Subgroups — C₃²×#547 #10596348

Order 576.

876 conjugacy classes of subgroups.

The subgroup lattice has 180 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	2	yes	maximal, complemented
2	C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#123	C₂	1	4	yes	maximal, complemented
4	C₃²×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₃²×#121	C₂	1	2	yes	maximal, complemented
6	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃²×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₃×#547	C₃	1	4	yes	maximal, direct product factor
10	C₃²×#41	—	2	8	no	—
11	C₂×C₃²×Q₈	C₂²	1	2	yes	—
12	C₂×C₃²×D₄	C₂²	1	4	yes	—
13	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	4	yes	—
16	C₂²×C₃²×C₄	C₂²	1	2	yes	—
17	C₂²×C₃²×C₄	C₂²	1	2	yes	—
18	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃²×C₄⋊C₄	C₂²	1	4	yes	—
21	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
23	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃²×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₃²×C₄²	C₂²	1	1	yes	characteristic
27	C₂×C₃×#41	C₂×C₃	1	8	yes	complemented
28	C₃×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
29	C₃×#123	C₂×C₃	1	16	yes	complemented
30	C₃×C₂²⋊Q₈	C₂×C₃	1	8	yes	complemented
31	C₃×#121	C₂×C₃	1	8	yes	complemented
32	C₃×C₄×D₄	C₂×C₃	1	4	yes	complemented
33	C₃×C₄×D₄	C₂×C₃	1	4	yes	complemented
34	C₂×C₃×C₄⋊C₄	C₂×C₃	1	8	yes	complemented
35	C₂³×C₃²	C₂³	1	2	yes	—
36	C₂³×C₃²	C₂³	1	2	yes	—
37	C₃²×Q₈	—	2	4	no	—
38	C₃²×D₄	—	2	8	no	—
39	C₃²×D₄	—	2	2	no	—
40	C₃²×D₄	D₄	1	4	yes	complemented
41	C₂×C₃²×C₄	—	2	8	no	—
42	C₂×C₃²×C₄	—	2	4	no	—
43	C₂×C₃²×C₄	—	2	4	no	—
44	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂³	1	2	yes	—
46	C₂×C₃²×C₄	C₂³	1	2	yes	—
47	C₂×C₃²×C₄	C₂³	1	4	yes	—
48	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
50	#547	C₃²	1	1	yes	characteristic, direct product factor, Sylow
51	C₃×#41	—	2	32	no	—
52	C₂×C₃×Q₈	C₂²×C₃	1	8	yes	—
53	C₂×C₃×D₄	C₂²×C₃	1	4	yes	—
54	C₂×C₃×D₄	C₂²×C₃	1	16	yes	—
55	C₂×C₃×D₄	C₂²×C₃	1	4	yes	—
56	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	—
57	C₂²×C₃×C₄	C₂²×C₃	1	16	yes	—
58	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	—
59	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
60	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
61	C₃×C₄⋊C₄	C₂²×C₃	1	16	yes	—
62	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
63	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
64	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
65	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
66	C₃×C₂²⋊C₄	C₂²×C₃	1	16	yes	—
67	C₃×C₄²	C₂²×C₃	1	4	yes	—
68	C₂²×C₃²	—	2	2	no	—
69	C₂²×C₃²	—	2	2	no	—
70	C₂²×C₃²	—	2	2	no	—
71	C₂²×C₃²	—	2	2	no	—
72	C₂²×C₃²	—	2	2	no	—
73	C₂²×C₃²	C₂×D₄	1	4	yes	—
74	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
75	C₃²×C₄	—	2	2	no	—
76	C₃²×C₄	—	2	4	no	—
77	C₃²×C₄	—	2	2	no	—
78	C₃²×C₄	—	2	1	no	—
79	C₃²×C₄	C₂×D₄	1	2	yes	—
80	C₃²×C₄	#41	1	2	yes	complemented
81	C₂×#41	C₂×C₃²	1	2	yes	complemented
82	C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
83	#123	C₂×C₃²	1	4	yes	complemented
84	C₂²⋊Q₈	C₂×C₃²	1	2	yes	complemented
85	#121	C₂×C₃²	1	2	yes	complemented
86	C₄×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
87	C₄×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
88	C₂×C₄⋊C₄	C₂×C₃²	1	2	yes	complemented
89	C₂³×C₃	C₂³×C₃	1	8	yes	—
90	C₂³×C₃	C₂³×C₃	1	8	yes	—
91	C₃×Q₈	—	2	16	no	—
92	C₃×D₄	—	2	32	no	—
93	C₃×D₄	—	2	8	no	—
94	C₃×D₄	C₃×D₄	1	16	yes	complemented
95	C₂×C₃×C₄	—	2	32	no	—
96	C₂×C₃×C₄	—	2	16	no	—
97	C₂×C₃×C₄	—	2	16	no	—
98	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
99	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
