Finite Group Databasebeta

← #13048748

Subgroups — #13048748

Order 1296.

493 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C4⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
2C4×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C3×#6229C211yesmaximal, characteristic, complemented
4#10619902C212yesmaximal, complemented
5#10619901C212yesmaximal, complemented
6#89018—31nomaximal
7#89017—31nomaximal
8#88648—31nomaximal
9C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
10C2×C3⋊((C3⋊D3)⋊C3)C2212yes—
11C3×C4×H(3,𝔽3)C2211yescharacteristic
12C3×#1084C2212yes—
13C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
15C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
16C3×C4⋊(C3⋊D3)—31no—
17C3×C4×C3⋊D3—31no—
18C32×Dic6—31no—
19C32×Dic6—31no—
20C32×Dic6—61no—
21#6292—32no—
22#6292—32no—
23#6289—32no—
24Dic3×C3⋊D3—32no—
25#6229D311yescharacteristic, complemented
26C4⋊((C3⋊D3)⋊C3)—31no—
27C4⋊((C3⋊D3)⋊C3)—61no—
28C4×(C3⋊D3)⋊C3—31no—
29C4×(C3⋊D3)⋊C3—61no—
30#6198—32no—
31((C3⋊D3)⋊C3)⋊C4—32no—
32C2×C3×H(3,𝔽3)C2311yescharacteristic
33C3⋊((C3⋊D3)⋊C3)—21no—
34C3⋊((C3⋊D3)⋊C3)—22no—
35#3832—91no—
36#3799—91no—
37#3799—93no—
38#3796—91no—
39#3796—93no—
40C2×C3⋊(C3⋊D3)C2×D312yes—
41C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
42C2×C3×C3⋊D3—32no—
43C2×C3×C3⋊D3—31no—
44C33×C4—21no—
45C33×C4C2×D311yescharacteristic
46C33×C4C2×D311yescharacteristic
47C3⋊(C3⋊Dic3)C2×D311yescharacteristic
48C3×C3⋊Dic3—31no—
49C32×Dic3—32no—
50C32×Dic3—32no—
51C32×Dic3—62no—
52C2×(C3⋊D3)⋊C3—32no—
53C2×(C3⋊D3)⋊C3—31no—
54C2×(C3⋊D3)⋊C3—61no—
55C2×(C3⋊D3)⋊C3—62no—
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)C2×D311yescharacteristic
60#1084C2×D312yes—
61(C3⋊Dic3)⋊C3—31no—
62(C3⋊Dic3)⋊C3—61no—
63C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
64C32×Q8—91no—
65C4⋊(C3⋊D3)—31no—
66C4⋊(C3⋊D3)—31no—
67C4⋊(C3⋊D3)—61no—
68C4⋊(C3⋊D3)—93no—
69C4⋊(C3⋊D3)—92no—
70C4×C3⋊D3—31no—
71C4×C3⋊D3—31no—
72C4×C3⋊D3—61no—
73C4×C3⋊D3—93no—
74C4×C3⋊D3—92no—
75C3×D12—93no—
76C3×D12—91no—
77C3×C4×D3—93no—
78C3×C4×D3—91no—
79C3×Dic6—33no—
80C3×Dic6—33no—
81C3×Dic6—31no—
82C3×Dic6—31no—
83C3×Dic6—61no—
84C3×Dic6—63no—
85#626—92no—
86#626—92no—
87#626—96no—
88#626—96no—
89#624—92no—
90#624—96no—
91D3×Dic3—92no—
92D3×Dic3—96no—
93C2×C33—21no—
94C2×C33C22×D311yescharacteristic
95C2×C33C22×D311yescharacteristic
96C3⋊(C3⋊D3)—21no—
97C3⋊(C3⋊D3)—22no—
98C3×C3⋊D3—62no—
99C3×C3⋊D3—61no—
100C2×H(3,𝔽3)—21no—
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)—41no—
103C2×H(3,𝔽3)C22×D311yescharacteristic
104(C3⋊D3)⋊C3—62no—
105(C3⋊D3)⋊C3—61no—
106(C3⋊D3)⋊C3—62no—
107(C3⋊D3)⋊C3—122no—
108#239—273no—
109#239—271no—
110#237—271no—
111#235—271no—
112C2×C3⋊D3—31no—
113C2×C3⋊D3—32no—
114C2×C3⋊D3—32no—
115C2×C3⋊D3—31no—
116C2×C3⋊D3—61no—
117C2×C3⋊D3—62no—
118C2×C3⋊D3—96no—
119C2×C3⋊D3—93no—
120C2×C3×D3—96no—
121C2×C3×D3—93no—
122C2×C3×D3—92no—
123C2×C3×D3—91no—
124C32×C4—21no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—33no—
128C32×C4—33no—
129C32×C4—41no—
130C32×C4—63no—
131C32×C4—92no—
132C32×C4D3211yescharacteristic
133C32×C4D3211yescharacteristic
134C32×C4D3211yescharacteristic
135C3⋊Dic3—31no—
