Finite Group Databasebeta

← #41014522

Subgroups — #41014522

Order 27008.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20269945C212yesmaximal, complemented
2C2×#20269892C212yesmaximal, complemented
3#27213001C211yesmaximal, characteristic, complemented
4#27212913C211yesmaximal, characteristic, complemented
5#534×C211C211yesmaximal, characteristic, complemented
6C24×D211C2211yescharacteristic
7C22×C22⋊D211—22no—
8#20269945—24no—
9#20269945—24no—
10C2×C22⋊Dic211C2211yescharacteristic
11C2×C22⋊Dic211C2212yes—
12C2×(C2×D211)⋊C4—22no—
13C2×(C2×D211)⋊C4—22no—
14#20269892—24no—
15C22×D4×C211C2212yes—
16#123×C211—22no—
17#121×C211—22no—
18#120×C211—22no—
19C2×C22⋊C4×C211C2211yescharacteristic
20C23×D211—21no—
21C23×D211—22no—
22C23×D211—22no—
23C23×D211—21no—
24C23×D211—44no—
25C23×D211D412yescomplemented
26C2×C22⋊D211—28no—
27C2×C22⋊D211—48no—
28C22×Dic211—22no—
29C22×Dic211—21no—
30C22⋊Dic211—22no—
31C22⋊Dic211—22no—
32C22⋊Dic211D414yescomplemented
33(C2×D211)⋊C4—28no—
34(C2×D211)⋊C4—44no—
35C24×C211D412yescomplemented
36C24×C211C2311yescharacteristic
37C2×D4×C211—22no—
38C2×D4×C211—22no—
39C2×D4×C211—22no—
40C2×D4×C211—22no—
41C2×D4×C211—22no—
42C2×D4×C211—22no—
43C2×D4×C211—22no—
44C22×C4×C211D412yescomplemented
45C22×C4×C211D412yescomplemented
46C4⋊C4×C211—22no—
47C22⋊C4×C211—24no—
48C22⋊C4×C211—22no—
49C22×D211—24no—
50C22×D211—28no—
51C22×D211—24no—
52C22×D211—22no—
53C22×D211—44no—
54C22×D211—48no—
55C22×D211—44no—
56C22×D211—42no—
57C22×D211—44no—
58C22×D211—41no—
59C22×D211—42no—
60C22×D211—41no—
61C22⋊D211—416no—
62C2×Dic211—24no—
63C2×Dic211—44no—
64C2×Dic211—42no—
65C2×Dic211—41no—
66C23×C211—22no—
67C23×C211—22no—
68C23×C211—22no—
69C23×C211—42no—
70C23×C211—44no—
71C23×C211C2×D411yescharacteristic
72C23×C211C2×D412yes—
73C23×C211C2×D412yes—
74C23×C211C2×D411yescharacteristic
75C23×C211#4111yescharacteristic
76D4×C211—42no—
77D4×C211—42no—
78D4×C211—42no—
79D4×C211—42no—
80D4×C211—82no—
81C2×C4×C211—22no—
82C2×C4×C211—22no—
83C2×C4×C211—22no—
84C2×C4×C211—22no—
85C2×C4×C211—42no—
86C22×C211—22no—
87C22×C211—24no—
88C22×C211—41no—
89C22×C211—42no—
90C22×C211—41no—
91C22×C211—44no—
92C22×C211—42no—
93C22×C211—44no—
94C22×C211—82no—
95C22×C211#12012yes—
96C22×C211#12011yescharacteristic
97C22×C211#12111yescharacteristic
98C22×C211#12112yes—
99C22×C211#12711yescharacteristic
100C2×D211—28no—
101C2×D211—24no—
102C2×D211—44no—
103C2×D211—44no—
104C2×D211—42no—
105C2×D211—48no—
106C2×D211—42no—
107C2×D211—44no—
108C4×C211—42no—
109C4×C211—42no—
110Dic211—44no—
111Dic211—81no—
112C2×C211—22no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211—22no—
116C2×C211—82no—
117C2×C211#39211yescharacteristic
118C2×C211B(4,𝔽2)12yes—
119D211—44no—
120C211#202111yescharacteristic, complemented, Sylow
121#2021—2111nomaximal, Sylow
122#534D21111yescharacteristic, complemented
123C2×#120—2112no—
124C2×#99—2112no—
125#379—2111no—
126#342—2111no—
127C25—2111no—
128C22×D4—4222no—
129C22×D4C2×D21112yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—4224no—
134#120—4224no—
135C2×C22⋊C4—4222no—
136C2×C22⋊C4—4222no—
137C2×C22⋊C4—2111no—
138C2×C22⋊C4—2112no—
139C2×C22⋊C4C2×D21111yescharacteristic
140#99—4224no—
141C24—8444no—
142C24—4222no—
143C24—4221no—
144C24—4221no—
145C24—4222no—
146C24—2112no—
147C24C22⋊D21112yescomplemented
148C24C22×D21111yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—8448no—
157C2×D4—4228no—
158C22×C4—4221no—
159C22×C4—4222no—
160C22×C4C22⋊D21112yescomplemented
161C22×C4C22⋊D21112yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—8444no—
166C22⋊C4—4228no—
167C22⋊C4—4222no—
168C22⋊C4—4222no—
169C22⋊C4—2114no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—8442no—
176C23—8444no—
177C23—8448no—
178C23—8444no—
179C23—8444no—
180C23—8441no—
181C23—8442no—
182C23—8441no—
183C23—4224no—
184C23—4228no—
185C23—4224no—
186C23—4222no—
187C23D4×D21111yescharacteristic
188C23D4×D21112yes—
189C23C2×C22⋊D21112yes—
190C23C2×C22⋊D21111yescharacteristic
191C23#1680996111yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—84416no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—8444no—
204C2×C4—8442no—
205C2×C4—8441no—
206C2×C4—4224no—
207C22—22no—
208C22—24no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—82no—
216C22—8444no—
217C22—8444no—
218C22—8448no—
219C22—8442no—
220C22—8444no—
221C22—8442no—
222C22—4228no—
223C22—4224no—
224C22#2026994512yes—
225C22#2026994611yescharacteristic
226C22#2026994811yescharacteristic
227C22#2026994912yes—
228C22#2026997311yescharacteristic
229C4—42no—
230C4—42no—
231C4—8444no—
232C4—16881no—
233C2—22no—
234C2—21no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—8444no—
239C2#2721305411yescharacteristic
240C2#2721333312yes—
subgroupG/H
centerC22#20269946
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#534×C211C2
socleC22×C211#120
hypercenter#534D211
hypocenterC211#2021
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#20269945C2semidirect
2. C2×#20269892C2semidirect
3. #27213001C2semidirect
4. #27212913C2semidirect
5. #534×C211C2semidirect
25. C23×D211D4semidirect
32. C22⋊Dic211D4semidirect
35. C24×C211D4semidirect
44. C22×C4×C211D4semidirect
45. C22×C4×C211D4semidirect
120. C211#2021semidirect
122. #534D211semidirect
147. C24C22⋊D211semidirect
160. C22×C4C22⋊D211semidirect
161. C22×C4C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699452
C2×#202698922
#272130012
#272129132
#534×C2112
#2021211