Finite Group Databasebeta

← #41014567

Subgroups — #41014567

Order 27008.

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

normal subgroups only

This group has 244 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
1#27213164C211yesmaximal, characteristic
2C2×#20269905C211yesmaximal, characteristic
3#27213119C211yesmaximal, characteristic
4#27212916C211yesmaximal, characteristic, complemented
5#27212906C211yesmaximal, characteristic, complemented
6#27212902C211yesmaximal, characteristic, complemented
7#533×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269905—21no—
13#20269905—21no—
14#20269905—22no—
15#20269891—22no—
16C4⋊Q8×C211—21no—
17#125×C211—21no—
18C22⋊Q8×C211—22no—
19C4×Q8×C211—22no—
20#117×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22C2×C42×C211C2211yescharacteristic
23#16809962—21no—
24#16809962—21no—
25#16809962—21no—
26#16809962—21no—
27C2×Dic422Q811yescharacteristic, complemented
28C2×D844Q811yescharacteristic, complemented
29C2×C22⋊D211—21no—
30C2×C4×D211—21no—
31#16809946—21no—
32#16809946—21no—
33#16809946—21no—
34#16809946—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C4⋊Dic211—21no—
38C22⋊Dic211—21no—
39C4×Dic211—21no—
40C2×Q8×C211—22no—
41C22×C4×C211—21no—
42C22×C4×C211C2311yescharacteristic
43C4⋊C4×C211—24no—
44C4⋊C4×C211—22no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211D412yes—
49C42×C211—21no—
50C42×C211—21no—
51C42×C211D411yescharacteristic
52C42×C211D411yescharacteristic
53C42×C211D412yes—
54C22×D211—21no—
55Dic422—21no—
56Dic422—21no—
57D844—21no—
58D844—21no—
59C22⋊D211—41no—
60C22⋊D211—41no—
61C2×Dic211—21no—
62C2×Dic211—41no—
63C4×D211—41no—
64C4×D211—41no—
65C211⋊C8—41no—
66C211⋊C8—41no—
67C23×C211C2×D411yescharacteristic
68Q8×C211—44no—
69C2×C4×C211—21no—
70C2×C4×C211—21no—
71C2×C4×C211—22no—
72C2×C4×C211—22no—
73C2×C4×C211—42no—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×Q811yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C22×C211—41no—
81C22×C211#12011yescharacteristic
82C22×C211C22⋊Q811yescharacteristic
83C22×C211#12411yescharacteristic
84C2×D211—41no—
85C2×D211—81no—
86C4×C211—22no—
87C4×C211—41no—
88C4×C211—42no—
89C4×C211—42no—
90C4×C211C22⋊Q811yescharacteristic
91C4×C211C22⋊Q811yescharacteristic
92C4×C211#12411yescharacteristic
93C4×C211#12411yescharacteristic
94Dic211—41no—
95Dic211—81no—
96C2×C211—21no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211#39311yescharacteristic
100D211—81no—
101C211#203111yescharacteristic, complemented, Sylow
102#2031—2111nomaximal, Sylow
103#533D21111yescharacteristic, complemented
104#421—2111no—
105C2×#104—2111no—
106#410—2111no—
107#344—2111no—
108#339—2111no—
109H(3,ℤ/4)—2111no—
110C2×#41—2111no—
111C2×M16—2111no—
112C2×M16—2111no—
113C4⋊Q8—21no—
114#125—21no—
115C22⋊Q8—22no—
116C4×Q8—22no—
117#117—2111no—
118#117C2×D21111yescharacteristic
119#117C2×D21111yescharacteristic
120C2×C42C2×D21111yescharacteristic
121#104—4221no—
122#104—4221no—
123#104—4222no—
124#99—4222no—
125#41—4221no—
126#41—4221no—
127#41—4221no—
128#41—4221no—
129C2×Q8—22no—
130C2×Q8—2111no—
131C2×D4—4221no—
132C2×D4—2111no—
133C22×C4—21no—
134C22×C4—4221no—
135C22×C4C22×D21111yescharacteristic
136M16—4221no—
137M16—4221no—
138M16—4221no—
139M16—4221no—
140C2×C8—4221no—
141C2×C8—4221no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—4221no—
147C22⋊C4—21no—
148C22⋊C4—4221no—
149C22⋊C4C22⋊D21112yes—
150C42—21no—
151C42—21no—
152C42—4221no—
153C42C22⋊D21111yescharacteristic
154C42C22⋊D21111yescharacteristic
155C42C22⋊D21112yes—
156C23—4221no—
157C23C2×C22⋊D21111yescharacteristic
158Q8—44no—
159Q8—4221no—
160Q8—4221no—
161D4—8441no—
162D4—8441no—
163D4—4221no—
164D4—4221no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—8441no—
171C2×C4—8441no—
172C2×C4—8441no—
173C2×C4—4221no—
174C2×C4C2×C22⋊D21111yescharacteristic
175C2×C4C2×C22⋊D21111yescharacteristic
176C2×C4Q8×D21111yescharacteristic
177C2×C4#1680996311yescharacteristic
178C2×C4#1680996311yescharacteristic
179C2×C4#1680996311yescharacteristic
180C8—8441no—
181C8—8441no—
182C22—41no—
183C22—8441no—
184C22—16881no—
185C22#2026994611yescharacteristic
186C22#2026995311yescharacteristic
187C22#2026996611yescharacteristic
188C4—22no—
189C4—41no—
190C4—42no—
191C4—42no—
192C4—8441no—
193C4—16881no—
194C4#2026995311yescharacteristic
195C4#2026995311yescharacteristic
196C4#2026996611yescharacteristic
197C4#2026996611yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—16881no—
202C2#2721305711yescharacteristic
subgroupG/H
centerC4#20269966
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#533×C211C2
socleC2×C211#393
prime radicalC2×#16809962C22
hypercenter#533D211
hypocenterC211#2031
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
4. #27212916C2semidirect
5. #27212906C2semidirect
6. #27212902C2semidirect
7. #533×C211C2semidirect
27. C2×Dic422Q8semidirect
28. C2×D844Q8semidirect
101. C211#2031semidirect
103. #533D211semidirect

Maximal subgroups

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

subgroupindex
#272131642
C2×#202699052
#272131192
#272129162
#272129062
#272129022
#533×C2112
#2031211