Subgroups — #10948818

Order 768.

556 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2931	C₂	1	1	yes	maximal, characteristic
2	#72004	C₂	1	4	yes	maximal, complemented
3	#70857	C₂	1	8	yes	maximal, complemented
4	#69745	C₂	1	2	yes	maximal, complemented
5	#33494	—	3	1	no	maximal, Sylow
6	C₂×C₃×#131	C₂²	1	2	yes	—
7	C₃×#532	C₄	1	1	yes	characteristic
8	C₃×#436	C₄	1	4	yes	—
9	C₃×C₈×D₄	C₂²	1	4	yes	—
10	C₃×#423	C₄	1	1	yes	characteristic
11	C₃×#408	C₄	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#812	C₂²	1	4	yes	—
14	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
15	#4549	C₂²	1	4	yes	—
16	#4549	C₂²	1	8	yes	—
17	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	—
18	#4461	C₂²	1	4	yes	—
19	#2931	D₃	1	1	yes	characteristic
20	#2168	—	3	8	no	—
21	#2163	—	3	2	no	—
22	#2113	—	3	4	no	—
23	C₂×C₃×#41	C₂×C₄	1	2	yes	—
24	C₃×#131	—	4	2	no	—
25	C₃×#131	—	4	2	no	—
26	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₈	C₂³	1	4	yes	—
31	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
32	C₃×#127	C₈	1	1	yes	characteristic
33	C₃×#124	C₈	1	2	yes	—
34	C₃×#121	C₈	1	4	yes	—
35	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
38	C₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
39	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
40	C₃×C₄×C₈	C₂³	1	2	yes	—
41	#812	—	4	4	no	—
42	C₂×C₃⋊C₁₆	—	4	4	no	—
43	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
44	C₂×#131	C₂×D₃	1	2	yes	—
45	#532	Dic₃	1	1	yes	characteristic
46	C₂×M₃₂	—	3	4	no	—
47	C₂²×C₁₆	—	3	4	no	—
48	#436	Dic₃	1	4	yes	—
49	C₈×D₄	C₂×D₃	1	4	yes	—
50	#423	Dic₃	1	1	yes	characteristic
51	#408	Dic₃	1	2	yes	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₄⋊C₁₆	—	3	4	no	—
54	C₂²⋊C₁₆	—	3	4	no	—
55	C₂²⋊C₁₆	—	3	8	no	—
56	#346	—	3	4	no	—
57	C₃×#41	—	4	2	no	—
58	C₃×#41	—	4	2	no	—
59	C₂×C₃×Q₈	C₂×C₈	1	2	yes	—
60	C₂×C₃×D₄	C₂×C₈	1	4	yes	—
61	C₂×C₃×D₄	C₂×C₈	1	4	yes	—
62	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
63	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
64	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
65	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃×M₁₆	—	4	2	no	—
67	C₃×M₁₆	—	4	4	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	—	2	4	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	—	4	1	no	—
72	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
75	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
76	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
77	C₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
78	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
79	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
80	C₃×C₄²	C₂²×C₄	1	2	yes	—
81	C₃⋊C₁₆	—	2	8	no	—
82	C₂×#41	C₂×Dic₃	1	2	yes	—
83	#131	—	4	2	no	—
84	#131	—	4	2	no	—
85	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
86	C₂×M₁₆	C₂×Dic₃	1	4	yes	—
87	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
88	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₃	1	4	yes	—
90	C₄⋊Q₈	C₃⋊C₈	1	1	yes	characteristic
91	#127	C₃⋊C₈	1	1	yes	characteristic
92	#124	C₃⋊C₈	1	2	yes	—
93	#121	C₃⋊C₈	1	4	yes	—
94	C₄×D₄	C₂×Dic₃	1	4	yes	—
95	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
96	M₃₂	—	12	4	no	—
97	C₂×C₁₆	—	3	8	no	—
98	C₂×C₁₆	—	12	4	no	—
99	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
100	C₂²⋊C₈	C₂×Dic₃	1	8	yes	—
101	C₄×C₈	C₂×Dic₃	1	2	yes	—
102	C₄×C₈	C₂²×D₃	1	2	yes	—
103	C₂³×C₃	C₂²×C₈	1	4	yes	—
104	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
105	C₃×Q₈	—	4	2	no	—
106	C₃×D₄	—	4	4	no	—
107	C₃×D₄	—	4	2	no	—
108	C₃×D₄	—	4	4	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	2	1	no	—
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	4	2	no	—
114	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
116	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
117	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
118	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
119	C₃×C₈	—	2	4	no	—
120	C₃×C₈	—	2	1	no	—
121	C₃×C₈	—	2	2	no	—
122	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
123	#41	—	4	2	no	—
124	#41	—	4	2	no	—
125	C₂×Q₈	C₂×C₃⋊C₈	1	2	yes	—
126	C₂×D₄	C₂×C₃⋊C₈	1	4	yes	—
127	C₂×D₄	C₂×C₃⋊C₈	1	4	yes	—
128	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
129	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
130	C₂²×C₄	C₂²×Dic₃	1	4	yes	—
131	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
132	M₁₆	—	4	4	no	—
133	M₁₆	—	4	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
139	C₂×C₈	C₂²×Dic₃	1	2	yes	—
140	C₂×C₈	C₂²×Dic₃	1	4	yes	—
141	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
142	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
143	C₂²⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
144	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
145	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
146	C₄²	C₂²×Dic₃	1	2	yes	—
147	C₁₆	—	6	8	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	8	no	—
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	—	2	4	no	—
152	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	2	1	no	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	#518	1	1	yes	characteristic
158	C₃×C₄	#568	1	1	yes	characteristic
159	C₂³	C₂²×C₃⋊C₈	1	4	yes	—
160	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
161	Q₈	—	4	2	no	—
162	D₄	—	4	2	no	—
163	D₄	—	4	4	no	—
164	D₄	—	4	4	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
172	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
173	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
174	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	4	no	—
177	C₈	—	2	2	no	—
178	C₈	#1004	1	2	yes	—
179	C₂×C₃	—	4	1	no	—
180	C₂×C₃	—	4	4	no	—
181	C₂×C₃	#2972	1	1	yes	characteristic
182	C₂×C₃	#3407	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	8	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	4	no	—
187	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	4	no	—
192	C₄	#5798	1	1	yes	characteristic
193	C₄	#5819	1	1	yes	characteristic
194	C₃	#33494	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	4	1	no	—
196	C₂	—	4	4	no	—
197	C₂	#82301	1	1	yes	characteristic
198	C₂	#83780	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2931	C₂
prime radical	C₃×#532	C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2931	D₃
hypocenter	C₃	#33494
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #72004	C₂	semidirect
3. #70857	C₂	semidirect
4. #69745	C₂	semidirect
194. C₃	#33494	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2931	2
#72004	2
#70857	2
#69745	2
#33494	3