Subgroups — C₃²×#548 #10596349

Order 576.

870 conjugacy classes of subgroups.

The subgroup lattice has 159 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#124	C₂	1	3	yes	maximal, complemented
4	C₃²×C₂²⋊Q₈	C₂	1	3	yes	maximal, complemented
5	C₃²×#121	C₂	1	3	yes	maximal, complemented
6	C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₄×D₄	C₂	1	3	yes	maximal, complemented
8	C₃×#548	C₃	1	4	yes	maximal, direct product factor
9	C₃²×#41	—	2	4	no	—
10	C₂×C₃²×Q₈	—	2	4	no	—
11	C₂×C₃²×Q₈	C₂²	1	3	yes	—
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃²×D₄	C₂²	1	3	yes	—
14	C₂×C₃²×D₄	C₂²	1	3	yes	—
15	C₂²×C₃²×C₄	C₂²	1	3	yes	—
16	C₂²×C₃²×C₄	C₂²	1	3	yes	—
17	C₃²×C₄⋊C₄	C₂²	1	3	yes	—
18	C₃²×C₄⋊C₄	C₂²	1	3	yes	—
19	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₃²×C₂²⋊C₄	C₂²	1	3	yes	—
21	C₃²×C₂²⋊C₄	C₂²	1	6	yes	—
22	C₃²×C₄²	C₂²	1	3	yes	—
23	C₂×C₃×#41	C₂×C₃	1	4	yes	complemented
24	C₂²×C₃×Q₈	C₂×C₃	1	4	yes	complemented
25	C₃×#124	C₂×C₃	1	12	yes	complemented
26	C₃×C₂²⋊Q₈	C₂×C₃	1	12	yes	complemented
27	C₃×#121	C₂×C₃	1	12	yes	complemented
28	C₃×C₄×Q₈	C₂×C₃	1	4	yes	complemented
29	C₃×C₄×D₄	C₂×C₃	1	12	yes	complemented
30	C₂³×C₃²	C₂³	1	3	yes	—
31	C₂³×C₃²	C₂³	1	1	yes	characteristic
32	C₃²×Q₈	—	2	6	no	—
33	C₃²×Q₈	D₄	1	4	yes	complemented
34	C₃²×D₄	—	2	6	no	—
35	C₃²×D₄	—	2	6	no	—
36	C₂×C₃²×C₄	—	2	6	no	—
37	C₂×C₃²×C₄	—	2	6	no	—
38	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₂³	1	3	yes	—
40	C₂×C₃²×C₄	C₂³	1	3	yes	—
41	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₂³	1	3	yes	—
43	#548	C₃²	1	1	yes	characteristic, direct product factor, Sylow
44	C₃×#41	—	2	16	no	—
45	C₂×C₃×Q₈	—	2	16	no	—
46	C₂×C₃×Q₈	C₂²×C₃	1	12	yes	—
47	C₂×C₃×Q₈	C₂²×C₃	1	4	yes	—
48	C₂×C₃×D₄	C₂²×C₃	1	12	yes	—
49	C₂×C₃×D₄	C₂²×C₃	1	12	yes	—
50	C₂²×C₃×C₄	C₂²×C₃	1	12	yes	—
51	C₂²×C₃×C₄	C₂²×C₃	1	12	yes	—
52	C₃×C₄⋊C₄	C₂²×C₃	1	12	yes	—
53	C₃×C₄⋊C₄	C₂²×C₃	1	12	yes	—
54	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
55	C₃×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
56	C₃×C₂²⋊C₄	C₂²×C₃	1	24	yes	—
57	C₃×C₄²	C₂²×C₃	1	12	yes	—
58	C₂²×C₃²	—	2	1	no	—
59	C₂²×C₃²	—	2	1	no	—
60	C₂²×C₃²	—	2	3	no	—
61	C₂²×C₃²	—	2	3	no	—
62	C₂²×C₃²	—	2	3	no	—
63	C₂²×C₃²	#41	1	2	yes	complemented
64	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
65	C₃²×C₄	—	2	1	no	—
66	C₃²×C₄	—	2	3	no	—
67	C₃²×C₄	—	2	3	no	—
68	C₃²×C₄	—	2	1	no	—
69	C₃²×C₄	C₂×D₄	1	6	yes	—
70	C₂×#41	C₂×C₃²	1	1	yes	characteristic, complemented
71	C₂²×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
72	#124	C₂×C₃²	1	3	yes	complemented
73	C₂²⋊Q₈	C₂×C₃²	1	3	yes	complemented
74	#121	C₂×C₃²	1	3	yes	complemented
75	C₄×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
76	C₄×D₄	C₂×C₃²	1	3	yes	complemented
77	C₂³×C₃	C₂³×C₃	1	12	yes	—
78	C₂³×C₃	C₂³×C₃	1	4	yes	—
79	C₃×Q₈	—	2	24	no	—
80	C₃×Q₈	C₃×D₄	1	16	yes	complemented
81	C₃×D₄	—	2	24	no	—
82	C₃×D₄	—	2	24	no	—
83	C₂×C₃×C₄	—	2	24	no	—
84	C₂×C₃×C₄	—	2	24	no	—
85	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
86	C₂×C₃×C₄	C₂³×C₃	1	12	yes	—
87	C₂×C₃×C₄	C₂³×C₃	1	12	yes	—
88	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
89	C₂×C₃×C₄	C₂³×C₃	1	12	yes	—
