Finite Group Databasebeta

← #41014473

Subgroups — #41014473

Order 27008.

426 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#16809950C211yesmaximal, characteristic, complemented
2C2×#20269964C211yesmaximal, characteristic, complemented
3C2×#20269902C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic, complemented
5#27213049C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C22×D8×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809950—22no—
10C2×#16809950—22no—
11C2×#16809950—22no—
12C22×C211⋊C8C2211yescharacteristic
13#20269964—22no—
14#20269964—22no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×C22⋊Dic211—22no—
17C2×C22⋊Dic211—22no—
18C2×C4×Dic211C2211yescharacteristic
19#20269902—22no—
20#20269898—22no—
21C22×D4×C211C2211yescharacteristic
22C22×D4×C211C2211yescharacteristic
23C2×D8×C211—22no—
24C2×D8×C211—22no—
25C2×D8×C211—22no—
26C22×C8×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422D412yes—
33C22×Dic211—21no—
34C22×Dic211—21no—
35C22×Dic211D412yes—
36#16809950—28no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8D412yes—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42C22⋊Dic211—28no—
43C22⋊Dic211—44no—
44C4×Dic211—21no—
45C4×Dic211—21no—
46C24×C211—21no—
47C24×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211—21no—
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211D412yes—
59C2×D4×C211D412yes—
60C22×C4×C211C2311yescharacteristic
61D8×C211—28no—
62C2×C8×C211—21no—
63C2×C8×C211—21no—
64C2×C8×C211D412yes—
65Dic422—22no—
66Dic422—22no—
67Dic422—41no—
68Dic422—41no—
69C2×Dic211—22no—
70C2×Dic211—22no—
71C2×Dic211—22no—
72C2×Dic211—22no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211—42no—
77C211⋊C8—22no—
78C23×C211—22no—
79C23×C211—22no—
80C23×C211—41no—
81C23×C211—41no—
82C23×C211—41no—
83C23×C211—41no—
84C23×C211—42no—
85C23×C211—42no—
86C23×C211—42no—
87C23×C211—42no—
88C23×C211C2×D411yescharacteristic
89D4×C211—22no—
90D4×C211—22no—
91D4×C211—22no—
92D4×C211—22no—
93D4×C211—41no—
94D4×C211—41no—
95D4×C211—41no—
96D4×C211—41no—
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C8×C211—22no—
104C22×C211—42no—
105C22×C211—42no—
106C22×C211—44no—
107C22×C211—44no—
108C22×C211—82no—
109C22×C211—82no—
110C22×C211—81no—
111C22×C211—81no—
112C22×C211—81no—
113C22×C211—81no—
114C22×C211#12011yescharacteristic
115C22×C211#12111yescharacteristic
116C22×C211#12711yescharacteristic
117C22×C211#13512yes—
118C22×C211AGL(1,ℤ/8)12yes—
119C4×C211#12011yescharacteristic
120C4×C211#12011yescharacteristic
121C4×C211#12111yescharacteristic
122C4×C211#12111yescharacteristic
123Dic211—42no—
124Dic211—42no—
125Dic211—81no—
126C2×C211—82no—
127C2×C211—82no—
128C2×C211#39211yescharacteristic
129C2×C211#44912yes—
130C2×C211#46312yes—
131C2×C211#49511yescharacteristic
132C2×C211#49611yescharacteristic
133C211#201211yescharacteristic, complemented, Sylow
134#2012—2111nomaximal, Sylow
135C22×SD16—2111no—
136C22×D8D21111yescharacteristic
137C2×#124—2111no—
138C2×#103—2111no—
139C2×#102—2111no—
140#390—2111no—
141#336—2111no—
142C22×Q8—2111no—
143C22×D4C2×D21111yescharacteristic
144C22×D4C2×D21111yescharacteristic
145C2×SD16—4222no—
146C2×SD16—4222no—
147C2×SD16—4222no—
148C2×D8—22no—
149C2×D8—22no—
150C2×D8—22no—
151C22×C8—2111no—
152C22×C8C2×D21111yescharacteristic
153#124—4222no—
154#124—4222no—
155C2×C4⋊C4—2111no—
156C2×C22⋊C4—4222no—
157C2×C22⋊C4—4222no—
158C2×C42—2111no—
159#103—4222no—
160#102—4222no—
161C24—21no—
162C24—21no—
163C2×Q8—4221no—
164C2×Q8—4221no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—2112no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4C22⋊D21112yes—
180C2×D4C22⋊D21112yes—
181C22×C4—4221no—
182C22×C4—4221no—
183C22×C4—2112no—
184C22×C4C22×D21111yescharacteristic
185SD16—4228no—
186D8—28no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—4221no—
190C2×C8—4221no—
191C2×C8—2112no—
192C2×C8C22⋊D21112yes—
193C4⋊C4—4221no—
194C4⋊C4—4221no—
195C22⋊C4—8444no—
196C22⋊C4—4228no—
197C42—4221no—
198C42—4221no—
199C23—22no—
200C23—22no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23D4×D21111yescharacteristic
210Q8—8441no—
211Q8—8441no—
212Q8—4222no—
213Q8—4222no—
214D4—22no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—41no—
222C2×C4—8441no—
223C2×C4—8441no—
224C2×C4—8441no—
225C2×C4—8442no—
226C2×C4—4222no—
227C2×C4—4222no—
228C2×C4—4222no—
229C2×C4—4222no—
230C2×C4D4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4C2×C22⋊D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4C2×C22⋊D21111yescharacteristic
235C2×C4#1680996111yescharacteristic
236C8—22no—
237C8—4222no—
238C22—44no—
239C22—44no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22#2026994511yescharacteristic
249C22#2026994911yescharacteristic
250C22#2026997311yescharacteristic
251C22#2026999612yes—
252C22#2027000112yes—
253C4—8442no—
254C4—8442no—
255C4—16881no—
256C4#2026994511yescharacteristic
257C4#2026994611yescharacteristic
258C4#2026994811yescharacteristic
259C4#2026994911yescharacteristic
260C2—82no—
261C2—82no—
262C2#2721305411yescharacteristic
263C2#2721328812yes—
264C2#2721335912yes—
265C2#2721352911yescharacteristic
266C2#2721353411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radicalC22×D8×C211C2
socleC23×C211C2×D4
hypercenterC22×D8D211
hypocenterC211#2012
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#16809950C2semidirect
2. C2×#20269964C2semidirect
3. C2×#20269902C2semidirect
4. C2×#20269898C2semidirect
5. #27213049C2semidirect
6. #27212900C2semidirect
133. C211#2012semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
C2×#202699642
C2×#202699022
C2×#202698982
#272130492
#272129002
C22×D8×C2112
#2012211