Finite Group Databasebeta

Subgroups — #11724194

Order 864.

278 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃⋊(#41⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(C₃⋊GL(2,𝔽₃))	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#64267	—	3	1	no	maximal
5	#64199	—	3	3	no	maximal
6	#63939	—	3	1	no	maximal
7	C₃²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
8	#6292	—	4	1	no	maximal
9	C₃×#239	D₃	1	1	yes	characteristic, complemented
10	C₃×#41⋊C₃	—	3	1	no	—
11	D₃×D₁₂	—	3	1	no	—
12	C₃⋊(#41⋊C₃)	D₃	1	3	yes	complemented
13	C₃⋊GL(2,𝔽₃)	—	3	1	no	—
14	C₃⋊GL(2,𝔽₃)	—	3	3	no	—
15	C₃⋊GL(2,𝔽₃)	—	6	1	no	—
16	C₃⋊GL(2,𝔽₃)	—	6	3	no	—
17	C₃⋊GL(2,𝔽₃)	D₃	1	1	yes	characteristic, complemented
18	C₃×GL(2,𝔽₃)	—	3	3	no	—
19	#3755	—	3	1	no	—
20	C₃×#215	—	3	1	no	—
21	#3714	—	3	1	no	—
22	#3713	—	3	1	no	—
23	#3711	—	3	1	no	—
24	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	—
25	C₂×C₃×C₃⋊D₃	—	4	1	no	—
26	C₃²×Dic₃	—	4	1	no	—
27	#986	—	9	3	no	—
28	#949	—	9	1	no	—
29	#914	—	9	1	no	—
30	C₂×D₃²	—	6	1	no	—
31	C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
32	C₄⋊(C₃⋊D₃)	—	3	1	no	—
33	C₄⋊(C₃⋊D₃)	—	12	1	no	—
34	C₃×D₁₂	—	3	1	no	—
35	C₃×D₁₂	—	3	1	no	—
36	C₃×C₄×D₃	—	3	1	no	—
37	C₃×SL(2,𝔽₃)	—	2	1	no	—
38	C₃×SL(2,𝔽₃)	—	2	3	no	—
39	C₃×SL(2,𝔽₃)	C₂×D₃	1	3	yes	complemented
40	C₃×SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic, complemented
41	#626	—	6	1	no	—
42	#626	—	12	3	no	—
43	C₃⋊(C₃⋊C₈)	—	3	1	no	—
44	C₃×C₃⋊C₈	—	3	1	no	—
45	C₂×C₃³	—	4	1	no	—
46	C₃⋊(C₃⋊D₃)	—	8	1	no	—
47	C₃×C₃⋊D₃	—	8	1	no	—
48	C₃×#41	—	3	1	no	—
49	#239	C₃⋊D₃	1	1	yes	characteristic, complemented
50	D₃×D₄	—	9	1	no	—
51	C₂×D₁₂	—	9	1	no	—
52	#41⋊C₃	—	3	3	no	—
53	GL(2,𝔽₃)	—	3	3	no	—
54	GL(2,𝔽₃)	—	9	3	no	—
55	GL(2,𝔽₃)	—	18	3	no	—
56	C₃×SD₁₆	—	9	1	no	—
57	#215	—	3	1	no	—
58	#215	—	9	1	no	—
59	#215	—	9	1	no	—
60	#215	—	18	1	no	—
61	#213	—	9	1	no	—
62	#213	—	9	1	no	—
63	#213	—	9	1	no	—
64	#208	—	9	1	no	—
65	D₂₄	—	9	1	no	—
66	ΓL(1,𝔽₂₅)	—	9	1	no	—
67	C₂×C₃⋊D₃	—	4	1	no	—
68	C₂×C₃⋊D₃	—	6	1	no	—
69	C₂×C₃⋊D₃	—	12	1	no	—
70	C₂×C₃⋊D₃	—	12	3	no	—
71	C₂×C₃⋊D₃	—	24	1	no	—
72	C₂×C₃⋊D₃	—	24	3	no	—
73	C₂×C₃×D₃	—	3	1	no	—
74	C₂×C₃×D₃	—	6	1	no	—
75	C₂×C₃×D₃	—	12	3	no	—
76	D₃²	—	12	2	no	—
77	C₃²×C₄	—	3	1	no	—
78	C₃²×C₄	—	12	1	no	—
79	C₃×Dic₃	—	4	3	no	—
80	C₃×Dic₃	S₄	1	1	yes	characteristic
81	AGL(1,ℤ/8)	—	27	1	no	Sylow
82	C₃³	—	4	1	no	Sylow
83	C₂²×D₃	—	18	1	no	—
84	C₂²×D₃	—	18	1	no	—
85	C₃×Q₈	—	2	1	no	—
86	C₃×Q₈	D₃²	1	1	yes	characteristic
87	C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
88	C₃×D₄	—	9	1	no	—
89	C₃×D₄	—	9	1	no	—
90	C₂×C₃×C₄	—	9	1	no	—
91	#67	—	18	1	no	—
92	D₁₂	—	3	1	no	—
93	D₁₂	—	3	1	no	—
94	D₁₂	—	9	1	no	—
95	D₁₂	—	9	1	no	—
96	D₁₂	—	18	1	no	—
97	D₁₂	—	18	1	no	—
98	D₁₂	—	36	3	no	—
99	C₄×D₃	—	3	1	no	—
100	SL(2,𝔽₃)	—	2	3	no	—
