Finite Group Databasebeta

← #41013700

Subgroups — #41013700

Order 27008.

376 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#27213657C211yesmaximal, characteristic
2C2×#20269903C212yesmaximal, complemented
3C4×C22⋊Dic211C211yesmaximal, characteristic, complemented
4#27212907C212yesmaximal, complemented
5C22×M16×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2211yescharacteristic
7#20269960—21no—
8#20269956C411yescharacteristic, complemented
9#20269956C411yescharacteristic, complemented
10#20269951C411yescharacteristic, complemented
11#20269951C411yescharacteristic, complemented
12#20269946—21no—
13C4×C22⋊D211—21no—
14C4×C22⋊D211—21no—
15#20269936C2211yescharacteristic
16C2×C22⋊Dic211—21no—
17C2×C4×Dic211C2212yes—
18#20269903—24no—
19C211⋊#95—21no—
20C23×C4×C211C2211yescharacteristic
21C2×M16×C211—21no—
22C2×M16×C211—24no—
23C2×M16×C211C2212yes—
24C22×C8×C211—21no—
25#16809962—22no—
26#16809962—22no—
27C2×Dic422C2×C411yescharacteristic
28C2×D844C2×C411yescharacteristic
29C2×C22⋊D211—21no—
30C2×C22⋊D211—21no—
31C2×C22⋊D211—21no—
32C22×Dic211—22no—
33C2×C4×D211—21no—
34C4⋊Dic211C2×C411yescharacteristic
35C4⋊Dic211C2×C411yescharacteristic
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38C22⋊Dic211—24no—
39C22⋊Dic211—21no—
40C22⋊Dic211—21no—
41C22⋊Dic211—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44C4×Dic211—21no—
45C4×Dic211—22no—
46C4×Dic211—22no—
47C24×C211D411yescharacteristic
48C22×C4×C211—22no—
49C22×C4×C211—22no—
50C22×C4×C211C2×C411yescharacteristic
51C22×C4×C211C2×C411yescharacteristic
52C22×C4×C211D411yescharacteristic
53C22×C4×C211D411yescharacteristic
54C22×C4×C211D411yescharacteristic
55C22×C4×C211C2311yescharacteristic
56M16×C211—24no—
57M16×C211—22no—
58M16×C211D414yescomplemented
59C2×C8×C211—24no—
60C2×C8×C211—22no—
61C22×D211—21no—
62Dic422—22no—
63D844—22no—
64C22⋊D211—42no—
65C22⋊D211—42no—
66C22⋊D211—42no—
67C2×Dic211—24no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—21no—
71C2×Dic211—42no—
72C2×Dic211—42no—
73C4×D211—41no—
74C4×D211—41no—
75C23×C211—22no—
76C23×C211—22no—
77C23×C211—41no—
78C23×C211C22⋊C411yescharacteristic
79C23×C211C22⋊C411yescharacteristic
80C23×C211C2×D411yescharacteristic
81C2×C4×C211—22no—
82C2×C4×C211—22no—
83C2×C4×C211—22no—
84C2×C4×C211—21no—
85C2×C4×C211—21no—
86C2×C4×C211—21no—
87C2×C4×C211—21no—
88C2×C4×C211C22⋊C411yescharacteristic
89C2×C4×C211C22⋊C411yescharacteristic
90C2×C4×C211C22×C411yescharacteristic
91C2×C4×C211C2×D412yes—
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211#4112yes—
94C8×C211—24no—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211—21no—
100C22×C211—21no—
101C22×C211—21no—
102C22×C211—44no—
103C22×C211#10412yescomplemented
104C22×C211C2×C22⋊C411yescharacteristic
105C22×C211#12112yes—
106C2×D211—41no—
107C2×D211—41no—
108C2×D211—41no—
109C4×C211—21no—
110C4×C211—21no—
111C4×C211C4×D412yes—
112C4×C211#12011yescharacteristic
113C4×C211#12311yescharacteristic
114Dic211—44no—
115Dic211—41no—
116Dic211—41no—
117Dic211—41no—
118C2×C211—22no—
119C2×C211—22no—
120C2×C211—41no—
121C2×C211#38611yescharacteristic
122C2×C211C2×#10411yescharacteristic
123C2×C211#42111yescharacteristic
124D211—81no—
