Finite Group Databasebeta

← #10595412⋊C3 #13166145

Subgroups — #10595412⋊C3 #13166145

Order 1728.

444 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 56 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⋊(#41⋊C3)C212yesmaximal, complemented
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#10595412C311yesmaximal, characteristic, complemented
4#5834⋊C3—32nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C3×C3⋊(#41⋊C3)—22no—
7C3×#3724—41nomaximal
8C2×C3×#239C2×C312yes—
9C2×C3×#41⋊C3—32no—
10C2×C3⋊(#41⋊C3)—22no—
11C2×C3⋊(#41⋊C3)D312yescomplemented
12Q8×C3⋊Dic3C2×C311yescharacteristic
13#63914—31no—
14#63908—32no—
15C4×#625—31no—
16Dic3×SL(2,𝔽3)—32no—
17Dic3×SL(2,𝔽3)—61no—
18C2×C3×C3⋊Dic3—41no—
19C2×C32×Dic3—42no—
20C32×SL(2,𝔽3)D411yescharacteristic, complemented
21#5834—32no—
22C4⋊(#41⋊C3)—91no—
23C2×C32×Q8C22×C311yescharacteristic
24C3×#239—22no—
25C3×#239—62no—
26C2×C3×D12—32no—
27C2×C3×C4×D3—32no—
28C3×#41⋊C3—62no—
29C2×C3×SL(2,𝔽3)—22no—
30C2×C3×SL(2,𝔽3)—21no—
31C2×C3×SL(2,𝔽3)—41no—
32C2×C3×SL(2,𝔽3)C2×D312yes—
33C2×#625—31no—
34C2×#625—32no—
35C3⋊(#41⋊C3)—24no—
36C3⋊(#41⋊C3)—22no—
37C4⋊(C3⋊Dic3)—31no—
38C4×C3⋊Dic3—31no—
39C3×Dic3⋊C4—122no—
40#3724A411yescharacteristic
41(C2×D3)⋊Dic3—32no—
42C22×C33—41no—
43C3×C3⋊Dic3—42no—
44C32×Dic3—82no—
45C2×C3×#41—32no—
46C2×#239C3×D312yes—
47C2×#41⋊C3—32no—
48C2×#41⋊C3—62no—
49Q8×Dic3—32no—
50Q8×Dic3—61no—
51#940—92no—
52#938—92no—
53#930—92no—
54C4×#67—92no—
55C4×SL(2,𝔽3)—91no—
56C22×C3×D3—32no—
57C32×Q8—31no—
58C32×Q8C3×D411yescharacteristic
59C2×C32×C4—31no—
60C2×C32×C4—122no—
61C2×C3⋊Dic3—31no—
62C2×C3⋊Dic3C2×A411yescharacteristic
63C2×C3×Dic3—42no—
64C2×C3×Dic3—82no—
65C2×C3×Dic3—122no—
66C2×C3×Dic3—241no—
67C2×C3×Dic3C2×A412yes—
68C3×D12—62no—
69C3×D12—62no—
70C3×C4×D3—62no—
71C3×C4×D3—62no—
72C3×SL(2,𝔽3)—22no—
73C3×SL(2,𝔽3)—21no—
74C3×SL(2,𝔽3)—41no—
75C3×SL(2,𝔽3)#6712yescomplemented
76#625—62no—
77#625—122no—
78#550—91noSylow
79C2×C33—41no—
80C2×C33—41no—
81C2×C33—41no—
82C3×#41—62no—
83C3×#41—182no—
84C2×C3×Q8—21no—
85C2×C3×Q8C2×C3×D312yes—
86C2×C3×D4—92no—
87C22×C3×C4—92no—
88C2×#67—92no—
89C2×#67—94no—
90#239—22no—
91#239—62no—
92C2×D12—32no—
93C2×C4×D3—32no—
94#41⋊C3—64no—
95#41⋊C3—62no—
96C2×SL(2,𝔽3)—22no—
97C2×SL(2,𝔽3)—41no—
98C2×SL(2,𝔽3)D3211yescharacteristic
99C3×C4⋊C4—361no—
100#217—92no—
101(C2×D3)⋊C4—92no—
102C4⋊Dic3—92no—
103C4⋊Dic3—181no—
104Dic3⋊C4—32no—
105C4×Dic3—92no—
106C4×Dic3—181no—
107C22×C32—42no—
108C22×C32—81no—
109C22×C32—82no—
110C22×C32—161no—
111C22×C32C22×A411yescharacteristic
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—62no—
115C32×C4—31no—
116C32×C4—31no—
117C32×C4—242no—
118C3⋊Dic3—61no—
119C3⋊Dic3#41⋊C312yescomplemented
120C3×Dic3—22no—
121C3×Dic3—82no—
122C3×Dic3—84no—
123C3×Dic3—124no—
124C3×Dic3—242no—
125C2×#41—32no—
126#127—271no—
127#121—272no—
128C4×Q8—91no—
129C4×D4—271no—
130C33—41noSylow
