Subgroups — C₃×#34552 #10653159

Order 768.

1962 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2402	C₂	1	6	yes	maximal, complemented
2	C₃×#2399	C₂	1	6	yes	maximal, complemented
3	C₃×#2396	C₂	1	12	yes	maximal
4	C₃×C₄²×Q₈	C₂	1	4	yes	maximal, complemented
5	C₃×C₄×#117	C₂	1	3	yes	maximal
6	#34552	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×C₄×Q₈	C₂²	1	12	yes	—
8	C₂×C₃×#117	C₂²	1	6	yes	—
9	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₃×#400	C₂²	1	12	yes	—
11	C₃×#399	C₂²	1	4	yes	—
12	C₃×#398	C₂²	1	24	yes	—
13	C₃×#396	C₂²	1	8	yes	—
14	C₃×#395	C₂²	1	12	yes	—
15	C₃×#393	C₂²	1	4	yes	—
16	C₃×C₄×C₄⋊C₄	C₂²	1	12	yes	—
17	C₃×C₄×C₄⋊C₄	C₂²	1	24	yes	—
18	C₃×C₄×C₄⋊C₄	C₂²	1	12	yes	—
19	C₃×C₄×C₂²⋊C₄	C₂²	1	12	yes	—
20	C₃×#376	C₂²	1	3	yes	—
21	C₃×#376	C₂²	1	3	yes	—
22	C₃×C₄³	C₂²	1	6	yes	—
23	#2402	C₂×C₃	1	6	yes	complemented
24	#2399	C₂×C₃	1	6	yes	complemented
25	#2396	C₂×C₃	1	12	yes	—
26	C₄²×Q₈	C₂×C₃	1	4	yes	complemented
27	C₄×#117	C₂×C₃	1	3	yes	—
28	C₂²×C₃×Q₈	C₂³	1	4	yes	—
29	C₂³×C₃×C₄	C₂³	1	3	yes	—
30	C₃×C₄×Q₈	—	2	48	no	—
31	C₃×#117	—	2	24	no	—
32	C₂×C₃×C₄⋊C₄	C₂³	1	24	yes	—
33	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
34	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
35	C₂×C₃×C₂²⋊C₄	C₂³	1	12	yes	—
36	C₂×C₃×C₄²	Q₈	1	8	yes	complemented
37	C₂×C₃×C₄²	C₂³	1	6	yes	—
38	C₂×C₃×C₄²	C₂³	1	24	yes	—
39	C₂×C₃×C₄²	C₂³	1	3	yes	—
40	C₂×C₃×C₄²	C₂³	1	6	yes	—
41	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₃×#95	C₂³	1	12	yes	—
43	C₃×#95	C₂³	1	12	yes	—
44	C₃×#95	C₂³	1	12	yes	—
45	C₃×#95	C₂³	1	4	yes	—
46	C₃×#95	C₂³	1	8	yes	—
47	C₂×C₄×Q₈	C₂²×C₃	1	12	yes	—
48	C₂×#117	C₂²×C₃	1	6	yes	—
49	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
50	#400	C₂²×C₃	1	12	yes	—
51	#399	C₂²×C₃	1	4	yes	—
52	#398	C₂²×C₃	1	24	yes	—
53	#396	C₂²×C₃	1	8	yes	—
54	#395	C₂²×C₃	1	12	yes	—
55	#393	C₂²×C₃	1	4	yes	—
56	C₄×C₄⋊C₄	C₂²×C₃	1	12	yes	—
57	C₄×C₄⋊C₄	C₂²×C₃	1	24	yes	—
58	C₄×C₄⋊C₄	C₂²×C₃	1	12	yes	—
59	C₄×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
60	#376	C₂²×C₃	1	3	yes	—
61	#376	C₂²×C₃	1	3	yes	—
62	C₄³	C₂²×C₃	1	6	yes	—
63	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₃×Q₈	—	2	24	no	—
65	C₂²×C₃×C₄	—	2	12	no	—
66	C₂²×C₃×C₄	C₂×Q₈	1	12	yes	—
67	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
68	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
69	C₂²×C₃×C₄	C₂⁴	1	12	yes	—
70	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
71	C₃×C₄⋊C₄	—	2	24	no	—
72	C₃×C₄⋊C₄	—	2	48	no	—
73	C₃×C₄⋊C₄	—	2	24	no	—
74	C₃×C₂²⋊C₄	—	2	24	no	—
75	C₃×C₄²	—	2	12	no	—
76	C₃×C₄²	—	2	48	no	—
77	C₃×C₄²	C₂×Q₈	1	12	yes	—
78	C₃×C₄²	C₂×Q₈	1	4	yes	—
79	C₃×C₄²	#41	1	24	yes	—
80	C₂²×Q₈	C₂³×C₃	1	4	yes	—
81	C₂³×C₄	C₂³×C₃	1	3	yes	—
82	C₄×Q₈	—	2	48	no	—
83	#117	—	2	24	no	—
84	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
85	C₂×C₄⋊C₄	C₂³×C₃	1	24	yes	—
86	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
87	C₂×C₂²⋊C₄	C₂³×C₃	1	12	yes	—
88	C₂×C₄²	C₃×Q₈	1	8	yes	complemented
