Finite Group Databasebeta

Subgroups — #11757730

Order 972.

304 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 28 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	#92338	C₂	1	1	yes	maximal, characteristic, complemented
2	#92300	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₃⋊D₃)⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1113	—	3	1	no	maximal
5	D₃×#272	—	3	1	no	maximal
6	C₃×((C₃⋊D₃)⋊C₃)⋊C₂	—	3	3	no	maximal
7	C₃²⋊H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented, Sylow
8	C₃²×C₃⋊D₃	—	3	1	no	—
9	C₃²×C₃⋊D₃	—	3	1	no	—
10	C₃²×C₃⋊D₃	D₃	1	1	yes	characteristic, complemented
11	C₃⋊#272	D₃	1	1	yes	characteristic, complemented
12	C₃×#272	—	3	1	no	—
13	C₃×#272	—	3	3	no	—
14	C₃×#272	—	6	1	no	—
15	C₃×#272	—	6	3	no	—
16	D₃×H(3,𝔽₃)	—	3	1	no	—
17	C₃×(C₃⋊D₃)⋊C₃	—	3	3	no	—
18	C₃×(C₃⋊D₃)⋊C₃	D₃	1	3	yes	complemented
19	#1113	—	3	1	no	—
20	C₃×D₃²	—	9	3	no	—
21	C₃×D₃²	—	9	1	no	—
22	C₃×D₃²	—	9	1	no	—
23	C₃×D₃²	—	9	1	no	—
24	C₂×#272	—	9	1	no	—
25	((C₃⋊D₃)⋊C₃)⋊C₂	—	3	3	no	—
26	C₃⁴	C₂×D₃	1	1	yes	characteristic, complemented
27	C₃×H(3,𝔽₃)	—	2	1	no	—
28	C₃×H(3,𝔽₃)	—	2	3	no	—
29	C₃×H(3,𝔽₃)	C₂×D₃	1	3	yes	complemented
30	C₃×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic, complemented
31	C₃×C₃⋊D₃	—	3	1	no	—
32	C₃×C₃⋊D₃	—	3	1	no	—
33	C₃×C₃⋊D₃	—	3	1	no	—
34	C₃×C₃⋊D₃	—	3	1	no	—
35	C₃×C₃⋊D₃	—	3	1	no	—
36	C₃×C₃⋊D₃	—	3	3	no	—
37	C₃×C₃⋊D₃	—	6	1	no	—
38	C₃×C₃⋊D₃	—	6	1	no	—
39	C₃×C₃⋊D₃	C₃⋊D₃	1	1	yes	characteristic
40	C₃²×D₃	—	3	1	no	—
41	C₃²×D₃	—	3	3	no	—
42	C₃²×D₃	—	9	1	no	—
43	C₃²×D₃	—	9	1	no	—
44	C₃²×D₃	—	9	1	no	—
45	C₃²×D₃	—	9	3	no	—
46	C₃²×D₃	—	9	1	no	—
47	C₃²×D₃	—	9	1	no	—
48	C₃²×D₃	—	18	3	no	—
49	C₃²×D₃	—	18	1	no	—
50	C₃²×D₃	—	18	1	no	—
51	C₂×H(3,𝔽₃)	—	9	1	no	—
52	#272	—	3	1	no	—
53	#272	—	3	3	no	—
54	#272	—	6	1	no	—
55	#272	—	6	3	no	—
56	#272	—	9	1	no	—
57	#272	—	18	1	no	—
58	(C₃⋊D₃)⋊C₃	—	3	3	no	—
59	(C₃⋊D₃)⋊C₃	—	3	3	no	—
60	(C₃⋊D₃)⋊C₃	—	6	3	no	—
61	C₂×C₃×D₃	—	27	1	no	—
62	C₂×C₃×D₃	—	27	1	no	—
63	C₂×C₃×D₃	—	27	1	no	—
64	C₂×C₃×D₃	—	27	3	no	—
65	D₃²	—	9	3	no	—
66	D₃²	—	9	1	no	—
67	D₃²	—	9	1	no	—
68	D₃²	—	9	1	no	—
69	C₃³	—	2	1	no	—
70	C₃³	—	2	3	no	—
71	C₃³	—	3	1	no	—
72	C₃³	—	3	1	no	—
73	C₃³	—	6	1	no	—
74	C₃³	—	6	1	no	—
75	C₃³	—	6	1	no	—
76	C₃³	—	12	1	no	—
77	C₃³	D₃²	1	1	yes	characteristic
78	C₃³	D₃²	1	3	yes	complemented
79	C₃³	C₂×C₃⋊D₃	1	1	yes	characteristic
80	H(3,𝔽₃)	—	3	1	no	—
81	H(3,𝔽₃)	—	3	3	no	—
82	H(3,𝔽₃)	—	6	1	no	—
83	H(3,𝔽₃)	—	6	3	no	—
84	H(3,𝔽₃)	—	6	3	no	—
85	H(3,𝔽₃)	—	6	1	no	—
