Finite Group Databasebeta

← #41013461

Subgroups — #41013461

Order 27008.

264 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 85 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×#20269985C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3C2×#20269910C211yesmaximal, characteristic, complemented
4#27213148C211yesmaximal, characteristic
5#27213145C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic, complemented
7#343×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269985—22no—
10#20269985C414yescomplemented
11C2×#16809946C2211yescharacteristic
12C2×#16809945—21no—
13C22×C211⋊C8C2211yescharacteristic
14C2×C8×D211—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16#20269912—22no—
17#20269910—21no—
18#20269910—21no—
19#20269902—21no—
20#20269900—21no—
21#20269899—21no—
22#20269897—21no—
23C2×M16×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26#16809962—21no—
27#16809962—21no—
28#16809962C2×C412yes—
29#16809962Q812yes—
30C2×Dic422D411yescharacteristic
31C2×D844D411yescharacteristic
32C2×C22⋊D211—21no—
33C22×Dic211—21no—
34C2×C4×D211—21no—
35#16809946—21no—
36#16809946—21no—
37#16809945—24no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8C2×C412yes—
40C2×C211⋊C8D411yescharacteristic
41C2×C211⋊C8D411yescharacteristic
42C2×C211⋊C8D412yes—
43C8×D211—24no—
44C4⋊Dic211—21no—
45C4⋊Dic211—21no—
46C22×C4×C211—21no—
47C22×C4×C211C2311yescharacteristic
48M16×C211—21no—
49M16×C211—21no—
50M16×C211—21no—
51M16×C211C2×C412yes—
52C2×C8×C211—21no—
53C2×C8×C211—21no—
54C4⋊C4×C211—21no—
55C4⋊C4×C211—21no—
56C22×D211—21no—
57Dic422—21no—
58Dic422C4⋊C412yes—
59D844—21no—
60D844C4⋊C412yes—
61C22⋊D211—22no—
62C22⋊D211—22no—
63C2×Dic211—21no—
64C2×Dic211—41no—
65C2×Dic211—81no—
66C4×D211—22no—
67C4×D211—22no—
68C211⋊C8—41no—
69C211⋊C8C22⋊C412yes—
70C211⋊C8C22⋊C412yes—
71C23×C211#4111yescharacteristic
72C2×C4×C211—41no—
73C2×C4×C211—81no—
74C2×C4×C211C22×C411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×Q811yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C8×C211—22no—
81C8×C211—41no—
82C22×C211—21no—
83C22×C211—21no—
84C22×C211#11711yescharacteristic
85C22×C211#12111yescharacteristic
86C22×C211C22⋊Q811yescharacteristic
87C2×D211—22no—
88C2×D211—41no—
89C2×D211—41no—
90C4×C211—81no—
91C4×C211C2×C22⋊C411yescharacteristic
92C4×C211C2×C4⋊C411yescharacteristic
93C4×C211#12111yescharacteristic
94C4×C211C22⋊Q811yescharacteristic
95Dic211—22no—
96Dic211—81no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211#38011yescharacteristic
100C2×C211#47211yescharacteristic
101C2×C211#47311yescharacteristic
102D211—42no—
103C211#181611yescharacteristic, complemented, Sylow
104#1816—2111nomaximal, Sylow
105C2×#131—2111no—
106C2×#108—2111no—
107C2×#107—2111no—
108#417—2111no—
109#417—2111no—
110#343—2111no—
111#343D21111yescharacteristic, complemented
112C2×#41—2111no—
113#131—4222no—
114#131—2114no—
115C2×M16—4221no—
116C2×M16—2111no—
117C2×M16C2×D21111yescharacteristic
118C2×M16C2×D21111yescharacteristic
119C22×C8—4221no—
120C22×C8—2111no—
121C2×C4⋊C4—2111no—
122C2×C4⋊C4C2×D21111yescharacteristic
123#108—4222no—
124#107—4221no—
125#107—4221no—
126#103—4221no—
127#103—4221no—
128#102—4221no—
129#102—4221no—
130#41—4221no—
131#41—4221no—
132#41—2112no—
133#41—2112no—
134C2×Q8—2111no—
135C2×D4—4221no—
136C2×D4—2111no—
137C22×C4—21no—
138C22×C4—4221no—
139C22×C4—4221no—
140C22×C4C22×D21111yescharacteristic
141M16—21no—
142M16—21no—
143M16—21no—
144M16—4224no—
145M16—4221no—
146M16—4221no—
147M16C4×D21112yes—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—4224no—
151C2×C8—4221no—
152C2×C8—2112no—
153C2×C8—2111no—
154C2×C8—2111no—
155C2×C8—2112no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C23—4221no—
161C23#1680996211yescharacteristic
162Q8—4221no—
163Q8—2112no—
164D4—4222no—
165D4—4222no—
166D4—4221no—
167D4—2112no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—8441no—
171C2×C4—16881no—
172C2×C4—4221no—
173C2×C4—4222no—
174C2×C4—4222no—
175C2×C4C2×C4×D21111yescharacteristic
176C2×C4D4×D21111yescharacteristic
177C2×C4D4×D21111yescharacteristic
178C2×C4D4×D21111yescharacteristic
179C2×C4Q8×D21111yescharacteristic
180C2×C4#1680996211yescharacteristic
181C8—22no—
182C8—41no—
183C8—8441no—
184C8—2112no—
185C8—2112no—
186C22—21no—
187C22—21no—
188C22—8441no—
189C22—8441no—
190C22—4222no—
191C22#2026993411yescharacteristic
192C22#2026995011yescharacteristic
193C22#2026995511yescharacteristic
194C4—81no—
195C4—16881no—
196C4—4222no—
197C4C22⋊C4×D21111yescharacteristic
198C4C4⋊C4×D21111yescharacteristic
199C4#2026995011yescharacteristic
200C4#2026995511yescharacteristic
201C2—21no—
202C2—21no—
203C2—8442no—
204C2#2721300511yescharacteristic
205C2#2721341511yescharacteristic
206C2#2721342011yescharacteristic
subgroupG/H
centerC22#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#121
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#343D211
hypocenterC211#1816
LCSG ▷ C2×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
3. C2×#20269910C2semidirect
6. #27212914C2semidirect
7. #343×C211C2semidirect
10. #20269985C4semidirect
103. C211#1816semidirect
111. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699852
C2×#202699122
C2×#202699102
#272131482
#272131452
#272129142
#343×C2112
#1816211