
143C2×C4⋊C4C2×D21111yescharacteristic
144C2×C22⋊C4—4221no—
145C2×C22⋊C4—4221no—
146#102—4221no—
147#102—4221no—
148C22⋊C8C2×D21111yescharacteristic
149C22⋊C8C2×D21111yescharacteristic
150#95—2111no—
151C24—4221no—
152C2×Q8—2111no—
153C2×D4—8442no—
154C2×D4—4221no—
155C2×D4—4221no—
156C2×D4—4221no—
157C2×D4—2111no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4D84411yescharacteristic
161C22×C4D84411yescharacteristic
162C22×C4C22×D21111yescharacteristic
163SD16—8442no—
164D8—8442no—
165C2×C8—21no—
166C2×C8—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—4221no—
170C4⋊C4—4221no—
171C4⋊C4—4221no—
172C4⋊C4—4221no—
173C4⋊C4—2111no—
174C4⋊C4—2111no—
175C22⋊C4—8442no—
176C22⋊C4—8442no—
177C22⋊C4—4221no—
178C22⋊C4—4221no—
179C22⋊C4—4221no—
180C22⋊C4—4221no—
181C22⋊C4—4221no—
182C42—4221no—
183C23—8441no—
184C23—8441no—
185C23—8441no—
186C23—8441no—
187C23—8441no—
188C23—8441no—
189C23—4222no—
190C23—4221no—
191C23C2×D84411yescharacteristic
192Q8—8441no—
193D4—8441no—
194D4—8441no—
195D4—8441no—
196D4—8441no—
197D4—8441no—
198D4—4222no—
199D4—4222no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—8441no—
207C2×C4—8441no—
208C2×C4—8441no—
209C2×C4—16881no—
210C2×C4—4221no—
211C2×C4—4221no—
212C2×C4—4221no—
213C2×C4D4×D21111yescharacteristic
214C2×C4D4×D21111yescharacteristic
215C8—41no—
216C8—41no—
217C22—21no—
218C22—21no—
219C22—8442no—
220C22—8442no—
221C22—8441no—
222C22—8441no—
223C22—8442no—
224C22—8441no—
225C22—16881no—
226C22—16881no—
227C22—16881no—
228C22—16881no—
229C22#2026994711yescharacteristic
230C22#2026999911yescharacteristic
231C22#2027000511yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—8441no—
237C4—8441no—
238C4—8441no—
239C4—16881no—
240C2—21no—
241C2—21no—
242C2—16882no—
243C2—16881no—
244C2#2721329311yescharacteristic
245C2#2721332111yescharacteristic
246C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211#120
prime radical#27213282C2
hypercenter#324D211
hypocenterC211#1597
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #27213743C2semidirect
2. #27213299C2semidirect
4. #27213063C2semidirect
5. #27212884C2semidirect
6. #27212881C2semidirect
7. #324×C211C2semidirect
123. C211#1597semidirect
131. #324D211semidirect

Maximal subgroups

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

subgroupindex
#272137432
#272132992
#272132822
#272130632
#272128842
#272128812
#324×C2112
#1597211