Subgroups — #10947849

Order 768.

716 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2875	C₂	1	1	yes	maximal, characteristic
2	#83780	C₂	1	2	yes	maximal
3	#69884	C₂	1	6	yes	maximal, complemented
4	C₂×#4466	C₂	1	6	yes	maximal, complemented
5	#33329	—	3	1	no	maximal, Sylow
6	C₂×C₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₃×#131	C₂²	1	1	yes	characteristic
8	C₃×C₄×#41	C₄	1	1	yes	characteristic
9	C₃×#405	C₄	1	3	yes	—
10	C₃×#405	C₂²	1	3	yes	—
11	C₃×C₄×M₁₆	C₄	1	3	yes	—
12	C₂×C₃×#97	C₂²	1	3	yes	—
13	#5140	C₄	1	16	yes	complemented
14	C₂×#812	C₂²	1	12	yes	—
15	#4466	C₂²	1	4	yes	—
16	#4466	C₂²	1	12	yes	—
17	#3407	—	3	2	no	—
18	#2875	D₃	1	1	yes	characteristic
19	#2110	—	3	6	no	—
20	C₂×#347	—	3	6	no	—
21	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₃×#131	—	2	4	no	—
23	C₃×#131	C₂×C₄	1	8	yes	—
24	C₂×C₃×M₁₆	C₂×C₄	1	3	yes	—
25	C₂×C₃×M₁₆	C₂×C₄	1	3	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
27	C₂²×C₃×C₈	C₂³	1	3	yes	—
28	C₃×C₄×Q₈	—	2	1	no	—
29	C₃×C₄×D₄	—	2	3	no	—
30	C₃×#117	C₂×C₄	1	3	yes	—
31	C₂×C₃×C₄²	C₂×C₄	1	3	yes	—
32	C₃×#97	C₂×C₄	1	3	yes	—
33	C₃×#97	C₂×C₄	1	3	yes	—
34	C₃×#97	C₂³	1	1	yes	characteristic
35	C₃×#97	C₂³	1	3	yes	—
36	C₃×C₄×C₈	C₂×C₄	1	6	yes	—
37	#812	C₂×C₄	1	24	yes	complemented
38	#812	C₂³	1	8	yes	—
39	C₂×C₃⋊C₁₆	C₂×C₄	1	24	yes	complemented
40	C₂×#131	Dic₃	1	1	yes	characteristic
41	C₂×#131	C₂×D₃	1	1	yes	characteristic
42	C₄×#41	Dic₃	1	1	yes	characteristic
43	#504	—	3	16	no	—
44	C₂×M₃₂	—	3	12	no	—
45	#405	Dic₃	1	3	yes	—
46	#405	C₂×D₃	1	3	yes	—
47	C₄×M₁₆	Dic₃	1	3	yes	—
48	C₂×#97	C₂×D₃	1	3	yes	—
49	#347	—	3	4	no	—
50	#347	—	3	12	no	—
51	C₃×#41	C₄²	1	4	yes	—
52	C₃×#41	M₁₆	1	4	yes	—
53	C₂×C₃×Q₈	M₁₆	1	1	yes	characteristic
54	C₂×C₃×D₄	M₁₆	1	3	yes	—
55	C₂²×C₃×C₄	—	2	3	no	—
56	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
57	C₃×M₁₆	—	2	6	no	—
58	C₃×M₁₆	C₄²	1	12	yes	—
59	C₂×C₃×C₈	—	2	6	no	—
60	C₂×C₃×C₈	C₂²×C₄	1	12	yes	—
61	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
64	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
65	C₃×C₄⋊C₄	—	2	3	no	—
66	C₃×C₂²⋊C₄	—	2	3	no	—
67	C₃×C₄²	—	2	3	no	—
68	C₃×C₄²	C₂²×C₄	1	3	yes	—
69	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
70	C₃⋊C₁₆	C₂²×C₄	1	16	yes	—
71	C₂×#41	C₂×Dic₃	1	1	yes	characteristic
72	#131	—	2	4	no	—
73	#131	C₄×D₃	1	8	yes	—
74	C₂×M₁₆	C₂×Dic₃	1	3	yes	—
75	C₂×M₁₆	C₂×Dic₃	1	3	yes	—
76	C₂²×C₈	C₂×Dic₃	1	3	yes	—
77	C₂²×C₈	C₂²×D₃	1	3	yes	—
78	C₄×Q₈	—	2	1	no	—
79	C₄×D₄	—	2	3	no	—
80	#117	C₂×Dic₃	1	3	yes	—
81	C₂×C₄²	C₂×Dic₃	1	3	yes	—
82	M₃₂	—	3	24	no	—
83	M₃₂	—	3	8	no	—
84	C₂×C₁₆	—	3	24	no	—
85	#97	C₂×Dic₃	1	3	yes	—
86	#97	C₂×Dic₃	1	3	yes	—
87	#97	C₂²×D₃	1	1	yes	characteristic
88	#97	C₂²×D₃	1	3	yes	—
89	C₄×C₈	C₂×Dic₃	1	6	yes	—
90	C₂³×C₃	C₂×M₁₆	1	3	yes	—
