Finite Group Databasebeta

← #13173176

Subgroups — #13173176

Order 1728.

708 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10596288C311yesmaximal, characteristic, complemented
5#5817⋊C3—34nomaximal
6C2×C32×SL(2,𝔽3)—21no—
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊(#41⋊C3))—21no—
9C3×#3757—41nomaximal
10C22×C32×Q8C2×C311yescharacteristic
11C2×#3832C2×C311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—24no—
13C22×C3×SL(2,𝔽3)D314yescomplemented
14C2×C3⋊(#41⋊C3)—34no—
15#64156—31no—
16#64155—31no—
17Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
18#64139—31no—
19C4×#638—31no—
20Dic3×SL(2,𝔽3)—34no—
21C23×C33—41no—
22C2×C3×C3⋊Dic3—41no—
23C2×C3×C3⋊Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#5817—34no—
26#548⋊C3—91no—
27C2×C32×Q8—21no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C2×C32×Q8C22×C311yescharacteristic
31C22×C32×C4—31no—
32C2×#638—31no—
33#3832—21no—
34#3832—61no—
35C2×C4⋊(C3⋊D3)—31no—
36C2×C4×C3⋊D3—31no—
37C2×C3×SL(2,𝔽3)—28no—
38C2×C3×SL(2,𝔽3)—24no—
39C2×C3×SL(2,𝔽3)—24no—
40C2×C3×SL(2,𝔽3)C2×D314yes—
41C3⋊(#41⋊C3)—64no—
42#3757A411yescharacteristic
43#3752—32no—
44#3752—31no—
45C4⋊(C3⋊Dic3)—31no—
46#3750—31no—
47C4×C3⋊Dic3—31no—
48C3×#217—124no—
49C22×C33—41no—
50C22×C33—42no—
51C22×C33—81no—
52C22×C33—81no—
53C3×C3⋊Dic3—81no—
54C3×C3⋊Dic3—81no—
55C22×C3×Q8C3×D314yes—
56C2×#239—34no—
57C2×#41⋊C3—91no—
58C22×SL(2,𝔽3)—24no—
59C22×SL(2,𝔽3)C3⋊D311yescharacteristic, complemented
60#947—94no—
61#946—94no—
62Q8×Dic3—34no—
63#930—94no—
64C4×#67—94no—
65C4×SL(2,𝔽3)—91no—
66C23×C32—44no—
67C23×C32—84no—
68C23×C32C2×A411yescharacteristic
69C22×C3⋊D3—31no—
70C32×Q8—31no—
71C32×Q8—61no—
72C32×Q8—61no—
73C32×Q8C3×D411yescharacteristic
74C2×C32×C4—31no—
75C2×C32×C4—31no—
76C2×C32×C4—61no—
77C2×C32×C4—61no—
78#638—62no—
79C2×C3⋊Dic3—31no—
80C2×C3⋊Dic3C2×A411yescharacteristic
81C2×C3⋊Dic3C2×A411yescharacteristic
82C4⋊(C3⋊D3)—61no—
83C4⋊(C3⋊D3)—61no—
84C4×C3⋊D3—61no—
85C4×C3⋊D3—61no—
86C2×C3×Dic3—124no—
87C2×C3×Dic3—124no—
88C3×SL(2,𝔽3)—24no—
89C3×SL(2,𝔽3)#6714yescomplemented
90#548—91noSylow
91C2×C33—41no—
92C2×C33—41no—
93C2×C33—41no—
94C2×C33—82no—
95C2×C3×Q8—24no—
96C2×C3×Q8—34no—
97C2×C3×Q8—64no—
98C2×C3×Q8C2×C3×D314yes—
99C22×C3×C4—34no—
100C2×#67—94no—
101#239—64no—
102#239—184no—
103C2×D12—94no—
104C2×C4×D3—94no—
105#41⋊C3—181no—
106C2×SL(2,𝔽3)—28no—
107C2×SL(2,𝔽3)—21no—
108C2×SL(2,𝔽3)—24no—
109C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
110C3×C22⋊C4—361no—
111#217—34no—
112(C2×D3)⋊C4—94no—
113(C2×D3)⋊C4—98no—
114C4⋊Dic3—94no—
115Dic3⋊C4—94no—
116C4×Dic3—94no—
117C22×C32—21no—
118C22×C32—21no—
119C22×C32—44no—
120C22×C32—48no—
121C22×C32—84no—
122C22×C32—84no—
123C22×C32—88no—
124C22×C32—88no—
125C22×C32—84no—
126C22×C32—88no—
127C22×C32#41⋊C312yescomplemented
128C22×C32C22×A411yescharacteristic
129C2×C3⋊D3—61no—
130C2×C3⋊D3—61no—
131C2×C3⋊D3—61no—
132C32×C4—31no—
133C32×C4—31no—
134C32×C4—61no—
135C3⋊Dic3—21no—
136C3⋊Dic3—21no—
137C3⋊Dic3—61no—
138C3×Dic3—244no—
139C3×Dic3—244no—
140C2×#41—91no—
141C22×Q8C3×C3⋊D311yescharacteristic
142#124—271no—
143C22⋊Q8—271no—
144#121—271no—
145C4×Q8—91no—
146C4×D4—271no—