89	C₂×C₄²	C₂³×C₃	1	6	yes	—
90	C₂×C₄²	C₂³×C₃	1	24	yes	—
91	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₄²	C₂³×C₃	1	3	yes	—
93	C₂×C₄²	C₂³×C₃	1	6	yes	—
94	#95	C₂³×C₃	1	12	yes	—
95	#95	C₂³×C₃	1	12	yes	—
96	#95	C₂³×C₃	1	12	yes	—
97	#95	C₂³×C₃	1	4	yes	—
98	#95	C₂³×C₃	1	8	yes	—
99	C₂³×C₃	—	2	6	no	—
100	C₂³×C₃	C₂²×Q₈	1	2	yes	—
101	C₂³×C₃	C₂⁵	1	1	yes	characteristic
102	C₃×Q₈	—	2	32	no	—
103	C₂×C₃×C₄	—	2	24	no	—
104	C₂×C₃×C₄	—	2	6	no	—
105	C₂×C₃×C₄	—	2	24	no	—
106	C₂×C₃×C₄	—	2	12	no	—
107	C₂×C₃×C₄	—	2	24	no	—
108	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
109	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
110	C₂×C₃×C₄	C₂×#41	1	24	yes	—
111	C₂×C₃×C₄	C₂×#41	1	12	yes	—
112	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
113	C₂×Q₈	—	2	24	no	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	C₂×C₃×Q₈	1	12	yes	—
116	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
117	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
118	C₂²×C₄	C₂⁴×C₃	1	12	yes	—
119	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
120	C₄⋊C₄	—	2	24	no	—
121	C₄⋊C₄	—	2	48	no	—
122	C₄⋊C₄	—	2	24	no	—
123	C₂²⋊C₄	—	2	24	no	—
124	C₄²	—	2	12	no	—
125	C₄²	—	2	48	no	—
126	C₄²	C₂×C₃×Q₈	1	12	yes	—
127	C₄²	C₂×C₃×Q₈	1	4	yes	—
128	C₄²	C₃×#41	1	24	yes	—
129	C₂²×C₃	—	2	6	no	—
130	C₂²×C₃	—	4	4	no	—
131	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
132	C₂²×C₃	C₂²×#41	1	6	yes	—
133	C₃×C₄	—	2	6	no	—
134	C₃×C₄	—	2	24	no	—
135	C₃×C₄	—	4	12	no	—
136	C₃×C₄	#533	1	12	yes	—
137	C₂³	—	2	6	no	—
138	C₂³	C₂²×C₃×Q₈	1	2	yes	—
139	C₂³	C₂⁵×C₃	1	1	yes	characteristic
140	Q₈	—	2	32	no	—
141	C₂×C₄	—	2	24	no	—
142	C₂×C₄	—	2	6	no	—
143	C₂×C₄	—	2	24	no	—
144	C₂×C₄	—	2	12	no	—
145	C₂×C₄	—	2	24	no	—
146	C₂×C₄	C₂²×C₃×Q₈	1	6	yes	—
147	C₂×C₄	C₂²×C₃×Q₈	1	6	yes	—
148	C₂×C₄	C₂×C₃×#41	1	24	yes	—
149	C₂×C₄	C₂×C₃×#41	1	12	yes	—
150	C₂×C₃	—	4	2	no	—
151	C₂×C₃	C₂×#533	1	3	yes	—
152	C₂×C₃	#3444	1	4	yes	—
153	C₂²	—	2	6	no	—
154	C₂²	—	4	4	no	—
155	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
156	C₂²	C₂²×C₃×#41	1	6	yes	—
157	C₄	—	2	6	no	—
158	C₄	—	2	24	no	—
159	C₄	—	4	12	no	—
160	C₄	C₃×#533	1	12	yes	—
161	C₃	#34552	1	1	yes	characteristic, direct product factor, Sylow
162	C₂	—	4	2	no	—
163	C₂	C₂×C₃×#533	1	3	yes	—
164	C₂	C₃×#3444	1	4	yes	—

	subgroup	G/H
center	C₂×C₃×C₄²	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
socle	C₂³×C₃	C₂⁵
prime radical	C₂⁴×C₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#2402	C₂	semidirect
2. C₃×#2399	C₂	semidirect
4. C₃×C₄²×Q₈	C₂	semidirect
6. #34552	C₃	direct (N × H)
23. #2402	C₂×C₃	semidirect
24. #2399	C₂×C₃	semidirect
26. C₄²×Q₈	C₂×C₃	semidirect
36. C₂×C₃×C₄²	Q₈	semidirect
88. C₂×C₄²	C₃×Q₈	semidirect
161. C₃	#34552	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#2402	2
C₃×#2399	2
C₃×#2396	2
C₃×C₄²×Q₈	2
C₃×C₄×#117	2
#34552	3

Subgroups — C3×#34552 #10653159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#34552 #10653159