Finite Group Databasebeta

← #13165955

Subgroups — #13165955

Order 1728.

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

normal subgroups only

This group has 542 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
1C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
2C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10593205—34nomaximal
5#10592865—31nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(C3⋊#226)—21no—
8C32×GL(2,𝔽3)—21no—
9(C2×D3)⋊(C3⋊Dic3)—41nomaximal
10C2×C3⋊#226—34no—
11C2×C3⋊#226—64no—
12C2×C3×GL(2,𝔽3)D314yes—
13C2×C32×SD16—31no—
14Q8×C3⋊Dic3D311yescharacteristic, complemented
15C2×#3756—31no—
16#64145—31no—
17Dic3×SL(2,𝔽3)—34no—
18#63659—31no—
19#63643—31no—
20#63641—31no—
21C22×C32×D3—41no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C2×C3×C3⋊Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic
25#5395—91no—
26#5166—94no—
27C2×C32×Q8C2×D311yescharacteristic
28C2×C32×D4—31no—
29C2×#634—31no—
30C2×C3×SL(2,𝔽3)—24no—
31C2×C3×SL(2,𝔽3)C2×D314yes—
32C3⋊#226—68no—
33C3⋊#226—64no—
34C3×GL(2,𝔽3)—24no—
35C32×SD16—61no—
36C32×SD16—61no—
37C2×C32×C8—31no—
38#3757—61no—
39#3757—61no—
40#3756—61no—
41#3756—61no—
42C4⋊(C3⋊Dic3)—31no—
43C4×C3⋊Dic3—31no—
44C2×C3⋊(C3⋊C8)—31no—
45(C2×D3)⋊Dic3—124no—
46C22×C33—41no—
47C2×C32×D3—42no—
48C2×C32×D3—81no—
49C2×C32×D3—81no—
50C3⋊(C3⋊Dic3)—81no—
51C3×C3⋊Dic3—81no—
52C2×GL(2,𝔽3)C3⋊D311yescharacteristic
53C2×#226—91no—
54C2×#226—184no—
55C2×C3×SD16—34no—
56Q8×Dic3—34no—
57C2×#216—94no—
58#936—94no—
59C4×SL(2,𝔽3)—91no—
60#832—94no—
61#816—94no—
62Dic3⋊C8—94no—
63C23×C32—61no—
64C22×C3×D3—44no—
65C32×Q8—31no—
66C32×Q8D1211yescharacteristic
67C32×D4—61no—
68C32×D4—61no—
69C2×C32×C4—31no—
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—124no—
73C2×C3⋊Dic3—244no—
74C2×C3⋊Dic3S411yescharacteristic
75#634—61no—
76#634—61no—
77C2×C3×Dic3—124no—
78C3×SL(2,𝔽3)—24no—
79C3×SL(2,𝔽3)#6714yes—
80C32×C8—61no—
81C3⋊(C3⋊C8)—61no—
82#464—271noSylow
83C2×C33—41no—
84C2×C33—41no—
85C2×C33—41no—
86C32×D3—82no—
87C2×C3×Q8D3214yes—
88C2×C3×D4—34no—
89C2×Dic6—94no—
90C2×SL(2,𝔽3)—24no—
91C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
92GL(2,𝔽3)—21no—
93#226—188no—
94#226—181no—
95C3×SD16—64no—
96C3×SD16—64no—
97C2×C3×C8—34no—
98#217—184no—
99#217—184no—
100#216—184no—
101#216—184no—
102(C2×D3)⋊C4—361no—
103C4⋊Dic3—94no—
104C4×Dic3—94no—
105C2×C3⋊C8—94no—
106C22×C32—44no—
107C22×C32—84no—
108C22×C32—121no—
109C22×C32—121no—
110C22×C32—121no—
111C22×C32C2×S411yescharacteristic
112C2×C3×D3—48no—
113C2×C3×D3—84no—
114C2×C3×D3—84no—
115C32×C4—31no—
116C32×C4—31no—
117C3⋊Dic3—21no—
118C3⋊Dic3—61no—
119C3⋊Dic3—121no—
120C3⋊Dic3—244no—
121C3⋊Dic3—248no—
122C3×Dic3—244no—
123C2×Q16—271no—
124C2×SD16—31no—
125#124—271no—
126C4×Q8—91no—
127C4⋊C8—271no—
128#103—271no—
129#102—271no—
130C33—41noSylow
131C23×C3—64no—
132C22×D3—41no—
133C3×Q8—34no—
134C3×Q8#62614yes—
135C3×D4—64no—
136C3×D4—64no—
137C2×C3×C4—34no—
138C2×C3×C4—361no—
139C2×Dic3—34no—
140C2×Dic3—94no—
141C2×Dic3—184no—
142C2×Dic3—361no—
143C2×Dic3—724no—
144Dic6—184no—
145Dic6—184no—
146SL(2,𝔽3)—24no—
147SL(2,𝔽3)#63811yescharacteristic
148C3×C8—64no—
149C3⋊C8—184no—
150C2×C32—44no—
151C2×C32—44no—
152C2×C32—44no—
153C2×C32—84no—
154C2×C32—84no—
155C2×C32—84no—
156C2×C32—241no—
157C2×C32C4⋊S411yescharacteristic
158C2×C32#98311yescharacteristic
159C2×C32#98511yescharacteristic
160C3×D3—88no—
161C2×Q8—271no—
162C2×Q8D3×C3⋊D311yescharacteristic
163C2×D4—31no—
164Q16—541no—
165Q16—541no—
166SD16—61no—
167SD16—61no—
168C2×C8—31no—
169C2×C8—271no—
170C4⋊C4—271no—
171C22⋊C4—541no—
172C22⋊C4—541no—
173C42—271no—
174C22×C3—41no—
175C22×C3—84no—
176C22×C3—124no—
177C22×C3—124no—
178C22×C3—124no—
179C22×C3D3×S414yes—
180C2×D3—42no—
181C2×D3—81no—
182C2×D3—81no—
183C3×C4—34no—
184C3×C4—34no—
185C3×C4—721no—
186Dic3—64no—
187Dic3—184no—
188Dic3—364no—
189Dic3—721no—
190Dic3—728no—
191C32—44no—
192C32—84no—
193C32#539511yescharacteristic, complemented
194C23—61no—
195Q8—31no—
196Q8—541no—
197Q8—541no—
198Q8#629111yescharacteristic
199D4—61no—
200D4—61no—
201C2×C4—31no—
202C2×C4—91no—
203C2×C4—271no—
204C2×C4—541no—
205C8—61no—
206C8—541no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—84no—
211C2×C3—84no—
212C2×C3—84no—
213C2×C3—244no—
214C2×C3#6419714yes—
215C2×C3#6420114yes—
216C2×C3#6420714yes—
217D3—82no—
218C22—121no—
219C22—121no—
220C22—121no—
221C22C3⋊D3×S411yescharacteristic
222C4—31no—
223C4—31no—
224C4—181no—
225C4—541no—
226C4—1081no—
227C3—41no—
228C3—84no—
229C3#1059320514yescomplemented
230C2—241no—
231C2#1172417711yescharacteristic
232C2#1172418111yescharacteristic
233C2#1172418711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C32×GL(2,𝔽3)C2
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊(C3⋊#226)C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
14. Q8×C3⋊Dic3D3semidirect
193. C32#5395semidirect
229. C3#10593205semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊#226)2
C2×C32×GL(2,𝔽3)2
SL(2,𝔽3)×C3⋊Dic32
#105932053
#105928653
(C2×D3)⋊(C3⋊Dic3)4