Finite Group Databasebeta

← #41014568

Subgroups — #41014568

Order 27008.

244 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 65 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#27213699C211yesmaximal, characteristic, complemented
2C2×#20269904C211yesmaximal, characteristic
3#27213120C211yesmaximal, characteristic
4#27212917C211yesmaximal, characteristic, complemented
5#27212907C211yesmaximal, characteristic, complemented
6#27212902C211yesmaximal, characteristic, complemented
7#421×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9#20269976—21no—
10#20269968—21no—
11#20269954—22no—
12C4×Dic422—22no—
13#20269936C2211yescharacteristic
14#20269936C2211yescharacteristic
15C2×C4×Dic211C2211yescharacteristic
16#20269904—21no—
17#20269904—21no—
18#20269892—22no—
19C2×#41×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22#104×C211—22no—
23C2×Dic422—22no—
24C22×Dic211—21no—
25#16809946—21no—
26#16809946—21no—
27C2×C211⋊C8—21no—
28C4⋊Dic211—22no—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31Dic211⋊C4—22no—
32Dic211⋊C4—22no—
33C22⋊Dic211—21no—
34C22⋊Dic211D412yes—
35C4×Dic211—21no—
36C4×Dic211—21no—
37C4×Dic211—21no—
38C4×Dic211D411yescharacteristic
39C4×Dic211D411yescharacteristic
40#41×C211—21no—
41#41×C211—21no—
42#41×C211—21no—
43#41×C211—21no—
44C2×Q8×C211Q811yescharacteristic, complemented
45C2×D4×C211—21no—
46C2×D4×C211Q811yescharacteristic, complemented
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49M16×C211—21no—
50M16×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C22⋊C4×C211—21no—
54C42×C211D412yes—
55Dic422—44no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—22no—
59C2×Dic211—42no—
60C2×Dic211—41no—
61C211⋊C8—41no—
62C23×C211—21no—
63C23×C211C2×D411yescharacteristic
64Q8×C211—21no—
65Q8×C211—21no—
66D4×C211—21no—
67D4×C211—21no—
68D4×C211—41no—
69D4×C211—41no—
70C2×C4×C211—22no—
71C2×C4×C211—21no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×Q811yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C8×C211—41no—
81C22×C211—41no—
82C22×C211—41no—
83C22×C211—81no—
84C22×C211#12011yescharacteristic
85C22×C211C22⋊Q811yescharacteristic
86C22×C211#12411yescharacteristic
87C4×C211—42no—
88C4×C211—41no—
89C4×C211C22⋊Q811yescharacteristic
90C4×C211C22⋊Q811yescharacteristic
91C4×C211#12411yescharacteristic
92C4×C211#12411yescharacteristic
93Dic211—22no—
94Dic211—41no—
95Dic211—42no—
96Dic211—81no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211—21no—
100C2×C211—81no—
101C2×C211#39311yescharacteristic
102C211#203111yescharacteristic, complemented, Sylow
103#2031—2111nomaximal, Sylow
104#533—2111no—
105#421D21111yescharacteristic
106C2×#104—2111no—
107#410—2111no—
108#344—2111no—
109#339—2111no—
110H(3,ℤ/4)—2111no—
111C2×#41C2×D21111yescharacteristic
112C2×M16—2111no—
113C2×M16C2×D21111yescharacteristic
114C4⋊Q8—4221no—
115#125—4221no—
116C22⋊Q8—4222no—
117C4×Q8—4222no—
118#117—2111no—
119#117—2111no—
120#117C2×D21111yescharacteristic
121C2×C42—2111no—
122#104—22no—
123#104—4221no—
124#104—4221no—
125#99—4222no—
126#41—21no—
127#41—21no—
128#41—21no—
129#41—21no—
130C2×Q8—4222no—
131C2×Q8Dic42211yescharacteristic, complemented
132C2×D4—21no—
133C2×D4Dic42211yescharacteristic, complemented
134C22×C4—21no—
135C22×C4—4221no—
136C22×C4C22×D21111yescharacteristic
137M16—21no—
138M16—21no—
139M16—4221no—
140M16—4221no—
141C2×C8—21no—
142C2×C8—4221no—
143C4⋊C4—21no—
144C4⋊C4—4222no—
145C4⋊C4—4222no—
146C4⋊C4—4222no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C22⋊C4—21no—
150C22⋊C4—4221no—
151C22⋊C4—2112no—
152C42—4221no—
153C42—4221no—
154C42—4221no—
155C42—2111no—
156C42—2111no—
157C42C22⋊D21112yes—
158C23—21no—
159C23D4×D21111yescharacteristic
160Q8—21no—
161Q8—21no—
162Q8—8444no—
163D4—21no—
164D4—21no—
165D4—41no—
166D4—41no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—8442no—
172C2×C4—8441no—
173C2×C4—4221no—
174C2×C4—4222no—
175C2×C4—4221no—
176C2×C4D4×D21111yescharacteristic
177C2×C4C2×C22⋊D21111yescharacteristic
178C2×C4C2×Dic42211yescharacteristic
179C2×C4#1680996111yescharacteristic
180C2×C4#1680996111yescharacteristic
181C2×C4#1680996211yescharacteristic
182C8—41no—
183C8—8441no—
184C22—41no—
185C22—41no—
186C22—81no—
187C22#2026994511yescharacteristic
188C22#2026995411yescharacteristic
189C22#2026996311yescharacteristic
190C4—42no—
191C4—41no—
192C4—8441no—
193C4—8442no—
194C4—16881no—
195C4—4222no—
196C4#2026995411yescharacteristic
197C4#2026995611yescharacteristic
198C4#2026996311yescharacteristic
199C4#2026996411yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—81no—
204C2#2721305811yescharacteristic
subgroupG/H
centerC4#20269964
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#421×C211C2
socleC2×C211#393
prime radicalC2×#41×C211C22
hypercenter#421D211
hypocenterC211#2031
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #27213699C2semidirect
4. #27212917C2semidirect
5. #27212907C2semidirect
6. #27212902C2semidirect
44. C2×Q8×C211Q8semidirect
46. C2×D4×C211Q8semidirect
102. C211#2031semidirect
131. C2×Q8Dic422semidirect
133. C2×D4Dic422semidirect

Maximal subgroups

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

subgroupindex
#272136992
C2×#202699042
#272131202
#272129172
#272129072
#272129022
#421×C2112
#2031211