Subgroups — #10710563

Order 768.

296 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	C₃×#1772	C₂	1	1	yes	maximal, characteristic
2	#69884	C₂	1	2	yes	maximal
3	C₂×#4463	C₂	1	1	yes	maximal, characteristic
4	C₄×#812	C₂	1	1	yes	maximal, characteristic
5	#67672	C₂	1	2	yes	maximal
6	#12667	—	3	1	no	maximal, Sylow
7	C₂×C₃×#108	C₄	1	1	yes	characteristic
8	C₃×#427	C₄	1	1	yes	characteristic
9	C₃×#405	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#812	C₂²	1	2	yes	—
12	C₂×#812	C₂²	1	1	yes	characteristic
13	C₂×#812	C₂²	1	1	yes	characteristic
14	C₃×#339	C₄	1	2	yes	—
15	#4466	—	2	4	no	—
16	#4463	C₄	1	2	yes	complemented
17	#4463	C₄	1	2	yes	complemented
18	#4461	C₄	1	2	yes	complemented
19	C₄×C₃⋊C₁₆	C₄	1	2	yes	complemented
20	C₂×#364	—	3	1	no	—
21	#2110	—	3	2	no	—
22	C₄×M₃₂	—	3	1	no	—
23	#1772	D₃	1	1	yes	characteristic
24	#1378	—	3	2	no	—
25	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×#117	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃×#108	—	2	1	no	—
31	C₃×#108	—	2	1	no	—
32	C₃×#107	—	2	1	no	—
33	C₃×#106	—	2	1	no	—
34	C₃×#97	—	2	2	no	—
35	C₃×C₄×C₈	—	2	2	no	—
36	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	#812	—	2	2	no	—
41	#812	—	2	1	no	—
42	#812	—	2	1	no	—
43	#812	C₂×C₄	1	4	yes	—
44	#812	D₄	1	2	yes	—
45	#812	Q₈	1	2	yes	—
46	C₂×C₃⋊C₁₆	—	2	2	no	—
47	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
48	C₂×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
49	C₂×M₃₂	—	3	1	no	—
50	C₂×M₃₂	—	3	1	no	—
51	C₂×M₃₂	—	3	2	no	—
52	C₂×#108	Dic₃	1	1	yes	characteristic
53	#427	Dic₃	1	1	yes	characteristic
54	#405	C₂×D₃	1	2	yes	—
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#364	—	3	2	no	—
57	#364	—	3	2	no	—
58	#347	—	6	4	no	—
59	#346	—	3	2	no	—
60	C₄×C₁₆	—	3	2	no	—
61	#339	Dic₃	1	2	yes	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₃×M₁₆	—	2	2	no	—
65	C₃×M₁₆	—	2	2	no	—
66	C₂×C₃×C₈	—	2	2	no	—
67	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
68	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
69	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
70	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
71	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
72	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
77	C₂×C₃×C₈	#41	1	1	yes	characteristic
78	C₂×C₃×C₈	#41	1	1	yes	characteristic
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₄²	—	2	1	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	C₄²	1	1	yes	characteristic
84	C₃×C₄²	C₄²	1	1	yes	characteristic
85	C₃⋊C₁₆	—	2	2	no	—
86	C₃⋊C₁₆	—	2	4	no	—
87	C₃⋊C₁₆	C₄⋊C₄	1	4	yes	—
88	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
89	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
91	#117	C₂×Dic₃	1	2	yes	—
92	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
93	M₃₂	—	3	2	no	—
94	M₃₂	—	3	2	no	—
95	M₃₂	—	3	4	no	—
96	M₃₂	—	6	2	no	—
97	M₃₂	—	6	1	no	—
98	M₃₂	—	6	1	no	—
99	C₂×C₁₆	—	3	2	no	—
100	C₂×C₁₆	—	3	2	no	—
101	C₂×C₁₆	—	6	2	no	—
102	#108	—	2	1	no	—
103	#108	—	2	1	no	—
104	#107	—	2	1	no	—
105	#106	—	2	1	no	—
106	#97	—	2	2	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	C₄×D₃	1	1	yes	characteristic
109	C₄×C₈	C₄×D₃	1	1	yes	characteristic
110	C₄×C₈	C₄×D₃	1	1	yes	characteristic
111	C₄×C₈	C₄×D₃	1	1	yes	characteristic
