Finite Group Databasebeta

← #13163817

Subgroups — #13163817

Order 1728.

1456 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 349 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×C33×SD16C211yesmaximal, characteristic, complemented
2#11723983C211yesmaximal, characteristic, complemented
3#11723973C211yesmaximal, characteristic
4#11723942C211yesmaximal, characteristic, complemented
5#11722206C211yesmaximal, characteristic, complemented
6#11722203C211yesmaximal, characteristic, complemented
7#11722189C211yesmaximal, characteristic, complemented
8#10592876—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11#89076C2211yescharacteristic
12C33×SD16—21no—
13C33×SD16—21no—
14C2×C33×C8C2211yescharacteristic
15#88863—21no—
16#88862—21no—
17#88861—21no—
18#88860—21no—
19#88855C2211yescharacteristic
20#88855C2211yescharacteristic
21#88855C2211yescharacteristic
22#88850—21no—
23C8×C3⋊(C3⋊D3)—21no—
24C2×C32×SD16D3113yescomplemented
25#64151—313no—
26#64141—313no—
27#64110—313no—
28#63663—313no—
29#63660—313no—
30#63646—313no—
31C23×C33—21no—
32C33×Q8—21no—
33C33×Q8—21no—
34C33×D4—21no—
35C33×D4—21no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38#6312—21no—
39C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
40C4×C3⋊(C3⋊D3)—21no—
41#6308D411yescharacteristic
42C33×C8D411yescharacteristic
43C33×C8D411yescharacteristic
44C3⋊(C3⋊(C3⋊C8))—21no—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47#5177—913no—
48C2×C32×Q8C2×D3113yes—
49C2×C32×D4C2×D3113yes—
50#3828—313no—
51C32×SD16—213no—
52C32×SD16—213no—
53C2×C32×C8C2×D3113yes—
54#3756—613no—
55#3755—613no—
56#3754—613no—
57#3753—613no—
58#3748—313no—
59#3748—313no—
60#3748—313no—
61#3743—613no—
62C8×C3⋊D3—613no—
63C22×C33—41no—
64C22×C33—81no—
65C22×C33#4111yescharacteristic
66C2×C3⋊(C3⋊D3)—21no—
67C33×C4—41no—
68C33×C4C2×D411yescharacteristic
69C33×C4C2×D411yescharacteristic
70C3⋊(C3⋊Dic3)—21no—
71C2×C3×SD16C3⋊D3113yescomplemented
72#942—913no—
73#932—913no—
74#901—913no—
75#836—913no—
76#833—913no—
77#819—913no—
78C23×C32—213no—
79C32×Q8—213no—
80C32×Q8—213no—
81C32×D4—213no—
82C32×D4—213no—
83C2×C32×C4—213no—
84C2×C32×C4C22×D3113yes—
85#638—613no—
86C4⋊(C3⋊D3)—313no—
87C4×C3⋊D3—613no—
88#634—313no—
89C32×C8#67113yes—
90C32×C8#67113yes—
91C3⋊(C3⋊C8)—613no—
92C3⋊(C3⋊C8)—613no—
93C3⋊(C3⋊C8)—613no—
94#471—271noSylow
95C2×C33—21no—
96C2×C33—81no—
97C2×C33#12111yescharacteristic
98C3⋊(C3⋊D3)—41no—
99C2×C3×Q8C2×C3⋊D3113yes—
100C2×C3×D4C2×C3⋊D3113yes—
101#235—913no—
102C3×SD16—213no—
103C3×SD16—213no—
104C2×C3×C8C2×C3⋊D3113yes—
105#216—1813no—
106#215—1813no—
107#214—1813no—
108#213—1813no—
109#208—913no—
110#208—913no—
111#208—913no—
112ΓL(1,𝔽25)—1813no—
113D3×C8—1813no—
114C22×C32—413no—
115C22×C32—813no—
116C22×C32#237113yes—
117C2×C3⋊D3—613no—
118C32×C4—413no—
119C32×C4D3×D4113yes—
120C32×C4C2×#67113yes—
121C3⋊Dic3—613no—
122#137—271no—
123AGL(1,ℤ/8)—271no—
124C2×SD16C3⋊(C3⋊D3)11yescharacteristic, complemented
125#131—271no—
126#108—271no—
127#101—271no—
128#100—271no—
129C33#47111yescharacteristic, complemented, Sylow
130C23×C3—213no—
131C3×Q8—213no—
132C3×Q8—213no—
133C3×D4—213no—
134C3×D4—213no—
135C2×C3×C4—213no—
136C2×C3×C4C22×C3⋊D3113yes—
137#67—1813no—
138D12—913no—
139C4×D3—1813no—
140Dic6—913no—
141C3×C8#638113yes—
142C3×C8#638113yes—
143C3⋊C8—1813no—
144C3⋊C8—1813no—
145C3⋊C8—1813no—
146C2×C32—213no—
147C2×C32—813no—
148C2×C32#938113yes—
149C3⋊D3—1213no—
150#41—271no—
151C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
152C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
153Q16—541no—
154SD16—21no—
155SD16—21no—
156SD16—541no—
157SD16—541no—
158D8—541no—
159M16—271no—
160M16—271no—
161M16—271no—
162M16—541no—
163C2×C8—541no—
164C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
165C22×C3—413no—
166C22×C3—813no—
167C22×C3#3830113yes—
168C2×D3—1813no—
169C3×C4—413no—
170C3×C4D4×C3⋊D3113yes—
171C3×C4C2×#638113yes—
172Dic3—1813no—
173C32#5177113yescomplemented
174C23—21no—
175Q8—21no—
176Q8—21no—
177Q8—271no—
178D4—21no—
179D4—21no—
180D4—271no—
181D4—541no—
182C2×C4—21no—
183C2×C4—541no—
184C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
185C8—541no—
186C8—541no—
187C8—541no—
188C8#631211yescharacteristic
189C8#631211yescharacteristic
190C2×C3—213no—
191C2×C3—813no—
192C2×C3#64147113yes—
193D3—3613no—
194C22—41no—
195C22—81no—
196C22—541no—
197C22#8907811yescharacteristic
198C4—41no—
199C4—541no—
200C4D4×C3⋊(C3⋊D3)11yescharacteristic
201C4C2×#631211yescharacteristic
202C3#10592876113yescomplemented
203C2—21no—
204C2—81no—
205C2—1081no—
206C2#1172397911yescharacteristic
subgroupG/H
centerC2#11723979
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC2×C33×SD16C2
socleC2×C33#121
prime radical#11723973C2
hypercenterC2×SD16C3⋊(C3⋊D3)
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C33×SD16C2semidirect
2. #11723983C2semidirect
4. #11723942C2semidirect
5. #11722206C2semidirect
6. #11722203C2semidirect
7. #11722189C2semidirect
24. C2×C32×SD16D3semidirect
71. C2×C3×SD16C3⋊D3semidirect
124. C2×SD16C3⋊(C3⋊D3)semidirect
129. C33#471semidirect
173. C32#5177semidirect
202. C3#10592876semidirect

Maximal subgroups

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

subgroupindex
C2×C33×SD162
#117239832
#117239732
#117239422
#117222062
#117222032
#117221892
#105928763