125C211#187111yescharacteristic, complemented, Sylow
126#1871—2111nomaximal, Sylow
127C22×M16D21111yescharacteristic, complemented
128#525—2111no—
129C2×#104—2112no—
130C4×C22⋊C4—2111no—
131#339—2112no—
132C2×#41—2111no—
133C23×C4C2×D21111yescharacteristic
134C2×M16—21no—
135C2×M16—24no—
136C2×M16C2×D21112yes—
137C22×C8—21no—
138#123—4221no—
139C22⋊Q8—2111no—
140C22⋊Q8—2111no—
141#121—2111no—
142#121—2111no—
143#120—4221no—
144C4×D4—4221no—
145C4×D4—4221no—
146#117—2111no—
147C2×C22⋊C4—4221no—
148C2×C42—2112no—
149#104—4224no—
150#95—4221no—
151C24D84411yescharacteristic
152#41—4222no—
153#41—4222no—
154C2×Q8—2111no—
155C2×D4—4221no—
156C2×D4—4221no—
157C2×D4—4221no—
158C2×D4—2111no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—4222no—
162C22×C4—4221no—
163C22×C4C4×D21111yescharacteristic
164C22×C4C4×D21111yescharacteristic
165C22×C4C22⋊D21111yescharacteristic
166C22×C4C22⋊D21111yescharacteristic
167C22×C4D84411yescharacteristic
168C22×C4C22×D21111yescharacteristic
169M16—24no—
170M16—22no—
171M16C22⋊D21114yescomplemented
172C2×C8—24no—
173C2×C8—22no—
174C4⋊C4—4221no—
175C4⋊C4—4221no—
176C4⋊C4—2111no—
177C4⋊C4—2111no—
178C22⋊C4—4224no—
179C22⋊C4—4221no—
180C22⋊C4—4221no—
181C22⋊C4—4221no—
182C22⋊C4—4221no—
183C22⋊C4—4221no—
184C42—4222no—
185C42—4222no—
186C42—4221no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—4221no—
191C23(C2×D211)⋊C411yescharacteristic
192C23(C2×D211)⋊C411yescharacteristic
193C23C2×D84411yescharacteristic
194Q8—4222no—
195D4—8442no—
196D4—8442no—
197D4—8442no—
198D4—4222no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—8442no—
207C2×C4—8442no—
208C2×C4—8441no—
209C2×C4—8441no—
210C2×C4—4224no—
211C2×C4—4221no—
212C2×C4—4221no—
213C2×C4—4221no—
214C2×C4(C2×D211)⋊C411yescharacteristic
215C2×C4(C2×D211)⋊C411yescharacteristic
216C2×C4C2×C4×D21111yescharacteristic
217C2×C4C2×C22⋊D21112yes—
218C2×C4C2×C22⋊D21111yescharacteristic
219C2×C4#1680996212yes—
220C8—24no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—8441no—
230C22—8441no—
231C22—8441no—
232C22#2026990312yescomplemented
233C22C2×(C2×D211)⋊C411yescharacteristic
234C22#2026995112yes—
235C4—21no—
236C4—21no—
237C4—8444no—
238C4—8441no—
239C4—8441no—
240C4—8441no—
241C4C4×C22⋊D21112yes—
242C4#2026994611yescharacteristic
243C4#2026996011yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—16881no—
248C2#2721302511yescharacteristic
249C2C2×#2026990311yescharacteristic
250C2#2721316211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×M16×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213657C2
cosocleC22×C4×C211C23
hypercenterC22×M16D211
hypocenterC211#1871
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#20269903C2semidirect
3. C4×C22⋊Dic211C2semidirect
4. #27212907C2semidirect
5. C22×M16×C211C2semidirect
8. #20269956C4semidirect
9. #20269956C4semidirect
10. #20269951C4semidirect
11. #20269951C4semidirect
58. M16×C211D4semidirect
103. C22×C211#104semidirect
125. C211#1871semidirect
127. C22×M16D211semidirect
171. M16C22⋊D211semidirect
232. C22#20269903semidirect

Maximal subgroups

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

subgroupindex
#272136572
C2×#202699032
C4×C22⋊Dic2112
#272129072
C22×M16×C2112
#1871211