Subgroups — #10705142

Order 768.

296 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2960	C₂	1	1	yes	maximal, characteristic
2	#69745	C₂	1	1	yes	maximal, characteristic
3	#69744	C₂	1	1	yes	maximal, characteristic
4	#67768	C₂	1	4	yes	maximal, complemented
5	#11640	—	3	1	no	maximal, Sylow
6	C₃×#533	C₄	1	1	yes	characteristic
7	C₃×#446	—	2	1	no	—
8	C₃×#446	C₄	1	2	yes	—
9	C₃×C₈×Q₈	—	2	1	no	—
10	C₃×C₈×Q₈	C₂²	1	2	yes	—
11	C₃×#424	—	2	1	no	—
12	C₃×#423	C₄	1	1	yes	characteristic
13	C₃×#405	—	2	1	no	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	#4549	—	2	1	no	—
16	#4549	—	2	1	no	—
17	C₄⋊(C₃⋊C₁₆)	—	2	1	no	—
18	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
19	#4461	—	2	1	no	—
20	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
21	#2960	D₃	1	1	yes	characteristic
22	#2163	—	3	1	no	—
23	#2162	—	3	1	no	—
24	Q₈⋊C₁₆	—	3	4	no	—
25	C₂×C₃×M₁₆	—	2	1	no	—
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
30	C₃×#125	—	2	1	no	—
31	C₃×C₂²⋊Q₈	—	2	1	no	—
32	C₃×C₂²⋊Q₈	C₈	1	2	yes	—
33	C₃×C₄×Q₈	—	2	1	no	—
34	C₃×C₄×Q₈	C₂×C₄	1	2	yes	—
35	C₃×#117	—	2	1	no	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄⋊C₈	—	2	2	no	—
38	C₃×C₄⋊C₈	—	2	2	no	—
39	C₃×C₄⋊C₈	—	2	1	no	—
40	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₃×#97	—	2	2	no	—
42	C₃×C₄×C₈	—	2	2	no	—
43	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×C₈	D₄	1	1	yes	characteristic
45	C₃×C₄×C₈	D₄	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₃⋊C₁₆	—	2	2	no	—
48	C₂×C₃⋊C₁₆	—	2	2	no	—
49	#533	Dic₃	1	1	yes	characteristic
50	#446	—	2	1	no	—
51	#446	Dic₃	1	2	yes	—
52	C₈×Q₈	—	2	1	no	—
53	C₈×Q₈	C₂×D₃	1	2	yes	—
54	#424	—	2	1	no	—
55	#423	Dic₃	1	1	yes	characteristic
56	#405	—	2	1	no	—
57	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₄⋊C₁₆	—	3	2	no	—
59	C₄⋊C₁₆	—	6	1	no	—
60	C₂²⋊C₁₆	—	6	1	no	—
61	C₂²⋊C₁₆	—	6	1	no	—
62	#346	—	6	1	no	—
63	C₄×C₁₆	—	3	2	no	—
64	C₂×C₃×Q₈	—	2	1	no	—
65	C₂×C₃×Q₈	C₂×C₈	1	2	yes	—
66	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
69	C₃×M₁₆	—	4	2	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	—	4	1	no	—
73	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄⋊C₄	—	2	1	no	—
78	C₃×C₄⋊C₄	—	2	1	no	—
79	C₃×C₄⋊C₄	—	2	1	no	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
83	C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
87	C₃×C₄²	M₁₆	1	1	yes	characteristic
88	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₃⋊C₁₆	—	4	2	no	—
90	C₃⋊C₁₆	—	4	2	no	—
91	C₂×M₁₆	—	2	1	no	—
92	C₂²×C₈	#67	1	1	yes	characteristic
93	C₂²×C₈	#67	1	1	yes	characteristic
94	C₄⋊Q₈	C₃⋊C₈	1	1	yes	characteristic
95	C₄⋊Q₈	C₃⋊C₈	1	1	yes	characteristic
96	#125	—	2	1	no	—
97	C₂²⋊Q₈	—	2	1	no	—
98	C₂²⋊Q₈	C₃⋊C₈	1	2	yes	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×Q₈	C₂×Dic₃	1	2	yes	—
101	#117	—	2	1	no	—
102	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₁₆	—	6	2	no	—
104	C₂×C₁₆	—	6	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
109	#97	—	2	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
112	C₄×C₈	#67	1	1	yes	characteristic
113	C₄×C₈	#67	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₂³×C₃	C₂²⋊C₈	1	1	yes	characteristic
116	C₃×Q₈	—	2	4	no	—
117	C₃×Q₈	—	4	2	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	4	1	no	—
124	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	—	4	2	no	—
134	C₃×C₈	#137	1	2	yes	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	C₂×C₃⋊C₈	1	2	yes	—
137	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
138	C₂²×C₄	#208	1	1	yes	characteristic
139	C₂²×C₄	#217	1	1	yes	characteristic
140	M₁₆	—	4	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	#217	1	1	yes	characteristic
145	C₂×C₈	#217	1	1	yes	characteristic
146	C₂×C₈	C₂×#67	1	1	yes	characteristic
147	C₂×C₈	C₂×#67	1	1	yes	characteristic
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
157	C₄²	#208	1	1	yes	characteristic
158	C₄²	#217	1	1	yes	characteristic
159	C₄²	C₂²×Dic₃	1	1	yes	characteristic
160	C₁₆	—	12	2	no	—
161	C₁₆	—	12	2	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	4	2	no	—
171	C₃×C₄	—	4	2	no	—
172	C₃×C₄	#419	1	1	yes	characteristic
173	C₃×C₄	#421	1	1	yes	characteristic
174	C₃×C₄	#504	1	2	yes	—
175	C₂³	#830	1	1	yes	characteristic
176	Q₈	—	2	4	no	—
177	Q₈	—	4	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	#830	1	1	yes	characteristic
185	C₂×C₄	#830	1	1	yes	characteristic
186	C₂×C₄	#830	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
188	C₂×C₄	C₂×#208	1	1	yes	characteristic
189	C₂×C₄	C₂×#217	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	2	no	—
194	C₈	#942	1	2	yes	—
195	C₂×C₃	—	4	1	no	—
196	C₂×C₃	#1479	1	1	yes	characteristic
197	C₂×C₃	#2113	1	1	yes	characteristic
198	C₂×C₃	#2117	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	C₂×#830	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	#5140	1	2	yes	—
210	C₄	#5225	1	1	yes	characteristic
211	C₄	#5236	1	1	yes	characteristic
212	C₃	#11640	1	1	yes	characteristic, complemented, Sylow
213	C₂	—	4	1	no	—
214	C₂	#71621	1	1	yes	characteristic
215	C₂	#72004	1	1	yes	characteristic
216	C₂	#72006	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#2960	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₃	C₂²⋊C₈
cosocle	C₃×C₄×C₈	C₂³
hypercenter	#2960	D₃
hypocenter	C₃	#11640
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
4. #67768	C₂	semidirect
212. C₃	#11640	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2960	2
#69745	2
#69744	2
#67768	2
#11640	3