Subgroups — #10713702

Order 768.

462 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4699	C₂	1	1	yes	maximal, characteristic
2	C₃×#1772	C₂	1	1	yes	maximal, characteristic
3	C₂×#4496	C₂	1	2	yes	maximal
4	#70654	C₂	1	2	yes	maximal
5	C₂×#4463	C₂	1	1	yes	maximal, characteristic
6	#13262	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	—	2	1	no	—
8	C₂²×Dic₁₂	C₂²	1	2	yes	—
9	C₂×#869	—	2	1	no	—
10	C₂×C₃×#108	C₂²	1	1	yes	characteristic
11	C₃×#427	C₂²	1	1	yes	characteristic
12	C₃×#405	—	2	1	no	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×#812	C₂²	1	2	yes	—
15	#4699	—	2	2	no	—
16	#4699	C₄	1	4	yes	complemented
17	C₃×#339	—	2	1	no	—
18	#4496	—	2	4	no	—
19	#4492	—	2	4	no	—
20	#4463	—	2	2	no	—
21	C₂×#494	—	3	1	no	—
22	C₂×#364	—	3	1	no	—
23	C₂×#362	—	3	2	no	—
24	#2141	—	3	2	no	—
25	#1772	D₃	1	1	yes	characteristic
26	C₂²×Dic₆	—	2	2	no	—
27	C₂×C₃×M₁₆	—	2	1	no	—
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
30	C₃×#117	—	2	1	no	—
31	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₃	—	4	2	no	—
33	C₂×Dic₁₂	—	2	2	no	—
34	C₂×Dic₁₂	—	2	2	no	—
35	C₂×Dic₁₂	—	2	2	no	—
36	C₂×Dic₁₂	—	2	2	no	—
37	C₂×Dic₁₂	—	2	4	no	—
38	C₂×Dic₁₂	—	2	2	no	—
39	C₂×Dic₁₂	—	4	1	no	—
40	C₂×Dic₁₂	—	4	1	no	—
41	C₂×Dic₁₂	—	4	2	no	—
42	C₂×Dic₁₂	C₂×C₄	1	4	yes	—
43	#869	—	2	4	no	—
44	#869	—	4	2	no	—
45	C₃×#108	—	2	1	no	—
46	C₃×#108	D₄	1	2	yes	—
47	C₃×#107	D₄	1	2	yes	—
48	C₃×#106	—	2	1	no	—
49	C₃×#97	—	4	1	no	—
50	C₃×C₄×C₈	—	4	1	no	—
51	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
52	C₃×C₄×C₈	D₄	1	2	yes	—
53	#812	—	2	2	no	—
54	#812	—	2	2	no	—
55	C₂×C₃⋊C₁₆	—	2	2	no	—
56	C₂²×Q₁₆	—	3	2	no	—
57	C₂²×Q₁₆	—	6	1	no	—
58	C₂×C₄⋊Q₈	—	6	1	no	—
59	C₂×M₃₂	—	3	2	no	—
60	#494	—	3	4	no	—
61	#494	—	6	2	no	—
62	C₂×#108	C₂×D₃	1	1	yes	characteristic
63	#427	C₂×D₃	1	1	yes	characteristic
64	#405	—	2	1	no	—
65	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
66	#364	—	6	2	no	—
67	#362	—	6	4	no	—
68	#358	—	6	4	no	—
69	#339	—	2	1	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₃	—	4	2	no	—
73	C₂×Dic₆	—	2	4	no	—
74	C₂×Dic₆	—	4	2	no	—
75	C₂×Dic₆	—	4	2	no	—
76	C₂×Dic₆	—	4	2	no	—
77	C₂×Dic₆	—	4	2	no	—
78	C₂×Dic₆	—	4	2	no	—
79	C₃×M₁₆	—	2	2	no	—
80	C₃×M₁₆	—	4	1	no	—
81	C₂×C₃×C₈	—	4	1	no	—
82	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃×C₈	D₈	1	1	yes	characteristic
84	C₂×C₃×C₈	D₈	1	1	yes	characteristic
85	C₂×C₃×C₈	SD₁₆	1	2	yes	—
86	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₈	#41	1	1	yes	characteristic
91	C₂×C₃×C₈	#41	1	1	yes	characteristic
92	C₃×C₄⋊C₄	—	2	2	no	—
93	C₃×C₂²⋊C₄	—	4	1	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	—	4	1	no	—
96	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
97	C₄⋊Dic₃	—	4	8	no	—
98	Dic₁₂	—	2	4	no	—
99	Dic₁₂	—	2	4	no	—
100	Dic₁₂	—	4	4	no	—
101	Dic₁₂	—	4	4	no	—
102	Dic₁₂	—	4	2	no	—
103	Dic₁₂	—	4	2	no	—
104	C₃⋊C₁₆	—	4	2	no	—
105	C₂²×Q₈	—	6	2	no	—
106	C₂×Q₁₆	—	3	4	no	—
107	C₂×Q₁₆	—	6	4	no	—
108	C₂×Q₁₆	—	6	2	no	—
109	C₂×Q₁₆	—	6	2	no	—
110	C₂×Q₁₆	—	6	2	no	—
111	C₂×Q₁₆	—	6	2	no	—
112	C₂×Q₁₆	—	6	2	no	—
113	C₂×Q₁₆	—	12	1	no	—
114	C₂×Q₁₆	—	12	1	no	—
115	C₂×Q₁₆	—	12	2	no	—
116	C₂×M₁₆	—	2	1	no	—
117	C₂²×C₈	#67	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
119	C₄⋊Q₈	—	6	4	no	—
120	C₄⋊Q₈	—	12	2	no	—
121	#117	—	2	1	no	—
122	C₂×C₄⋊C₄	—	12	2	no	—
123	C₂×C₄²	#67	1	1	yes	characteristic
124	M₃₂	—	6	2	no	—
125	M₃₂	—	6	2	no	—
126	C₂×C₁₆	—	6	2	no	—
127	#108	—	2	1	no	—
128	#108	D₁₂	1	2	yes	—
129	#107	D₁₂	1	2	yes	—
130	#106	—	2	1	no	—
131	#97	—	4	1	no	—
132	C₄×C₈	—	4	1	no	—
133	C₄×C₈	C₄×D₃	1	2	yes	—
134	C₄×C₈	D₁₂	1	2	yes	—
