Subgroups — #10764359

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#4926	C₂	1	2	yes	maximal
3	#70752	C₂	1	8	yes	maximal, complemented
4	#69732	C₂	1	4	yes	maximal, complemented
5	#17818	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₃×#431	C₄	1	4	yes	—
8	C₃×#431	C₂²	1	4	yes	—
9	C₂×C₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₂×#830	—	2	2	no	—
13	#4926	—	2	2	no	—
14	#4926	—	2	2	no	—
15	#4926	C₂²	1	4	yes	—
16	#4926	C₂²	1	4	yes	—
17	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
18	C₂×#803	—	2	2	no	—
19	#4458	C₂²	1	8	yes	complemented
20	#4457	C₂²	1	8	yes	complemented
21	C₂×#432	—	3	2	no	—
22	C₂×#431	D₃	1	1	yes	characteristic
23	#1573	—	3	8	no	—
24	#1568	—	3	4	no	—
25	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
27	C₂³×C₃×C₄	Q₈	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄²	D₄	1	2	yes	—
32	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄²	Q₈	1	2	yes	—
34	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₄²	C₂³	1	2	yes	—
36	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
37	C₂²×C₃⋊C₈	—	2	4	no	—
38	C₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
39	C₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
40	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
41	C₃×C₄×C₈	C₂³	1	4	yes	—
42	#830	—	2	4	no	—
43	#830	—	2	4	no	—
44	C₄⋊(C₃⋊C₈)	—	2	4	no	—
45	C₄⋊(C₃⋊C₈)	C₂³	1	8	yes	—
46	#803	—	2	4	no	—
47	#803	—	2	4	no	—
48	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
49	#432	—	3	4	no	—
50	#432	—	3	4	no	—
51	#432	—	6	2	no	—
52	#432	—	6	2	no	—
53	#431	Dic₃	1	4	yes	—
54	#431	C₂×D₃	1	4	yes	—
55	C₂×C₄⋊C₈	—	3	4	no	—
56	C₂×C₄⋊C₈	Dic₃	1	2	yes	—
57	C₂×C₂²⋊C₈	—	6	2	no	—
58	C₂×C₂²⋊C₈	Dic₃	1	2	yes	—
59	C₂×#97	—	6	2	no	—
60	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
61	#335	—	3	8	no	—
62	#334	—	3	8	no	—
63	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	—	2	4	no	—
65	C₂²×C₃×C₄	—	4	2	no	—
66	C₂²×C₃×C₄	—	4	2	no	—
67	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
69	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
71	C₂²×C₃×C₄	C₄⋊C₄	1	2	yes	—
72	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
74	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
77	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
78	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃×C₈	—	4	4	no	—
80	C₂×C₃×C₈	C₂²×C₄	1	8	yes	—
81	C₃×C₄²	C₄⋊C₄	1	2	yes	—
82	C₃×C₄²	C₄⋊C₄	1	2	yes	—
83	C₃×C₄²	C₄⋊C₄	1	2	yes	—
84	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₃×C₄²	C₂²×C₄	1	2	yes	—
86	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₃×C₄²	C₂×D₄	1	2	yes	—
88	C₃×C₄²	C₂×Q₈	1	2	yes	—
89	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
90	C₃×C₄²	C₂⁴	1	1	yes	characteristic
91	C₂×C₃⋊C₈	—	2	8	no	—
92	C₂×C₃⋊C₈	—	4	8	no	—
93	C₂³×C₄	Dic₆	1	1	yes	characteristic
94	C₂³×C₄	D₁₂	1	1	yes	characteristic
95	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
96	C₂²×C₈	—	6	4	no	—
97	C₂²×C₈	C₂×Dic₃	1	4	yes	—
98	C₂×C₄²	Dic₆	1	2	yes	—
99	C₂×C₄²	Dic₆	1	1	yes	characteristic
100	C₂×C₄²	D₁₂	1	2	yes	—
101	C₂×C₄²	D₁₂	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₃	1	2	yes	—
104	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₄²	C₂²×D₃	1	2	yes	—
