Subgroups — #10948767

Order 768.

548 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	#70858	C₂	1	6	yes	maximal, complemented
4	#69745	C₂	1	3	yes	maximal, complemented
5	C₄×#812	C₂	1	3	yes	maximal, complemented
6	#33485	—	3	1	no	maximal, Sylow
7	C₃×C₄×#41	C₄	1	1	yes	characteristic
8	C₃×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₈×D₄	C₄	1	3	yes	—
11	C₃×C₈×D₄	C₂²	1	3	yes	—
12	C₃×#431	C₄	1	3	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
14	C₂×#812	C₂²	1	6	yes	—
15	#4549	C₂²	1	6	yes	—
16	C₄⋊(C₃⋊C₁₆)	C₂²	1	6	yes	—
17	#4461	C₂²	1	1	yes	characteristic
18	#4461	C₂²	1	3	yes	—
19	C₄×C₃⋊C₁₆	C₂²	1	6	yes	—
20	C₈×#41	D₃	1	1	yes	characteristic
21	Q₈×C₁₆	—	3	2	no	—
22	#2169	—	3	6	no	—
23	#2163	—	3	3	no	—
24	C₄×M₃₂	—	3	3	no	—
25	C₂×C₃×#41	C₈	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
27	C₂²×C₃×C₈	C₂³	1	3	yes	—
28	C₃×C₄×Q₈	C₈	1	1	yes	characteristic
29	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×D₄	C₈	1	3	yes	—
31	C₃×C₄×D₄	C₂×C₄	1	3	yes	—
32	C₃×#117	C₈	1	3	yes	—
33	C₂×C₃×C₄²	C₂×C₄	1	3	yes	—
34	C₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
36	C₃×C₂²⋊C₈	C₂×C₄	1	6	yes	—
37	C₃×C₄×C₈	C₂×C₄	1	3	yes	—
38	C₃×C₄×C₈	C₂×C₄	1	3	yes	—
39	C₃×C₄×C₈	C₂³	1	3	yes	—
40	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
41	#812	—	2	12	no	—
42	C₂×C₃⋊C₁₆	C₂³	1	6	yes	—
43	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
44	C₄×#41	Dic₃	1	1	yes	characteristic
45	C₂×M₃₂	—	3	6	no	—
46	C₈×Q₈	Dic₃	1	1	yes	characteristic
47	C₈×Q₈	C₂×D₃	1	1	yes	characteristic
48	C₈×D₄	Dic₃	1	3	yes	—
49	C₈×D₄	C₂×D₃	1	3	yes	—
50	#431	Dic₃	1	3	yes	—
51	C₂×C₄×C₈	C₂×D₃	1	3	yes	—
52	C₄⋊C₁₆	—	3	6	no	—
53	C₂²⋊C₁₆	—	3	6	no	—
54	#346	—	3	1	no	—
55	#346	—	3	3	no	—
56	C₄×C₁₆	—	3	6	no	—
57	C₃×#41	—	2	4	no	—
58	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
59	C₂×C₃×D₄	C₂×C₈	1	3	yes	—
60	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
61	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
62	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
63	C₂×C₃×C₈	—	2	6	no	—
64	C₂×C₃×C₈	—	2	6	no	—
65	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
66	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
67	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
69	C₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
70	C₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
71	C₃×C₂²⋊C₄	C₂×C₈	1	6	yes	—
72	C₃×C₄²	C₂×C₈	1	3	yes	—
73	C₃×C₄²	C₂×C₈	1	3	yes	—
74	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄²	C₂²×C₄	1	3	yes	—
76	C₃⋊C₁₆	—	2	6	no	—
77	C₃⋊C₁₆	#41	1	4	yes	complemented
78	C₂×#41	C₃⋊C₈	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₃	1	3	yes	—
80	C₂²×C₈	C₂²×D₃	1	3	yes	—
81	C₄×Q₈	C₃⋊C₈	1	1	yes	characteristic
82	C₄×Q₈	C₂×Dic₃	1	1	yes	characteristic
83	C₄×D₄	C₃⋊C₈	1	3	yes	—
84	C₄×D₄	C₂×Dic₃	1	3	yes	—
85	#117	C₃⋊C₈	1	3	yes	—
86	C₂×C₄²	C₂×Dic₃	1	3	yes	—
87	M₃₂	—	6	12	no	—
88	C₂×C₁₆	—	3	2	no	—
89	C₂×C₁₆	—	3	6	no	—
90	C₄⋊C₈	C₂×Dic₃	1	3	yes	—
91	C₄⋊C₈	C₂×Dic₃	1	3	yes	—
92	C₂²⋊C₈	C₂×Dic₃	1	6	yes	—
93	C₄×C₈	C₂×Dic₃	1	3	yes	—
94	C₄×C₈	C₂×Dic₃	1	3	yes	—
95	C₄×C₈	C₂²×D₃	1	3	yes	—
96	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
