Finite Group Databasebeta

Subgroups — #13167659

Order 1728.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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	#11724365	C₂	1	1	yes	maximal, characteristic
2	#11724361	C₂	1	1	yes	maximal, characteristic
3	#11723094	C₂	1	1	yes	maximal, characteristic
4	#11723093	C₂	1	1	yes	maximal, characteristic
5	C₃×#63767	C₂	1	1	yes	maximal, characteristic
6	#11722808	C₂	1	1	yes	maximal, characteristic
7	#11722806	C₂	1	1	yes	maximal, characteristic
8	#10593537	—	3	1	no	maximal
9	C₂×C₃⋊PSU(3,𝔽₂)	—	2	1	no	—
10	C₄⋊(C₃⋊#157)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊#157)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊#157)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊#157)	C₂²	1	1	yes	characteristic
14	C₃×C₄×#157	C₂²	1	1	yes	characteristic
15	C₃×#3787	C₂²	1	1	yes	characteristic
16	#88946	—	2	1	no	—
17	#88945	—	2	1	no	—
18	#88945	—	2	1	no	—
19	C₃×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
20	C₄⋊PSU(3,𝔽₂)	—	3	1	no	—
21	#64243	—	3	1	no	—
22	#63769	—	3	1	no	—
23	#63768	—	3	1	no	—
24	#63767	D₃	1	1	yes	characteristic
25	#63746	—	3	1	no	—
26	#63745	—	3	1	no	—
27	C₂×C₃⋊#157	—	2	1	no	—
28	C₂×C₃⋊#157	—	2	1	no	—
29	C₂×C₃⋊#157	—	2	1	no	—
30	C₂×C₃⋊#157	—	2	1	no	—
31	C₂×C₃×#157	D₄	1	1	yes	characteristic
32	C₂×C₃×#157	D₄	1	1	yes	characteristic
33	C₃⋊PSU(3,𝔽₂)	—	4	2	no	—
34	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
35	C₃×#622	Q₈	1	2	yes	complemented
36	C₃³×C₈	Q₈	1	1	yes	characteristic, complemented
37	C₃×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic, complemented
38	#4702	—	9	1	no	maximal
39	C₂×PSU(3,𝔽₂)	—	6	1	no	—
40	C₄⋊#157	—	3	1	no	—
41	C₄⋊#157	—	3	1	no	—
42	C₄⋊#157	—	3	1	no	—
43	C₄⋊#157	—	3	1	no	—
44	C₄×#157	C₂×D₃	1	1	yes	characteristic
45	#3787	C₂×D₃	1	1	yes	characteristic
46	#3777	—	6	1	no	—
47	#3777	—	6	1	no	—
48	#3777	—	6	1	no	—
49	C₈×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
50	C₃×D₃×C₈	—	12	1	no	—
51	C₂×C₃×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
52	C₃⋊#157	—	4	1	no	—
53	C₃⋊#157	—	4	1	no	—
54	C₃⋊#157	—	4	1	no	—
55	C₃⋊#157	—	4	1	no	—
56	C₃×#157	—	2	1	no	—
57	C₃×#157	—	2	1	no	—
58	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
59	C₃×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
