Subgroups — #10713726

Order 768.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4679	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#69743	C₂	1	1	yes	maximal, characteristic
4	#67812	C₂	1	4	yes	maximal
5	#13268	—	3	1	no	maximal, Sylow
6	C₂×C₄×Dic₆	C₄	1	1	yes	characteristic
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#815	C₂²	1	2	yes	—
10	C₂×Dic₃⋊C₈	C₄	1	2	yes	—
11	C₂×#812	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
13	#4679	—	2	1	no	—
14	#4679	—	2	1	no	—
15	#4679	—	2	1	no	—
16	#4679	—	2	1	no	—
17	C₃×#346	—	2	1	no	—
18	C₃×C₄×C₁₆	—	2	1	no	—
19	C₄⋊(C₃⋊C₁₆)	—	2	1	no	—
20	C₄⋊(C₃⋊C₁₆)	—	2	1	no	—
21	C₂×#446	—	3	1	no	—
22	#2150	—	3	1	no	—
23	C₄×M₃₂	D₃	1	1	yes	characteristic
24	#1377	—	3	4	no	—
25	C₂²×Dic₆	C₈	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	C₈	1	1	yes	characteristic
30	C₂×Dic₃⋊C₄	C₈	1	2	yes	—
31	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
32	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₄×Dic₆	—	2	1	no	—
34	C₄×Dic₆	—	2	1	no	—
35	C₄×Dic₆	—	2	1	no	—
36	C₄×Dic₆	—	2	1	no	—
37	C₃×M₃₂	—	2	2	no	—
38	C₃×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₃×C₁₆	—	2	2	no	—
40	C₃×C₄×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄×C₈	D₄	1	1	yes	characteristic
42	C₃×C₄×C₈	D₄	1	1	yes	characteristic
43	C₃×C₄×C₈	D₄	1	1	yes	characteristic
44	#815	—	2	2	no	—
45	#815	—	2	2	no	—
46	Dic₃⋊C₈	—	2	2	no	—
47	Dic₃⋊C₈	—	2	2	no	—
48	#812	—	2	4	no	—
49	C₂×C₃⋊C₁₆	—	2	2	no	—
50	C₄⋊(C₃⋊C₈)	—	2	1	no	—
51	C₄⋊(C₃⋊C₈)	—	2	1	no	—
52	C₂×C₄×Q₈	—	3	1	no	—
53	C₂×M₃₂	—	3	2	no	—
54	C₂×M₃₂	C₂×D₃	1	2	yes	—
55	#446	—	6	1	no	—
56	#446	—	6	1	no	—
57	#446	—	6	1	no	—
58	#446	—	6	1	no	—
59	C₂×C₄⋊C₈	—	3	2	no	—
60	C₂×C₄⋊C₈	—	3	1	no	—
61	C₂×#97	—	3	2	no	—
62	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	C₄⋊C₁₆	—	6	1	no	—
64	C₄⋊C₁₆	—	6	1	no	—
65	#346	—	2	1	no	—
66	C₄×C₁₆	—	2	1	no	—
67	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₃	C₂×C₈	1	2	yes	—
71	C₂²×Dic₃	C₂×C₈	1	2	yes	—
72	C₂×Dic₆	—	2	1	no	—
73	C₂×Dic₆	—	2	1	no	—
74	C₂×Dic₆	—	4	2	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
81	C₂×C₃×C₈	#41	1	2	yes	—
82	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₄²	M₁₆	1	1	yes	characteristic
85	C₃×C₄²	M₁₆	1	1	yes	characteristic
86	C₄⋊Dic₃	—	2	1	no	—
87	C₄⋊Dic₃	—	2	1	no	—
88	Dic₃⋊C₄	—	2	2	no	—
89	Dic₃⋊C₄	—	2	2	no	—
90	C₄×Dic₃	—	2	2	no	—
91	C₄×Dic₃	—	2	2	no	—
92	C₂×C₃⋊C₈	—	2	2	no	—
93	C₂×C₃⋊C₈	—	2	4	no	—
94	C₃×C₁₆	—	2	4	no	—
95	C₃⋊C₁₆	—	4	2	no	—
96	C₂²×Q₈	—	3	1	no	—
97	C₂²×C₈	—	3	2	no	—
98	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
100	C₄×Q₈	—	6	1	no	—
101	C₄×Q₈	—	6	1	no	—
102	C₄×Q₈	—	6	1	no	—
103	C₄×Q₈	—	6	1	no	—
104	C₂×C₄⋊C₄	—	3	2	no	—
105	C₂×C₄⋊C₄	—	3	1	no	—
106	C₂×C₄²	—	3	2	no	—
107	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	6	4	no	—
110	M₃₂	Dic₆	1	4	yes	complemented
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	6	2	no	—
113	C₄⋊C₈	—	6	2	no	—
114	C₄⋊C₈	—	6	2	no	—
115	C₄⋊C₈	—	6	1	no	—
116	C₄⋊C₈	—	6	1	no	—
117	#97	—	6	2	no	—
118	#97	—	6	2	no	—
119	C₄×C₈	D₁₂	1	1	yes	characteristic
120	C₄×C₈	D₁₂	1	1	yes	characteristic
121	C₄×C₈	#67	1	1	yes	characteristic
122	C₄×C₈	#67	1	1	yes	characteristic
123	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	4	1	no	—
126	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
