Finite Group Databasebeta

Subgroups — #13164464

Order 1728.

560 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₉×#994	C₂	1	1	yes	maximal, characteristic, complemented
2	#11722373	C₂	1	3	yes	maximal, complemented
3	#11722372	C₂	1	3	yes	maximal, complemented
4	#10596587	—	3	1	no	maximal
5	#10593099	—	3	2	no	maximal
6	#10589938	—	3	1	no	maximal
7	Q₈×C₃⋊D₉	—	4	1	no	maximal
8	C₉×#41⋊C₃	C₂²	1	3	yes	—
9	C₉⋊GL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
10	C₉⋊#226	C₂²	1	3	yes	complemented
11	C₃×#994	D₃	1	1	yes	characteristic
12	#64266	—	3	3	no	—
13	#64265	—	3	3	no	—
14	C₉×H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
15	#63716	—	3	1	no	—
16	#63700	D₃	1	2	yes	complemented
17	#63691	—	3	6	no	—
18	#63690	—	3	6	no	—
19	#63513	—	3	3	no	—
20	#63511	—	3	3	no	—
21	#63510	—	3	1	no	—
22	#63495	—	3	3	no	—
23	C₂×#3672	—	3	3	no	—
24	C₃×Q₈×C₉	—	4	1	no	—
25	C₄×C₃⋊D₉	—	4	3	no	—
26	#6226	—	4	3	no	—
27	C₉×SL(2,𝔽₃)	C₂³	1	1	yes	characteristic
28	#5924	—	9	3	no	—
29	#5836	—	9	1	no	—
30	Q₈×C₃⋊D₃	—	12	1	no	—
31	C₃×#41⋊C₃	C₂×D₃	1	3	yes	—
32	C₃⋊GL(2,𝔽₃)	—	3	1	no	—
33	C₃⋊#226	—	3	3	no	—
34	C₉×#41	—	3	1	no	—
35	C₉×#41	C₂×D₃	1	3	yes	—
36	C₂×D₄×C₉	—	3	3	no	—
37	Q₈×D₉	—	6	1	no	—
38	Q₈×D₉	—	12	2	no	—
39	#3699	—	6	3	no	—
40	#3697	—	3	3	no	—
41	C₂×Dic₁₈	—	3	3	no	—
42	#41⋊C₉	C₂×D₃	1	6	yes	—
43	#3689	—	3	2	no	—
44	#3688	—	3	6	no	—
45	#3675	—	3	1	no	—
46	#3674	—	3	3	no	—
47	#3673	—	3	3	no	—
48	#3672	—	3	3	no	—
49	#3672	—	3	6	no	—
50	#3667	—	3	3	no	—
51	C₂×C₉⋊C₈	—	3	3	no	—
52	C₂×C₃⋊D₉	—	4	1	no	—
53	C₃×C₄×C₉	—	4	3	no	—
54	C₃⋊Dic₉	—	4	3	no	—
55	C₃×H(5,𝔽₂)	C₃⋊D₃	1	1	yes	characteristic
56	#1010	—	9	1	no	—
57	#994	D₉	1	3	yes	complemented
58	#985	—	9	9	no	—
59	#984	—	9	9	no	—
60	#951	—	9	3	no	—
61	#950	—	9	3	no	—
62	#948	—	9	1	no	—
63	C₂×#214	—	9	3	no	—
64	#932	—	9	3	no	—
65	C₃²×Q₈	—	4	1	no	—
66	C₄×C₃⋊D₃	—	12	3	no	—
67	#634	—	12	3	no	—
68	C₃×SL(2,𝔽₃)	C₂²×D₃	1	1	yes	characteristic
69	C₂³×C₉	—	6	1	no	—
70	Q₈×C₉	—	4	2	no	—
