Finite Group Databasebeta

← #41014477

Subgroups — #41014477

Order 27008.

426 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16809951C211yesmaximal, characteristic, complemented
2C22×#16809947C211yesmaximal, characteristic
3C2×#20269964C211yesmaximal, characteristic, complemented
4C2×#20269900C211yesmaximal, characteristic, complemented
5#27213047C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C2×#16809951—22no—
11C2×#16809951—22no—
12C2×#16809951—22no—
13C2×#16809947—22no—
14C2×#16809947—22no—
15C2×#16809947—22no—
16C22×C211⋊C8C2211yescharacteristic
17#20269964—22no—
18#20269964—22no—
19C2×C22⋊Dic211—22no—
20C2×(C2×D211)⋊C4—22no—
21C2×C4×Dic211C2211yescharacteristic
22#20269900—22no—
23C22×D4×C211C2211yescharacteristic
24C22×C8×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26#102×C211—22no—
27C23×D211—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422D412yes—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844D412yes—
40C22×Dic211—21no—
41C22×Dic211D412yes—
42#16809951—28no—
43#16809947—28no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8D412yes—
47C22⋊Dic211—28no—
48(C2×D211)⋊C4—44no—
49C4×Dic211—21no—
50C4×Dic211—21no—
51C24×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211D412yes—
58C22×C4×C211—21no—
59C22×C4×C211C2311yescharacteristic
60C2×C8×C211—21no—
61C2×C8×C211—21no—
62C2×C8×C211D412yes—
63C4⋊C4×C211—21no—
64C4⋊C4×C211—21no—
65C22×D211—22no—
66C22×D211—41no—
67C22×D211—41no—
68C22×D211—42no—
69C22×D211—42no—
70Dic422—22no—
71Dic422—22no—
72Dic422—41no—
73Dic422—41no—
74D844—22no—
75D844—22no—
76D844—41no—
77D844—41no—
78C2×Dic211—22no—
79C2×Dic211—22no—
80C2×Dic211—22no—
81C2×Dic211—22no—
82C2×Dic211—41no—
83C2×Dic211—41no—
84C211⋊C8—22no—
85C23×C211—22no—
86C23×C211—41no—
87C23×C211—41no—
88C23×C211—42no—
89C23×C211—42no—
90C23×C211C2×D411yescharacteristic
91D4×C211—22no—
92D4×C211—22no—
93D4×C211—41no—
94D4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—42no—
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—44no—
106C22×C211—82no—
107C22×C211—81no—
108C22×C211—81no—
109C22×C211#12011yescharacteristic
110C22×C211#12111yescharacteristic
111C22×C211#12711yescharacteristic
112C22×C211#13512yes—
113C22×C211AGL(1,ℤ/8)12yes—
114C2×D211—42no—
115C2×D211—44no—
116C2×D211—82no—
117C2×D211—81no—
118C2×D211—81no—
119C4×C211—81no—
120C4×C211#12011yescharacteristic
121C4×C211#12011yescharacteristic
122C4×C211#12111yescharacteristic
123C4×C211#12111yescharacteristic
124Dic211—42no—
125Dic211—42no—
126C2×C211—82no—
127C2×C211#39211yescharacteristic
128C2×C211#44912yes—
129C2×C211#46312yes—
130C2×C211#49511yescharacteristic
131C2×C211#49611yescharacteristic
132D211—82no—
133C211#201211yescharacteristic, complemented, Sylow
134#2012—2111nomaximal, Sylow
135C22×SD16—2111no—
136C22×D8—2111no—
137C2×#124—2111no—
138C2×#103—2111no—
139C2×#102D21111yescharacteristic, complemented
140#390—2111no—
141#336—2111no—
142C22×Q8—2111no—
143C22×D4—2111no—
144C22×D4C2×D21111yescharacteristic
145C2×SD16—4222no—
146C2×SD16—4222no—
147C2×SD16—4222no—
148C2×D8—4222no—
149C2×D8—4222no—
150C2×D8—4222no—
151C22×C8—2111no—
152C22×C8C2×D21111yescharacteristic
153#124—4222no—
154#124—4222no—
155C2×C4⋊C4C2×D21111yescharacteristic
156C2×C22⋊C4—4222no—
157C2×C22⋊C4—4222no—
158C2×C42—2111no—
159#103—4222no—
160#102—22no—
161C24—21no—
162C24—4221no—
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—4221no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—4221no—
179C2×D4—2112no—
180C2×D4D84412yes—
181C22×C4—21no—
182C22×C4—4221no—
183C22×C4—2112no—
184C22×C4C22×D21111yescharacteristic
185SD16—4228no—
186D8—4228no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—4221no—
190C2×C8—4221no—
191C2×C8—2112no—
192C2×C8C22⋊D21112yes—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C22⋊C4—8444no—
196C22⋊C4—4228no—
197C42—4221no—
198C42—4221no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—8441no—
205C23—8441no—
206C23—8442no—
207C23—8442no—
208C23—4222no—
209C23D4×D21111yescharacteristic
210Q8—8441no—
211Q8—8441no—
212Q8—4222no—
213Q8—4222no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218D4—8441no—
219D4—8441no—
220D4—4222no—
221D4—4222no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—8441no—
225C2×C4—8441no—
226C2×C4—4222no—
227C2×C4—4222no—
228C2×C4—4222no—
229C2×C4—4222no—
230C2×C4D4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4C2×D84411yescharacteristic
235C2×C4#1680996211yescharacteristic
236C8—22no—
237C8—4222no—
238C22—44no—
239C22—42no—
240C22—82no—
241C22—81no—
242C22—81no—
243C22—8444no—
244C22—8442no—
245C22—16881no—
246C22—16881no—
247C22—16882no—
248C22#2026994711yescharacteristic
249C22#2026995011yescharacteristic
250C22#2026997311yescharacteristic
251C22#2026999812yes—
252C22#2027000112yes—
253C4—81no—
254C4—8442no—
255C4—8442no—
256C4#2026994511yescharacteristic
257C4#2026994711yescharacteristic
258C4#2026995011yescharacteristic
259C4#2026995111yescharacteristic
260C2—82no—
261C2—16882no—
262C2#2721305511yescharacteristic
263C2#2721329212yes—
264C2#2721336012yes—
265C2#2721352711yescharacteristic
266C2#2721353411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×#16809947C2
hypercenterC2×#102D211
hypocenterC211#2012
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#16809951C2semidirect
3. C2×#20269964C2semidirect
4. C2×#20269900C2semidirect
5. #27213047C2semidirect
6. #27212900C2semidirect
7. C2×#102×C211C2semidirect
133. C211#2012semidirect
139. C2×#102D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099512
C22×#168099472
C2×#202699642
C2×#202699002
#272130472
#272129002
C2×#102×C2112
#2012211