Subgroups — #10948852

Order 768.

444 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78445	C₂	1	1	yes	maximal, characteristic
2	C₃×#2163	C₂	1	1	yes	maximal, characteristic
3	#69745	C₂	1	1	yes	maximal, characteristic
4	#69282	C₂	1	4	yes	maximal, complemented
5	#69094	C₂	1	4	yes	maximal
6	#69068	C₂	1	4	yes	maximal, complemented
7	#33501	—	3	1	no	maximal, Sylow
8	#5470	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#5102	C₄	1	2	yes	—
11	#5101	C₂²	1	2	yes	—
12	#4924	C₄	1	1	yes	characteristic
13	#4679	C₄	1	2	yes	—
14	#4679	C₄	1	2	yes	—
15	C₈×Dic₆	C₂²	1	2	yes	—
16	C₈×Dic₆	C₂²	1	2	yes	—
17	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
19	C₃×#346	C₂²	1	2	yes	—
20	#4549	C₂²	1	2	yes	—
21	#4502	C₂²	1	4	yes	—
22	Dic₃⋊C₁₆	C₂²	1	8	yes	—
23	Dic₃×C₁₆	C₂²	1	4	yes	—
24	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
25	#4461	C₂²	1	2	yes	—
26	#2960	—	3	1	no	—
27	#2183	—	3	4	no	—
28	#2183	—	3	4	no	—
29	#2164	—	3	4	no	—
30	#2163	—	3	1	no	—
31	#2163	D₃	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
35	#926	C₂×C₄	1	2	yes	—
36	#924	C₈	1	2	yes	—
37	#924	C₈	1	2	yes	—
38	C₂×#208	C₂×C₄	1	2	yes	—
39	#870	C₈	1	2	yes	—
40	#869	C₈	1	1	yes	characteristic
41	#869	C₈	1	1	yes	characteristic
42	C₄×Dic₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₆	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₁₆	C₂³	1	4	yes	—
45	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
49	#815	C₂×C₄	1	4	yes	—
50	Dic₃⋊C₈	C₂×C₄	1	4	yes	—
51	Dic₃⋊C₈	C₂×C₄	1	4	yes	—
52	C₈×Dic₃	C₂³	1	4	yes	—
53	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
54	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
55	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
56	#533	—	3	1	no	—
57	#446	—	3	2	no	—
58	#446	—	3	2	no	—
59	C₈×Q₈	—	3	2	no	—
60	C₈×Q₈	—	3	2	no	—
61	#424	—	3	2	no	—
62	#423	—	3	1	no	—
63	#405	—	3	2	no	—
64	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
65	C₄⋊C₁₆	—	3	8	no	—
66	C₄⋊C₁₆	—	3	2	no	—
67	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
68	C₂²⋊C₁₆	—	3	2	no	—
69	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
70	#346	—	3	4	no	—
71	#346	—	3	2	no	—
72	#346	C₂×D₃	1	2	yes	—
73	C₄×C₁₆	—	3	4	no	—
74	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₆	C₂×C₈	1	2	yes	—
78	C₂×Dic₆	C₂×C₈	1	2	yes	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	4	1	no	—
81	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
85	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
86	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
87	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
89	#217	C₂×C₈	1	4	yes	—
90	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
94	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
95	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
96	C₄×Dic₃	C₂²×C₄	1	4	yes	—
97	#208	—	4	2	no	—
98	C₂×C₃⋊C₈	C₂²×C₄	1	4	yes	—
99	C₃×C₁₆	—	2	4	no	—
100	C₃⋊C₁₆	—	2	4	no	—
101	C₂×M₁₆	—	3	2	no	—
102	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
104	C₄⋊Q₈	—	3	1	no	—
105	C₄⋊Q₈	—	3	1	no	—
106	#125	—	3	2	no	—
107	C₂²⋊Q₈	—	3	2	no	—
108	C₂²⋊Q₈	—	3	2	no	—
109	C₄×Q₈	—	3	2	no	—
110	C₄×Q₈	—	3	2	no	—
111	#117	—	3	2	no	—
112	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
113	C₂×C₁₆	—	3	4	no	—
114	C₂×C₁₆	C₂²×D₃	1	4	yes	—
115	C₄⋊C₈	—	3	4	no	—
116	C₄⋊C₈	—	3	4	no	—
117	C₄⋊C₈	—	3	2	no	—
118	C₄⋊C₈	—	3	2	no	—
119	#97	—	3	4	no	—
120	C₄×C₈	—	3	4	no	—
121	C₄×C₈	C₄×D₃	1	1	yes	characteristic
122	C₄×C₈	C₄×D₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	4	1	no	—
129	C₂×C₃×C₄	—	4	1	no	—
130	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₃	C₂²×C₈	1	4	yes	—
137	C₂×Dic₃	C₂²×C₈	1	4	yes	—
138	Dic₆	—	4	2	no	—
139	Dic₆	—	4	2	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	2	1	no	—
143	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
144	C₃⋊C₈	—	2	4	no	—
145	C₂×Q₈	—	3	2	no	—
146	C₂×Q₈	—	3	2	no	—
147	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
148	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
150	M₁₆	—	12	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	3	4	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
158	C₄⋊C₄	—	3	2	no	—
159	C₄⋊C₄	—	3	2	no	—
160	C₄⋊C₄	—	3	4	no	—
161	C₄⋊C₄	—	3	4	no	—
162	C₄⋊C₄	—	3	2	no	—
163	C₄⋊C₄	—	3	2	no	—
164	C₂²⋊C₄	—	3	4	no	—
165	C₄²	—	3	4	no	—
166	C₄²	D₃×C₈	1	1	yes	characteristic
167	C₄²	D₃×C₈	1	1	yes	characteristic
168	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
169	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
170	C₁₆	—	2	4	no	—
171	C₁₆	—	6	4	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	#519	1	1	yes	characteristic
182	C₃×C₄	#568	1	1	yes	characteristic
183	Dic₃	—	2	4	no	—
184	Dic₃	—	2	4	no	—
185	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
186	Q₈	—	12	2	no	—
187	Q₈	—	12	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	3	4	no	—
191	C₂×C₄	—	3	4	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
195	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
197	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	6	4	no	—
204	C₈	#1010	1	2	yes	—
205	C₂×C₃	—	4	1	no	—
206	C₂×C₃	#2982	1	1	yes	characteristic
207	C₂×C₃	#3407	1	1	yes	characteristic
208	C₂×C₃	#3407	1	1	yes	characteristic
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	6	4	no	—
219	C₄	—	6	4	no	—
220	C₄	#5516	1	1	yes	characteristic
221	C₄	#5745	1	1	yes	characteristic
222	C₃	#33501	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	4	1	no	—
224	C₂	#79263	1	1	yes	characteristic
225	C₂	#82007	1	1	yes	characteristic
226	C₂	#82007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2163	C₂
prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2163	D₃
hypocenter	C₃	#33501
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
4. #69282	C₂	semidirect
6. #69068	C₂	semidirect
222. C₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
#78445	2
C₃×#2163	2
#69745	2
#69282	2
#69094	2
#69068	2
#33501	3