97	C₂³×C₃	C₂²×C₈	1	3	yes	—
98	C₃×Q₈	M₃₂	1	4	yes	complemented
99	C₃×D₄	—	2	6	no	—
100	C₂×C₃×C₄	—	2	3	no	—
101	C₂×C₃×C₄	—	2	3	no	—
102	C₂×C₃×C₄	—	2	6	no	—
103	C₂×C₃×C₄	—	4	3	no	—
104	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
107	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
108	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
109	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
110	C₃×C₈	—	2	3	no	—
111	C₃×C₈	—	2	3	no	—
112	C₃×C₈	#131	1	2	yes	—
113	C₃×C₈	C₂×#41	1	2	yes	—
114	#41	—	2	4	no	—
115	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
116	C₂×D₄	C₂×C₃⋊C₈	1	3	yes	—
117	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
118	C₂²×C₄	C₂×C₃⋊C₈	1	3	yes	—
119	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	—	2	6	no	—
122	C₂×C₈	C₂²×Dic₃	1	3	yes	—
123	C₂×C₈	C₂²×Dic₃	1	3	yes	—
124	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
125	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
126	C₄⋊C₄	C₂×C₃⋊C₈	1	3	yes	—
127	C₄⋊C₄	C₂×C₃⋊C₈	1	3	yes	—
128	C₂²⋊C₄	C₂×C₃⋊C₈	1	6	yes	—
129	C₄²	C₂×C₃⋊C₈	1	3	yes	—
130	C₄²	C₂×C₃⋊C₈	1	3	yes	—
131	C₄²	C₂²×Dic₃	1	1	yes	characteristic
132	C₄²	C₂²×Dic₃	1	3	yes	—
133	C₁₆	—	3	4	no	—
134	C₁₆	—	6	6	no	—
135	C₂²×C₃	—	2	3	no	—
136	C₂²×C₃	—	2	3	no	—
137	C₂²×C₃	—	2	3	no	—
138	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
139	C₃×C₄	—	2	1	no	—
140	C₃×C₄	—	2	1	no	—
141	C₃×C₄	—	2	3	no	—
142	C₃×C₄	—	2	3	no	—
143	C₃×C₄	C₂×M₃₂	1	6	yes	—
144	C₃×C₄	C₄×#41	1	1	yes	characteristic
145	C₃×C₄	C₂×#131	1	1	yes	characteristic
146	C₂³	C₂²×C₃⋊C₈	1	3	yes	—
147	Q₈	#812	1	4	yes	complemented
148	D₄	—	2	6	no	—
149	C₂×C₄	—	2	3	no	—
150	C₂×C₄	—	2	3	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	4	3	no	—
153	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
154	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
155	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
156	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
157	C₂×C₄	C₂²×C₃⋊C₈	1	3	yes	—
158	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
159	C₈	—	2	3	no	—
160	C₈	—	2	3	no	—
161	C₈	#948	1	2	yes	—
162	C₈	D₃×#41	1	2	yes	—
163	C₂×C₃	—	4	3	no	—
164	C₂×C₃	C₈×#41	1	1	yes	characteristic
165	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
166	C₂×C₃	#3407	1	1	yes	characteristic
167	C₂²	—	2	3	no	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	3	no	—
174	C₄	—	2	3	no	—
175	C₄	C₂×#812	1	6	yes	—
176	C₄	C₂×#948	1	1	yes	characteristic
177	C₄	Dic₃×#41	1	1	yes	characteristic
178	C₃	#33485	1	1	yes	characteristic, complemented, Sylow
179	C₂	—	4	3	no	—
180	C₂	#41×C₃⋊C₈	1	1	yes	characteristic
181	C₂	C₂²×#812	1	1	yes	characteristic
182	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×C₈×#41	C₂
prime radical	C₂×C₃×#41	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₈×#41	D₃
hypocenter	C₃	#33485
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊C₁₆	C₂	semidirect
3. #70858	C₂	semidirect
4. #69745	C₂	semidirect
5. C₄×#812	C₂	semidirect
77. C₃⋊C₁₆	#41	semidirect
98. C₃×Q₈	M₃₂	semidirect
147. Q₈	#812	semidirect
178. C₃	#33485	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×#41	2
Q₈×C₃⋊C₁₆	2
#70858	2
#69745	2
C₄×#812	2
#33485	3