Finite Group Databasebeta

← #41014700

Subgroups — #41014700

Order 27008.

262 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 67 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×#20270010C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213145C211yesmaximal, characteristic
4C2×#20269900C211yesmaximal, characteristic, complemented
5C2×#20269895C211yesmaximal, characteristic, complemented
6#27213114C211yesmaximal, characteristic
7#343×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10#20270010—22no—
11#20270010—22no—
12C2×Dic844—21no—
13C2×#16809951—21no—
14C2×#16809946C2211yescharacteristic
15C22×C211⋊C8C2211yescharacteristic
16#20269912—22no—
17#20269900—21no—
18#20269900—22no—
19#20269897—21no—
20#20269895—22no—
21(C2×D211)⋊C8—22no—
22C2×M16×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25#16809962—21no—
26#16809962—21no—
27#16809962—21no—
28#16809962—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422D412yes—
35C2×Dic422D411yescharacteristic
36C2×D844D411yescharacteristic
37C2×C22⋊D211—21no—
38C22×Dic211—21no—
39C2×C4×D211—21no—
40Dic844—24no—
41#16809951—24no—
42#16809946—21no—
43#16809946—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8D412yes—
48C22×C4×C211—21no—
49C22×C4×C211C2311yescharacteristic
50M16×C211—21no—
51M16×C211—21no—
52M16×C211—21no—
53M16×C211D412yes—
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58C22×D211—21no—
59Dic422—22no—
60Dic422—21no—
61Dic422—22no—
62Dic422—21no—
63Dic422—41no—
64Dic422—41no—
65D844—21no—
66D844—21no—
67C22⋊D211—41no—
68C22⋊D211—41no—
69C2×Dic211—21no—
70C2×Dic211—22no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C4×D211—41no—
74C4×D211—41no—
75C211⋊C8—22no—
76C211⋊C8—41no—
77C23×C211#4111yescharacteristic
78C2×C4×C211—41no—
79C2×C4×C211—81no—
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—22no—
87C8×C211—41no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211#12111yescharacteristic
91C22×C211#12311yescharacteristic
92C22×C211#12411yescharacteristic
93C2×D211—41no—
94C2×D211—41no—
95C2×D211—41no—
96C4×C211—81no—
97C4×C211#12011yescharacteristic
98C4×C211#12111yescharacteristic
99C4×C211#12111yescharacteristic
100C4×C211#12311yescharacteristic
101Dic211—41no—
102Dic211—42no—
103C2×C211—21no—
104C2×C211—21no—
105C2×C211#39411yescharacteristic
106C2×C211#47111yescharacteristic
107C2×C211#47311yescharacteristic
108D211—81no—
109C211#205211yescharacteristic, complemented, Sylow
110#2052—2111nomaximal, Sylow
111C2×#137—2111no—
112C2×#108—2111no—
113#417—2111no—
114C2×#103—2111no—
115C2×#101—2111no—
116#408—2111no—
117#343D21111yescharacteristic, complemented
118C2×#41—2111no—
119C22×Q8—2111no—
120#137—4222no—
121#137—4222no—
122C2×Q16—4221no—
123C2×SD16—4221no—
124C2×M16—2111no—
125C2×M16C2×D21111yescharacteristic
126C2×M16C2×D21111yescharacteristic
127C22×C8—2111no—
128C2×C4⋊C4C2×D21111yescharacteristic
129#108—4222no—
130#103—4221no—
131#103—4222no—
132#102—4221no—
133#101—4222no—
134C22⋊C8—4222no—
135#41—4221no—
136#41—4221no—
137#41—4221no—
138#41—4221no—
139C2×Q8—4221no—
140C2×Q8—4221no—
141C2×Q8—4221no—
142C2×Q8—4221no—
143C2×Q8—4221no—
144C2×Q8—2112no—
145C2×Q8—2111no—
146C2×D4—4221no—
147C2×D4—2111no—
148C22×C4—21no—
149C22×C4—4221no—
150C22×C4—4221no—
151C22×C4C22×D21111yescharacteristic
152Q16—4224no—
153SD16—4224no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—4221no—
158M16—4221no—
159M16D84412yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—4221no—
163C2×C8—4221no—
164C2×C8—4221no—
165C2×C8—2112no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C23—4221no—
169C23#1680996211yescharacteristic
170Q8—8441no—
171Q8—8441no—
172Q8—4222no—
173Q8—4221no—
174Q8—4222no—
175Q8—4221no—
176D4—8441no—
177D4—8441no—
178D4—4221no—
179D4—4221no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—8441no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—4221no—
187C2×C4—4222no—
188C2×C4D4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C2×C4C2×D84411yescharacteristic
192C2×C4#1680996211yescharacteristic
193C2×C4#1680996311yescharacteristic
194C8—22no—
195C8—41no—
196C8—8441no—
197C8—4222no—
198C22—21no—
199C22—21no—
200C22—8441no—
201C22—8441no—
202C22—8441no—
203C22#2026995011yescharacteristic
204C22#2026996211yescharacteristic
205C22#2026996511yescharacteristic
206C4—81no—
207C4—8441no—
208C4—8442no—
209C4#2026994711yescharacteristic
210C4#2026995011yescharacteristic
211C4#2026995211yescharacteristic
212C4#2026996211yescharacteristic
213C2—21no—
214C2—21no—
215C2—16881no—
216C2#2721306411yescharacteristic
217C2#2721340811yescharacteristic
218C2#2721342011yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#121
prime radicalC2×#16809962C22
hypercenter#343D211
hypocenterC211#2052
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#20269912C2semidirect
4. C2×#20269900C2semidirect
5. C2×#20269895C2semidirect
7. #343×C211C2semidirect
109. C211#2052semidirect
117. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700102
C2×#202699122
#272131452
C2×#202699002
C2×#202698952
#272131142
#343×C2112
#2052211