Subgroups — #10713343

Order 768.

534 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#4549	C₂	1	4	yes	maximal, complemented
3	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#13189	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄×D₄	C₄	1	1	yes	characteristic
8	C₃×C₈×D₄	—	2	2	no	—
9	C₃×C₈×D₄	—	2	2	no	—
10	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₃×C₂²⋊C₈	C₄	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
15	#4549	—	2	8	no	—
16	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
17	C₄×C₃⋊C₁₆	—	2	2	no	—
18	C₂×C₈×D₄	D₃	1	1	yes	characteristic
19	C₂×C₄⋊C₁₆	—	3	1	no	—
20	C₂×C₂²⋊C₁₆	—	3	4	no	—
21	C₂×C₄×C₁₆	—	3	1	no	—
22	C₂²×C₃×D₄	C₈	1	1	yes	characteristic
23	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₈	—	2	2	no	—
25	C₂²×C₃×C₈	—	2	2	no	—
26	C₂²×C₃×C₈	—	2	4	no	—
27	C₂²×C₃×C₈	—	2	4	no	—
28	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₄×D₄	—	2	2	no	—
32	C₃×C₄×D₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊C₄	C₈	1	2	yes	—
35	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄⋊C₈	—	2	2	no	—
37	C₃×C₂²⋊C₈	—	2	4	no	—
38	C₃×C₄×C₈	D₄	1	2	yes	—
39	C₃×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₃⋊C₁₆	—	2	2	no	—
41	C₂×C₃⋊C₁₆	—	2	2	no	—
42	C₂×C₃⋊C₁₆	—	2	2	no	—
43	C₂×C₃⋊C₁₆	—	2	4	no	—
44	C₂×C₃⋊C₁₆	D₄	1	4	yes	complemented
45	C₂³×C₈	C₂×D₃	1	2	yes	—
46	C₂×C₄×D₄	Dic₃	1	1	yes	characteristic
47	C₂²×C₁₆	—	3	2	no	—
48	C₂²×C₁₆	—	3	2	no	—
49	C₈×D₄	—	2	2	no	—
50	C₈×D₄	—	2	2	no	—
51	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
52	C₂×C₂²⋊C₈	Dic₃	1	2	yes	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄⋊C₁₆	—	6	2	no	—
55	C₂²⋊C₁₆	—	6	8	no	—
56	C₄×C₁₆	—	6	2	no	—
57	C₂⁴×C₃	C₂×C₈	1	2	yes	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	C₁₆	1	4	yes	complemented
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	—	2	4	no	—
63	C₂²×C₃×C₄	—	2	2	no	—
64	C₂²×C₃×C₄	—	2	4	no	—
65	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
66	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₈	—	2	2	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	—	4	4	no	—
73	C₂×C₃×C₈	—	4	2	no	—
74	C₂×C₃×C₈	—	4	2	no	—
75	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₈	#41	1	2	yes	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₂²⋊C₄	—	2	4	no	—
82	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
83	C₃×C₄²	M₁₆	1	2	yes	—
84	C₃⋊C₁₆	—	2	4	no	—
85	C₃⋊C₁₆	—	4	2	no	—
86	C₂²×D₄	C₃⋊C₈	1	1	yes	characteristic
87	C₂³×C₄	C₂×Dic₃	1	2	yes	—
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	—	2	4	no	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	C₂×Dic₃	1	2	yes	—
93	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
95	C₄×D₄	—	2	2	no	—
96	C₄×D₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
99	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
100	C₂×C₁₆	—	3	4	no	—
101	C₂×C₁₆	—	6	2	no	—
102	C₂×C₁₆	—	6	2	no	—
103	C₂×C₁₆	—	6	4	no	—
104	C₂×C₁₆	—	6	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₂²⋊C₈	—	2	4	no	—
107	C₄×C₈	#67	1	2	yes	—
108	C₄×C₈	#67	1	2	yes	—
109	C₂³×C₃	—	2	2	no	—
110	C₂³×C₃	—	2	4	no	—
111	C₂³×C₃	—	2	2	no	—
112	C₂³×C₃	—	2	4	no	—
113	C₂³×C₃	C₂×C₁₆	1	4	yes	—
114	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
115	C₃×D₄	—	2	8	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	—	4	2	no	—
121	C₂×C₃×C₄	—	4	4	no	—
122	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
123	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