106	C₄⋊C₈	—	3	8	no	—
107	C₄⋊C₈	—	6	4	no	—
108	C₄⋊C₈	C₂×Dic₃	1	8	yes	—
109	C₂²⋊C₈	—	6	4	no	—
110	C₂²⋊C₈	—	6	4	no	—
111	C₂²⋊C₈	C₂×Dic₃	1	8	yes	—
112	#97	—	6	4	no	—
113	#97	—	6	4	no	—
114	C₄×C₈	C₂×Dic₃	1	4	yes	—
115	C₄×C₈	C₂²×D₃	1	4	yes	—
116	C₂³×C₃	—	4	1	no	—
117	C₂³×C₃	—	4	1	no	—
118	C₂³×C₃	#108	1	4	yes	—
119	C₂³×C₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	—	4	1	no	—
131	C₂×C₃×C₄	—	4	1	no	—
132	C₂×C₃×C₄	—	4	2	no	—
133	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
142	C₃×C₈	—	2	8	no	—
143	C₃⋊C₈	—	4	8	no	—
144	C₂⁴	C₄⋊Dic₃	1	1	yes	characteristic
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	C₄⋊Dic₃	1	2	yes	—
149	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
150	C₂²×C₄	C₄⋊Dic₃	1	2	yes	—
151	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
153	C₂²×C₄	C₄⋊Dic₃	1	2	yes	—
154	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₆	1	2	yes	—
156	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₁₂	1	2	yes	—
158	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
159	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
160	C₂×C₈	—	4	4	no	—
161	C₂×C₈	—	6	8	no	—
162	C₂×C₈	—	12	8	no	—
163	C₂×C₈	C₂²×Dic₃	1	8	yes	—
164	C₄²	C₄⋊Dic₃	1	2	yes	—
165	C₄²	C₄⋊Dic₃	1	2	yes	—
166	C₄²	C₄⋊Dic₃	1	2	yes	—
167	C₄²	C₂×Dic₆	1	1	yes	characteristic
168	C₄²	C₂×Dic₆	1	2	yes	—
169	C₄²	C₂×D₁₂	1	1	yes	characteristic
170	C₄²	C₂×D₁₂	1	2	yes	—
171	C₄²	C₂²×Dic₃	1	1	yes	characteristic
172	C₄²	C₂²×Dic₃	1	2	yes	—
173	C₄²	C₂³×D₃	1	1	yes	characteristic
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	4	4	no	—
179	C₂²×C₃	C₂×#108	1	4	yes	—
180	C₂²×C₃	C₂×#108	1	2	yes	—
181	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	#568	1	2	yes	—
190	C₃×C₄	#575	1	2	yes	—
191	C₂³	—	4	1	no	—
192	C₂³	—	4	1	no	—
193	C₂³	#819	1	4	yes	—
194	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
195	C₂³	C₂×C₄⋊Dic₃	1	2	yes	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
212	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
214	C₂×C₄	C₂²×Dic₆	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
216	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
217	C₈	—	2	8	no	—
218	C₈	—	12	8	no	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	#2905	1	1	yes	characteristic
222	C₂×C₃	#2911	1	1	yes	characteristic
223	C₂×C₃	C₂²×#108	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	C₂×#819	1	2	yes	—
230	C₂²	C₂×#819	1	4	yes	—
231	C₂²	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	#5748	1	2	yes	—
240	C₄	#5819	1	2	yes	—
241	C₃	#17818	1	1	yes	characteristic, complemented, Sylow
242	C₂	—	2	2	no	—
243	C₂	—	2	4	no	—
244	C₂	#82204	1	1	yes	characteristic
245	C₂	C₂²×#819	1	1	yes	characteristic
246	C₂	#83752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₄²	C₂³×D₃
Fitting	C₂×C₃×#431	C₂
socle	C₂²×C₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂⁴
hypercenter	C₂×#431	D₃
hypocenter	C₃	#17818
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #70752	C₂	semidirect
4. #69732	C₂	semidirect
19. #4458	C₂²	semidirect
20. #4457	C₂²	semidirect
241. C₃	#17818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#431	2
C₂×#4926	2
#70752	2
#69732	2
#17818	3