112	C₂³×C₃	#131	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₃×C₄	#131	1	1	yes	characteristic
123	C₃×C₈	—	4	2	no	—
124	C₃×C₈	#97	1	1	yes	characteristic
125	C₃×C₈	#97	1	1	yes	characteristic
126	C₃×C₈	#97	1	1	yes	characteristic
127	C₃×C₈	#97	1	1	yes	characteristic
128	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₃×C₈	#117	1	1	yes	characteristic
130	C₃×C₈	C₄×D₄	1	1	yes	characteristic
131	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	#208	1	1	yes	characteristic
138	C₂×C₈	#208	1	1	yes	characteristic
139	C₂×C₈	#208	1	1	yes	characteristic
140	C₂×C₈	#208	1	1	yes	characteristic
141	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
142	C₂×C₈	C₄×Dic₃	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₈	D₃×D₄	1	1	yes	characteristic
146	C₂×C₈	#237	1	1	yes	characteristic
147	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
148	C₂×C₈	#239	1	1	yes	characteristic
149	C₄⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₄×Dic₃	1	1	yes	characteristic
154	C₄²	C₄×Dic₃	1	1	yes	characteristic
155	C₁₆	—	3	4	no	—
156	C₁₆	—	6	2	no	—
157	C₁₆	—	6	4	no	—
158	C₂²×C₃	—	4	1	no	—
159	C₂²×C₃	#405	1	1	yes	characteristic
160	C₂²×C₃	#436	1	1	yes	characteristic
161	C₂²×C₃	#446	1	1	yes	characteristic
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	4	1	no	—
164	C₃×C₄	—	8	2	no	—
165	C₃×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
166	C₃×C₄	C₂×#97	1	1	yes	characteristic
167	C₃×C₄	#436	1	1	yes	characteristic
168	C₃×C₄	#446	1	1	yes	characteristic
169	C₂³	#948	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂×#208	1	1	yes	characteristic
175	C₂×C₄	C₂×#208	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
177	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
178	C₂×C₄	Q₈×Dic₃	1	1	yes	characteristic
179	C₂×C₄	#948	1	1	yes	characteristic
180	C₈	—	4	2	no	—
181	C₈	#803	1	1	yes	characteristic
182	C₈	#803	1	1	yes	characteristic
183	C₈	#803	1	1	yes	characteristic
184	C₈	#803	1	1	yes	characteristic
185	C₈	#887	1	1	yes	characteristic
186	C₈	D₃×C₄⋊C₄	1	1	yes	characteristic
187	C₈	#892	1	1	yes	characteristic
188	C₈	#893	1	1	yes	characteristic
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	#1770	1	1	yes	characteristic
193	C₂²	—	4	1	no	—
194	C₂²	#4997	1	1	yes	characteristic
195	C₂²	#5012	1	1	yes	characteristic
196	C₂²	#5024	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	4	1	no	—
199	C₄	—	8	2	no	—
200	C₄	C₂×#803	1	1	yes	characteristic
201	C₄	Dic₃×C₄⋊C₄	1	1	yes	characteristic
202	C₄	#5012	1	1	yes	characteristic
203	C₄	#5024	1	1	yes	characteristic
204	C₃	#12667	1	1	yes	characteristic, complemented, Sylow
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#69477	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#892
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1772	C₂
socle	C₂×C₃	#1770
prime radical	C₂³×C₃	#131
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1772	D₃
hypocenter	C₃	#12667
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
16. #4463	C₄	semidirect
17. #4463	C₄	semidirect
18. #4461	C₄	semidirect
19. C₄×C₃⋊C₁₆	C₄	semidirect
204. C₃	#12667	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1772	2
#69884	2
C₂×#4463	2
C₄×#812	2
#67672	2
#12667	3