Subgroups — #10693904

Order 768.

322 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×#1848	C₂	1	1	yes	maximal, characteristic
2	#70494	C₂	1	2	yes	maximal
3	C₈×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#69729	C₂	1	1	yes	maximal, characteristic
5	#67715	C₂	1	2	yes	maximal
6	#9627	—	3	1	no	maximal, Sylow
7	C₂×C₃×#108	—	2	1	no	—
8	C₂×C₃×#108	—	2	1	no	—
9	C₃×#423	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₃	—	2	1	no	—
12	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
13	#4925	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	#4550	—	2	1	no	—
16	#4479	—	2	4	no	—
17	#4458	C₄	1	2	yes	complemented
18	#4457	C₄	1	2	yes	complemented
19	#1848	D₃	1	1	yes	characteristic
20	C₈×C₄⋊C₄	—	3	1	no	—
21	#1571	—	3	2	no	—
22	#1564	—	3	1	no	—
23	#1303	—	3	2	no	—
24	C₂×C₃×M₁₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₃	—	2	1	no	—
29	C₂×C₄×Dic₃	—	2	1	no	—
30	C₂²×C₃⋊C₈	—	2	1	no	—
31	C₃×#108	—	4	2	no	—
32	C₃×#108	—	4	2	no	—
33	C₃×C₄⋊C₈	—	2	2	no	—
34	C₃×C₄⋊C₈	—	2	2	no	—
35	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
38	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₃⋊#95	Q₈	1	2	yes	—
40	#830	—	2	2	no	—
41	C₈×Dic₃	—	4	2	no	—
42	C₄⋊(C₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
44	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
45	C₄×C₃⋊C₈	—	2	2	no	—
46	#431	—	3	2	no	—
47	C₂×#108	—	2	1	no	—
48	C₂×#108	—	2	1	no	—
49	#423	C₂×D₃	1	2	yes	—
50	C₂×C₄⋊C₈	—	3	1	no	—
51	C₂×C₄×C₈	—	6	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₄×C₄⋊C₄	—	3	1	no	—
54	#336	—	6	1	no	—
55	#335	—	3	2	no	—
56	#335	—	6	4	no	—
57	#334	—	3	2	no	—
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₄	#41	1	1	yes	characteristic
61	C₂²×Dic₃	—	2	2	no	—
62	C₃×M₁₆	—	4	2	no	—
63	C₃×M₁₆	—	4	2	no	—
64	C₂×C₃×C₈	—	2	1	no	—
65	C₂×C₃×C₈	—	2	1	no	—
66	C₂×C₃×C₈	—	4	2	no	—
67	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₈	D₈	1	2	yes	—
72	C₂×C₃×C₈	Q₁₆	1	2	yes	—
73	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
76	C₃×C₄²	#41	1	1	yes	characteristic
77	C₄⋊Dic₃	—	2	4	no	—
78	C₄×Dic₃	—	4	1	no	—
79	C₄×Dic₃	—	4	1	no	—
80	C₂×C₃⋊C₈	—	2	4	no	—
81	C₂×C₃⋊C₈	—	2	2	no	—
82	C₂×C₃⋊C₈	—	4	1	no	—
83	C₂×C₃⋊C₈	—	4	1	no	—
84	C₂×M₁₆	—	2	2	no	—
85	C₂²×C₈	—	6	1	no	—
86	C₂²×C₈	#67	1	1	yes	characteristic
87	C₂²×C₈	#67	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	6	1	no	—
89	C₂×C₄²	—	6	1	no	—
90	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
91	#108	—	4	2	no	—
92	#108	—	4	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	3	4	no	—
96	C₄⋊C₈	—	3	2	no	—
97	C₄⋊C₈	—	6	1	no	—
98	C₂²⋊C₈	—	6	2	no	—
99	C₄×C₈	—	6	2	no	—
100	C₄×C₈	—	12	2	no	—
101	C₄×C₈	Dic₆	1	1	yes	characteristic
102	C₄×C₈	Dic₆	1	1	yes	characteristic
103	C₄×C₈	C₄×D₃	1	1	yes	characteristic
104	C₄×C₈	C₄×D₃	1	1	yes	characteristic
105	#95	—	3	2	no	—
106	C₂³×C₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	4	1	no	—
109	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₃×C₄	#121	1	1	yes	characteristic
113	C₂×C₃×C₄	#125	1	1	yes	characteristic
114	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₃×C₄	#137	1	1	yes	characteristic
119	C₂×Dic₃	—	2	4	no	—
120	C₂×Dic₃	—	4	2	no	—
121	C₂×Dic₃	—	4	2	no	—
