Subgroups — #10705138

Order 768.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Q₈	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	#11639	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₃×C₈×Q₈	—	2	1	no	—
8	C₃×C₈×Q₈	—	2	1	no	—
9	C₃×C₈×Q₈	C₄	1	2	yes	—
10	C₃×C₈×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×C₈	—	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	C₂×C₈×Q₈	D₃	1	1	yes	characteristic
22	#2163	—	3	1	no	—
23	#2162	—	3	1	no	—
24	Q₈⋊C₁₆	—	3	4	no	—
25	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
26	C₂²×C₃×C₈	—	2	1	no	—
27	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₃×C₄×Q₈	—	2	1	no	—
30	C₃×C₄×Q₈	—	2	1	no	—
31	C₃×C₄×Q₈	C₈	1	2	yes	—
32	C₃×C₄×Q₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₃×C₄²	—	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×C₄×C₈	—	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	C₂×C₄×Q₈	Dic₃	1	1	yes	characteristic
50	C₈×Q₈	—	2	1	no	—
51	C₈×Q₈	—	2	1	no	—
52	C₈×Q₈	Dic₃	1	2	yes	—
53	C₈×Q₈	C₂×D₃	1	2	yes	—
54	C₂×C₄⋊C₈	—	2	1	no	—
55	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
56	C₂×C₄×C₈	—	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	4	no	—
65	C₂×C₃×Q₈	—	2	1	no	—
66	C₂×C₃×Q₈	C₂×C₈	1	2	yes	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	—	4	1	no	—
75	C₂×C₃×C₈	—	4	2	no	—
76	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	1	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	2	no	—
86	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
87	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
88	C₃×C₄²	M₁₆	1	1	yes	characteristic
89	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₃⋊C₁₆	—	4	2	no	—
91	C₃⋊C₁₆	—	4	2	no	—
92	C₂²×Q₈	C₃⋊C₈	1	1	yes	characteristic
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	#67	1	1	yes	characteristic
95	C₂²×C₈	#67	1	1	yes	characteristic
96	C₄×Q₈	—	2	1	no	—
97	C₄×Q₈	—	2	1	no	—
98	C₄×Q₈	C₃⋊C₈	1	2	yes	—
99	C₄×Q₈	C₂×Dic₃	1	2	yes	—
100	C₂×C₄⋊C₄	—	2	1	no	—
101	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
102	C₂×C₄²	—	2	1	no	—
103	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₁₆	—	6	2	no	—
105	C₂×C₁₆	—	6	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
113	C₄×C₈	#67	1	1	yes	characteristic
114	C₄×C₈	#67	1	1	yes	characteristic
115	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
116	C₂³×C₃	C₂²⋊C₈	1	1	yes	characteristic
117	C₃×Q₈	—	2	4	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	M₃₂	1	4	yes	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
134	C₃×C₈	—	2	1	no	—
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	—	4	2	no	—
138	C₃×C₈	#137	1	2	yes	—
139	C₂×Q₈	—	2	4	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	C₂×C₃⋊C₈	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
145	C₂²×C₄	#208	1	1	yes	characteristic
146	C₂²×C₄	#217	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	#217	1	1	yes	characteristic
152	C₂×C₈	#217	1	1	yes	characteristic
153	C₂×C₈	C₂×#67	1	1	yes	characteristic
154	C₂×C₈	C₂×#67	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
162	C₄²	#208	1	1	yes	characteristic
163	C₄²	#217	1	1	yes	characteristic
164	C₄²	C₂²×Dic₃	1	1	yes	characteristic
165	C₁₆	—	12	2	no	—
166	C₁₆	—	12	2	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	#350	1	2	yes	—
170	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₃×C₄	—	2	4	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	#419	1	1	yes	characteristic
178	C₃×C₄	#421	1	1	yes	characteristic
179	C₃×C₄	C₂×M₃₂	1	2	yes	—
180	C₂³	#830	1	1	yes	characteristic
181	Q₈	—	2	4	no	—
182	Q₈	—	2	2	no	—
183	Q₈	#812	1	4	yes	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	#830	1	1	yes	characteristic
193	C₂×C₄	#830	1	1	yes	characteristic
194	C₂×C₄	#830	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×#208	1	1	yes	characteristic
197	C₂×C₄	C₂×#217	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	#942	1	2	yes	—
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	#1479	1	1	yes	characteristic
205	C₂×C₃	#2112	1	1	yes	characteristic
206	C₂×C₃	C₂×#350	1	1	yes	characteristic
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	#4561	1	2	yes	—
210	C₂²	C₂×#830	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	C₂×#812	1	2	yes	—
218	C₄	#5225	1	1	yes	characteristic
219	C₄	#5236	1	1	yes	characteristic
220	C₃	#11639	1	1	yes	characteristic, complemented, Sylow
221	C₂	—	2	2	no	—
222	C₂	#71621	1	1	yes	characteristic
223	C₂	#71963	1	1	yes	characteristic
224	C₂	C₂×#4561	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₂×C₃×C₈×Q₈	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₃	C₂²⋊C₈
cosocle	C₃×C₄×C₈	C₂³
hypercenter	C₂×C₈×Q₈	D₃
hypocenter	C₃	#11639
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
4. #67768	C₂	semidirect
220. C₃	#11639	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Q₈	2
#69745	2
#69744	2
#67768	2
#11639	3