Subgroups — #10694216

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#73311	C₂	1	1	yes	maximal, characteristic
2	C₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×#4481	C₂	1	4	yes	maximal
4	C₄×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#9691	—	3	1	no	maximal, Sylow
6	C₂×#869	C₄	1	4	yes	—
7	C₂×#869	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊C₈	—	2	2	no	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	#4935	—	2	1	no	—
12	#4935	—	2	1	no	—
13	#4935	—	2	2	no	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄×C₃⋊C₈	—	2	1	no	—
16	#4524	—	2	1	no	—
17	#4481	—	2	8	no	—
18	#2868	—	3	1	no	—
19	#1842	D₃	1	1	yes	characteristic
20	C₄×C₄⋊C₈	—	3	1	no	—
21	C₂×#333	—	3	4	no	—
22	C₂²×Dic₆	C₂×C₄	1	4	yes	—
23	C₂²×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄²	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	4	no	—
30	C₂×C₄⋊Dic₃	—	2	4	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	—	2	2	no	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	#869	—	2	4	no	—
35	#869	—	2	4	no	—
36	#869	—	4	8	no	—
37	C₃×C₄⋊C₈	—	2	4	no	—
38	C₃×C₄⋊C₈	—	4	4	no	—
39	C₄⋊(C₃⋊C₈)	—	2	2	no	—
40	C₄⋊(C₃⋊C₈)	D₄	1	4	yes	—
41	C₄×C₃⋊C₈	—	4	2	no	—
42	C₂×C₄⋊Q₈	—	3	4	no	—
43	C₂×C₄⋊Q₈	—	3	2	no	—
44	C₂×C₄⋊C₈	—	2	2	no	—
45	C₂×C₄⋊C₈	—	3	2	no	—
46	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
47	C₂×C₄×C₈	—	6	1	no	—
48	#384	—	6	1	no	—
49	#384	—	6	1	no	—
50	#384	—	6	2	no	—
51	C₄³	C₂×D₃	1	1	yes	characteristic
52	#336	—	6	1	no	—
53	#333	—	6	8	no	—
54	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₃	—	2	4	no	—
60	C₂×Dic₆	—	2	8	no	—
61	C₂×Dic₆	—	4	8	no	—
62	C₂×C₃×C₈	—	4	2	no	—
63	C₂×C₃×C₈	—	4	4	no	—
64	C₃×C₄²	—	2	4	no	—
65	C₃×C₄²	—	2	2	no	—
66	C₃×C₄²	SD₁₆	1	2	yes	—
67	C₃×C₄²	SD₁₆	1	4	yes	—
68	C₃×C₄²	Q₁₆	1	2	yes	—
69	C₃×C₄²	Q₁₆	1	4	yes	—
70	C₃×C₄²	C₂×D₄	1	2	yes	—
71	C₃×C₄²	#41	1	2	yes	—
72	C₄⋊Dic₃	—	4	8	no	—
73	C₄⋊Dic₃	—	4	8	no	—
74	C₄⋊Dic₃	—	4	4	no	—
75	C₄⋊Dic₃	—	4	2	no	—
76	C₄⋊Dic₃	—	4	2	no	—
77	C₂×C₃⋊C₈	—	2	4	no	—
78	C₂×C₃⋊C₈	—	4	2	no	—
79	C₂×C₃⋊C₈	—	4	2	no	—
80	C₂²×Q₈	—	3	4	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	6	2	no	—
83	C₄⋊Q₈	—	6	4	no	—
84	C₄⋊Q₈	—	6	4	no	—
85	C₄⋊Q₈	—	12	8	no	—
86	C₂×C₄⋊C₄	—	6	4	no	—
87	C₂×C₄⋊C₄	—	6	4	no	—
88	C₂×C₄⋊C₄	—	6	2	no	—
89	C₂×C₄⋊C₄	—	6	2	no	—
90	C₂×C₄²	C₄×D₃	1	2	yes	—
91	C₂×C₄²	D₁₂	1	1	yes	characteristic
92	C₂×C₄²	D₁₂	1	1	yes	characteristic
93	C₂×C₄²	#67	1	2	yes	—
94	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
95	C₄⋊C₈	—	2	4	no	—
96	C₄⋊C₈	—	3	4	no	—
97	C₄⋊C₈	—	4	4	no	—
98	C₄⋊C₈	—	6	2	no	—
99	C₄×C₈	—	12	2	no	—
100	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	#100	1	4	yes	—
105	C₂×C₃×C₄	#103	1	4	yes	—
106	C₂×C₃×C₄	#103	1	4	yes	—
107	C₂×C₃×C₄	#103	1	4	yes	—
108	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
109	C₂×C₃×C₄	#121	1	2	yes	—
110	C₂×C₃×C₄	#124	1	1	yes	characteristic
111	C₂×C₃×C₄	#127	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
114	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
116	C₂×C₃×C₄	#137	1	2	yes	—
