Finite Group Databasebeta

← #41014476

Subgroups — #41014476

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×#16809949C211yesmaximal, characteristic, complemented
2C22×D1688C211yesmaximal, characteristic
3C2×#20269966C211yesmaximal, characteristic, complemented
4C2×#20269897C211yesmaximal, characteristic, complemented
5#27213047C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#103×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C2×#16809949—22no—
11C2×#16809949—22no—
12C2×#16809949—22no—
13C2×D1688—22no—
14C2×D1688—22no—
15C2×D1688—22no—
16C22×C211⋊C8C2211yescharacteristic
17#20269966—22no—
18#20269966—22no—
19C2×(C2×D211)⋊C4—22no—
20C2×(C2×D211)⋊C4—22no—
21C2×C4×Dic211C2211yescharacteristic
22#20269897—22no—
23C22×Q8×C211C2211yescharacteristic
24C22×C8×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26#103×C211—22no—
27C23×D211—21no—
28C23×D211—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844D412yes—
40C2×D844D412yes—
41C22×Dic211D412yes—
42#16809949—28no—
43D1688—28no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8D412yes—
47(C2×D211)⋊C4—28no—
48(C2×D211)⋊C4—44no—
49C4×Dic211—21no—
50C4×Dic211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211D412yes—
57C22×C4×C211—21no—
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—22no—
67C22×D211—41no—
68C22×D211—41no—
69C22×D211—41no—
70C22×D211—41no—
71C22×D211—42no—
72C22×D211—42no—
73C22×D211—42no—
74C22×D211—42no—
75D844—22no—
76D844—22no—
77D844—22no—
78D844—22no—
79D844—41no—
80D844—41no—
81D844—41no—
82D844—41no—
83C2×Dic211—22no—
84C2×Dic211—22no—
85C2×Dic211—22no—
86C211⋊C8—22no—
87C23×C211C2×D411yescharacteristic
88Q8×C211—22no—
89Q8×C211—22no—
90Q8×C211—41no—
91Q8×C211—41no—
92C2×C4×C211—22no—
93C2×C4×C211—41no—
94C2×C4×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#12011yescharacteristic
105C22×C211#12111yescharacteristic
106C22×C211#12711yescharacteristic
107C22×C211#13512yes—
108C22×C211AGL(1,ℤ/8)12yes—
109C2×D211—42no—
110C2×D211—42no—
111C2×D211—44no—
112C2×D211—44no—
113C2×D211—82no—
114C2×D211—82no—
115C2×D211—81no—
116C2×D211—81no—
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#39211yescharacteristic
127C2×C211#44912yes—
128C2×C211#46312yes—
129C2×C211#49511yescharacteristic
130C2×C211#49611yescharacteristic
131D211—82no—
132D211—82no—
133C211#201211yescharacteristic, complemented, Sylow
134#2012—2111nomaximal, Sylow
135C22×SD16—2111no—
136C22×D8—2111no—
137C2×#124—2111no—
138C2×#103D21111yescharacteristic, complemented
139C2×#102—2111no—
140#390—2111no—
141#336—2111no—
142C22×Q8C2×D21111yescharacteristic
143C22×D4—2111no—
144C22×D4—2111no—
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—22no—
160#102—4222no—
161C24—4221no—
162C24—4221no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8D84412yes—
169C2×D4—4221no—
170C2×D4—4221no—
171C2×D4—4221no—
172C2×D4—4221no—
173C2×D4—4221no—
174C2×D4—4221no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—4221no—
179C2×D4—2112no—
180C2×D4—2112no—
181C22×C4—21no—
182C22×C4—21no—
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—8441no—
200C23—8441no—
201C23—8441no—
202C23—8441no—
203C23—8442no—
204C23—8442no—
205C23—8442no—
206C23—8442no—
207C23—4222no—
208C23—4222no—
209C23D4×D21111yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
214D4—8441no—
215D4—8441no—
216D4—8441no—
217D4—8441no—
218D4—4222no—
219D4—4222no—
220D4—4222no—
221D4—4222no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
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—8444no—
239C22—8444no—
240C22—8442no—
241C22—8442no—
242C22—16882no—
243C22—16881no—
244C22—16881no—
245C22—16881no—
246C22—16881no—
247C22—16882no—
248C22#2026994711yescharacteristic
249C22#2026995011yescharacteristic
250C22#2026997311yescharacteristic
251C22#2027000012yes—
252C22#2027000412yes—
253C4—42no—
254C4—81no—
255C4—8442no—
256C4#2026994511yescharacteristic
257C4#2026994711yescharacteristic
258C4#2026995011yescharacteristic
259C4#2026995111yescharacteristic
260C2—16882no—
261C2—16882no—
262C2#2721305511yescharacteristic
263C2#2721329412yes—
264C2#2721336412yes—
265C2#2721353111yescharacteristic
266C2#2721353211yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socleC23×C211C2×D4
prime radicalC22×D1688C2
hypercenterC2×#103D211
hypocenterC211#2012
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#16809949C2semidirect
3. C2×#20269966C2semidirect
4. C2×#20269897C2semidirect
5. #27213047C2semidirect
6. #27212900C2semidirect
7. C2×#103×C211C2semidirect
133. C211#2012semidirect
138. C2×#103D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099492
C22×D16882
C2×#202699662
C2×#202698972
#272130472
#272129002
C2×#103×C2112
#2012211