Finite Group Databasebeta

← #41013813

Subgroups — #41013813

Order 27008.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#27213912C211yesmaximal, characteristic, complemented
2#27213657C211yesmaximal, characteristic
3#27213128C211yesmaximal, characteristic, complemented
4#27213110C211yesmaximal, characteristic, complemented
5#27212918C211yesmaximal, characteristic, complemented
6#27212917C211yesmaximal, characteristic, complemented
7#407×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269985—22no—
10#20269983—22no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2211yescharacteristic
13M16×D211—22no—
14#20269960C412yes—
15#20269956—21no—
16#20269951—21no—
17#20269946C412yes—
18C4×C22⋊D211—22no—
19#20269936C2211yescharacteristic
20#20269895—21no—
21#20269893—21no—
22C22⋊(C211⋊C8)—22no—
23C23×C4×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26C22⋊C8×C211—22no—
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31C2×Dic422D411yescharacteristic
32C2×D844D411yescharacteristic
33C2×C22⋊D211—22no—
34C2×C22⋊D211C2×C412yes—
35C2×C4×D211D412yes—
36#16809946—21no—
37#16809946—21no—
38#16809946—21no—
39#16809946—21no—
40#16809945—24no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8D412yescomplemented
43C8×D211—24no—
44C4⋊Dic211—21no—
45Dic211⋊C4—22no—
46C22⋊Dic211—22no—
47C22⋊Dic211C2×C412yes—
48(C2×D211)⋊C4—22no—
49C4×Dic211—21no—
50C24×C211C2×C411yescharacteristic
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211—21no—
54C22×C4×C211—21no—
55C22×C4×C211—21no—
56C22×C4×C211—21no—
57C22×C4×C211C2×C411yescharacteristic
58C22×C4×C211C2311yescharacteristic
59M16×C211—21no—
60M16×C211—21no—
61M16×C211—21no—
62M16×C211—21no—
63C2×C8×C211—21no—
64C2×C8×C211D412yescomplemented
65C22×D211C22⋊C412yes—
66Dic422—21no—
67Dic422—21no—
68D844—21no—
69D844—21no—
70C22⋊D211—22no—
71C22⋊D211—22no—
72C22⋊D211—44no—
73C2×Dic211—22no—
74C2×Dic211C22⋊C412yes—
75C4×D211—22no—
76C4×D211—22no—
77C211⋊C8—22no—
78C211⋊C8—41no—
79C23×C211—21no—
80C23×C211—21no—
81C23×C211—21no—
82C23×C211—42no—
83C23×C211C22×C411yescharacteristic
84C2×C4×C211—21no—
85C2×C4×C211—21no—
86C2×C4×C211—21no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211#4111yescharacteristic
96C2×C4×C211#4111yescharacteristic
97C8×C211—22no—
98C8×C211—41no—
99C22×C211—22no—
100C22×C211—22no—
101C22×C211—22no—
102C22×C211—41no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211—41no—
106C22×C211—41no—
107C22×C211C2×C22⋊C411yescharacteristic
108C22×C211C4×D411yescharacteristic
109C22×C211C4×D411yescharacteristic
110C2×D211—22no—
111C2×D211—42no—
112C4×C211—41no—
113C4×C211#12011yescharacteristic
114C4×C211#12111yescharacteristic
115C4×C211#12111yescharacteristic
116C4×C211#12311yescharacteristic
117Dic211—22no—
118Dic211—42no—
119C2×C211—21no—
120C2×C211—21no—
121C2×C211—21no—
122C2×C211—42no—
123C2×C211#38611yescharacteristic
124D211—42no—
125C211#189111yescharacteristic, complemented, Sylow
126#1891—2111nomaximal, Sylow
127#568—2111no—
128#525—2111no—
129#413—2111no—
130#407—2111no—
131#407D21111yescharacteristic, complemented
132#344—2111no—
133#344—2111no—
134C2×#41—2111no—
135C23×C4C2×D21111yescharacteristic
136#131—4222no—
137#131—4222no—
138C2×M16—4222no—
139C2×M16—2111no—
140C2×M16—2111no—
141C2×M16C2×D21111yescharacteristic
142C2×M16C2×D21111yescharacteristic
143#123—2112no—
144C22⋊Q8—4221no—
145#121—4221no—
146#120—2112no—
147C4×D4—4222no—
148#117—2111no—
149#101—4221no—
150#100—4221no—
151C22⋊C8—22no—
152C22⋊C8—4222no—
153C24C4×D21111yescharacteristic
154#41—4221no—
155#41—4221no—
156#41—4221no—
157#41—4221no—
158C2×Q8—2111no—
159C2×D4—4222no—
160C2×D4—2112no—
161C2×D4—2111no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—2112no—
169C22×C4C4×D21111yescharacteristic
170C22×C4C22×D21111yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—4224no—
176M16—4221no—
177M16—4221no—
178M16—4221no—
179M16—4221no—
180C2×C8—21no—
181C2×C8—4224no—
182C2×C8—4221no—
183C2×C8—2112no—
184C2×C8C22⋊D21112yescomplemented
185C4⋊C4—4222no—
186C4⋊C4—4221no—
187C22⋊C4—4222no—
188C22⋊C4—4222no—
189C22⋊C4—2112no—
190C42—4221no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—2112no—
196C23C2×C4×D21111yescharacteristic
197Q8—4221no—
198Q8—4221no—
199D4—8444no—
200D4—4222no—
201D4—4222no—
202D4—4221no—
203D4—4221no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—4222no—
212C2×C4—4222no—
213C2×C4—4222no—
214C2×C4—2112no—
215C2×C4D4×D21111yescharacteristic
216C2×C4D4×D21111yescharacteristic
217C2×C4D4×D21111yescharacteristic
218C2×C4C2×C22⋊D21111yescharacteristic
219C2×C4#1680996111yescharacteristic
220C2×C4#1680996211yescharacteristic
221C8—22no—
222C8—41no—
223C8—8441no—
224C8—4222no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—8442no—
234C22—4222no—
235C22C22⋊C4×D21111yescharacteristic
236C22(C22⋊D211)⋊C411yescharacteristic
237C22C4×C22⋊D21111yescharacteristic
238C4—41no—
239C4—8442no—
240C4—4222no—
241C4#2026994511yescharacteristic
242C4#2026994811yescharacteristic
243C4#2026995011yescharacteristic
244C4#2026995911yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—8442no—
250C2#2721302211yescharacteristic
subgroupG/H
centerC4#20269945
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC2×C211#386
prime radical#27213657C2
cosocleC22×C4×C211C23
hypercenter#407D211
hypocenterC211#1891
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #27213912C2semidirect
3. #27213128C2semidirect
4. #27213110C2semidirect
5. #27212918C2semidirect
6. #27212917C2semidirect
7. #407×C211C2semidirect
42. C2×C211⋊C8D4semidirect
64. C2×C8×C211D4semidirect
125. C211#1891semidirect
131. #407D211semidirect
184. C2×C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272139122
#272136572
#272131282
#272131102
#272129182
#272129172
#407×C2112
#1891211