124	C₂×C₃×C₄	C₂×C₁₆	1	2	yes	—
125	C₂×C₃×C₄	M₃₂	1	2	yes	—
126	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
128	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₃×C₄	#131	1	2	yes	—
130	C₃×C₈	—	2	2	no	—
131	C₃×C₈	—	4	2	no	—
132	C₃×C₈	#121	1	2	yes	—
133	C₃×C₈	#124	1	1	yes	characteristic
134	C₃×C₈	#127	1	1	yes	characteristic
135	C₂⁴	C₂×C₃⋊C₈	1	2	yes	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	C₃⋊C₁₆	1	4	yes	complemented
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
144	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	#217	1	2	yes	—
154	C₂×C₈	D₃×D₄	1	2	yes	—
155	C₂×C₈	#237	1	2	yes	—
156	C₂×C₈	C₂×#67	1	1	yes	characteristic
157	C₂×C₈	C₂×#67	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₄²	#208	1	2	yes	—
161	C₄²	#217	1	2	yes	—
162	C₁₆	—	6	4	no	—
163	C₁₆	—	12	2	no	—
164	C₂²×C₃	—	2	4	no	—
165	C₂²×C₃	—	2	8	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	4	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₂²×C₃	C₈×D₄	1	2	yes	—
171	C₂²×C₃	C₂²×C₁₆	1	1	yes	characteristic
172	C₂²×C₃	C₂×M₃₂	1	1	yes	characteristic
173	C₂²×C₃	#504	1	2	yes	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	C₂²⋊C₁₆	1	4	yes	complemented
177	C₃×C₄	#390	1	1	yes	characteristic
178	C₃×C₄	#408	1	1	yes	characteristic
179	C₃×C₄	#436	1	2	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	2	no	—
184	C₂³	C₂×C₃⋊C₁₆	1	4	yes	—
185	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
186	D₄	—	2	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×C₃⋊C₁₆	1	2	yes	—
194	C₂×C₄	#812	1	2	yes	—
195	C₂×C₄	#830	1	2	yes	—
196	C₂×C₄	#830	1	2	yes	—
197	C₂×C₄	C₂×#208	1	1	yes	characteristic
198	C₂×C₄	D₄×Dic₃	1	2	yes	—
199	C₂×C₄	#948	1	2	yes	—
200	C₂×C₄	C₂×#217	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	4	2	no	—
203	C₈	#936	1	1	yes	characteristic
204	C₈	#938	1	2	yes	—
205	C₈	#940	1	1	yes	characteristic
206	C₂×C₃	—	4	4	no	—
207	C₂×C₃	#1954	1	1	yes	characteristic
208	C₂×C₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
209	C₂×C₃	#2113	1	1	yes	characteristic
210	C₂×C₃	D₄×C₁₆	1	2	yes	—
211	C₂×C₃	#2169	1	2	yes	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	8	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	4	no	—
217	C₂²	D₄×C₃⋊C₈	1	2	yes	—
218	C₂²	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
219	C₂²	C₂×#812	1	1	yes	characteristic
220	C₂²	#5140	1	2	yes	—
221	C₂²	C₂×#830	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	4	2	no	—
224	C₄	#4549	1	4	yes	complemented
225	C₄	#5012	1	2	yes	—
226	C₄	#5221	1	1	yes	characteristic
227	C₄	#5234	1	1	yes	characteristic
228	C₃	#13189	1	1	yes	characteristic, complemented, Sylow
229	C₂	—	4	4	no	—
230	C₂	D₄×C₃⋊C₁₆	1	2	yes	—
231	C₂	#70858	1	2	yes	—
232	C₂	#71614	1	1	yes	characteristic
233	C₂	C₂×#4549	1	1	yes	characteristic
234	C₂	#72004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₁₆
Fitting	C₂×C₃×C₈×D₄	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂²×C₃×D₄	C₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₈×D₄	D₃
hypocenter	C₃	#13189
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#4549	C₂	semidirect
3. C₂×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
4. C₂×C₄×C₃⋊C₁₆	C₂	semidirect
44. C₂×C₃⋊C₁₆	D₄	semidirect
60. C₂×C₃×D₄	C₁₆	semidirect
138. C₂×D₄	C₃⋊C₁₆	semidirect
176. C₃×C₄	C₂²⋊C₁₆	semidirect
224. C₄	#4549	semidirect
228. C₃	#13189	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×D₄	2
C₂×#4549	2
C₂×C₄⋊(C₃⋊C₁₆)	2
C₂×C₄×C₃⋊C₁₆	2
#13189	3