Finite Group Databasebeta

← #41013910

Subgroups — #41013910

Order 27008.

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

normal subgroups only

This group has 312 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×#20270008C211yesmaximal, characteristic
2C2×#20269985C211yesmaximal, characteristic
3#27213148C211yesmaximal, characteristic
4C2×#20269901C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7C2×#101×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—22no—
10#20270008C414yescomplemented
11C2×#16809952—21no—
12C2×#16809949—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#20269901—21no—
22#20269901—21no—
23#20269899—21no—
24C22×Q8×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27#101×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#16809952—24no—
38#16809949—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—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211C2×C412yes—
56C22×C4×C211—21no—
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#4111yescharacteristic
80Q8×C211—22no—
81Q8×C211—22no—
82Q8×C211—41no—
83Q8×C211—41no—
84C2×C4×C211—22no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211C22×C411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211#4111yescharacteristic
93C8×C211—22no—
94C8×C211—41no—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211C4×D411yescharacteristic
98C22×C211#12111yescharacteristic
99C22×C211#12311yescharacteristic
100C2×D211—22no—
101C2×D211—41no—
102C2×D211—41no—
103C4×C211—42no—
104C4×C211C2×C22⋊C411yescharacteristic
105C4×C211C4×D411yescharacteristic
106C4×C211#12011yescharacteristic
107C4×C211#12111yescharacteristic
108Dic211—22no—
109Dic211—81no—
110C2×C211—21no—
111C2×C211—21no—
112C2×C211#38611yescharacteristic
113C2×C211#47111yescharacteristic
114C2×C211#47311yescharacteristic
115D211—42no—
116C211#190811yescharacteristic, complemented, Sylow
117#1908—2111nomaximal, Sylow
118C2×#137—2111no—
119C2×#131—2111no—
120#417—2111no—
121C2×#103—2111no—
122C2×#101D21111yescharacteristic, complemented
123#343—2111no—
124#341—2111no—
125C2×#41—2111no—
126C22×Q8C2×D21111yescharacteristic
127#137—4222no—
128#137—2114no—
129C2×Q16—4221no—
130C2×SD16—4221no—
131#131—4222no—
132#131—4222no—
133C2×M16—4221no—
134C2×M16—2111no—
135C2×M16C2×D21111yescharacteristic
136C2×M16C2×D21111yescharacteristic
137C22×C8—4221no—
138C22×C8—2111no—
139C2×C4⋊C4—2111no—
140#103—4221no—
141#103—4221no—
142#103—4221no—
143#102—4221no—
144#101—22no—
145#41—4221no—
146#41—4221no—
147#41—2112no—
148#41—2112no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—2111no—
155C2×Q8C4×D21112yes—
156C2×D4—4221no—
157C2×D4—2111no—
158C22×C4—21no—
159C22×C4—4221no—
160C22×C4—4221no—
161C22×C4C22×D21111yescharacteristic
162Q16—4224no—
163SD16—4224no—
164M16—21no—
165M16—21no—
166M16—21no—
167M16—4224no—
168M16—4221no—
169M16—2112no—
170M16C22⋊D21112yes—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—4224no—
174C2×C8—4221no—
175C2×C8—4221no—
176C2×C8—4221no—
177C2×C8—2111no—
178C2×C8—2111no—
179C4⋊C4—4221no—
180C4⋊C4—4221no—
181C23—4221no—
182C23#1680996211yescharacteristic
183Q8—22no—
184Q8—22no—
185Q8—41no—
186Q8—41no—
187Q8—4221no—
188Q8—2112no—
189D4—4222no—
190D4—4222no—
191D4—4221no—
192D4—2112no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—8441no—
197C2×C4—16881no—
198C2×C4—4221no—
199C2×C4—4222no—
200C2×C4—4222no—
201C2×C4C2×C4×D21111yescharacteristic
202C2×C4D4×D21111yescharacteristic
203C2×C4D4×D21111yescharacteristic
204C2×C4D4×D21111yescharacteristic
205C2×C4C2×C22⋊D21111yescharacteristic
206C2×C4#1680996111yescharacteristic
207C8—22no—
208C8—41no—
209C8—4222no—
210C8—4221no—
211C8—4221no—
212C22—21no—
213C22—21no—
214C22—8441no—
215C22—8441no—
216C22—4222no—
217C22C4×C22⋊D21111yescharacteristic
218C22#2026995011yescharacteristic
219C22#2026995911yescharacteristic
220C4—42no—
221C4—16881no—
222C4—4222no—
223C4C22⋊C4×D21111yescharacteristic
224C4(C22⋊D211)⋊C411yescharacteristic
225C4#2026994511yescharacteristic
226C4#2026994811yescharacteristic
227C2—21no—
228C2—21no—
229C2—8442no—
230C2#2721302211yescharacteristic
231C2#2721341211yescharacteristic
232C2#2721342011yescharacteristic
subgroupG/H
centerC22#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211#121
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenterC2×#101D211
hypocenterC211#1908
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
4. C2×#20269901C2semidirect
5. #27212914C2semidirect
6. #27212910C2semidirect
7. C2×#101×C211C2semidirect
10. #20270008C4semidirect
116. C211#1908semidirect
122. C2×#101D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700082
C2×#202699852
#272131482
C2×#202699012
#272129142
#272129102
C2×#101×C2112
#1908211