60	#870	—	9	1	no	—
61	#869	—	9	1	no	—
62	C₃×#97	—	9	1	no	—
63	#818	—	9	1	no	—
64	#818	—	9	1	no	—
65	#817	—	9	1	no	—
66	#817	—	9	1	no	—
67	C₂×#157	—	6	1	no	—
68	C₂×#157	—	6	1	no	—
69	C₂×#157	—	6	1	no	—
70	C₂×#157	—	6	1	no	—
71	C₂×#157	D₁₂	1	1	yes	characteristic
72	C₂×#157	D₁₂	1	1	yes	characteristic
73	PSU(3,𝔽₂)	—	12	2	no	—
74	C₄×C₃⋊D₃	C₂²×D₃	1	1	yes	characteristic
75	C₃×C₄×D₃	—	12	1	no	—
76	#622	Dic₆	1	2	yes	complemented
77	C₃²×C₈	—	4	1	no	—
78	C₃²×C₈	—	8	1	no	—
79	C₃²×C₈	Dic₆	1	1	yes	characteristic, complemented
80	C₃⋊(C₃⋊C₈)	Dic₆	1	1	yes	characteristic, complemented
81	C₃×C₃⋊C₈	—	12	1	no	—
82	#501	—	27	1	no	Sylow
83	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
84	C₃×C₃⋊D₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₃×C₃⋊D₃	#137	1	1	yes	characteristic
86	C₂×Dic₆	—	18	1	no	—
87	C₂×C₃×C₈	—	9	1	no	—
88	C₂×C₃×C₈	—	9	1	no	—
89	C₃×C₄²	—	9	1	no	—
90	C₄⋊Dic₃	—	9	1	no	—
91	C₄⋊Dic₃	—	9	1	no	—
92	C₄⋊Dic₃	—	9	1	no	—
93	C₄⋊Dic₃	—	9	1	no	—
94	C₄⋊Dic₃	—	18	1	no	—
95	Dic₃⋊C₄	—	18	1	no	—
96	Dic₃⋊C₄	—	18	1	no	—
97	D₃×C₈	—	12	1	no	—
98	C₂×C₃⋊D₃	C₂×D₁₂	1	1	yes	characteristic
99	C₂×C₃×D₃	—	12	1	no	—
100	#157	—	2	1	no	—
101	#157	—	2	1	no	—
102	#157	—	12	1	no	—
103	#157	—	12	1	no	—
104	#157	—	12	1	no	—
105	#157	—	12	1	no	—
106	C₃²×C₄	—	4	1	no	—
107	C₃²×C₄	—	8	1	no	—
108	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
109	C₃⋊Dic₃	C₂×Dic₆	1	1	yes	characteristic
110	C₃×Dic₃	—	12	1	no	—
111	C₄⋊Q₈	—	27	1	no	—
112	#125	—	27	1	no	—
113	#107	—	27	1	no	—
114	#107	—	27	1	no	—
115	#106	—	27	1	no	—
116	#106	—	27	1	no	—
117	#97	—	9	1	no	—
118	C₃³	#501	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₃×C₄	—	9	1	no	—
120	C₂×C₃×C₄	—	9	1	no	—
121	C₂×C₃×C₄	—	9	1	no	—
122	C₂×Dic₃	—	18	1	no	—
123	C₂×Dic₃	—	18	1	no	—
124	C₂×Dic₃	—	18	1	no	—
125	C₂×Dic₃	—	18	1	no	—
126	C₄×D₃	—	12	1	no	—
127	Dic₆	—	36	2	no	—
128	C₃×C₈	—	4	1	no	—
129	C₃×C₈	—	8	1	no	—
130	C₃×C₈	—	9	2	no	—
131	C₃×C₈	—	9	1	no	—
132	C₃×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
133	C₃⋊C₈	—	12	1	no	—
134	C₂×C₃²	—	4	1	no	—
135	C₂×C₃²	—	8	1	no	—
136	C₂×C₃²	#869	1	1	yes	characteristic
137	C₃⋊D₃	#908	1	1	yes	characteristic
138	C₃⋊D₃	#909	1	1	yes	characteristic
139	C₃×D₃	—	12	1	no	—
140	C₃×D₃	—	12	1	no	—
141	C₂×Q₈	—	54	1	no	—