147C33—41noSylow
148C23×C3—41no—
149C23×C3—84no—
150C23×C3D3×A414yes—
151C22×D3—94no—
152C3×Q8—34no—
153C3×Q8—64no—
154C3×Q8—64no—
155C3×Q8C3×#6714yes—
156C2×C3×C4—34no—
157C2×C3×C4—34no—
158C2×C3×C4—64no—
159C2×C3×C4—64no—
160C2×C3×C4—361no—
161C2×C3×C4—361no—
162#67—188no—
163C2×Dic3—34no—
164C2×Dic3—34no—
165C2×Dic3—94no—
166D12—184no—
167D12—184no—
168C4×D3—184no—
169C4×D3—184no—
170SL(2,𝔽3)—24no—
171SL(2,𝔽3)#63811yescharacteristic, complemented
172C2×C32—22no—
173C2×C32—44no—
174C2×C32—44no—
175C2×C32—44no—
176C2×C32—88no—
177C2×C32—816no—
178C2×C32—84no—
179C2×C32—84no—
180C2×C32—84no—
181C2×C32D4×A411yescharacteristic
182C2×C32C2×#41⋊C311yescharacteristic
183C2×C322−1+4⋊C311yescharacteristic
184C3⋊D3—121no—
185#41—181no—
186#41—541no—
187C2×Q8—21no—
188C2×Q8—31no—
189C2×Q8—61no—
190C2×Q8C2×C3×C3⋊D311yescharacteristic
191C2×D4—271no—
192C2×D4—271no—
193C22×C4—31no—
194C22×C4—271no—
195C4⋊C4—271no—
196C4⋊C4—271no—
197C22⋊C4—91no—
198C22⋊C4—271no—
199C22⋊C4—272no—
200C42—271no—
201C22×C3—24no—
202C22×C3—24no—
203C22×C3—41no—
204C22×C3—42no—
205C22×C3—81no—
206C22×C3—81no—
207C22×C3—88no—
208C22×C3—88no—
209C22×C3—84no—
210C22×C3—88no—
211C22×C3C3⋊(#41⋊C3)18yescomplemented
212C22×C3C2×D3×A414yes—
213C2×D3—184no—
214C2×D3—184no—
215C2×D3—184no—
216C3×C4—34no—
217C3×C4—34no—
218C3×C4—64no—
219C3×C4—721no—
220C3×C4—721no—
221Dic3—64no—
222Dic3—64no—
223Dic3—184no—
224C32—44no—
225C32—84no—
226C32#548⋊C311yescharacteristic, complemented
227C23—271no—
228C23A4×C3⋊D311yescharacteristic
229Q8—31no—
230Q8—61no—
231Q8—61no—
232Q8C3×#63811yescharacteristic
233D4—541no—
234D4—541no—
235D4—542no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—91no—
241C2×C4—91no—
242C2×C4—271no—
243C2×C4—541no—
244C2×C4—541no—
245C2×C3—28no—
246C2×C3—41no—
247C2×C3—41no—
248C2×C3—41no—
249C2×C3—82no—
250C2×C3—816no—
251C2×C3—84no—
252C2×C3—84no—
253C2×C3—84no—
254C2×C3C2×C3⋊(#41⋊C3)14yes—
255C2×C3#1007⋊C314yes—
256C2×C3A4×#6714yes—
257D3—364no—
258C22—21no—
259C22—21no—
260C22—541no—
261C22—541no—
262C22—541no—
263C22C3⋊(C3⋊(#41⋊C3))12yescomplemented
264C22C2×A4×C3⋊D311yescharacteristic
265C4—31no—
266C4—31no—
267C4—61no—
268C4—181no—
269C4—181no—
270C4—541no—
271C3—41no—
272C3—84no—
273C3#5817⋊C314yescomplemented
274C2—22no—
275C2—1081no—
276C2C2×C3⋊(C3⋊(#41⋊C3))11yescharacteristic
277C2#1172449011yescharacteristic
278C2A4×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutator, cosocleC2×C32×Q8C22×C3
FittingC22×C32×Q8C2×C3
socleC22×C32C22×A4
hypercenterC23A4×C3⋊D3
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C2×C3⋊(C3⋊(#41⋊C3))C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
4. #10596288C3semidirect
11. C2×#3832C2×C3semidirect
13. C22×C3×SL(2,𝔽3)D3semidirect
17. Q8×C3⋊Dic3C2×C3semidirect
24. C32×SL(2,𝔽3)D4semidirect
59. C22×SL(2,𝔽3)C3⋊D3semidirect
89. C3×SL(2,𝔽3)#67semidirect
127. C22×C32#41⋊C3semidirect
171. SL(2,𝔽3)#638semidirect
211. C22×C3C3⋊(#41⋊C3)semidirect
226. C32#548⋊C3semidirect
263. C22C3⋊(C3⋊(#41⋊C3))semidirect
273. C3#5817⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C2×C3⋊(C3⋊(#41⋊C3))2
SL(2,𝔽3)×C3⋊Dic32
#105962883
#5817⋊C33
C3×#37574