135	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	#121	1	1	yes	characteristic
141	C₂×C₃×C₄	#121	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×Dic₃	—	4	4	no	—
146	C₂×Dic₃	—	8	2	no	—
147	C₂×Dic₃	—	8	2	no	—
148	Dic₆	—	4	4	no	—
149	Dic₆	—	4	4	no	—
150	Dic₆	—	8	2	no	—
151	Dic₆	—	8	2	no	—
152	C₃×C₈	—	4	2	no	—
153	C₃×C₈	#102	1	2	yes	—
154	C₃×C₈	#102	1	2	yes	—
155	C₃×C₈	C₄×D₄	1	1	yes	characteristic
156	C₃×C₈	C₄×D₄	1	1	yes	characteristic
157	C₃×C₈	#124	1	1	yes	characteristic
158	C₃×C₈	#127	1	1	yes	characteristic
159	C₂×Q₈	—	6	4	no	—
160	C₂×Q₈	—	12	2	no	—
161	C₂×Q₈	—	12	2	no	—
162	C₂×Q₈	—	12	2	no	—
163	C₂×Q₈	—	12	2	no	—
164	C₂×Q₈	—	12	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	12	2	no	—
167	C₂²×C₄	C₂×#67	1	1	yes	characteristic
168	Q₁₆	—	6	4	no	—
169	Q₁₆	—	6	4	no	—
170	Q₁₆	—	12	4	no	—
171	Q₁₆	—	12	4	no	—
172	Q₁₆	—	12	2	no	—
173	Q₁₆	—	12	2	no	—
174	M₁₆	—	2	2	no	—
175	M₁₆	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
178	C₂×C₈	#213	1	1	yes	characteristic
179	C₂×C₈	#213	1	1	yes	characteristic
180	C₂×C₈	#215	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
183	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
185	C₂×C₈	#235	1	1	yes	characteristic
186	C₂×C₈	#235	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	12	8	no	—
189	C₂²⋊C₄	—	4	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	4	1	no	—
192	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
193	C₁₆	—	12	2	no	—
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	#418	1	1	yes	characteristic
196	C₂²×C₃	#459	1	1	yes	characteristic
197	C₂²×C₃	#460	1	1	yes	characteristic
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	8	2	no	—
200	C₃×C₄	#390	1	1	yes	characteristic
201	C₃×C₄	C₂×#102	1	1	yes	characteristic
202	C₃×C₄	#459	1	1	yes	characteristic
203	C₃×C₄	#460	1	1	yes	characteristic
204	Dic₃	—	8	4	no	—
205	C₂³	#949	1	1	yes	characteristic
206	Q₈	—	12	4	no	—
207	Q₈	—	12	4	no	—
208	Q₈	—	24	2	no	—
209	Q₈	—	24	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	12	4	no	—
214	C₂×C₄	—	24	2	no	—
215	C₂×C₄	—	24	2	no	—
216	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
217	C₂×C₄	#930	1	1	yes	characteristic
218	C₂×C₄	#930	1	1	yes	characteristic
219	C₂×C₄	C₂×#213	1	1	yes	characteristic
220	C₂×C₄	C₂×#215	1	1	yes	characteristic
221	C₂×C₄	#949	1	1	yes	characteristic
222	C₈	—	4	2	no	—
223	C₈	#809	1	2	yes	—
224	C₈	#809	1	2	yes	—
225	C₈	C₄×D₁₂	1	1	yes	characteristic
226	C₈	C₄×D₁₂	1	1	yes	characteristic
227	C₈	#874	1	1	yes	characteristic
228	C₈	#875	1	1	yes	characteristic
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	#1982	1	1	yes	characteristic
232	C₂×C₃	#2214	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	#5001	1	1	yes	characteristic
235	C₂²	#5015	1	1	yes	characteristic
236	C₂²	#5027	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	8	2	no	—
239	C₄	—	24	4	no	—
240	C₄	#4939	1	1	yes	characteristic
241	C₄	C₂×#809	1	1	yes	characteristic
242	C₄	#5015	1	1	yes	characteristic
243	C₄	#5027	1	1	yes	characteristic
244	C₃	#13262	1	1	yes	characteristic, complemented, Sylow
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#69487	1	1	yes	characteristic
248	C₂	#70927	1	2	yes	—

	subgroup	G/H
center	C₂²	#5015
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1772	C₂
socle	C₂²×C₃	#459
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1772	D₃
hypocenter	C₃	#13262
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
16. #4699	C₄	semidirect
244. C₃	#13262	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4699	2
C₃×#1772	2
C₂×#4496	2
#70654	2
C₂×#4463	2
#13262	3