Finite Group Databasebeta

Subgroups — #64297

Order 288.

312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 108 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	#3828	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3804	C₂	1	4	yes	maximal, complemented
5	#3799	C₂	1	2	yes	maximal, complemented
6	#3797	C₂	1	2	yes	maximal, complemented
7	#3796	C₂	1	2	yes	maximal, complemented
8	D₃×Dic₆	C₂	1	2	yes	maximal, complemented
9	#1010	—	3	1	no	maximal
10	#1007	—	3	1	no	maximal
11	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
12	#638	C₂²	1	2	yes	—
13	C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
14	C₄×C₃⋊D₃	C₂²	1	2	yes	—
15	#634	C₂²	1	1	yes	characteristic
16	C₃×#67	C₂²	1	2	yes	—
17	C₂×C₃×Dic₃	C₂²	1	2	yes	—
18	C₃×D₁₂	C₂²	1	1	yes	characteristic
19	C₃×C₄×D₃	C₂²	1	2	yes	—
20	C₃×Dic₆	C₂²	1	2	yes	—
21	C₃×Dic₆	C₂²	1	2	yes	—
22	C₃×Dic₆	C₂²	1	1	yes	characteristic
23	#627	C₂²	1	4	yes	—
24	#626	C₂²	1	4	yes	—
25	#624	C₂²	1	4	yes	—
26	D₃×Dic₃	C₂²	1	4	yes	—
27	C₃×#41	—	3	1	no	—
28	C₂×C₃×Q₈	—	3	1	no	—
29	#239	—	3	2	no	—
30	#239	—	3	2	no	—
31	D₃×Q₈	—	3	2	no	—
32	D₃×Q₈	—	3	2	no	—
33	D₃×Q₈	—	3	2	no	—
34	#237	—	3	4	no	—
35	#237	—	3	2	no	—
36	#235	—	3	1	no	—
37	#235	—	3	1	no	—
38	#235	—	3	4	no	—
39	#235	—	3	2	no	—
40	#235	—	6	1	no	—
41	#235	D₃	1	1	yes	characteristic, complemented
42	C₂×Dic₆	—	3	2	no	—
43	C₂×Dic₆	D₃	1	1	yes	characteristic, complemented
44	C₂²×C₃²	C₂³	1	1	yes	characteristic
45	C₂×C₃⋊D₃	C₂³	1	2	yes	—
46	C₂×C₃×D₃	C₂³	1	2	yes	—
47	C₃²×C₄	C₂³	1	1	yes	characteristic
48	C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₃⋊Dic₃	C₂³	1	2	yes	—
50	C₃×Dic₃	C₂³	1	4	yes	—
51	C₃×Dic₃	C₂³	1	2	yes	—
52	2₋¹⁺⁴	—	9	1	no	Sylow
53	C₃×Q₈	—	3	1	no	—
54	C₃×Q₈	—	3	2	no	—
55	C₃×Q₈	—	3	2	no	—
56	C₃×D₄	—	3	2	no	—
57	C₃×D₄	—	3	1	no	—
58	C₂×C₃×C₄	—	2	1	no	—
59	C₂×C₃×C₄	—	3	2	no	—
60	C₂×C₃×C₄	—	3	2	no	—
61	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
63	#67	—	3	2	no	—
64	#67	—	3	2	no	—
65	#67	—	3	4	no	—
66	#67	—	6	2	no	—
67	#67	C₂×D₃	1	2	yes	—
68	C₂×Dic₃	—	3	4	no	—
69	C₂×Dic₃	C₂×D₃	1	2	yes	—
70	D₁₂	—	3	4	no	—
71	D₁₂	—	3	1	no	—
72	D₁₂	—	3	1	no	—
73	D₁₂	—	6	1	no	—
74	D₁₂	C₂×D₃	1	1	yes	characteristic
75	C₄×D₃	—	3	4	no	—
76	C₄×D₃	—	3	2	no	—
77	C₄×D₃	—	3	2	no	—
78	C₄×D₃	—	3	4	no	—
79	C₄×D₃	—	3	4	no	—
80	C₄×D₃	—	6	2	no	—
81	C₄×D₃	C₂×D₃	1	2	yes	—
82	Dic₆	—	3	1	no	—
83	Dic₆	—	3	1	no	—
84	Dic₆	—	3	4	no	—
85	Dic₆	—	3	4	no	—
86	Dic₆	—	6	1	no	—
87	Dic₆	C₂×D₃	1	2	yes	—
88	Dic₆	C₂×D₃	1	2	yes	—
89	Dic₆	C₂×D₃	1	1	yes	characteristic
90	C₂×C₃²	—	2	1	no	—
