Finite Group Databasebeta

Subgroups — #64307

Order 288.

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

normal subgroups only

This group has 352 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×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×#67	C₂	1	4	yes	maximal, complemented
5	#3802	C₂	1	2	yes	maximal, complemented
6	D₃×D₁₂	C₂	1	2	yes	maximal, complemented
7	#3796	C₂	1	2	yes	maximal, complemented
8	#3795	C₂	1	2	yes	maximal, complemented
9	#1009	—	3	1	no	maximal
10	#1004	—	3	1	no	maximal
11	C₂²×C₃×D₃	C₂²	1	2	yes	complemented
12	C₂×D₃²	C₂²	1	4	yes	complemented
13	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic, complemented
14	#638	C₂²	1	2	yes	complemented
15	C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₃⋊D₃	C₂²	1	2	yes	complemented
17	#634	C₂²	1	1	yes	characteristic, complemented
18	C₃×#67	C₂²	1	2	yes	complemented
19	C₃×D₁₂	C₂²	1	2	yes	complemented
20	C₃×D₁₂	C₂²	1	2	yes	complemented
21	C₃×D₁₂	C₂²	1	1	yes	characteristic, complemented
22	C₃×C₄×D₃	C₂²	1	2	yes	complemented
23	C₃×Dic₆	C₂²	1	1	yes	characteristic, complemented
24	#626	C₂²	1	4	yes	complemented
25	#625	C₂²	1	4	yes	complemented
26	D₃×Dic₃	C₂²	1	4	yes	complemented
27	C₃×#41	—	3	1	no	—
28	C₂×C₃×D₄	—	3	1	no	—
29	C₂×#67	—	3	4	no	—
30	#239	—	3	2	no	—
31	#237	—	3	2	no	—
32	#237	—	3	2	no	—
33	D₃×D₄	—	3	4	no	—
34	D₃×D₄	—	3	2	no	—
35	D₃×D₄	—	3	2	no	—
36	D₃×D₄	—	3	2	no	—
37	#235	—	3	1	no	—
38	#235	—	3	1	no	—
39	#235	—	3	2	no	—
40	#235	—	6	1	no	—
41	#235	D₃	1	1	yes	characteristic, complemented
42	C₂×D₁₂	—	3	2	no	—
43	C₂×D₁₂	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	—	4	2	no	—
47	C₂×C₃×D₃	C₂³	1	4	yes	—
48	C₂×C₃×D₃	C₂³	1	2	yes	—
49	D₃²	—	4	4	no	—
50	C₃²×C₄	C₂³	1	1	yes	characteristic
51	C₃²×C₄	C₂³	1	1	yes	characteristic
52	C₃⋊Dic₃	C₂³	1	2	yes	—
53	C₃×Dic₃	C₂³	1	2	yes	—
54	H(5,𝔽₂)	—	9	1	no	Sylow
55	C₂³×C₃	—	3	2	no	—
56	C₂²×D₃	—	3	4	no	—
57	C₂²×D₃	—	3	4	no	—
58	C₂²×D₃	C₂×D₃	1	2	yes	complemented
59	C₃×Q₈	—	3	1	no	—
60	C₃×D₄	—	3	2	no	—
61	C₃×D₄	—	3	1	no	—
62	C₃×D₄	—	3	2	no	—
63	C₃×D₄	—	3	2	no	—
64	C₂×C₃×C₄	—	2	1	no	—
65	C₂×C₃×C₄	—	3	2	no	—
66	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic, complemented
67	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic, complemented
68	#67	—	3	2	no	—
69	#67	—	3	2	no	—
70	#67	—	3	4	no	—
71	#67	—	3	4	no	—
72	#67	—	3	4	no	—
73	#67	—	6	2	no	—
74	#67	C₂×D₃	1	2	yes	complemented
75	C₂×Dic₃	—	3	4	no	—
76	D₁₂	—	3	4	no	—
77	D₁₂	—	3	1	no	—
78	D₁₂	—	3	1	no	—
79	D₁₂	—	6	1	no	—
80	D₁₂	C₂×D₃	1	2	yes	complemented
81	D₁₂	C₂×D₃	1	2	yes	complemented
82	D₁₂	C₂×D₃	1	1	yes	characteristic, complemented
83	C₄×D₃	—	3	4	no	—
84	C₄×D₃	—	3	2	no	—
85	C₄×D₃	—	3	2	no	—
86	C₄×D₃	—	6	2	no	—
87	C₄×D₃	C₂×D₃	1	2	yes	complemented
88	Dic₆	—	3	1	no	—
89	Dic₆	—	3	1	no	—
90	Dic₆	—	6	1	no	—
91	Dic₆	C₂×D₃	1	1	yes	characteristic, complemented
92	C₂×C₃²	—	2	1	no	—
93	C₂×C₃²	C₂⁴	1	1	yes	characteristic
94	C₃⋊D₃	—	2	2	no	—
95	C₃×D₃	—	2	2	no	—
96	C₃×D₃	—	2	4	no	—
97	#41	—	3	1	no	—
98	#41	—	9	2	no	—
99	#41	—	9	2	no	—
100	#41	—	9	1	no	—
101	C₂×D₄	—	3	1	no	—
102	C₂×D₄	—	9	2	no	—
103	C₂×D₄	—	9	2	no	—
104	C₂×D₄	—	9	4	no	—
