Finite Group Databasebeta

← #13173187

Subgroups — #13173187

Order 1728.

526 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C32×#994C211yesmaximal, characteristic, complemented
2#11724493C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
4#10596307C311yesmaximal, characteristic, complemented
5#5836⋊C3—34nomaximal
6C32×#41⋊C3—21no—
7C32×#41⋊C3C2211yescharacteristic
8C3⋊D3×SL(2,𝔽3)—21no—
9C3×#3755—41nomaximal
10C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
11#64367C2×C311yescharacteristic
12C3×#994—24no—
13C3×#994D314yescomplemented
14#1010⋊C3—34no—
15#64160—31no—
16#64159—31no—
17#64157C2×C311yescharacteristic, complemented
18C2×#3754—31no—
19#64141—31no—
20C3⋊(#131⋊C3)—34no—
21C33×Q8—41no—
22C3×C4⋊(C3⋊D3)—41no—
23C32×SL(2,𝔽3)D411yescharacteristic
24C3×C3⋊(C3⋊C8)—41no—
25#5836—34no—
26#577⋊C3—91no—
27C32×#41—21no—
28C32×#41—31no—
29C32×#41C22×C311yescharacteristic
30C2×C32×D4—31no—
31C2×C32×D4—61no—
32Q8×C3⋊D3—21no—
33#3830—61no—
34#3828—31no—
35C2×#634—31no—
36C3×#41⋊C3—28no—
37C3×#41⋊C3—24no—
38C3×#41⋊C3—24no—
39C3×#41⋊C3C2×D314yes—
40D3×SL(2,𝔽3)—64no—
41#3756—31no—
42#3755A411yescharacteristic
43#3754—32no—
44#3754—31no—
45#3753—31no—
46#3748—31no—
47C2×C3⋊(C3⋊C8)—31no—
48C3×#215—124no—
49C2×C3×C3⋊D3—81no—
50C33×C4—41no—
51C33×C4—81no—
52C3×H(5,𝔽2)C3×D314yescomplemented
53#1010—34no—
542−1+4⋊C3—91no—
55#994—24no—
56#994C3⋊D311yescharacteristic, complemented
57#951—94no—
58#950—94no—
59#948—34no—
60C2×#214—94no—
61#932—94no—
62#131⋊C3—91no—
63C23×C32—61no—
64C32×Q8—44no—
65C32×Q8—84no—
66C32×Q8C3×D411yescharacteristic
67C32×Q8C2×A411yescharacteristic
68C32×D4—31no—
69C32×D4—31no—
70C32×D4—61no—
71C32×D4—61no—
72C2×C32×C4—31no—
73C2×C32×C4—61no—
74#638—61no—
75C2×C3⋊Dic3—61no—
76C4⋊(C3⋊D3)C2×A411yescharacteristic
77C4×C3⋊D3—61no—
78#634—31no—
79#634—61no—
80C3×D12—124no—
81C3×SL(2,𝔽3)—24no—
82C3×SL(2,𝔽3)#6714yes—
83C3⋊(C3⋊C8)—31no—
84C3⋊(C3⋊C8)C2×A411yescharacteristic
85C3×C3⋊C8—124no—
86#577—91noSylow
87C2×C33—41no—
88C3×C3⋊D3—161no—
89C3×#41—24no—
90C3×#41—34no—
91C3×#41C2×C3×D314yes—
92C2×C3×D4—34no—
93C2×C3×D4—64no—
94D3×Q8—64no—
95#237—184no—
96#235—94no—
97C2×Dic6—94no—
98#41⋊C3—28no—
99#41⋊C3—21no—
100#41⋊C3—24no—
101#41⋊C3C2×C3⋊D311yescharacteristic
102C2×SL(2,𝔽3)—181no—
103C3×SD16—361no—
104#216—94no—
105#215—34no—
106#214—94no—
107#214—98no—
108#213—94no—
109#208—94no—
110C2×C3⋊C8—94no—
111C22×C32—31no—
112C22×C32—61no—
113C22×C32—241no—
114C2×C3⋊D3—21no—
115C2×C3×D3—244no—
116C32×C4—21no—
117C32×C4—31no—
118C32×C4—44no—
119C32×C4—84no—
120C32×C4—88no—
121C32×C4—84no—