71	Q₈×C₉	S₄	1	1	yes	characteristic
72	Q₈×C₉	C₂²×D₃	1	1	yes	characteristic
73	D₄×C₉	—	3	3	no	—
74	D₄×C₉	—	3	3	no	—
75	C₂×C₄×C₉	—	3	3	no	—
76	#611	—	6	3	no	—
77	C₂×Dic₉	—	6	3	no	—
78	D₃₆	—	3	1	no	—
79	C₄×D₉	—	6	3	no	—
80	C₄×D₉	—	12	6	no	—
81	Dic₁₈	—	3	3	no	—
82	Dic₁₈	—	6	3	no	—
83	Dic₁₈	—	12	6	no	—
84	Q₈⋊C₉	C₂²×D₃	1	2	yes	—
85	C₉⋊C₈	—	3	1	no	—
86	C₉⋊C₈	—	3	3	no	—
87	#577	—	27	1	no	Sylow
88	C₂×C₃×C₉	—	4	1	no	—
89	C₃⋊D₉	—	4	2	no	—
90	C₃×#41	—	3	1	no	—
91	C₃×#41	C₂×C₃⋊D₃	1	3	yes	—
92	C₂×C₃×D₄	—	3	3	no	—
93	D₃×Q₈	—	18	1	no	—
94	D₃×Q₈	—	36	3	no	—
95	#237	—	18	3	no	—
96	#235	—	9	3	no	—
97	C₂×Dic₆	—	9	3	no	—
98	#41⋊C₃	C₂×D₉	1	9	yes	—
99	GL(2,𝔽₃)	—	9	3	no	—
100	#226	—	9	9	no	—
101	#216	—	9	3	no	—
102	#215	—	9	1	no	—
103	#214	—	9	6	no	—
104	#214	—	9	3	no	—
105	#213	—	9	3	no	—
106	#208	—	9	3	no	—
107	C₂×C₃⋊C₈	—	9	3	no	—
108	C₂×C₃⋊D₃	—	12	1	no	—
109	C₃²×C₄	—	4	3	no	—
110	C₃⋊Dic₃	—	12	3	no	—
111	C₂²×C₉	—	3	3	no	—
112	C₂²×C₉	—	24	1	no	—
113	C₂×D₉	—	6	1	no	—
114	C₂×D₉	—	12	2	no	—
115	C₄×C₉	—	3	1	no	—
116	C₄×C₉	—	4	6	no	—
117	C₄×C₉	C₂×S₄	1	3	yes	—
118	Dic₉	—	6	3	no	—
119	Dic₉	—	12	6	no	—
120	2₋¹⁺⁴	—	27	1	no	—
121	H(5,𝔽₂)	C₃⋊D₉	1	1	yes	characteristic, complemented
122	#137	—	27	3	no	—
123	AGL(1,ℤ/8)	—	27	3	no	—
124	#135	—	27	3	no	—
125	C₂×SD₁₆	—	27	3	no	—
126	#131	—	27	1	no	—
127	C₃×C₉	—	4	1	no	Sylow
128	C₂³×C₃	—	6	1	no	—
129	C₃×Q₈	—	4	3	no	—
130	C₃×Q₈	C₃⋊S₄	1	1	yes	characteristic
131	C₃×Q₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
132	C₃×D₄	—	3	3	no	—
133	C₃×D₄	—	3	3	no	—
134	C₂×C₃×C₄	—	3	3	no	—
135	#67	—	18	3	no	—
136	C₂×Dic₃	—	18	3	no	—
137	D₁₂	—	9	1	no	—
138	C₄×D₃	—	18	3	no	—
139	C₄×D₃	—	36	9	no	—
140	Dic₆	—	9	3	no	—
141	Dic₆	—	18	3	no	—
142	Dic₆	—	36	9	no	—
143	SL(2,𝔽₃)	C₂²×D₉	1	3	yes	—
144	C₃⋊C₈	—	9	1	no	—
145	C₃⋊C₈	—	9	3	no	—
146	C₂×C₃²	—	4	1	no	—
147	C₃⋊D₃	—	12	2	no	—
148	C₂×C₉	—	4	2	no	—
149	C₂×C₉	—	6	3	no	—
150	C₂×C₉	C₂²×S₄	1	1	yes	characteristic
151	D₉	—	12	4	no	—
152	D₉	—	12	1	no	—