131C23×C3—92no—
132C22×D3—32no—
133C3×Q8—21no—
134C3×Q8—32no—
135C3×Q8—61no—
136C3×Q8C3×#6712yes—
137C3×D4—182no—
138C3×D4—182no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—61no—
142C2×C3×C4—122no—
143C2×C3×C4—182no—
144C2×C3×C4—182no—
145C2×C3×C4—242no—
146C2×C3×C4—361no—
147#67—184no—
148#67—364no—
149C2×Dic3—32no—
150C2×Dic3—61no—
151C2×Dic3—92no—
152C2×Dic3—181no—
153C2×Dic3D3×A412yes—
154D12—62no—
155D12—62no—
156C4×D3—62no—
157C4×D3—62no—
158SL(2,𝔽3)—22no—
159SL(2,𝔽3)—41no—
160SL(2,𝔽3)#62511yescharacteristic, complemented
161C2×C32—42no—
162C2×C32—42no—
163C2×C32—42no—
164C2×C32—81no—
165C2×C32—81no—
166C2×C32—81no—
167C2×C32—82no—
168C2×C32—82no—
169C2×C32—82no—
170C2×C32—161no—
171C2×C32—161no—
172C2×C32—161no—
173C2×C32D4×A411yescharacteristic
174C2×C32C2×#41⋊C311yescharacteristic
175C2×C32#99411yescharacteristic
176C3×D3—122no—
177#41—62no—
178#41—182no—
179C2×Q8C3×D3211yescharacteristic
180C2×D4—92no—
181C2×D4—271no—
182C2×D4—272no—
183C22×C4—92no—
184C4⋊C4—91no—
185C4⋊C4—271no—
186C22⋊C4—272no—
187C42—271no—
188C22×C3—21no—
189C22×C3—41no—
190C22×C3—82no—
191C22×C3—161no—
192C22×C3—182no—
193C22×C3—182no—
194C22×C3—182no—
195C22×C3C2×D3×A412yes—
196C2×D3—62no—
197C2×D3—62no—
198C2×D3—62no—
199C3×C4—32no—
200C3×C4—32no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—62no—
204C3×C4—242no—
205C3×C4—244no—
206C3×C4—362no—
207Dic3—22no—
208Dic3—34no—
209Dic3—62no—
210Dic3—182no—
211Dic3—361no—
212C32—42no—
213C32—81no—
214C32—82no—
215C32—161no—
216C32C4⋊(#41⋊C3)11yescharacteristic, complemented
217C23—92no—
218Q8—31no—
219Q8C3×#62511yescharacteristic
220D4—182no—
221D4—182no—
222D4—542no—
223D4—1082no—
224C2×C4—31no—
225C2×C4—32no—
226C2×C4—91no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4—271no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—82no—
237C2×C3—82no—
238C2×C3—82no—
239C2×C3—161no—
240C2×C3—161no—
241C2×C3—161no—
242C2×C3—362no—
243C2×C3C3⋊#99412yes—
244C2×C3D3×#41⋊C312yes—
245C2×C3A4×#6712yes—
246D3—122no—
247C22—182no—
248C22—182no—
249C22—182no—
250C22D32×A411yescharacteristic
251C4—31no—
252C4—31no—
253C4—62no—
254C4—92no—
255C4—541no—
256C3—21no—
257C3—41no—
258C3—82no—
259C3—161no—
260C3#5834⋊C312yescomplemented
261C2—362no—
262C2#64319⋊C311yescharacteristic
263C2#64320⋊C311yescharacteristic
264C2A4×#62511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×A4
commutator, Fitting, cosocleC2×C32×Q8C22×C3
socleC22×C32C22×A4
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊(#41⋊C3)C2semidirect
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. #10595412C3semidirect
11. C2×C3⋊(#41⋊C3)D3semidirect
20. C32×SL(2,𝔽3)D4semidirect
75. C3×SL(2,𝔽3)#67semidirect
119. C3⋊Dic3#41⋊C3semidirect
160. SL(2,𝔽3)#625semidirect
216. C32C4⋊(#41⋊C3)semidirect
260. C3#5834⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
SL(2,𝔽3)×C3⋊Dic32
#105954123
#5834⋊C33
C3×#37244