86	H(3,𝔽₃)	—	12	1	no	—
87	H(3,𝔽₃)	—	12	3	no	—
88	C₂×C₃²	—	9	3	no	—
89	C₂×C₃²	—	9	1	no	—
90	C₂×C₃²	—	27	1	no	—
91	C₂×C₃²	—	27	1	no	—
92	C₃⋊D₃	—	3	1	no	—
93	C₃⋊D₃	—	3	1	no	—
94	C₃⋊D₃	—	3	3	no	—
95	C₃⋊D₃	#272	1	1	yes	characteristic, complemented
96	C₃×D₃	—	3	1	no	—
97	C₃×D₃	—	9	3	no	—
98	C₃×D₃	—	9	1	no	—
99	C₃×D₃	—	9	1	no	—
100	C₃×D₃	—	9	3	no	—
101	C₃×D₃	—	9	3	no	—
102	C₃×D₃	—	9	1	no	—
103	C₃×D₃	—	9	1	no	—
104	C₃×D₃	—	9	1	no	—
105	C₃×D₃	—	9	1	no	—
106	C₃×D₃	—	9	1	no	—
107	C₃×D₃	—	9	1	no	—
108	C₃×D₃	—	9	1	no	—
109	C₃×D₃	—	9	1	no	—
110	C₃×D₃	—	9	1	no	—
111	C₃×D₃	—	18	1	no	—
112	C₃×D₃	—	18	1	no	—
113	C₃×D₃	—	18	3	no	—
114	C₃×D₃	—	18	1	no	—
115	C₃×D₃	—	18	1	no	—
116	C₃×D₃	—	18	1	no	—
117	C₃×D₃	—	18	1	no	—
118	C₃×D₃	—	18	3	no	—
119	C₃×D₃	—	18	3	no	—
120	C₃×D₃	—	18	1	no	—
121	C₃×D₃	—	18	1	no	—
122	C₃×D₃	—	18	1	no	—
123	C₃×D₃	—	18	1	no	—
124	C₃×D₃	—	18	6	no	—
125	C₃×D₃	—	18	1	no	—
126	C₃×D₃	—	18	1	no	—
127	C₃×D₃	—	18	1	no	—
128	C₃×D₃	—	27	3	no	—
129	C₂²×C₃	—	81	1	no	—
130	C₂×D₃	—	27	1	no	—
131	C₂×D₃	—	27	1	no	—
132	C₂×D₃	—	27	1	no	—
133	C₂×D₃	—	27	3	no	—
134	C₃²	—	2	1	no	—
135	C₃²	—	2	1	no	—
136	C₃²	—	2	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	3	no	—
141	C₃²	—	3	3	no	—
142	C₃²	—	4	1	no	—
143	C₃²	—	6	1	no	—
144	C₃²	—	6	1	no	—
145	C₃²	—	6	1	no	—
146	C₃²	—	6	3	no	—
147	C₃²	—	6	6	no	—
148	C₃²	—	6	1	no	—
149	C₃²	—	6	1	no	—
150	C₃²	—	6	1	no	—
151	C₃²	—	6	1	no	—
152	C₃²	—	6	1	no	—
153	C₃²	—	6	1	no	—
154	C₃²	—	6	1	no	—
155	C₃²	—	6	3	no	—
156	C₃²	—	6	3	no	—
157	C₃²	—	12	1	no	—
158	C₃²	—	12	1	no	—
159	C₃²	—	12	1	no	—
160	C₃²	—	12	1	no	—
161	C₃²	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic, complemented
162	C₃²	C₂×#272	1	1	yes	characteristic, complemented
163	C₃²	D₃×C₃⋊D₃	1	1	yes	characteristic
164	C₂×C₃	—	9	1	no	—
165	C₂×C₃	—	27	1	no	—
166	C₂×C₃	—	27	3	no	—
167	C₂×C₃	—	27	1	no	—
168	C₂×C₃	—	27	1	no	—
169	C₂×C₃	—	27	1	no	—
170	C₂×C₃	—	27	1	no	—
171	C₂×C₃	—	54	1	no	—
172	C₂×C₃	—	54	1	no	—
173	D₃	—	3	1	no	—
174	D₃	—	9	1	no	—
175	D₃	—	9	1	no	—
176	D₃	—	9	1	no	—
177	D₃	—	9	1	no	—
178	D₃	—	9	3	no	—
179	D₃	—	9	1	no	—
180	D₃	—	18	1	no	—
181	D₃	—	18	1	no	—
182	D₃	—	18	3	no	—
183	D₃	—	27	3	no	—
184	C₂²	—	81	1	no	Sylow
185	C₃	—	2	1	no	—
186	C₃	—	3	1	no	—
187	C₃	—	3	1	no	—
188	C₃	—	6	1	no	—
189	C₃	—	6	1	no	—
190	C₃	—	6	1	no	—
191	C₃	—	9	3	no	—
192	C₃	—	12	1	no	—
193	C₃	—	18	3	no	—
194	C₃	((C₃⋊D₃)⋊C₃)⋊D₃	1	1	yes	characteristic
195	C₃	D₃×#272	1	1	yes	characteristic
196	C₂	—	9	1	no	—
197	C₂	—	27	1	no	—
198	C₂	—	27	1	no	—