91	C₂×C₃²	C₂⁴	1	1	yes	characteristic
92	C₃⋊D₃	—	2	2	no	—
93	C₃×D₃	—	2	2	no	—
94	#41	—	3	1	no	—
95	#41	—	9	4	no	—
96	#41	—	9	2	no	—
97	#41	—	9	2	no	—
98	#41	—	9	1	no	—
99	C₂×Q₈	—	3	1	no	—
100	C₂×Q₈	—	9	2	no	—
101	C₂×Q₈	—	9	2	no	—
102	C₂²×C₃	—	2	1	no	—
103	C₂²×C₃	—	3	2	no	—
104	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
106	C₂×D₃	—	3	2	no	—
107	C₂×D₃	—	3	2	no	—
108	C₂×D₃	—	6	2	no	—
109	C₂×D₃	C₂²×D₃	1	2	yes	—
110	C₃×C₄	—	2	1	no	—
111	C₃×C₄	—	2	1	no	—
112	C₃×C₄	—	3	2	no	—
113	C₃×C₄	—	3	4	no	—
114	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
115	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
118	Dic₃	—	3	2	no	—
119	Dic₃	—	3	2	no	—
120	Dic₃	—	6	2	no	—
121	Dic₃	C₂²×D₃	1	2	yes	—
122	Dic₃	C₂²×D₃	1	4	yes	—
123	C₃²	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
124	Q₈	—	3	2	no	—
125	Q₈	—	3	2	no	—
126	Q₈	—	3	1	no	—
127	Q₈	—	9	4	no	—
128	Q₈	—	9	1	no	—
129	D₄	—	3	2	no	—
130	D₄	—	3	1	no	—
131	D₄	—	9	4	no	—
132	D₄	—	9	2	no	—
133	D₄	—	9	1	no	—
134	C₂×C₄	—	3	2	no	—
135	C₂×C₄	—	3	2	no	—
136	C₂×C₄	—	9	4	no	—
137	C₂×C₄	—	9	4	no	—
138	C₂×C₄	—	9	2	no	—
139	C₂×C₄	D₃²	1	1	yes	characteristic
140	C₂×C₃	—	2	1	no	—
141	C₂×C₃	—	2	1	no	—
142	C₂×C₃	—	2	1	no	—
143	C₂×C₃	—	4	1	no	—
144	C₂×C₃	—	6	2	no	—
145	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
146	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
147	D₃	—	2	2	no	—
148	D₃	—	6	2	no	—
149	D₃	—	6	2	no	—
150	D₃	—	12	2	no	—
151	C₂²	—	3	2	no	—
152	C₂²	—	9	2	no	—
153	C₂²	C₂×D₃²	1	1	yes	characteristic
154	C₄	—	3	2	no	—
155	C₄	—	3	4	no	—
156	C₄	—	9	2	no	—
157	C₄	C₂×D₃²	1	1	yes	characteristic
158	C₄	C₂×D₃²	1	1	yes	characteristic
159	C₃	—	2	1	no	—
160	C₃	#1007	1	1	yes	characteristic, complemented
161	C₃	#1010	1	1	yes	characteristic, complemented
162	C₂	—	2	1	no	—
163	C₂	—	6	2	no	—
164	C₂	—	18	2	no	—
165	C₂	C₂²×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃²
commutator, socle, cosocle	C₂×C₃²	C₂⁴
Fitting	C₂×C₃²×C₄	C₂²
hypercenter	C₂×C₄	D₃²
hypocenter	C₃²	2₋¹⁺⁴
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. #3828	C₂	semidirect
2. C₃×#235	C₂	semidirect
3. C₂×C₃×Dic₆	C₂	semidirect
4. #3804	C₂	semidirect
5. #3799	C₂	semidirect
6. #3797	C₂	semidirect
7. #3796	C₂	semidirect
8. D₃×Dic₆	C₂	semidirect
41. #235	D₃	semidirect
43. C₂×Dic₆	D₃	semidirect
123. C₃²	2₋¹⁺⁴	semidirect
160. C₃	#1007	semidirect
161. C₃	#1010	semidirect

Maximal subgroups

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

subgroup	index
#3828	2
C₃×#235	2
C₂×C₃×Dic₆	2
#3804	2
#3799	2
#3797	2
#3796	2
D₃×Dic₆	2
#1010	3
#1007	3