Finite Group Databasebeta

← #41014420

Subgroups — #41014420

Order 27008.

382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20270003C211yesmaximal, characteristic
2#27213720C211yesmaximal, characteristic, complemented
3C2×#20269905C211yesmaximal, characteristic, complemented
4#27213151C211yesmaximal, characteristic, complemented
5C2×#20269896C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#124×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270003—22no—
10#20270003—22no—
11C2×#16809950—21no—
12C2×#16809947—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15#20269975—21no—
16#20269966—21no—
17#20269960—22no—
18#20269953—22no—
19#20269936C2211yescharacteristic
20#20269905—22no—
21#20269898—22no—
22#20269896—22no—
23C22×Q8×C211C2211yescharacteristic
24C22×D4×C211C2211yescharacteristic
25#124×C211—22no—
26#124×C211—22no—
27C2×C22⋊C4×C211—22no—
28C2×C42×C211C2211yescharacteristic
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33C2×Dic422D411yescharacteristic
34C2×D844D411yescharacteristic
35C2×C22⋊D211—21no—
36C2×C4×D211—21no—
37#16809950—24no—
38#16809947—24no—
39#16809946—21no—
40#16809946—21no—
41#16809946—21no—
42#16809946D412yes—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C4⋊Dic211D412yes—
46Dic211⋊C4—24no—
47C22⋊Dic211—21no—
48(C2×D211)⋊C4—24no—
49C4×Dic211—21no—
50C24×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211D412yes—
63C22×C4×C211—21no—
64C22×C4×C211D412yes—
65C22×C4×C211C2311yescharacteristic
66C22⋊C4×C211—28no—
67C42×C211—21no—
68C42×C211D412yes—
69C22×D211—21no—
70Dic422—21no—
71Dic422—21no—
72D844—21no—
73D844—21no—
74C22⋊D211—41no—
75C22⋊D211—41no—
76C2×Dic211—22no—
77C2×Dic211—21no—
78C4×D211—41no—
79C4×D211—41no—
80C211⋊C8—22no—
81C211⋊C8—41no—
82C23×C211—22no—
83C23×C211—44no—
84C23×C211—41no—
85C23×C211—41no—
86C23×C211C2×D411yescharacteristic
87Q8×C211—41no—
88Q8×C211—41no—
89Q8×C211—41no—
90Q8×C211—41no—
91D4×C211—22no—
92D4×C211—22no—
93D4×C211—41no—
94D4×C211—41no—
95C2×C4×C211—22no—
96C2×C4×C211—22no—
97C2×C4×C211—22no—
98C2×C4×C211—22no—
99C2×C4×C211—42no—
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211#4111yescharacteristic
106C22×C211—22no—
107C22×C211—44no—
108C22×C211—42no—
109C22×C211—82no—
110C22×C211—82no—
111C22×C211#12011yescharacteristic
112C22×C211#12011yescharacteristic
113C22×C211#12111yescharacteristic
114C2×D211—41no—
115C2×D211—42no—
116C4×C211—42no—
117C4×C211—42no—
118C4×C211#12011yescharacteristic
119C4×C211#12111yescharacteristic
120C4×C211#12111yescharacteristic
121C4×C211#12711yescharacteristic
122Dic211—41no—
123Dic211—42no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211—82no—
127C2×C211#39211yescharacteristic
128C2×C211#45412yes—
129D211—81no—
130C211#200311yescharacteristic, complemented, Sylow
131#2003—2111nomaximal, Sylow
132C2×AGL(1,ℤ/8)—2111no—
133#536—2111no—
134C2×#124D21111yescharacteristic, complemented
135C2×#104—2111no—
136#418—2111no—
137C2×#101—2111no—
138#339—2111no—
139C2×#41—2111no—
140C22×Q8C2×D21111yescharacteristic
141C22×D4C2×D21111yescharacteristic
142AGL(1,ℤ/8)—4222no—
143AGL(1,ℤ/8)—4222no—
144C2×SD16—4221no—
145C2×D8—4221no—
146C2×M16—2111no—
147C2×M16—2111no—
148C4⋊Q8—4221no—
149#124—22no—
150#124—22no—
151#124—4221no—
152#123—4222no—
153C22⋊Q8—4222no—
154#117—2111no—
155C2×C22⋊C4—22no—
156C2×C42C2×D21111yescharacteristic
157#104—4222no—
158#102—4222no—
159#101—4222no—
160C24—21no—
161#41—4221no—
162#41—4221no—
163#41—4221no—
164#41—4221no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—2111no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—4221no—
178C2×D4—2111no—
179C2×D4C22⋊D21112yes—
180C22×C4—21no—
181C22×C4—4221no—
182C22×C4C22⋊D21112yes—
183C22×C4C22×D21111yescharacteristic
184SD16—4224no—
185D8—4224no—
186M16—4221no—
187M16—4221no—
188M16—4221no—
189M16—2112no—
190C2×C8—4221no—
191C2×C8—4221no—
192C4⋊C4—4224no—
193C4⋊C4—2112no—
194C22⋊C4—28no—
195C22⋊C4—4224no—
196C22⋊C4—4221no—
197C42—21no—
198C42—4221no—
199C42C22⋊D21112yes—
200C23—22no—
201C23—44no—
202C23—41no—
203C23—41no—
204C23—4221no—
205C23C2×C22⋊D21111yescharacteristic
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—4221no—
211Q8—4221no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—8441no—
217D4—8441no—
218D4—4221no—
219D4—4221no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—8441no—
226C2×C4—8441no—
227C2×C4—4222no—
228C2×C4—4221no—
229C2×C4D4×D21111yescharacteristic
230C2×C4D4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4C2×C22⋊D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4#1680996111yescharacteristic
235C8—8441no—
236C8—4222no—
237C22—22no—
238C22—44no—
239C22—42no—
240C22—82no—
241C22—82no—
242C22—8442no—
243C22—8441no—
244C22#2026994511yescharacteristic
245C22#2026994611yescharacteristic
246C22#2026994911yescharacteristic
247C4—42no—
248C4—42no—
249C4—8442no—
250C4—8441no—
251C4#2026994511yescharacteristic
252C4#2026994811yescharacteristic
253C4#2026994911yescharacteristic
254C4#2026997311yescharacteristic
255C2—21no—
256C2—21no—
257C2—82no—
258C2—16881no—
259C2#2721305411yescharacteristic
260C2#2721331812yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#124×C211C2
socleC22×C211#120
prime radicalC2×#20270003C2
hypercenterC2×#124D211
hypocenterC211#2003
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #27213720C2semidirect
3. C2×#20269905C2semidirect
4. #27213151C2semidirect
5. C2×#20269896C2semidirect
6. #27212906C2semidirect
7. C2×#124×C211C2semidirect
130. C211#2003semidirect
134. C2×#124D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700032
#272137202
C2×#202699052
#272131512
C2×#202698962
#272129062
C2×#124×C2112
#2003211