122C32×C4C22×A411yescharacteristic
123C3⋊Dic3—61no—
1242−1+4—91no—
125H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
126#137—271no—
127AGL(1,ℤ/8)—271no—
128#135—271no—
129C2×SD16—271no—
130#131—91no—
131C33—41noSylow
132C23×C3—64no—
133C3×Q8—41no—
134C3×Q8—84no—
135C3×Q8C3×#6714yes—
136C3×Q8D3×A414yes—
137C3×D4—34no—
138C3×D4—34no—
139C3×D4—64no—
140C3×D4—64no—
141C3×D4—361no—
142C2×C3×C4—34no—
143C2×C3×C4—64no—
144#67—184no—
145C2×Dic3—184no—
146D12—34no—
147C4×D3—184no—
148Dic6—94no—
149Dic6—184no—
150SL(2,𝔽3)—24no—
151SL(2,𝔽3)#63811yescharacteristic
152C3×C8—361no—
153C3⋊C8—34no—
154C3⋊C8—94no—
155C2×C32—44no—
156C2×C32—61no—
157C2×C32—84no—
158C2×C32—121no—
159C2×C32D4×A411yescharacteristic
160C3⋊D3—41no—
161C3×D3—484no—
162#41—21no—
163#41—31no—
164#41—271no—
165#41—541no—
166#41C2×C3×C3⋊D311yescharacteristic
167C2×Q8—181no—
168C2×Q8—271no—
169C2×D4—31no—
170C2×D4—61no—
171Q16—271no—
172SD16—91no—
173SD16—271no—
174SD16—272no—
175D8—271no—
176M16—271no—
177C2×C8—271no—
178C22×C3—34no—
179C22×C3—64no—
180C22×C3—244no—
181C22×C3—721no—
182C2×D3—64no—
183C3×C4—24no—
184C3×C4—34no—
185C3×C4—41no—
186C3×C4—81no—
187C3×C4—88no—
188C3×C4—84no—
189C3×C4C2×D3×A414yes—
190Dic3—184no—
191C32—44no—
192C32—84no—
193C32#577⋊C311yescharacteristic, complemented
194C23—61no—
195Q8—271no—
196Q8—541no—
197Q8C3×#63811yescharacteristic
198Q8A4×C3⋊D311yescharacteristic
199D4—31no—
200D4—31no—
201D4—61no—
202D4—61no—
203D4—91no—
204D4—541no—
205C2×C4—31no—
206C2×C4—61no—
207C2×C4—541no—
208C2×C4—541no—
209C8—91no—
210C8—271no—
211C2×C3—41no—
212C2×C3—64no—
213C2×C3—84no—
214C2×C3—124no—
215C2×C3—1441no—
216C2×C3A4×#6714yes—
217D3—124no—
218C22—31no—
219C22—61no—
220C22—181no—
221C22—241no—
222C4—21no—
223C4—31no—
224C4—541no—
225C4C2×A4×C3⋊D311yescharacteristic
226C3—41no—
227C3—84no—
228C3#5836⋊C314yescomplemented
229C2—61no—
230C2—121no—
231C2—361no—
232C2A4×#63811yescharacteristic
subgroupG/H
centerC2A4×#638
commutator, cosocleC32×#41C22×C3
FrattiniC4C2×A4×C3⋊D3
FittingC32×H(5,𝔽2)C2×C3
socleC2×C32D4×A4
hypercenterQ8A4×C3⋊D3
hypocenterC32×Q8C3×D4
LCSG ▷ C32×#41 ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×#994C2semidirect
2. #11724493C2semidirect
3. C3⋊(C3⋊(#131⋊C3))C2semidirect
4. #10596307C3semidirect
10. C32×H(5,𝔽2)C2×C3semidirect
13. C3×#994D3semidirect
17. #64157C2×C3semidirect
52. C3×H(5,𝔽2)C3×D3semidirect
56. #994C3⋊D3semidirect
125. H(5,𝔽2)C3×C3⋊D3semidirect
193. C32#577⋊C3semidirect
228. C3#5836⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×#9942
#117244932
C3⋊(C3⋊(#131⋊C3))2
#105963073
#5836⋊C33
C3×#37554