Subgroups — #10948844

Order 768.

452 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78445	C₂	1	1	yes	maximal, characteristic
2	C₃×#2163	C₂	1	1	yes	maximal, characteristic
3	#69744	C₂	1	1	yes	maximal, characteristic
4	#69281	C₂	1	2	yes	maximal, complemented
5	#69280	C₂	1	2	yes	maximal, complemented
6	#69094	C₂	1	4	yes	maximal
7	#69068	C₂	1	4	yes	maximal, complemented
8	#33499	—	3	1	no	maximal, Sylow
9	#5470	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#5102	C₄	1	2	yes	—
12	#5101	C₂²	1	2	yes	—
13	#4924	C₄	1	1	yes	characteristic
14	#4679	C₄	1	2	yes	—
15	#4679	C₄	1	2	yes	—
16	C₈×Dic₆	C₂²	1	2	yes	—
17	C₈×Dic₆	C₂²	1	2	yes	—
18	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
20	C₃×#346	C₂²	1	2	yes	—
21	#4549	C₂²	1	2	yes	—
22	#4502	C₂²	1	4	yes	—
23	Dic₃⋊C₁₆	C₂²	1	4	yes	—
24	Dic₃⋊C₁₆	C₂²	1	4	yes	—
25	Dic₃×C₁₆	C₂²	1	4	yes	—
26	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
27	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
29	#2960	—	3	1	no	—
30	#2183	—	3	2	no	—
31	#2183	—	3	4	no	—
32	Q₈×C₁₆	—	3	2	no	—
33	#2164	—	3	4	no	—
34	#2163	D₃	1	1	yes	characteristic
35	#2162	—	3	1	no	—
36	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
39	#926	C₂×C₄	1	2	yes	—
40	#924	C₈	1	2	yes	—
41	#924	C₈	1	2	yes	—
42	C₂×#208	C₂×C₄	1	2	yes	—
43	#870	C₈	1	2	yes	—
44	#869	C₈	1	1	yes	characteristic
45	#869	C₈	1	1	yes	characteristic
46	C₄×Dic₆	C₂×C₄	1	2	yes	—
47	C₄×Dic₆	C₂×C₄	1	2	yes	—
48	C₂×C₃×C₁₆	C₂³	1	4	yes	—
49	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
53	#815	C₂×C₄	1	4	yes	—
54	Dic₃⋊C₈	C₂×C₄	1	4	yes	—
55	Dic₃⋊C₈	C₂×C₄	1	4	yes	—
56	C₈×Dic₃	C₂³	1	4	yes	—
57	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
59	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
61	#533	—	3	1	no	—
62	#446	—	3	2	no	—
63	#446	—	3	2	no	—
64	C₈×Q₈	—	3	2	no	—
65	C₈×Q₈	—	3	2	no	—
66	#424	—	3	2	no	—
67	#423	—	3	1	no	—
68	#405	—	3	2	no	—
69	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
70	C₄⋊C₁₆	—	3	4	no	—
71	C₄⋊C₁₆	—	3	4	no	—
72	C₄⋊C₁₆	—	3	2	no	—
73	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
74	C₂²⋊C₁₆	—	3	2	no	—
75	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
76	#346	—	3	4	no	—
77	#346	C₂×D₃	1	2	yes	—
78	C₄×C₁₆	—	3	4	no	—
79	C₄×C₁₆	—	3	1	no	—
80	C₄×C₁₆	—	3	1	no	—
81	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₆	C₂×C₈	1	2	yes	—
85	C₂×Dic₆	C₂×C₈	1	2	yes	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₂×C₃×C₈	—	4	1	no	—
88	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
92	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
93	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
94	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
96	#217	C₂×C₈	1	4	yes	—
97	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₃	C₂×C₈	1	2	yes	—
101	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
102	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
103	C₄×Dic₃	C₂²×C₄	1	4	yes	—
104	#208	—	4	2	no	—
105	C₂×C₃⋊C₈	C₂²×C₄	1	4	yes	—
106	C₃×C₁₆	—	2	4	no	—
107	C₃⋊C₁₆	—	2	2	no	—
108	C₃⋊C₁₆	—	2	2	no	—
109	C₂×M₁₆	—	3	2	no	—
110	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
112	C₄⋊Q₈	—	3	1	no	—