153	#41	—	3	1	no	—
154	#41	—	27	3	no	—
155	#41	—	54	3	no	—
156	#41	C₂×C₃⋊D₉	1	3	yes	—
157	C₂×Q₈	—	27	3	no	—
158	C₂×Q₈	—	54	1	no	—
159	C₂×D₄	—	3	3	no	—
160	Q₁₆	—	27	3	no	—
161	SD₁₆	—	27	3	no	—
162	SD₁₆	—	27	6	no	—
163	SD₁₆	—	27	1	no	—
164	D₈	—	27	3	no	—
165	M₁₆	—	27	3	no	—
166	C₂×C₈	—	27	3	no	—
167	C₂²×C₃	—	3	3	no	—
168	C₂²×C₃	—	24	1	no	—
169	C₂×D₃	—	18	1	no	—
170	C₂×D₃	—	36	3	no	—
171	C₃×C₄	—	3	1	no	—
172	C₃×C₄	—	4	9	no	—
173	C₃×C₄	C₂×C₃⋊S₄	1	3	yes	—
174	Dic₃	—	18	3	no	—
175	Dic₃	—	36	9	no	—
176	C₃²	—	4	1	no	—
177	C₉	—	4	2	no	—
178	C₉	#5924	1	1	yes	characteristic, complemented
179	C₂³	—	6	1	no	—
180	Q₈	—	27	3	no	—
181	Q₈	—	54	3	no	—
182	Q₈	C₉⋊S₄	1	1	yes	characteristic
183	Q₈	C₂²×C₃⋊D₉	1	1	yes	characteristic
184	D₄	—	3	3	no	—
185	D₄	—	3	3	no	—
186	D₄	—	27	1	no	—
187	D₄	—	54	3	no	—
188	C₂×C₄	—	3	3	no	—
189	C₂×C₄	—	54	3	no	—
190	C₂×C₄	—	54	3	no	—
191	C₈	—	27	1	no	—
192	C₈	—	27	3	no	—
193	C₂×C₃	—	4	3	no	—
194	C₂×C₃	—	6	3	no	—
195	C₂×C₃	C₂²×C₃⋊S₄	1	1	yes	characteristic
196	D₃	—	36	6	no	—
197	D₃	—	36	1	no	—
198	C₂²	—	3	3	no	—
199	C₂²	—	24	1	no	—
200	C₂²	—	54	1	no	—
201	C₄	—	3	1	no	—
202	C₄	—	54	3	no	—
203	C₄	C₂×C₉⋊S₄	1	3	yes	—
204	C₃	—	4	3	no	—
205	C₃	#10596587	1	1	yes	characteristic
206	C₂	—	6	3	no	—
207	C₂	—	108	1	no	—
208	C₂	C₂²×C₉⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₉⋊S₄
commutator, cosocle, hypocenter	C₉×SL(2,𝔽₃)	C₂³
Frattini, socle	C₂×C₃	C₂²×C₃⋊S₄
Fitting	C₉×H(5,𝔽₂)	D₃
hypercenter	Q₈	C₉⋊S₄
LCS	G ▷ C₉×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₉×#994	C₂	semidirect
2. #11722373	C₂	semidirect
3. #11722372	C₂	semidirect
9. C₉⋊GL(2,𝔽₃)	C₂²	semidirect
10. C₉⋊#226	C₂²	semidirect
14. C₉×H(5,𝔽₂)	D₃	semidirect
16. #63700	D₃	semidirect
57. #994	D₉	semidirect
121. H(5,𝔽₂)	C₃⋊D₉	semidirect
178. C₉	#5924	semidirect

Maximal subgroups

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

subgroup	index
C₉×#994	2
#11722373	2
#11722372	2
#10596587	3
#10593099	3
#10589938	3
Q₈×C₃⋊D₉	4