105	C₂²×C₃	—	2	1	no	—
106	C₂²×C₃	—	3	4	no	—
107	C₂²×C₃	—	3	2	no	—
108	C₂²×C₃	—	12	2	no	—
109	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
111	C₂×D₃	—	3	2	no	—
112	C₂×D₃	—	3	2	no	—
113	C₂×D₃	—	4	2	no	—
114	C₂×D₃	—	6	2	no	—
115	C₂×D₃	—	12	4	no	—
116	C₂×D₃	—	12	4	no	—
117	C₂×D₃	C₂²×D₃	1	4	yes	—
118	C₂×D₃	C₂²×D₃	1	2	yes	—
119	C₃×C₄	—	2	1	no	—
120	C₃×C₄	—	2	1	no	—
121	C₃×C₄	—	3	2	no	—
122	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
123	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
124	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
125	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
126	Dic₃	—	3	2	no	—
127	Dic₃	—	3	2	no	—
128	Dic₃	—	6	2	no	—
129	Dic₃	C₂²×D₃	1	2	yes	—
130	C₃²	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
131	C₂³	—	3	2	no	—
132	C₂³	—	9	4	no	—
133	Q₈	—	3	1	no	—
134	Q₈	—	9	1	no	—
135	D₄	—	3	2	no	—
136	D₄	—	3	2	no	—
137	D₄	—	3	2	no	—
138	D₄	—	3	1	no	—
139	D₄	—	9	4	no	—
140	D₄	—	9	4	no	—
141	D₄	—	9	2	no	—
142	D₄	—	9	1	no	—
143	C₂×C₄	—	3	2	no	—
144	C₂×C₄	—	9	4	no	—
145	C₂×C₄	—	9	2	no	—
146	C₂×C₄	D₃²	1	1	yes	characteristic, complemented
147	C₂×C₃	—	2	1	no	—
148	C₂×C₃	—	2	1	no	—
149	C₂×C₃	—	2	1	no	—
150	C₂×C₃	—	4	1	no	—
151	C₂×C₃	—	6	2	no	—
152	C₂×C₃	—	6	4	no	—
153	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
154	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
155	D₃	—	2	4	no	—
156	D₃	—	2	2	no	—
157	D₃	—	6	2	no	—
158	D₃	—	6	2	no	—
159	D₃	—	12	2	no	—
160	C₂²	—	3	2	no	—
161	C₂²	—	3	4	no	—
162	C₂²	—	9	2	no	—
163	C₂²	—	12	2	no	—
164	C₂²	—	36	4	no	—
165	C₂²	C₂×D₃²	1	1	yes	characteristic
166	C₄	—	3	2	no	—
167	C₄	—	9	2	no	—
168	C₄	C₂×D₃²	1	1	yes	characteristic
169	C₄	C₂×D₃²	1	1	yes	characteristic
170	C₃	—	2	1	no	—
171	C₃	#1004	1	1	yes	characteristic, complemented
172	C₃	#1009	1	1	yes	characteristic, complemented
173	C₂	—	2	1	no	—
174	C₂	—	6	4	no	—
175	C₂	—	6	2	no	—
176	C₂	—	18	2	no	—
177	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	H(5,𝔽₂)
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×D₁₂	C₂	semidirect
4. D₃×#67	C₂	semidirect
5. #3802	C₂	semidirect
6. D₃×D₁₂	C₂	semidirect
7. #3796	C₂	semidirect
8. #3795	C₂	semidirect
11. C₂²×C₃×D₃	C₂²	semidirect
12. C₂×D₃²	C₂²	semidirect
13. C₂×C₃²×C₄	C₂²	semidirect
14. #638	C₂²	semidirect
15. C₄⋊(C₃⋊D₃)	C₂²	semidirect
16. C₄×C₃⋊D₃	C₂²	semidirect
17. #634	C₂²	semidirect
18. C₃×#67	C₂²	semidirect
19. C₃×D₁₂	C₂²	semidirect
20. C₃×D₁₂	C₂²	semidirect
21. C₃×D₁₂	C₂²	semidirect
22. C₃×C₄×D₃	C₂²	semidirect
23. C₃×Dic₆	C₂²	semidirect
24. #626	C₂²	semidirect
25. #625	C₂²	semidirect
26. D₃×Dic₃	C₂²	semidirect
41. #235	D₃	semidirect
43. C₂×D₁₂	D₃	semidirect
58. C₂²×D₃	C₂×D₃	semidirect
66. C₂×C₃×C₄	C₂×D₃	semidirect
67. C₂×C₃×C₄	C₂×D₃	semidirect
74. #67	C₂×D₃	semidirect
80. D₁₂	C₂×D₃	semidirect
81. D₁₂	C₂×D₃	semidirect
82. D₁₂	C₂×D₃	semidirect
87. C₄×D₃	C₂×D₃	semidirect
91. Dic₆	C₂×D₃	semidirect
130. C₃²	H(5,𝔽₂)	semidirect
146. C₂×C₄	D₃²	semidirect
171. C₃	#1004	semidirect
172. C₃	#1009	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×D₁₂	2
D₃×#67	2
#3802	2
D₃×D₁₂	2
#3796	2
#3795	2
#1009	3
#1004	3