101	SL(2,𝔽₃)	D₃²	1	3	yes	complemented
102	C₃×C₈	—	9	1	no	—
103	C₃⋊C₈	—	3	1	no	—
104	C₃⋊C₈	—	9	1	no	—
105	C₃⋊C₈	—	9	1	no	—
106	C₃⋊C₈	—	18	1	no	—
107	C₂×C₃²	—	4	1	no	—
108	C₂×C₃²	—	4	3	no	—
109	C₂×C₃²	—	8	3	no	—
110	C₂×C₃²	—	8	1	no	—
111	C₂×C₃²	C₂×S₄	1	1	yes	characteristic
112	C₃⋊D₃	—	8	1	no	—
113	C₃⋊D₃	—	12	1	no	—
114	C₃⋊D₃	—	24	3	no	—
115	C₃⋊D₃	—	24	1	no	—
116	C₃⋊D₃	—	24	2	no	—
117	C₃⋊D₃	—	24	6	no	—
118	C₃×D₃	—	6	1	no	—
119	C₃×D₃	—	12	1	no	—
120	C₃×D₃	—	24	3	no	—
121	#41	—	3	1	no	—
122	C₂×D₄	—	27	1	no	—
123	SD₁₆	—	9	1	no	—
124	SD₁₆	—	27	1	no	—
125	D₈	—	27	1	no	—
126	D₈	—	27	1	no	—
127	M₁₆	—	27	1	no	—
128	C₂²×C₃	—	9	1	no	—
129	C₂²×C₃	—	18	1	no	—
130	C₂×D₃	—	3	1	no	—
131	C₂×D₃	—	6	1	no	—
132	C₂×D₃	—	12	3	no	—
133	C₂×D₃	—	18	1	no	—
134	C₂×D₃	—	18	1	no	—
135	C₂×D₃	—	36	1	no	—
136	C₂×D₃	—	36	3	no	—
137	C₂×D₃	—	36	2	no	—
138	C₂×D₃	—	36	2	no	—
139	C₂×D₃	—	72	3	no	—
140	C₃×C₄	—	3	1	no	—
141	C₃×C₄	—	3	1	no	—
142	C₃×C₄	—	3	1	no	—
143	C₃×C₄	—	6	1	no	—
144	C₃×C₄	—	12	3	no	—
145	Dic₃	C₃⋊S₄	1	1	yes	characteristic
146	C₃²	—	4	1	no	—
147	C₃²	—	4	3	no	—
148	C₃²	—	8	3	no	—
149	C₃²	—	8	1	no	—
150	C₃²	#986	1	1	yes	characteristic, complemented
151	C₂³	—	54	1	no	—
152	Q₈	D₃×C₃⋊D₃	1	1	yes	characteristic
153	D₄	—	9	1	no	—
154	D₄	—	9	1	no	—
155	D₄	—	27	1	no	—
156	D₄	—	54	1	no	—
157	C₂×C₄	—	9	1	no	—
158	C₈	—	9	1	no	—
159	C₈	—	27	1	no	—
160	C₂×C₃	—	2	1	no	—
161	C₂×C₃	—	4	3	no	—
162	C₂×C₃	—	8	3	no	—
163	C₂×C₃	—	18	1	no	—
164	C₂×C₃	—	36	1	no	—
165	C₂×C₃	D₃×S₄	1	1	yes	characteristic
166	C₂×C₃	C₂×C₃⋊S₄	1	1	yes	characteristic
167	D₃	—	6	1	no	—
168	D₃	—	12	1	no	—
169	D₃	—	24	3	no	—
170	D₃	—	36	1	no	—
171	D₃	—	36	1	no	—
172	D₃	—	72	1	no	—
173	D₃	—	72	3	no	—
174	D₃	—	72	6	no	—
175	C₂²	—	9	1	no	—
176	C₂²	—	18	1	no	—
177	C₂²	—	54	1	no	—
178	C₂²	—	108	2	no	—
179	C₄	—	3	1	no	—
180	C₄	—	3	1	no	—
181	C₃	—	2	1	no	—
182	C₃	—	4	3	no	—
183	C₃	—	8	3	no	—
184	C₃	#64199	1	1	yes	characteristic, complemented
185	C₃	#64267	1	1	yes	characteristic, complemented
186	C₂	—	18	1	no	—
187	C₂	—	36	1	no	—
188	C₂	—	108	1	no	—
189	C₂	D₃×C₃⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊S₄
commutator, cosocle, hypocenter	C₃²×SL(2,𝔽₃)	C₂²
Fitting	C₃²×Q₈	C₂×D₃
socle	C₂×C₃²	C₂×S₄
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₃×C₃⋊(#41⋊C₃)	C₂	semidirect
2. C₃⋊(C₃⋊GL(2,𝔽₃))	C₂	semidirect
3. C₃×C₃⋊GL(2,𝔽₃)	C₂	semidirect
7. C₃²×SL(2,𝔽₃)	C₂²	semidirect
9. C₃×#239	D₃	semidirect
12. C₃⋊(#41⋊C₃)	D₃	semidirect
17. C₃⋊GL(2,𝔽₃)	D₃	semidirect
39. C₃×SL(2,𝔽₃)	C₂×D₃	semidirect
40. C₃×SL(2,𝔽₃)	C₂×D₃	semidirect
49. #239	C₃⋊D₃	semidirect
101. SL(2,𝔽₃)	D₃²	semidirect
150. C₃²	#986	semidirect
184. C₃	#64199	semidirect
185. C₃	#64267	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃⋊(#41⋊C₃)	2
C₃⋊(C₃⋊GL(2,𝔽₃))	2
C₃×C₃⋊GL(2,𝔽₃)	2
#64267	3
#64199	3
#63939	3
#6292	4