132	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₃×C₄	#131	1	2	yes	—
134	C₂×Dic₃	—	2	2	no	—
135	C₂×Dic₃	—	2	2	no	—
136	C₂×Dic₃	—	4	2	no	—
137	C₂×Dic₃	—	4	2	no	—
138	Dic₆	—	4	2	no	—
139	Dic₆	—	4	2	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
143	C₃×C₈	#124	1	1	yes	characteristic
144	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
145	C₃⋊C₈	—	4	2	no	—
146	C₂×Q₈	—	6	1	no	—
147	C₂×Q₈	—	6	1	no	—
148	C₂×Q₈	—	12	2	no	—
149	C₂²×C₄	—	3	2	no	—
150	C₂²×C₄	—	3	2	no	—
151	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
152	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	6	4	no	—
156	C₂×C₈	—	6	2	no	—
157	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₆	1	2	yes	—
160	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
161	C₂×C₈	#235	1	2	yes	—
162	C₂×C₈	C₂×#67	1	1	yes	characteristic
163	C₄⋊C₄	—	6	2	no	—
164	C₄⋊C₄	—	6	2	no	—
165	C₄⋊C₄	—	6	1	no	—
166	C₄⋊C₄	—	6	1	no	—
167	C₄²	—	6	2	no	—
168	C₄²	—	6	2	no	—
169	C₄²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
170	C₄²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
171	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₁₆	—	2	4	no	—
174	C₁₆	—	12	2	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
178	C₂²×C₃	C₈×Q₈	1	2	yes	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	#349	1	2	yes	—
183	C₃×C₄	#391	1	1	yes	characteristic
184	C₃×C₄	#408	1	1	yes	characteristic
185	C₃×C₄	#446	1	2	yes	—
186	Dic₃	—	4	2	no	—
187	Dic₃	—	4	2	no	—
188	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
189	Q₈	—	12	2	no	—
190	Q₈	—	12	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	6	2	no	—
194	C₂×C₄	—	6	2	no	—
195	C₂×C₄	—	12	2	no	—
196	C₂×C₄	—	12	2	no	—
197	C₂×C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
198	C₂×C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
199	C₂×C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
200	C₂×C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
201	C₂×C₄	C₄×Dic₆	1	2	yes	—
202	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
203	C₂×C₄	#901	1	2	yes	—
204	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	12	2	no	—
208	C₈	#869	1	1	yes	characteristic
209	C₈	#875	1	1	yes	characteristic
210	C₈	#924	1	2	yes	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	#1984	1	1	yes	characteristic
213	C₂×C₃	C₂×#349	1	1	yes	characteristic
214	C₂×C₃	#2115	1	1	yes	characteristic
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	C₈×Dic₆	1	2	yes	—
218	C₂²	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	12	2	no	—
223	C₄	—	12	2	no	—
224	C₄	#4514	1	2	yes	—
225	C₄	#4679	1	2	yes	—
226	C₄	#4929	1	1	yes	characteristic
227	C₄	#5110	1	1	yes	characteristic
228	C₃	#13268	1	1	yes	characteristic, complemented, Sylow
229	C₂	—	2	2	no	—
230	C₂	#69788	1	1	yes	characteristic
231	C₂	C₂×#4514	1	1	yes	characteristic
232	C₂	#71435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₃)⋊C₄
commutator, prime radical	C₂³×C₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×C₄×M₃₂	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₄×M₃₂	D₃
hypocenter	C₃	#13268
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
38. C₃×M₃₂	Q₈	semidirect
110. M₃₂	Dic₆	semidirect
228. C₃	#13268	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4679	2
C₃×C₄×M₃₂	2
#69743	2
#67812	2
#13268	3