136C3⋊Dic3—31no—
137C3⋊Dic3—61no—
138C3⋊Dic3—93no—
139C3×Dic3—36no—
140C3×Dic3—36no—
141C3×Dic3—32no—
142C3×Dic3—32no—
143C3×Dic3—62no—
144C3×Dic3—66no—
145C3×Dic3—93no—
146C3×Dic3—91no—
147C33—21no—
148C33#23511yescharacteristic, complemented
149C33#23711yescharacteristic, complemented
150H(3,𝔽3)—21no—
151H(3,𝔽3)—21no—
152H(3,𝔽3)—41no—
153H(3,𝔽3)#23911yescharacteristic, complemented
154C3×Q8—93no—
155C3×Q8—91no—
156C3×D4—271no—
157C2×C3×C4—271no—
158#67—272no—
159#67—272no—
160C2×Dic3—272no—
161D12—93no—
162D12—91no—
163D12—271no—
164D12—276no—
165D12—272no—
166D12—541no—
167C4×D3—93no—
168C4×D3—91no—
169C4×D3—271no—
170C4×D3—272no—
171C4×D3—276no—
172C4×D3—272no—
173C4×D3—541no—
174Dic6—31no—
175Dic6—31no—
176Dic6—61no—
177C2×C32—21no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—33no—
181C2×C32—33no—
182C2×C32—41no—
183C2×C32—63no—
184C2×C32C2×D3211yescharacteristic
185C2×C32C2×D3211yescharacteristic
186C2×C32C2×D3211yescharacteristic
187C3⋊D3—61no—
188C3⋊D3—62no—
189C3⋊D3—62no—
190C3⋊D3—62no—
191C3⋊D3—61no—
192C3⋊D3—122no—
193C3⋊D3—183no—
194C3⋊D3—186no—
195C3×D3—186no—
196C3×D3—183no—
197C3×D3—181no—
198C3×D3—182no—
199#41—811noSylow
200C22×C3—272no—
201C22×C3—271no—
202C2×D3—93no—
203C2×D3—96no—
204C2×D3—91no—
205C2×D3—92no—
206C2×D3—272no—
207C2×D3—271no—
208C2×D3—541no—
209C2×D3—542no—
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—61no—
213C3×C4—61no—
214C3×C4—61no—
215C3×C4—96no—
216C3×C4—92no—
217C3×C4—121no—
218C3×C4—271no—
219C3×C4((C3⋊D3)⋊C3)⋊C213yes—
220C3×C4#111311yescharacteristic
221Dic3—32no—
222Dic3—32no—
223Dic3—62no—
224Dic3—93no—
225Dic3—91no—
226Dic3—271no—
227Dic3—541no—
228C32—21no—
229C32—21no—
230C32—21no—
231C32—33no—
232C32—33no—
233C32—41no—
234C32—63no—
235C32#379511yescharacteristic
236C32#379611yescharacteristic, complemented
237C32#379911yescharacteristic, complemented
238Q8—91no—
239D4—271no—
240D4—812no—
241C2×C4—271no—
242C2×C4—812no—
243C2×C3—31no—
244C2×C3—31no—
245C2×C3—61no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3—121no—
249C2×C3—541no—
250C2×C3—542no—
251C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
252C2×C3C2×#111311yescharacteristic
253D3—183no—
254D3—186no—
255D3—181no—
256D3—182no—
257D3—541no—
258D3—542no—
259D3—542no—
260D3—1082no—
261C22—271no—
262C22—272no—
263C4—92no—
264C4—271no—
265C4#6643211yescharacteristic
266C3—31no—
267C3—31no—
268C3—61no—
269C3—61no—
270C3—61no—
271C3—121no—
272C3#8864813yescomplemented
273C3#8904111yescharacteristic
274C2—541no—
275C2—542no—
276C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC4#66432
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
2. C4×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. C3×#6229C2semidirect
4. #10619902C2semidirect
5. #10619901C2semidirect
14. C4⋊(C3⋊(C3⋊D3))D3semidirect
15. C4×C3⋊(C3⋊D3)D3semidirect
25. #6229D3semidirect
63. C3×H(3,𝔽3)#41semidirect
148. C33#235semidirect
149. C33#237semidirect
153. H(3,𝔽3)#239semidirect
236. C32#3796semidirect
237. C32#3799semidirect
272. C3#88648semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊((C3⋊D3)⋊C3))2
C4×C3⋊((C3⋊D3)⋊C3)2
C3×#62292
#106199022
#106199012
#890183
#890173
#886483