100	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
101	C₂×C₃×C₄	C₂³×C₃	1	16	yes	—
102	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
103	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
104	C₂×C₃²	—	2	4	no	—
105	C₂×C₃²	—	4	2	no	—
106	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₃²	C₂×#41	1	1	yes	characteristic
108	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
109	#41	—	2	8	no	—
110	C₂×Q₈	C₂²×C₃²	1	2	yes	—
111	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic
112	C₂×D₄	C₂²×C₃²	1	4	yes	—
113	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic
114	C₂²×C₄	C₂²×C₃²	1	2	yes	—
115	C₂²×C₄	C₂²×C₃²	1	4	yes	—
116	C₂²×C₄	C₂²×C₃²	1	2	yes	—
117	C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
118	C₄⋊C₄	C₂²×C₃²	1	2	yes	—
119	C₄⋊C₄	C₂²×C₃²	1	4	yes	—
120	C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
121	C₄⋊C₄	C₂²×C₃²	1	2	yes	—
122	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
123	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
124	C₂²⋊C₄	C₂²×C₃²	1	4	yes	—
125	C₄²	C₂²×C₃²	1	1	yes	characteristic
126	C₂²×C₃	—	2	8	no	—
127	C₂²×C₃	—	2	8	no	—
128	C₂²×C₃	—	2	8	no	—
129	C₂²×C₃	—	2	8	no	—
130	C₂²×C₃	—	2	8	no	—
131	C₂²×C₃	C₂×C₃×D₄	1	16	yes	—
132	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
133	C₃×C₄	—	2	8	no	—
134	C₃×C₄	—	2	16	no	—
135	C₃×C₄	—	2	8	no	—
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	C₂×C₃×D₄	1	8	yes	—
138	C₃×C₄	C₃×#41	1	8	yes	complemented
139	C₃²	#547	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³	C₂³×C₃²	1	2	yes	—
141	C₂³	C₂³×C₃²	1	2	yes	—
142	Q₈	—	2	4	no	—
143	D₄	—	2	8	no	—
144	D₄	—	2	2	no	—
145	D₄	C₃²×D₄	1	4	yes	complemented
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
150	C₂×C₄	C₂³×C₃²	1	2	yes	—
151	C₂×C₄	C₂³×C₃²	1	2	yes	—
152	C₂×C₄	C₂³×C₃²	1	4	yes	—
153	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
154	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
155	C₂×C₃	—	2	16	no	—
156	C₂×C₃	—	4	8	no	—
157	C₂×C₃	C₂²×C₃×D₄	1	4	yes	—
158	C₂×C₃	C₂×C₃×#41	1	4	yes	—
159	C₂×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂×C₃²×D₄	1	4	yes	—
166	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	C₂×C₃²×D₄	1	2	yes	—
172	C₄	C₃²×#41	1	2	yes	complemented
173	C₃	C₃×#547	1	4	yes	direct product factor
174	C₂	—	2	4	no	—
175	C₂	—	4	2	no	—
176	C₂	C₂²×C₃²×D₄	1	1	yes	characteristic
177	C₂	C₂×C₃²×#41	1	1	yes	characteristic
178	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. C₃²×C₄⋊Q₈	C₂	semidirect
3. C₃²×#123	C₂	semidirect
4. C₃²×C₂²⋊Q₈	C₂	semidirect
5. C₃²×#121	C₂	semidirect
6. C₃²×C₄×D₄	C₂	semidirect
7. C₃²×C₄×D₄	C₂	semidirect
8. C₂×C₃²×C₄⋊C₄	C₂	semidirect
9. C₃×#547	C₃	direct (N × H)
27. C₂×C₃×#41	C₂×C₃	semidirect
28. C₃×C₄⋊Q₈	C₂×C₃	semidirect
29. C₃×#123	C₂×C₃	semidirect
30. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
31. C₃×#121	C₂×C₃	semidirect
32. C₃×C₄×D₄	C₂×C₃	semidirect
33. C₃×C₄×D₄	C₂×C₃	semidirect
34. C₂×C₃×C₄⋊C₄	C₂×C₃	semidirect
40. C₃²×D₄	D₄	semidirect
50. #547	C₃²	direct (N × H)
80. C₃²×C₄	#41	semidirect
81. C₂×#41	C₂×C₃²	semidirect
82. C₄⋊Q₈	C₂×C₃²	semidirect
83. #123	C₂×C₃²	semidirect
84. C₂²⋊Q₈	C₂×C₃²	semidirect
85. #121	C₂×C₃²	semidirect
86. C₄×D₄	C₂×C₃²	semidirect
87. C₄×D₄	C₂×C₃²	semidirect
88. C₂×C₄⋊C₄	C₂×C₃²	semidirect
94. C₃×D₄	C₃×D₄	semidirect
138. C₃×C₄	C₃×#41	semidirect
139. C₃²	#547	direct (N × H)
145. D₄	C₃²×D₄	semidirect
172. C₄	C₃²×#41	semidirect
173. C₃	C₃×#547	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
C₃²×C₄⋊Q₈	2
C₃²×#123	2
C₃²×C₂²⋊Q₈	2
C₃²×#121	2
C₃²×C₄×D₄	2
C₂×C₃²×C₄⋊C₄	2
C₃×#547	3

Subgroups — C32×#547 #10596348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#547 #10596348