142	C₂×C₈	—	9	1	no	—
143	C₂×C₈	—	9	1	no	—
144	C₄⋊C₄	—	27	1	no	—
145	C₄⋊C₄	—	27	1	no	—
146	C₄⋊C₄	—	27	1	no	—
147	C₄⋊C₄	—	27	1	no	—
148	C₄⋊C₄	—	54	1	no	—
149	C₄⋊C₄	—	54	1	no	—
150	C₄⋊C₄	—	54	1	no	—
151	C₄²	—	9	1	no	—
152	C₂²×C₃	—	9	1	no	—
153	C₂×D₃	—	12	1	no	—
154	C₃×C₄	—	4	1	no	—
155	C₃×C₄	—	8	1	no	—
156	C₃×C₄	—	9	1	no	—
157	C₃×C₄	—	18	1	no	—
158	C₃×C₄	—	18	1	no	—
159	C₃×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
160	Dic₃	—	12	1	no	—
161	Dic₃	—	36	1	no	—
162	Dic₃	—	36	1	no	—
163	Dic₃	—	36	1	no	—
164	Dic₃	—	36	1	no	—
165	C₃²	—	4	1	no	—
166	C₃²	—	8	1	no	—
167	C₃²	#4702	1	1	yes	characteristic, complemented
168	Q₈	—	108	2	no	—
169	C₂×C₄	—	9	1	no	—
170	C₂×C₄	—	9	1	no	—
171	C₂×C₄	—	9	1	no	—
172	C₂×C₄	—	54	1	no	—
173	C₂×C₄	—	54	1	no	—
174	C₂×C₄	—	54	1	no	—
175	C₂×C₄	—	54	1	no	—
176	C₈	—	9	2	no	—
177	C₈	—	9	1	no	—
178	C₈	C₃⋊PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
179	C₂×C₃	—	4	1	no	—
180	C₂×C₃	—	8	1	no	—
181	C₂×C₃	—	9	1	no	—
182	C₂×C₃	—	9	1	no	—
183	C₂×C₃	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
184	D₃	—	12	1	no	—
185	D₃	—	12	1	no	—
186	C₂²	—	9	1	no	—
187	C₄	—	9	1	no	—
188	C₄	—	18	1	no	—
189	C₄	—	18	1	no	—
190	C₄	—	108	1	no	—
191	C₄	—	108	1	no	—
192	C₄	—	108	1	no	—
193	C₄	—	108	1	no	—
194	C₄	C₂×C₃⋊PSU(3,𝔽₂)	1	1	yes	characteristic
195	C₃	—	4	1	no	—
196	C₃	—	8	1	no	—
197	C₃	#10593537	1	1	yes	characteristic, complemented
198	C₂	—	9	1	no	—
199	C₂	—	9	1	no	—
200	C₂	#11724365	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11724365
commutator, cosocle	C₃×C₄×C₃⋊D₃	C₂³
Frattini	C₄	C₂×C₃⋊PSU(3,𝔽₂)
Fitting	C₃³×C₈	Q₈
socle	C₂×C₃³	C₄⋊Q₈
prime radical	C₂×C₃×C₃⋊D₃	C₂×D₄
hypercenter	C₈	C₃⋊PSU(3,𝔽₂)
hypocenter	C₃³	#501
LCS	G ▷ C₃×C₄×C₃⋊D₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
35. C₃×#622	Q₈	semidirect
36. C₃³×C₈	Q₈	semidirect
37. C₃×C₃⋊(C₃⋊C₈)	Q₈	semidirect
76. #622	Dic₆	semidirect
79. C₃²×C₈	Dic₆	semidirect
80. C₃⋊(C₃⋊C₈)	Dic₆	semidirect
118. C₃³	#501	semidirect
132. C₃×C₈	PSU(3,𝔽₂)	semidirect
167. C₃²	#4702	semidirect
178. C₈	C₃⋊PSU(3,𝔽₂)	semidirect
197. C₃	#10593537	semidirect

Maximal subgroups

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

subgroup	index
#11724365	2
#11724361	2
#11723094	2
#11723093	2
C₃×#63767	2
#11722808	2
#11722806	2
#10593537	3
#4702	9