	subgroup	G/H
center, hypercenter	C₃	((C₃⋊D₃)⋊C₃)⋊D₃
commutator, Fitting, cosocle, hypocenter	C₃²⋊H(3,𝔽₃)	C₂²
Frattini, socle	C₃²	D₃×C₃⋊D₃
LCS	G ▷ C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. #92338	C₂	semidirect
2. #92300	C₂	semidirect
3. (C₃⋊D₃)⋊H(3,𝔽₃)	C₂	semidirect
7. C₃²⋊H(3,𝔽₃)	C₂²	semidirect
10. C₃²×C₃⋊D₃	D₃	semidirect
11. C₃⋊#272	D₃	semidirect
18. C₃×(C₃⋊D₃)⋊C₃	D₃	semidirect
26. C₃⁴	C₂×D₃	semidirect
29. C₃×H(3,𝔽₃)	C₂×D₃	semidirect
30. C₃×H(3,𝔽₃)	C₂×D₃	semidirect
78. C₃³	D₃²	semidirect
95. C₃⋊D₃	#272	semidirect
161. C₃²	((C₃⋊D₃)⋊C₃)⋊C₂	semidirect
162. C₃²	C₂×#272	semidirect

Maximal subgroups

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

subgroup	index
#92338	2
#92300	2
(C₃⋊D₃)⋊H(3,𝔽₃)	2
C₃×#1113	3
D₃×#272	3
C₃×((C₃⋊D₃)⋊C₃)⋊C₂	3