Finite Group Databasebeta

← #41014474

Subgroups — #41014474

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×#16809950C211yesmaximal, characteristic, complemented
2C22×#16809947C211yesmaximal, characteristic
3C2×#20269965C211yesmaximal, characteristic, complemented
4C2×#20269900C211yesmaximal, characteristic, complemented
5C2×#20269897C211yesmaximal, characteristic, complemented
6#27212901C211yesmaximal, characteristic, complemented
7#390×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C2×#16809950—22no—
11C2×#16809950—22no—
12C2×#16809950—22no—
13C2×#16809947—22no—
14C2×#16809947—22no—
15C2×#16809947—22no—
16C22×C211⋊C8C2211yescharacteristic
17C22×C211⋊C8C2211yescharacteristic
18#20269965—22no—
19#20269965—22no—
20C2×(C2×D211)⋊C4—22no—
21#20269900—22no—
22#20269897—22no—
23C22×D4×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C22⋊C4×C211—22no—
26C2×C42×C211C2211yescharacteristic
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—
41#16809950—28no—
42#16809947—28no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8D412yes—
48C2×C211⋊C8D412yes—
49(C2×D211)⋊C4—28no—
50C24×C211—21no—
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211D412yes—
57C22×C4×C211—21no—
58C22×C4×C211D412yes—
59C22×C4×C211C2311yescharacteristic
60C4⋊C4×C211—21no—
61C4⋊C4×C211—21no—
62C22⋊C4×C211—44no—
63C42×C211—21no—
64C42×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—41no—
80C2×Dic211—41no—
81C211⋊C8—22no—
82C211⋊C8—22no—
83C23×C211—22no—
84C23×C211—41no—
85C23×C211—41no—
86C23×C211—42no—
87C23×C211—42no—
88C23×C211C2×D411yescharacteristic
89D4×C211—22no—
90D4×C211—22no—
91D4×C211—41no—
92D4×C211—41no—
93C2×C4×C211—22no—
94C2×C4×C211—22no—
95C2×C4×C211—22no—
96C2×C4×C211—41no—
97C2×C4×C211—42no—
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211#4111yescharacteristic
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—42no—
120C4×C211—81no—
121C4×C211#12011yescharacteristic
122C4×C211#12011yescharacteristic
123C4×C211#12111yescharacteristic
124C4×C211#12111yescharacteristic
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×#102—2111no—
140#390D21111yescharacteristic, complemented
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×C8—2111no—
153#124—4222no—
154#124—4222no—
155C2×C4⋊C4C2×D21111yescharacteristic
156C2×C22⋊C4—22no—
157C2×C22⋊C4—4222no—
158C2×C42C2×D21111yescharacteristic
159#103—4222no—
160#102—4222no—
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×C4C22⋊D21112yes—
184C22×C4C22×D21111yescharacteristic
185SD16—4228no—
186D8—4228no—
187C2×C8—4221no—
188C2×C8—4221no—
189C2×C8—4221no—
190C2×C8—4221no—
191C2×C8—2112no—
192C2×C8—2112no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C22⋊C4—44no—
196C22⋊C4—4228no—
197C42—21no—
198C42—21no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—8441no—
205C23—8441no—
206C23—8442no—
207C23—8442no—
208C23—4222no—
209C23C2×C22⋊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—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—8441no—
228C2×C4—8441no—
229C2×C4—4222no—
230C2×C4D4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4D4×D21111yescharacteristic
234C2×C4C2×D84411yescharacteristic
235C2×C4#1680996211yescharacteristic
236C8—4222no—
237C8—4222no—
238C22—44no—
239C22—42no—
240C22—81no—
241C22—81no—
242C22—82no—
243C22—8444no—
244C22—8442no—
245C22—16882no—
246C22—16881no—
247C22—16881no—
248C22#2026994511yescharacteristic
249C22#2026995111yescharacteristic
250C22#2026997311yescharacteristic
251C22#2026999712yes—
252C22#2027000312yes—
253C4—42no—
254C4—81no—
255C4—8442no—
256C4#2026994711yescharacteristic
257C4#2026994711yescharacteristic
258C4#2026995011yescharacteristic
259C4#2026995011yescharacteristic
260C2—82no—
261C2—16882no—
262C2#2721305511yescharacteristic
263C2#2721328912yes—
264C2#2721336112yes—
265C2#2721352811yescharacteristic
266C2#2721353311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#390×C211C2
socleC23×C211C2×D4
prime radicalC22×#16809947C2
hypercenter#390D211
hypocenterC211#2012
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×#16809950C2semidirect
3. C2×#20269965C2semidirect
4. C2×#20269900C2semidirect
5. C2×#20269897C2semidirect
6. #27212901C2semidirect
7. #390×C211C2semidirect
133. C211#2012semidirect
140. #390D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
C22×#168099472
C2×#202699652
C2×#202699002
C2×#202698972
#272129012
#390×C2112
#2012211