Finite Group Databasebeta

Subgroups — #13173054

Order 1728.

688 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 113 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	#11724470	C₂	1	1	yes	maximal, characteristic
2	C₃²×#131⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₃⋊GU(2,𝔽₃))	C₂	1	1	yes	maximal, characteristic, complemented
4	#10593638	—	3	12	no	maximal
5	#10592843	—	3	1	no	maximal
6	C₃²×#41⋊C₃	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊GL(2,𝔽₃))	C₄	1	1	yes	characteristic, complemented
8	C₃⋊(C₃⋊#226)	C₄	1	1	yes	characteristic, complemented
9	C₈×C₃⋊(C₃⋊D₃)	—	4	1	no	maximal
10	#64266	—	3	12	no	—
11	C₃²×#131	D₃	1	1	yes	characteristic, complemented
12	#64159	—	3	1	no	—
13	#64110	—	3	1	no	—
14	C₃×#131⋊C₃	D₃	1	12	yes	complemented
15	C₃⋊GU(2,𝔽₃)	—	3	12	no	—
16	#63664	—	3	1	no	—
17	#63652	—	3	1	no	—
18	#63649	—	3	1	no	—
19	C₈×C₃⋊Dic₃	—	3	1	no	—
20	C₄×C₃⋊(C₃⋊D₃)	—	4	1	no	—
21	C₃²×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
22	C₃³×C₈	—	4	1	no	—
23	C₃⋊(C₃⋊(C₃⋊C₈))	—	4	1	no	—
24	#5404	—	9	9	no	—
25	#5144	—	9	4	no	—
26	C₃²×#41	C₂×D₃	1	1	yes	characteristic
27	#3828	—	3	1	no	—
28	C₃×#41⋊C₃	C₂×D₃	1	12	yes	—
29	C₃⋊GL(2,𝔽₃)	—	3	12	no	—
30	C₃⋊#226	—	3	12	no	—
31	C₃²×M₁₆	—	3	1	no	—
32	C₂×C₃²×C₈	—	3	1	no	—
33	#3756	—	3	1	no	—
34	#3755	—	3	1	no	—
35	#3754	—	3	1	no	—
36	#3753	—	3	1	no	—
37	C₄×C₃⋊Dic₃	—	3	1	no	—
38	#3748	—	3	1	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
40	#3743	—	6	1	no	—
41	C₈×C₃⋊D₃	—	6	1	no	—
42	C₈×C₃⋊D₃	—	12	12	no	—
43	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	—
44	C₃³×C₄	—	4	1	no	—
45	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
46	#985	—	9	9	no	—
47	C₃×#131	C₃⋊D₃	1	4	yes	complemented
48	#950	—	9	4	no	—
49	#901	—	9	4	no	—
50	#131⋊C₃	C₃⋊D₃	1	9	yes	complemented
51	GU(2,𝔽₃)	—	9	9	no	—
52	#837	—	9	4	no	—
53	#825	—	9	4	no	—
54	#822	—	9	4	no	—
55	C₈×Dic₃	—	9	4	no	—
56	C₃²×Q₈	C₄×D₃	1	1	yes	characteristic
57	C₃²×D₄	—	3	1	no	—
58	C₂×C₃²×C₄	—	3	1	no	—
59	#638	—	6	1	no	—
60	C₂×C₃⋊Dic₃	—	6	1	no	—
61	C₄⋊(C₃⋊D₃)	—	3	1	no	—
62	C₄×C₃⋊D₃	—	6	1	no	—
63	C₄×C₃⋊D₃	—	12	12	no	—
64	#634	—	3	1	no	—
65	C₃×SL(2,𝔽₃)	C₄×D₃	1	12	yes	—
66	C₃²×C₈	—	3	1	no	—
67	C₃²×C₈	—	4	12	no	—
68	C₃²×C₈	S₄	1	1	yes	characteristic
69	C₃⋊(C₃⋊C₈)	—	3	1	no	—
70	C₃⋊(C₃⋊C₈)	—	3	1	no	—
71	C₃⋊(C₃⋊C₈)	—	6	1	no	—
72	C₃⋊(C₃⋊C₈)	—	12	12	no	—
73	#443	—	27	1	no	Sylow
74	C₂×C₃³	—	4	1	no	—
75	C₃⋊(C₃⋊D₃)	—	4	2	no	—
76	C₃×#41	C₂×C₃⋊D₃	1	4	yes	—
77	#235	—	9	4	no	—
78	#41⋊C₃	C₂×C₃⋊D₃	1	9	yes	—
79	GL(2,𝔽₃)	—	9	9	no	—
80	#226	—	9	9	no	—
81	C₃×M₁₆	—	3	4	no	—
82	C₂×C₃×C₈	—	3	4	no	—
83	#216	—	9	4	no	—
84	#215	—	9	4	no	—
85	#214	—	9	4	no	—
86	#213	—	9	4	no	—
87	C₄×Dic₃	—	9	4	no	—
88	#208	—	9	4	no	—
89	C₂×C₃⋊C₈	—	9	4	no	—
90	ΓL(1,𝔽₂₅)	—	18	4	no	—
91	D₃×C₈	—	18	4	no	—
92	D₃×C₈	—	36	9	no	—
93	C₂²×C₃²	—	3	1	no	—
94	C₂×C₃⋊D₃	—	6	1	no	—
95	C₂×C₃⋊D₃	—	12	12	no	—
96	C₃²×C₄	—	3	1	no	—
97	C₃²×C₄	—	4	12	no	—
98	C₃²×C₄	C₂×S₄	1	1	yes	characteristic
