Finite Group Databasebeta

← #41014475

Subgroups — #41014475

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×#16809949C211yesmaximal, characteristic, complemented
2C22×#16809947C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic, complemented
5#27213048C211yesmaximal, characteristic, complemented
6#27212901C211yesmaximal, characteristic, complemented
7C2×#124×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809949—22no—
10C2×#16809949—22no—
11C2×#16809949—22no—
12C2×#16809947—22no—
13C2×#16809947—22no—
14C2×#16809947—22no—
15C22×C211⋊C8C2211yescharacteristic
16C22×C211⋊C8C2211yescharacteristic
17C2×C4⋊Dic211C2211yescharacteristic
18C2×(C2×D211)⋊C4—22no—
19#20269901—22no—
20#20269898—22no—
21C22×Q8×C211C2211yescharacteristic
22C22×D4×C211C2211yescharacteristic
23#124×C211—22no—
24#124×C211—22no—
25C2×C22⋊C4×C211—22no—
26C2×C42×C211C2211yescharacteristic
27C23×D211—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844D412yes—
34C22×Dic211—21no—
35#16809949—28no—
36#16809947—28no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8D412yes—
42C2×C211⋊C8D412yes—
43C4⋊Dic211—21no—
44C4⋊Dic211—21no—
45(C2×D211)⋊C4—44no—
46C24×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211D412yes—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211D412yes—
59C22×C4×C211—21no—
60C22×C4×C211D412yes—
61C22×C4×C211C2311yescharacteristic
62C22⋊C4×C211—28no—
63C42×C211—21no—
64C42×C211—21no—
65C22×D211—22no—
66C22×D211—41no—
67C22×D211—41no—
68C22×D211—42no—
69C22×D211—42no—
70D844—22no—
71D844—22no—
72D844—41no—
73D844—41no—
74C2×Dic211—41no—
75C2×Dic211—42no—
76C211⋊C8—22no—
77C211⋊C8—22no—
78C23×C211—22no—
79C23×C211—41no—
80C23×C211—41no—
81C23×C211—42no—
82C23×C211—42no—
83C23×C211C2×D411yescharacteristic
84Q8×C211—22no—
85Q8×C211—22no—
86Q8×C211—41no—
87Q8×C211—41no—
88D4×C211—22no—
89D4×C211—22no—
90D4×C211—41no—
91D4×C211—41no—
92C2×C4×C211—22no—
93C2×C4×C211—22no—
94C2×C4×C211—22no—
95C2×C4×C211—22no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
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—42no—
121C4×C211#12011yescharacteristic
122C4×C211#12011yescharacteristic
123C4×C211#12111yescharacteristic
124C4×C211#12111yescharacteristic
125Dic211—81no—
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×#124D21111yescharacteristic, complemented
138C2×#103—2111no—
139C2×#102—2111no—
140#390—2111no—
141#336—2111no—
142C22×Q8C2×D21111yescharacteristic
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—22no—
154#124—22no—
155C2×C4⋊C4—2111no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—4222no—
158C2×C42C2×D21111yescharacteristic
159#103—4222no—
160#102—4222no—
161C24—21no—
162C24—4221no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8C22⋊D21112yes—
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×D4C22⋊D21112yes—
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—4221no—
194C4⋊C4—4221no—
195C22⋊C4—28no—
196C22⋊C4—8444no—
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—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
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—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—8441no—
229C2×C4—8442no—
230C2×C4D4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4C2×C22⋊D21111yescharacteristic
235C2×C4#1680996111yescharacteristic
236C8—4222no—
237C8—4222no—
238C22—44no—
239C22—42no—
240C22—82no—
241C22—81no—
242C22—81no—
243C22—8444no—
244C22—8442no—
245C22—16881no—
246C22—16881no—
247C22—16882no—
248C22#2026994611yescharacteristic
249C22#2026994811yescharacteristic
250C22#2026997311yescharacteristic
251C22#2026999712yes—
252C22#2027000212yes—
253C4—42no—
254C4—42no—
255C4—16881no—
256C4#2026994511yescharacteristic
257C4#2026994511yescharacteristic
258C4#2026994911yescharacteristic
259C4#2026994911yescharacteristic
260C2—82no—
261C2—16882no—
262C2#2721305411yescharacteristic
263C2#2721329112yes—
264C2#2721335812yes—
265C2#2721352811yescharacteristic
266C2#2721353611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#124×C211C2
socleC23×C211C2×D4
prime radicalC22×#16809947C2
hypercenterC2×#124D211
hypocenterC211#2012
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#16809949C2semidirect
3. C2×#20269901C2semidirect
4. C2×#20269898C2semidirect
5. #27213048C2semidirect
6. #27212901C2semidirect
7. C2×#124×C211C2semidirect
133. C211#2012semidirect
137. C2×#124D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099492
C22×#168099472
C2×#202699012
C2×#202698982
#272130482
#272129012
C2×#124×C2112
#2012211