117	C₂×Dic₃	—	4	4	no	—
118	C₂×Dic₃	—	4	8	no	—
119	Dic₆	—	8	8	no	—
120	C₃×C₈	—	8	2	no	—
121	C₃⋊C₈	—	4	4	no	—
122	C₂×Q₈	—	6	8	no	—
123	C₂×Q₈	—	12	8	no	—
124	C₂²×C₄	—	6	4	no	—
125	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
126	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
129	C₂²×C₄	C₂×#67	1	1	yes	characteristic
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	6	4	no	—
133	C₂×C₈	—	12	2	no	—
134	C₂×C₈	—	12	2	no	—
135	C₄⋊C₄	—	12	8	no	—
136	C₄⋊C₄	—	12	8	no	—
137	C₄⋊C₄	—	12	4	no	—
138	C₄⋊C₄	—	12	2	no	—
139	C₄⋊C₄	—	12	2	no	—
140	C₄²	—	2	4	no	—
141	C₄²	—	2	2	no	—
142	C₄²	#204	1	2	yes	—
143	C₄²	Dic₁₂	1	2	yes	—
144	C₄²	#214	1	4	yes	—
145	C₄²	#216	1	4	yes	—
146	C₄²	D₃×D₄	1	2	yes	—
147	C₄²	#239	1	2	yes	—
148	C₂²×C₃	#390	1	1	yes	characteristic
149	C₂²×C₃	C₂×#100	1	2	yes	—
150	C₂²×C₃	C₂×#103	1	2	yes	—
151	C₂²×C₃	C₂×#103	1	1	yes	characteristic
152	C₂²×C₃	#419	1	1	yes	characteristic
153	C₃×C₄	—	2	4	no	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	#333	1	8	yes	complemented
156	C₃×C₄	#461	1	2	yes	—
157	C₃×C₄	#462	1	2	yes	—
158	C₃×C₄	#487	1	2	yes	—
159	C₃×C₄	#492	1	1	yes	characteristic
160	C₃×C₄	#494	1	1	yes	characteristic
161	Dic₃	—	8	4	no	—
162	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
163	Q₈	—	24	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	12	8	no	—
168	C₂×C₄	—	12	4	no	—
169	C₂×C₄	#810	1	4	yes	—
170	C₂×C₄	#810	1	4	yes	—
171	C₂×C₄	#816	1	4	yes	—
172	C₂×C₄	#823	1	4	yes	—
173	C₂×C₄	#893	1	2	yes	—
174	C₂×C₄	#895	1	2	yes	—
175	C₂×C₄	C₂×#204	1	1	yes	characteristic
176	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
177	C₂×C₄	#909	1	2	yes	—
178	C₂×C₄	C₂×#214	1	2	yes	—
179	C₂×C₄	#940	1	1	yes	characteristic
180	C₂×C₄	C₂×#216	1	2	yes	—
181	C₂×C₄	#947	1	1	yes	characteristic
182	C₈	—	8	2	no	—
183	C₈	—	12	4	no	—
184	C₂×C₃	C₂×#333	1	2	yes	—
185	C₂×C₃	#1548	1	2	yes	—
186	C₂×C₃	#1956	1	1	yes	characteristic
187	C₂×C₃	#1965	1	1	yes	characteristic
188	C₂×C₃	#1981	1	1	yes	characteristic
189	C₂²	C₂×#810	1	2	yes	—
190	C₂²	#4988	1	1	yes	characteristic
191	C₂²	C₂×#816	1	1	yes	characteristic
192	C₂²	#5120	1	1	yes	characteristic
193	C₂²	C₂×#823	1	2	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	24	4	no	—
197	C₄	#4481	1	8	yes	complemented
198	C₄	#4841	1	2	yes	—
199	C₄	#4849	1	2	yes	—
200	C₄	#5076	1	1	yes	characteristic
201	C₄	#5092	1	1	yes	characteristic
202	C₄	#5093	1	2	yes	—
203	C₃	#9691	1	1	yes	characteristic, complemented, Sylow
204	C₂	#69579	1	1	yes	characteristic
205	C₂	C₂×#4481	1	2	yes	—
206	C₂	#70456	1	1	yes	characteristic
207	C₂	#70507	1	2	yes	—
208	C₂	#70521	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1842	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1842	D₃
hypocenter	C₃	#9691
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
155. C₃×C₄	#333	semidirect
197. C₄	#4481	semidirect
203. C₃	#9691	semidirect

Maximal subgroups

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

subgroup	index
#73311	2
C₃×#1842	2
C₂×#4481	2
C₄×C₄⋊(C₃⋊C₈)	2
#9691	3