Finite Group Databasebeta

← #41013903

Subgroups — #41013903

Order 27008.

332 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C2×#20269985C211yesmaximal, characteristic, complemented
3#27213148C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270003—22no—
10#20270003C414yescomplemented
11C2×#16809950—21no—
12C2×#16809947—21no—
13#20269985—22no—
14#20269985—22no—
15C2×#16809946C2211yescharacteristic
16C2×#16809945—21no—
17C22×C211⋊C8C2211yescharacteristic
18C2×C8×D211—21no—
19C2×C4⋊Dic211C2211yescharacteristic
20#20269902—21no—
21#20269899—21no—
22#20269898—21no—
23#20269898—21no—
24C22×D4×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27#100×C211—22no—
28#16809962—21no—
29#16809962—21no—
30#16809962C2×C412yes—
31#16809962D412yes—
32C2×Dic422D411yescharacteristic
33C2×D844D411yescharacteristic
34C2×C22⋊D211—21no—
35C22×Dic211—21no—
36C2×C4×D211—21no—
37#16809950—24no—
38#16809947—24no—
39#16809946—21no—
40#16809946C2×C412yes—
41#16809945—24no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8D411yescharacteristic
46C2×C211⋊C8D411yescharacteristic
47C8×D211—24no—
48C4⋊Dic211—21no—
49C4⋊Dic211—21no—
50C24×C211—21no—
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211C2×C412yes—
57C22×C4×C211C2311yescharacteristic
58M16×C211—21no—
59M16×C211—21no—
60M16×C211—21no—
61M16×C211D412yes—
62C2×C8×C211—21no—
63C2×C8×C211—21no—
64C22×D211—21no—
65Dic422—21no—
66Dic422C22⋊C412yes—
67D844—21no—
68D844C22⋊C412yes—
69C22⋊D211—22no—
70C22⋊D211—22no—
71C2×Dic211—21no—
72C2×Dic211—41no—
73C2×Dic211—81no—
74C4×D211—22no—
75C4×D211—22no—
76C211⋊C8—22no—
77C211⋊C8—21no—
78C211⋊C8—21no—
79C23×C211—22no—
80C23×C211—41no—
81C23×C211—41no—
82C23×C211—81no—
83C23×C211—81no—
84C23×C211#4111yescharacteristic
85D4×C211—22no—
86D4×C211—22no—
87D4×C211—41no—
88D4×C211—41no—
89C2×C4×C211C22×C411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C8×C211—22no—
96C8×C211—41no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211—42no—
100C22×C211—82no—
101C22×C211—82no—
102C22×C211—81no—
103C22×C211—81no—
104C22×C211C4×D411yescharacteristic
105C22×C211#12111yescharacteristic
106C22×C211#12311yescharacteristic
107C2×D211—22no—
108C2×D211—41no—
109C2×D211—41no—
110C4×C211C2×C22⋊C411yescharacteristic
111C4×C211C4×D411yescharacteristic
112C4×C211#12011yescharacteristic
113C4×C211#12111yescharacteristic
114Dic211—22no—
115Dic211—81no—
116C2×C211—21no—
117C2×C211—21no—
118C2×C211—82no—
119C2×C211#38611yescharacteristic
120C2×C211#47111yescharacteristic
121C2×C211#47211yescharacteristic
122D211—42no—
123C211#190711yescharacteristic, complemented, Sylow
124#1907—2111nomaximal, Sylow
125C2×AGL(1,ℤ/8)—2111no—
126C2×#131—2111no—
127#417—2111no—
128C2×#102—2111no—
129C2×#100D21111yescharacteristic, complemented
130#343—2111no—
131#341—2111no—
132C2×#41—2111no—
133C22×D4C2×D21111yescharacteristic
134AGL(1,ℤ/8)—4222no—
135AGL(1,ℤ/8)—2114no—
136C2×SD16—4221no—
137C2×D8—4221no—
138#131—4222no—
139#131—4222no—
140C2×M16—4221no—
141C2×M16—2111no—
142C2×M16C2×D21111yescharacteristic
143C2×M16C2×D21111yescharacteristic
144C22×C8—4221no—
145C22×C8—2111no—
146C2×C4⋊C4—2111no—
147#103—4221no—
148#102—4221no—
149#102—4221no—
150#102—4221no—
151#100—22no—
152C24—21no—
153#41—4221no—
154#41—4221no—
155#41—2112no—
156#41—2112no—
157C2×Q8—2111no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—4221no—
164C2×D4—2111no—
165C2×D4C4×D21112yes—
166C22×C4—4221no—
167C22×C4—4221no—
168C22×C4C22×D21111yescharacteristic
169SD16—4224no—
170D8—4224no—
171M16—21no—
172M16—21no—
173M16—21no—
174M16—4224no—
175M16—4221no—
176M16—2112no—
177M16C22⋊D21112yes—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—4224no—
181C2×C8—4221no—
182C2×C8—4221no—
183C2×C8—4221no—
184C2×C8—2111no—
185C2×C8—2111no—
186C4⋊C4—4221no—
187C4⋊C4—4221no—
188C23—22no—
189C23—41no—
190C23—41no—
191C23—81no—
192C23—81no—
193C23—4221no—
194C23#1680996211yescharacteristic
195Q8—4221no—
196Q8—2112no—
197D4—22no—
198D4—22no—
199D4—41no—
200D4—41no—
201D4—4222no—
202D4—4222no—
203D4—4221no—
204D4—2112no—
205C2×C4—8441no—
206C2×C4—16881no—
207C2×C4—4221no—
208C2×C4—4222no—
209C2×C4—4222no—
210C2×C4C2×C4×D21111yescharacteristic
211C2×C4D4×D21111yescharacteristic
212C2×C4D4×D21111yescharacteristic
213C2×C4D4×D21111yescharacteristic
214C2×C4C2×C22⋊D21111yescharacteristic
215C2×C4#1680996111yescharacteristic
216C8—22no—
217C8—41no—
218C8—4222no—
219C8—4221no—
220C8—4221no—
221C22—21no—
222C22—21no—
223C22—42no—
224C22—82no—
225C22—82no—
226C22—81no—
227C22—81no—
228C22—8441no—
229C22—8441no—
230C22—4222no—
231C22C4×C22⋊D21111yescharacteristic
232C22#2026995011yescharacteristic
233C22#2026995911yescharacteristic
234C4—16881no—
235C4—4222no—
236C4C22⋊C4×D21111yescharacteristic
237C4(C22⋊D211)⋊C411yescharacteristic
238C4#2026994511yescharacteristic
239C4#2026994811yescharacteristic
240C2—21no—
241C2—21no—
242C2—82no—
243C2—8442no—
244C2#2721302211yescharacteristic
245C2#2721340811yescharacteristic
246C2#2721341511yescharacteristic
subgroupG/H
centerC22#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211#121
prime radicalC2×#20270003C2
cosocleC22×C4×C211C23
hypercenterC2×#100D211
hypocenterC211#1907
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#20269985C2semidirect
3. #27213148C2semidirect
4. C2×#20269898C2semidirect
5. #27212914C2semidirect
6. #27212910C2semidirect
7. C2×#100×C211C2semidirect
10. #20270003C4semidirect
123. C211#1907semidirect
129. C2×#100D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700032
C2×#202699852
#272131482
C2×#202698982
#272129142
#272129102
C2×#100×C2112
#1907211