Finite Group Databasebeta

← #13173189

Subgroups — #13173189

Order 1728.

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

normal subgroups only

This group has 542 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
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#11724493C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
4#10596309C311yesmaximal, characteristic, complemented
5#5838⋊C3—34nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C2×C32×SL(2,𝔽3)—21no—
8C3⋊D3×SL(2,𝔽3)—21no—
9C3×#3753—41nomaximal
10C32×2−1+4C2×C311yescharacteristic, complemented
11#64367C2×C311yescharacteristic, complemented
12C3×2−1+4⋊C3—24no—
13C3×2−1+4⋊C3D314yescomplemented
14#1010⋊C3—34no—
15#64160—31no—
16#64159—31no—
17#64157C2×C311yescharacteristic, complemented
18C2×#3756—31no—
19#64151—31no—
20C3⋊(#131⋊C3)—34no—
21C33×D4—41no—
22C3×C4⋊(C3⋊D3)—41no—
23C32×SL(2,𝔽3)D411yescharacteristic
24C3×C3⋊(C3⋊C8)—41no—
25#5838—34no—
26#578⋊C3—91no—
27C32×#41—31no—
28C32×#41—61no—
29C32×#41C22×C311yescharacteristic
30C2×C32×Q8—21no—
31C2×C32×Q8—31no—
32Q8×C3⋊D3—21no—
33#3830—61no—
34#3828—31no—
35C2×#634—31no—
36C3×#41⋊C3—24no—
37C3×#41⋊C3C2×D314yes—
38C2×C3×SL(2,𝔽3)—28no—
39C2×C3×SL(2,𝔽3)—24no—
40D3×SL(2,𝔽3)—64no—
41#3756—32no—
42#3756—31no—
43#3755—31no—
44#3754—31no—
45#3753A411yescharacteristic
46#3748—31no—
47C2×C3⋊(C3⋊C8)—31no—
48C3×#213—124no—
49C22×C33—81no—
50C2×C3×C3⋊D3—81no—
51C33×C4—41no—
52C3×2−1+4C3×D314yescomplemented
53#1010—34no—
542−1+4⋊C3—24no—
552−1+4⋊C3—91no—
562−1+4⋊C3C3⋊D311yescharacteristic, complemented
57#951—94no—
58#950—94no—
59#948—34no—
60C2×#216—94no—
61#942—94no—
62#131⋊C3—91no—
63C32×Q8—31no—
64C32×Q8—61no—
65C32×Q8C3×D411yescharacteristic
66C32×D4—31no—
67C32×D4—44no—
68C32×D4—61no—
69C32×D4—84no—
70C32×D4C2×A411yescharacteristic
71C2×C32×C4—31no—
72C2×C32×C4—61no—
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#578—91noSylow
87C2×C33—41no—
88C2×C33—161no—
89C3×C3⋊D3—161no—
90C3×#41—34no—
91C3×#41—64no—
92C3×#41C2×C3×D314yes—
93C2×C3×Q8—24no—
94C2×C3×Q8—34no—
95D3×Q8—64no—
96#237—184no—
97#235—94no—
98C2×Dic6—94no—
99#41⋊C3—24no—
100#41⋊C3C2×C3⋊D311yescharacteristic
101C2×SL(2,𝔽3)—28no—
102C2×SL(2,𝔽3)—21no—
103C2×SL(2,𝔽3)—181no—
104C3×D8—361no—
105#216—94no—
106#216—98no—
107#215—94no—
108#214—94no—
109#213—34no—
110#208—94no—
111C2×C3⋊C8—94no—
112C22×C32—21no—
113C22×C32—31no—
114C22×C32—84no—
115C22×C32—88no—
116C2×C3⋊D3—21no—
117C2×C3×D3—244no—
118C32×C4—31no—
119C32×C4—44no—
120C32×C4—61no—
121C32×C4—84no—
122C32×C4C22×A411yescharacteristic
123C3⋊Dic3—61no—
1242−1+4—91no—
1252−1+4C3×C3⋊D311yescharacteristic, complemented
126#137—271no—
127#137—271no—
128#135—271no—
129C2×Q16—271no—
130#131—91no—
131C33—41noSylow
132C3×Q8—34no—
133C3×Q8—64no—
134C3×Q8C3×#6714yes—
135C3×D4—34no—
136C3×D4—41no—
137C3×D4—64no—
138C3×D4—84no—
139C3×D4—361no—
140C3×D4D3×A414yes—
141C2×C3×C4—34no—
142C2×C3×C4—64no—
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—41no—
156C2×C32—44no—
157C2×C32—61no—
158C2×C32—84no—
159C2×C32—164no—
160C2×C32—168no—
161C2×C32D4×A411yescharacteristic
162C3⋊D3—41no—
163C3×D3—484no—
164#41—31no—
165#41—61no—
166#41—271no—
167#41—541no—
168#41C2×C3×C3⋊D311yescharacteristic
169C2×Q8—21no—
170C2×Q8—31no—
171C2×Q8—181no—
172C2×Q8—271no—
173Q16—271no—
174Q16—272no—
175SD16—271no—
176SD16—271no—
177D8—91no—
178M16—271no—
179C2×C8—271no—
180C22×C3—24no—
181C22×C3—34no—
182C22×C3—81no—
183C22×C3—88no—
184C22×C3—721no—
185C2×D3—64no—
186C3×C4—34no—
187C3×C4—41no—
188C3×C4—64no—
189C3×C4—84no—
190C3×C4C2×D3×A414yes—
191Dic3—184no—
192C32—44no—
193C32—84no—
194C32#578⋊C311yescharacteristic, complemented
195Q8—31no—
196Q8—61no—
197Q8—271no—
198Q8—541no—
199Q8C3×#63811yescharacteristic
200D4—31no—
201D4—61no—
202D4—91no—
203D4—541no—
204D4A4×C3⋊D311yescharacteristic
205C2×C4—31no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—541no—
209C2×C4—541no—
210C8—91no—
211C8—271no—
212C2×C3—44no—
213C2×C3—41no—
214C2×C3—64no—
215C2×C3—84no—
216C2×C3—161no—
217C2×C3—168no—
218C2×C3—1441no—
219C2×C3A4×#6714yes—
220D3—124no—
221C22—21no—
222C22—31no—
223C22—181no—
224C4—31no—
225C4—61no—
226C4—541no—
227C4C2×A4×C3⋊D311yescharacteristic
228C3—41no—
229C3—84no—
230C3#5838⋊C314yescomplemented
231C2—41no—
232C2—61no—
233C2—361no—
234C2A4×#63811yescharacteristic
subgroupG/H
centerC2A4×#638
commutator, cosocleC32×#41C22×C3
FrattiniC4C2×A4×C3⋊D3
FittingC32×2−1+4C2×C3
socleC2×C32D4×A4
hypercenterD4A4×C3⋊D3
hypocenterC32×Q8C3×D4
LCSG ▷ C32×#41 ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. #11724493C2semidirect
3. C3⋊(C3⋊(#131⋊C3))C2semidirect
4. #10596309C3semidirect
10. C32×2−1+4C2×C3semidirect
11. #64367C2×C3semidirect
13. C3×2−1+4⋊C3D3semidirect
17. #64157C2×C3semidirect
52. C3×2−1+4C3×D3semidirect
56. 2−1+4⋊C3C3⋊D3semidirect
125. 2−1+4C3×C3⋊D3semidirect
194. C32#578⋊C3semidirect
230. C3#5838⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
#117244932
C3⋊(C3⋊(#131⋊C3))2
#105963093
#5838⋊C33
C3×#37534