90	C₂×C₃²	—	2	2	no	—
91	C₂×C₃²	—	4	3	no	—
92	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₃²	C₂×#41	1	1	yes	characteristic
94	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
95	#41	—	2	4	no	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×Q₈	C₂²×C₃²	1	3	yes	—
98	C₂×Q₈	C₂²×C₃²	1	1	yes	characteristic
99	C₂×D₄	C₂²×C₃²	1	3	yes	—
100	C₂×D₄	C₂²×C₃²	1	3	yes	—
101	C₂²×C₄	C₂²×C₃²	1	3	yes	—
102	C₂²×C₄	C₂²×C₃²	1	3	yes	—
103	C₄⋊C₄	C₂²×C₃²	1	3	yes	—
104	C₄⋊C₄	C₂²×C₃²	1	3	yes	—
105	C₂²⋊C₄	C₂²×C₃²	1	1	yes	characteristic
106	C₂²⋊C₄	C₂²×C₃²	1	3	yes	—
107	C₂²⋊C₄	C₂²×C₃²	1	6	yes	—
108	C₄²	C₂²×C₃²	1	3	yes	—
109	C₂²×C₃	—	2	4	no	—
110	C₂²×C₃	—	2	4	no	—
111	C₂²×C₃	—	2	12	no	—
112	C₂²×C₃	—	2	12	no	—
113	C₂²×C₃	—	2	12	no	—
114	C₂²×C₃	C₃×#41	1	8	yes	complemented
115	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
116	C₃×C₄	—	2	4	no	—
117	C₃×C₄	—	2	12	no	—
118	C₃×C₄	—	2	12	no	—
119	C₃×C₄	—	2	4	no	—
120	C₃×C₄	C₂×C₃×D₄	1	24	yes	—
121	C₃²	#548	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³	C₂³×C₃²	1	3	yes	—
123	C₂³	C₂³×C₃²	1	1	yes	characteristic
124	Q₈	—	2	6	no	—
125	Q₈	C₃²×D₄	1	4	yes	complemented
126	D₄	—	2	6	no	—
127	D₄	—	2	6	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
131	C₂×C₄	C₂³×C₃²	1	3	yes	—
132	C₂×C₄	C₂³×C₃²	1	3	yes	—
133	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
134	C₂×C₄	C₂³×C₃²	1	3	yes	—
135	C₂×C₃	—	2	8	no	—
136	C₂×C₃	—	4	12	no	—
137	C₂×C₃	C₂²×C₃×D₄	1	4	yes	—
138	C₂×C₃	C₂×C₃×#41	1	4	yes	—
139	C₂×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	3	no	—
143	C₂²	—	2	3	no	—
144	C₂²	—	2	3	no	—
145	C₂²	C₃²×#41	1	2	yes	complemented
146	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	3	no	—
149	C₄	—	2	3	no	—
150	C₄	—	2	1	no	—
151	C₄	C₂×C₃²×D₄	1	6	yes	—
152	C₃	C₃×#548	1	4	yes	direct product factor
153	C₂	—	2	2	no	—
154	C₂	—	4	3	no	—
155	C₂	C₂²×C₃²×D₄	1	1	yes	characteristic
156	C₂	C₂×C₃²×#41	1	1	yes	characteristic
157	C₂	C₃²×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#41	C₂	semidirect
2. C₂²×C₃²×Q₈	C₂	semidirect
3. C₃²×#124	C₂	semidirect
4. C₃²×C₂²⋊Q₈	C₂	semidirect
5. C₃²×#121	C₂	semidirect
6. C₃²×C₄×Q₈	C₂	semidirect
7. C₃²×C₄×D₄	C₂	semidirect
8. C₃×#548	C₃	direct (N × H)
23. C₂×C₃×#41	C₂×C₃	semidirect
24. C₂²×C₃×Q₈	C₂×C₃	semidirect
25. C₃×#124	C₂×C₃	semidirect
26. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
27. C₃×#121	C₂×C₃	semidirect
28. C₃×C₄×Q₈	C₂×C₃	semidirect
29. C₃×C₄×D₄	C₂×C₃	semidirect
33. C₃²×Q₈	D₄	semidirect
43. #548	C₃²	direct (N × H)
63. C₂²×C₃²	#41	semidirect
70. C₂×#41	C₂×C₃²	semidirect
71. C₂²×Q₈	C₂×C₃²	semidirect
72. #124	C₂×C₃²	semidirect
73. C₂²⋊Q₈	C₂×C₃²	semidirect
74. #121	C₂×C₃²	semidirect
75. C₄×Q₈	C₂×C₃²	semidirect
76. C₄×D₄	C₂×C₃²	semidirect
80. C₃×Q₈	C₃×D₄	semidirect
114. C₂²×C₃	C₃×#41	semidirect
121. C₃²	#548	direct (N × H)
125. Q₈	C₃²×D₄	semidirect
145. C₂²	C₃²×#41	semidirect
152. C₃	C₃×#548	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#41	2
C₂²×C₃²×Q₈	2
C₃²×#124	2
C₃²×C₂²⋊Q₈	2
C₃²×#121	2
C₃²×C₄×Q₈	2
C₃²×C₄×D₄	2
C₃×#548	3

Subgroups — C32×#548 #10596349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#548 #10596349