Subgroups — #10693891

Order 768.

298 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×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#69875	C₂	1	2	yes	maximal
3	#69731	C₂	1	1	yes	maximal, characteristic
4	C₂×#4458	C₂	1	1	yes	maximal, characteristic
5	#67745	C₂	1	2	yes	maximal
6	#9624	—	3	1	no	maximal, Sylow
7	C₃×#431	C₂²	1	1	yes	characteristic
8	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#822	—	2	2	no	—
11	#4924	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₃×#322	—	2	1	no	—
14	C₃×C₈²	—	2	1	no	—
15	#4558	—	2	2	no	—
16	#4458	—	2	1	no	—
17	#4458	—	2	1	no	—
18	#4458	C₄	1	2	yes	—
19	#4458	C₄	1	2	yes	—
20	#1950	—	3	2	no	—
21	#1567	—	3	1	no	—
22	C₂×#335	—	3	1	no	—
23	C₈×M₁₆	D₃	1	1	yes	characteristic
24	#1301	—	3	2	no	—
25	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
26	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
28	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×#208	—	2	2	no	—
31	C₂²×C₃⋊C₈	—	2	2	no	—
32	C₃×C₄⋊C₈	—	2	1	no	—
33	C₃×C₂²⋊C₈	Q₈	1	2	yes	—
34	C₃×#97	—	2	1	no	—
35	C₃×C₄×C₈	—	2	1	no	—
36	C₃×C₄×C₈	—	2	1	no	—
37	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×C₈	D₄	1	1	yes	characteristic
40	C₃×C₄×C₈	D₄	1	1	yes	characteristic
41	#822	—	4	4	no	—
42	C₄⋊(C₃⋊C₈)	—	2	2	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
46	#431	C₂×D₃	1	1	yes	characteristic
47	C₂×#108	—	6	2	no	—
48	#423	—	3	2	no	—
49	C₂×C₄⋊C₈	—	3	2	no	—
50	C₄×M₁₆	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	#341	—	6	2	no	—
53	#335	—	3	2	no	—
54	#335	—	3	2	no	—
55	#335	—	6	1	no	—
56	#335	—	6	1	no	—
57	#322	—	2	1	no	—
58	C₈²	—	2	1	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₃×M₁₆	—	2	1	no	—
63	C₃×M₁₆	—	2	1	no	—
64	C₃×M₁₆	D₈	1	2	yes	—
65	C₃×M₁₆	Q₁₆	1	2	yes	—
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	2	1	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	4	1	no	—
71	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄²	#41	1	1	yes	characteristic
78	C₃×C₄²	#41	1	1	yes	characteristic
79	#208	—	4	2	no	—
80	#208	—	4	2	no	—
81	C₂×C₃⋊C₈	—	2	4	no	—
82	C₂×C₃⋊C₈	—	4	2	no	—
83	C₂×C₃⋊C₈	—	4	2	no	—
84	C₂×C₃⋊C₈	—	4	2	no	—
85	C₂×M₁₆	—	6	2	no	—
86	C₂×M₁₆	D₁₂	1	1	yes	characteristic
87	C₂×M₁₆	D₁₂	1	1	yes	characteristic
88	C₂²×C₈	—	6	2	no	—
89	C₂²×C₈	Dic₆	1	1	yes	characteristic
90	C₂²×C₈	Dic₆	1	1	yes	characteristic
91	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
92	#108	—	12	4	no	—
93	C₄⋊C₈	—	2	1	no	—
94	C₄⋊C₈	—	3	4	no	—
95	C₄⋊C₈	—	6	2	no	—
96	C₄⋊C₈	—	6	2	no	—
97	C₄⋊C₈	—	6	2	no	—
98	C₂²⋊C₈	Dic₆	1	2	yes	—
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	C₄×D₃	1	1	yes	characteristic
103	C₄×C₈	C₄×D₃	1	1	yes	characteristic
104	C₄×C₈	#67	1	1	yes	characteristic
105	C₄×C₈	#67	1	1	yes	characteristic
106	C₂³×C₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	—	4	1	no	—
110	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₃×C₄	#121	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	#125	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₃×C₄	#135	1	1	yes	characteristic
119	C₂×C₃×C₄	#135	1	1	yes	characteristic
120	C₃×C₈	—	2	1	no	—
121	C₃×C₈	—	2	1	no	—
122	C₃×C₈	—	2	1	no	—
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	—	2	2	no	—
125	C₃×C₈	—	4	2	no	—
126	C₃×C₈	#108	1	2	yes	complemented
127	C₃⋊C₈	—	4	4	no	—
128	C₃⋊C₈	—	8	2	no	—
129	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
132	M₁₆	—	2	1	no	—
133	M₁₆	—	2	1	no	—
134	M₁₆	—	12	2	no	—
135	M₁₆	—	12	2	no	—
136	M₁₆	D₂₄	1	2	yes	—
137	M₁₆	Dic₁₂	1	2	yes	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	6	4	no	—
144	C₂×C₈	—	12	2	no	—
145	C₂×C₈	—	12	2	no	—
146	C₂×C₈	—	12	2	no	—
147	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
148	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
149	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
150	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
151	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
152	C₄²	#235	1	1	yes	characteristic
153	C₄²	#235	1	1	yes	characteristic
154	C₄²	C₂×#67	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	#499	1	1	yes	characteristic
159	C₂²×C₃	#500	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×#108	1	1	yes	characteristic
164	C₃×C₄	#430	1	1	yes	characteristic
165	C₃×C₄	C₄×D₈	1	1	yes	characteristic
166	C₃×C₄	C₄×Q₁₆	1	1	yes	characteristic
167	C₃×C₄	#466	1	1	yes	characteristic
168	C₃×C₄	#467	1	1	yes	characteristic
169	C₂³	#869	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	C₄×Dic₆	1	1	yes	characteristic
174	C₂×C₄	#870	1	1	yes	characteristic
175	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
176	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
177	C₂×C₄	#904	1	1	yes	characteristic
178	C₂×C₄	#904	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
180	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
181	C₂×C₄	#924	1	1	yes	characteristic
182	C₂×C₄	#930	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	—	12	4	no	—
190	C₈	—	24	2	no	—
191	C₈	#822	1	2	yes	complemented
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	#1942	1	1	yes	characteristic
195	C₂×C₃	#1950	1	1	yes	characteristic
196	C₂×C₃	#1953	1	1	yes	characteristic
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#4682	1	1	yes	characteristic
200	C₂²	#4683	1	1	yes	characteristic
201	C₂²	#4928	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₄×D₂₄	1	1	yes	characteristic
206	C₄	C₄×Dic₁₂	1	1	yes	characteristic
207	C₄	#5115	1	1	yes	characteristic
208	C₄	#5116	1	1	yes	characteristic
209	C₄	C₂×#822	1	1	yes	characteristic
210	C₄	#5124	1	1	yes	characteristic
211	C₃	#9624	1	1	yes	characteristic, complemented, Sylow
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#69770	1	1	yes	characteristic
215	C₂	#69875	1	1	yes	characteristic
216	C₂	#69901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#930
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₈×M₁₆	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₄⋊Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₈×M₁₆	D₃
hypocenter	C₃	#9624
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
126. C₃×C₈	#108	semidirect
191. C₈	#822	semidirect
211. C₃	#9624	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×M₁₆	2
#69875	2
#69731	2
C₂×#4458	2
#67745	2
#9624	3