99	C₃⋊Dic₃	—	6	2	no	—
100	C₃⋊Dic₃	—	6	1	no	—
101	C₃⋊Dic₃	—	12	12	no	—
102	#135	—	27	1	no	—
103	#131	—	27	1	no	—
104	#131	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
105	#108	—	27	1	no	—
106	#104	—	27	1	no	—
107	#104	—	27	1	no	—
108	C₄×C₈	—	27	1	no	—
109	C₃³	—	4	1	no	Sylow
110	C₃×Q₈	C₄×C₃⋊D₃	1	4	yes	—
111	C₃×D₄	—	3	4	no	—
112	C₂×C₃×C₄	—	3	4	no	—
113	#67	—	18	4	no	—
114	C₂×Dic₃	—	18	4	no	—
115	D₁₂	—	9	4	no	—
116	C₄×D₃	—	18	4	no	—
117	C₄×D₃	—	36	9	no	—
118	Dic₆	—	9	4	no	—
119	SL(2,𝔽₃)	C₄×C₃⋊D₃	1	9	yes	—
120	C₃×C₈	—	3	4	no	—
121	C₃×C₈	—	4	9	no	—
122	C₃×C₈	C₃⋊S₄	1	4	yes	—
123	C₃⋊C₈	—	9	4	no	—
124	C₃⋊C₈	—	9	4	no	—
125	C₃⋊C₈	—	18	4	no	—
126	C₃⋊C₈	—	36	9	no	—
127	C₂×C₃²	—	4	12	no	—
128	C₂×C₃²	—	6	1	no	—
129	C₂×C₃²	C₄×S₄	1	1	yes	characteristic
130	C₃⋊D₃	—	12	24	no	—
131	C₃⋊D₃	—	12	1	no	—
132	#41	—	27	1	no	—
133	#41	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
134	Q₁₆	—	27	1	no	—
135	SD₁₆	—	27	1	no	—
136	SD₁₆	—	27	1	no	—
137	D₈	—	27	1	no	—
138	M₁₆	—	3	1	no	—
139	M₁₆	—	27	1	no	—
140	M₁₆	—	54	1	no	—
141	C₂×C₈	—	3	1	no	—
142	C₂×C₈	—	27	1	no	—
143	C₂×C₈	—	54	1	no	—
144	C₄²	—	27	1	no	—
145	C₂²×C₃	—	3	4	no	—
146	C₂×D₃	—	18	4	no	—
147	C₂×D₃	—	36	9	no	—
148	C₃×C₄	—	3	4	no	—
149	C₃×C₄	—	4	9	no	—
150	C₃×C₄	C₂×C₃⋊S₄	1	4	yes	—
151	Dic₃	—	18	8	no	—
152	Dic₃	—	18	4	no	—
153	Dic₃	—	36	9	no	—
154	C₃²	—	4	12	no	—
155	C₃²	#5404	1	1	yes	characteristic, complemented
156	Q₈	—	27	1	no	—
157	Q₈	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
158	D₄	—	3	1	no	—
159	D₄	—	27	1	no	—
160	D₄	—	54	1	no	—
161	C₂×C₄	—	3	1	no	—
162	C₂×C₄	—	54	1	no	—
163	C₂×C₄	—	54	1	no	—
164	C₈	—	3	1	no	—
165	C₈	—	27	1	no	—
166	C₈	—	27	1	no	—
167	C₈	—	54	1	no	—
168	C₈	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic
169	C₂×C₃	—	4	9	no	—
170	C₂×C₃	—	6	4	no	—
171	C₂×C₃	C₄×C₃⋊S₄	1	4	yes	—
172	D₃	—	36	18	no	—
173	D₃	—	36	4	no	—
174	C₂²	—	3	1	no	—
175	C₂²	—	54	1	no	—
176	C₄	—	3	1	no	—
177	C₄	—	54	2	no	—
178	C₄	—	54	1	no	—
179	C₄	C₂×C₃⋊(C₃⋊S₄)	1	1	yes	characteristic
180	C₃	—	4	9	no	—
181	C₃	#10593638	1	4	yes	complemented
182	C₂	—	6	1	no	—
183	C₂	—	108	1	no	—
184	C₂	C₄×C₃⋊(C₃⋊S₄)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	C₃⋊(C₃⋊S₄)
commutator, hypocenter	C₃²×SL(2,𝔽₃)	C₂×C₄
Frattini	C₄	C₂×C₃⋊(C₃⋊S₄)
Fitting	C₃²×#131	D₃
socle	C₂×C₃²	C₄×S₄
prime radical	#11724470	C₂
cosocle	C₃²×#41⋊C₃	C₂²
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₃²×#131⋊C₃	C₂	semidirect
3. C₃⋊(C₃⋊GU(2,𝔽₃))	C₂	semidirect
7. C₃⋊(C₃⋊GL(2,𝔽₃))	C₄	semidirect
8. C₃⋊(C₃⋊#226)	C₄	semidirect
11. C₃²×#131	D₃	semidirect
14. C₃×#131⋊C₃	D₃	semidirect
47. C₃×#131	C₃⋊D₃	semidirect
50. #131⋊C₃	C₃⋊D₃	semidirect
104. #131	C₃⋊(C₃⋊D₃)	semidirect
155. C₃²	#5404	semidirect
181. C₃	#10593638	semidirect

Maximal subgroups

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

subgroup	index
#11724470	2
C₃²×#131⋊C₃	2
C₃⋊(C₃⋊GU(2,𝔽₃))	2
#10593638	3
#10592843	3
C₈×C₃⋊(C₃⋊D₃)	4