91	C₃×Q₈	#97	1	4	yes	—
92	C₃×D₄	#97	1	12	yes	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	3	no	—
95	C₂×C₃×C₄	—	8	3	no	—
96	C₂×C₃×C₄	C₂×C₄²	1	6	yes	—
97	C₂×C₃×C₄	C₂×M₁₆	1	6	yes	—
98	C₂×C₃×C₄	C₂×M₁₆	1	3	yes	—
99	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
100	C₃×C₈	—	2	1	no	—
101	C₃×C₈	—	2	3	no	—
102	C₃×C₈	C₂×C₄²	1	6	yes	—
103	C₃×C₈	C₂³×C₄	1	2	yes	—
104	#41	#208	1	4	yes	—
105	#41	C₄×Dic₃	1	4	yes	—
106	C₂×Q₈	#208	1	1	yes	characteristic
107	C₂×D₄	#208	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
110	M₁₆	—	2	6	no	—
111	M₁₆	C₄×Dic₃	1	12	yes	—
112	C₂×C₈	—	2	6	no	—
113	C₂×C₈	C₂×C₄×D₃	1	12	yes	—
114	C₂×C₈	C₂²×Dic₃	1	3	yes	—
115	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
116	C₂×C₈	C₂²×Dic₃	1	3	yes	—
117	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
118	C₄⋊C₄	—	2	3	no	—
119	C₂²⋊C₄	—	2	3	no	—
120	C₄²	—	2	3	no	—
121	C₄²	C₂²×Dic₃	1	3	yes	—
122	C₄²	C₂²×Dic₃	1	1	yes	characteristic
123	C₁₆	—	3	16	no	—
124	C₂²×C₃	—	4	3	no	—
125	C₂²×C₃	C₂×#97	1	6	yes	—
126	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
127	C₃×C₄	—	4	3	no	—
128	C₃×C₄	—	4	1	no	—
129	C₃×C₄	C₂×#97	1	6	yes	—
130	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
131	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
132	C₂³	C₂×#208	1	3	yes	—
133	Q₈	#803	1	4	yes	—
134	D₄	#803	1	12	yes	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	3	no	—
137	C₂×C₄	—	8	3	no	—
138	C₂×C₄	C₂×#208	1	3	yes	—
139	C₂×C₄	C₂×#208	1	6	yes	—
140	C₂×C₄	C₂×C₄×Dic₃	1	6	yes	—
141	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	3	no	—
144	C₈	C₂×C₄×Dic₃	1	6	yes	—
145	C₈	C₂²×C₄×D₃	1	2	yes	—
146	C₂×C₃	—	2	1	no	—
147	C₂×C₃	—	2	6	no	—
148	C₂×C₃	C₂²×#97	1	1	yes	characteristic
149	C₂²	—	4	3	no	—
150	C₂²	C₂×#803	1	6	yes	—
151	C₂²	C₂²×#208	1	1	yes	characteristic
152	C₄	—	4	1	no	—
153	C₄	—	4	3	no	—
154	C₄	C₂×#803	1	6	yes	—
155	C₄	C₂²×#208	1	1	yes	characteristic
156	C₄	C₂²×C₄×Dic₃	1	1	yes	characteristic
157	C₃	#33329	1	1	yes	characteristic, complemented, Sylow
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×#803	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₃
commutator	C₃×C₄	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#2875	C₂
socle	C₂×C₃	C₂²×#97
prime radical	C₂×C₃×#41	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2875	D₃
hypocenter	C₃	#33329
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #2875

Complements

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

N	H ≅ G/N	type
3. #69884	C₂	semidirect
4. C₂×#4466	C₂	semidirect
13. #5140	C₄	semidirect
37. #812	C₂×C₄	semidirect
39. C₂×C₃⋊C₁₆	C₂×C₄	semidirect
157. C₃	#33329	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2875	2
#83780	2
#69884	2
C₂×#4466	2
#33329	3