113	C₄⋊Q₈	—	3	1	no	—
114	#125	—	3	2	no	—
115	C₂²⋊Q₈	—	3	2	no	—
116	C₂²⋊Q₈	—	3	2	no	—
117	C₄×Q₈	—	3	2	no	—
118	C₄×Q₈	—	3	2	no	—
119	#117	—	3	2	no	—
120	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
121	C₂×C₁₆	—	3	2	no	—
122	C₂×C₁₆	—	3	2	no	—
123	C₂×C₁₆	C₂²×D₃	1	4	yes	—
124	C₄⋊C₈	—	3	4	no	—
125	C₄⋊C₈	—	3	4	no	—
126	C₄⋊C₈	—	3	2	no	—
127	C₄⋊C₈	—	3	2	no	—
128	#97	—	3	4	no	—
129	C₄×C₈	—	3	4	no	—
130	C₄×C₈	C₄×D₃	1	1	yes	characteristic
131	C₄×C₈	C₄×D₃	1	1	yes	characteristic
132	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
133	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
134	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	4	1	no	—
139	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
145	C₂×Dic₃	C₂²×C₈	1	4	yes	—
146	C₂×Dic₃	C₂²×C₈	1	4	yes	—
147	Dic₆	—	2	4	no	—
148	Dic₆	—	4	2	no	—
149	C₃×C₈	—	2	1	no	—
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	2₋¹⁺⁴	1	2	yes	—
153	C₃⋊C₈	—	2	4	no	—
154	C₂×Q₈	—	3	2	no	—
155	C₂×Q₈	—	3	2	no	—
156	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
157	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
159	M₁₆	—	12	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	3	4	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
167	C₄⋊C₄	—	3	2	no	—
168	C₄⋊C₄	—	3	2	no	—
169	C₄⋊C₄	—	3	4	no	—
170	C₄⋊C₄	—	3	4	no	—
171	C₄⋊C₄	—	3	2	no	—
172	C₄⋊C₄	—	3	2	no	—
173	C₂²⋊C₄	—	3	4	no	—
174	C₄²	—	3	4	no	—
175	C₄²	D₃×C₈	1	1	yes	characteristic
176	C₄²	D₃×C₈	1	1	yes	characteristic
177	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
178	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
179	C₁₆	—	2	4	no	—
180	C₁₆	—	6	2	no	—
181	C₁₆	—	6	2	no	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	#504	1	2	yes	—
191	C₃×C₄	#519	1	1	yes	characteristic
192	C₃×C₄	#568	1	1	yes	characteristic
193	Dic₃	—	2	4	no	—
194	Dic₃	—	2	4	no	—
195	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
196	Q₈	—	6	4	no	—
197	Q₈	—	12	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	3	4	no	—
202	C₂×C₄	—	3	4	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
205	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
206	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
208	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	6	4	no	—
214	C₈	#1010	1	2	yes	—
215	C₂×C₃	—	4	1	no	—
216	C₂×C₃	#2982	1	1	yes	characteristic
217	C₂×C₃	C₂×#504	1	1	yes	characteristic
218	C₂×C₃	#3407	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	6	4	no	—
228	C₄	—	6	4	no	—
229	C₄	#4907	1	2	yes	—
230	C₄	#5516	1	1	yes	characteristic
231	C₄	#5745	1	1	yes	characteristic
232	C₃	#33499	1	1	yes	characteristic, complemented, Sylow
233	C₂	—	4	1	no	—
234	C₂	#79263	1	1	yes	characteristic
235	C₂	C₂×#4907	1	1	yes	characteristic
236	C₂	#82007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2163	C₂
prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2163	D₃
hypocenter	C₃	#33499
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
4. #69281	C₂	semidirect
5. #69280	C₂	semidirect
7. #69068	C₂	semidirect
232. C₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#78445	2
C₃×#2163	2
#69744	2
#69281	2
#69280	2
#69094	2
#69068	2
#33499	3