122	C₃×C₈	—	2	1	no	—
123	C₃×C₈	—	2	1	no	—
124	C₃×C₈	—	8	2	no	—
125	C₃×C₈	#107	1	2	yes	—
126	C₃×C₈	#107	1	2	yes	—
127	C₃⋊C₈	—	4	4	no	—
128	C₃⋊C₈	—	4	2	no	—
129	C₂²×C₄	—	6	2	no	—
130	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
131	C₂²×C₄	#239	1	1	yes	characteristic
132	C₂²×C₄	C₂×#67	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	M₁₆	—	4	2	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	6	4	no	—
139	C₂×C₈	—	6	2	no	—
140	C₂×C₈	—	12	1	no	—
141	C₂×C₈	—	12	1	no	—
142	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
143	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
145	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
146	C₂×C₈	#213	1	2	yes	—
147	C₂×C₈	#216	1	2	yes	—
148	C₄⋊C₄	—	6	4	no	—
149	C₄²	—	12	1	no	—
150	C₄²	—	12	1	no	—
151	C₄²	C₂×Dic₆	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
153	C₄²	D₃×D₄	1	1	yes	characteristic
154	C₄²	#237	1	1	yes	characteristic
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	#389	1	1	yes	characteristic
158	C₂²×C₃	#474	1	1	yes	characteristic
159	C₂²×C₃	#477	1	1	yes	characteristic
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	C₂×#107	1	1	yes	characteristic
164	C₃×C₄	#428	1	1	yes	characteristic
165	C₃×C₄	#443	1	2	yes	—
166	C₃×C₄	#459	1	1	yes	characteristic
167	C₃×C₄	#462	1	1	yes	characteristic
168	Dic₃	—	4	4	no	—
169	C₂³	#946	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	6	4	no	—
173	C₂×C₄	—	12	2	no	—
174	C₂×C₄	—	12	2	no	—
175	C₂×C₄	#887	1	1	yes	characteristic
176	C₂×C₄	#888	1	1	yes	characteristic
177	C₂×C₄	#890	1	1	yes	characteristic
178	C₂×C₄	#893	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₂×#213	1	1	yes	characteristic
181	C₂×C₄	#932	1	1	yes	characteristic
182	C₂×C₄	#938	1	1	yes	characteristic
183	C₂×C₄	#942	1	1	yes	characteristic
184	C₂×C₄	C₂×#216	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	8	2	no	—
188	C₈	—	12	4	no	—
189	C₈	—	12	2	no	—
190	C₈	#807	1	2	yes	—
191	C₈	#807	1	2	yes	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	#1945	1	1	yes	characteristic
195	C₂×C₃	#1947	1	1	yes	characteristic
196	C₂×C₃	#1953	1	1	yes	characteristic
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#4972	1	1	yes	characteristic
200	C₂²	#5087	1	1	yes	characteristic
201	C₂²	#5091	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	12	4	no	—
206	C₄	#4895	1	2	yes	—
207	C₄	C₂×#807	1	1	yes	characteristic
208	C₄	#4964	1	1	yes	characteristic
209	C₄	#5072	1	1	yes	characteristic
210	C₄	#5090	1	1	yes	characteristic
211	C₃	#9627	1	1	yes	characteristic, complemented, Sylow
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#69567	1	1	yes	characteristic
215	C₂	#70704	1	1	yes	characteristic
216	C₂	#70706	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#938
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1848	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₂²⋊Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1848	D₃
hypocenter	C₃	#9627
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
17. #4458	C₄	semidirect
18. #4457	C₄	semidirect
211. C₃	#9627	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1848	2
#70494	2
C₈×C₄⋊Dic₃	2
#69729	2
#67715	2
#9627	3