Subgroups — #10693724

Order 768.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈²	C₂	1	1	yes	maximal, characteristic
2	#69786	C₂	1	2	yes	maximal
3	C₄×#817	C₂	1	1	yes	maximal, characteristic
4	C₂×#4457	C₂	1	1	yes	maximal, characteristic
5	#67655	C₂	1	2	yes	maximal
6	#9593	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#817	C₄	1	1	yes	characteristic
11	C₂×#817	C₄	1	1	yes	characteristic
12	C₂×Dic₃⋊C₈	—	2	2	no	—
13	C₄×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃×C₈²	—	2	2	no	—
16	#4551	C₄	1	2	yes	—
17	#4550	—	2	2	no	—
18	#4457	—	2	2	no	—
19	#1939	—	3	2	no	—
20	C₄×#106	—	3	1	no	—
21	C₂×#334	—	3	1	no	—
22	C₂×C₈²	D₃	1	1	yes	characteristic
23	#1279	—	3	2	no	—
24	C₂²×C₃×C₈	—	2	1	no	—
25	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₃	—	2	2	no	—
32	C₂²×C₃⋊C₈	—	2	2	no	—
33	C₃×C₄×C₈	—	2	1	no	—
34	C₃×C₄×C₈	—	2	1	no	—
35	C₃×C₄×C₈	—	2	1	no	—
36	C₃×C₄×C₈	D₄	1	2	yes	—
37	C₃×C₄×C₈	Q₈	1	2	yes	—
38	C₃×C₄×C₈	Q₈	1	2	yes	—
39	C₃⋊#95	—	2	2	no	—
40	#817	—	2	2	no	—
41	#817	C₈	1	4	yes	complemented
42	Dic₃⋊C₈	—	4	4	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₂×#106	—	3	1	no	—
46	C₂×#106	—	3	1	no	—
47	C₂×C₄⋊C₈	—	3	2	no	—
48	C₂×C₄⋊C₈	—	6	2	no	—
49	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	C₄×C₄⋊C₄	—	3	2	no	—
53	#340	—	3	2	no	—
54	#336	—	6	2	no	—
55	#334	—	6	2	no	—
56	C₈²	—	2	2	no	—
57	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	#41	1	1	yes	characteristic
60	C₂²×Dic₃	—	2	2	no	—
61	C₂²×Dic₃	—	2	2	no	—
62	C₂×C₃×C₈	—	2	4	no	—
63	C₂×C₃×C₈	—	2	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	—	2	1	no	—
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
69	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
71	C₂×C₃×C₈	M₁₆	1	2	yes	—
72	C₂×C₃×C₈	SD₁₆	1	4	yes	—
73	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
75	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
76	C₃×C₄²	#41	1	1	yes	characteristic
77	C₄⋊Dic₃	—	2	2	no	—
78	C₄⋊Dic₃	—	2	2	no	—
79	C₄⋊Dic₃	—	2	2	no	—
80	C₄⋊Dic₃	C₂×C₈	1	4	yes	—
81	C₄×Dic₃	—	4	2	no	—
82	C₄×Dic₃	—	4	2	no	—
83	C₂×C₃⋊C₈	—	4	2	no	—
84	C₂×C₃⋊C₈	—	4	2	no	—
85	C₂×C₃⋊C₈	—	4	2	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	—	6	2	no	—
88	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
89	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
90	C₂²×C₈	D₁₂	1	2	yes	—
91	C₂×C₄⋊C₄	—	3	2	no	—
92	C₂×C₄⋊C₄	—	3	2	no	—
93	C₂×C₄²	—	6	2	no	—
94	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
95	#106	—	3	4	no	—
96	#106	—	6	2	no	—
97	C₄⋊C₈	—	6	2	no	—
98	C₄⋊C₈	—	6	2	no	—
99	C₄⋊C₈	—	12	4	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	Dic₆	1	2	yes	—
104	C₄×C₈	Dic₆	1	2	yes	—
105	C₄×C₈	#67	1	2	yes	—
106	#95	—	6	2	no	—
107	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₃×C₄	#121	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₃×C₄	#125	1	1	yes	characteristic
114	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₃×C₄	#131	1	1	yes	characteristic
118	C₂×C₃×C₄	#131	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
120	C₂×C₃×C₄	#135	1	2	yes	—
121	C₂×Dic₃	—	2	4	no	—
122	C₂×Dic₃	—	4	2	no	—
123	C₂×Dic₃	—	4	2	no	—
124	C₂×Dic₃	—	4	2	no	—
125	C₂×Dic₃	—	4	2	no	—
126	C₂×Dic₃	—	4	2	no	—
127	C₃×C₈	—	2	4	no	—
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	—	2	4	no	—
130	C₃×C₈	C₄⋊C₈	1	4	yes	complemented
131	C₃⋊C₈	—	8	2	no	—
132	C₂²×C₄	—	6	2	no	—
133	C₂²×C₄	—	6	2	no	—
134	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
136	C₂²×C₄	#235	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	12	2	no	—
144	C₂×C₈	—	12	2	no	—
145	C₂×C₈	—	12	2	no	—
146	C₂×C₈	D₃×C₈	1	2	yes	—
147	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
148	C₂×C₈	#204	1	4	yes	—
149	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
150	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
151	C₄⋊C₄	—	3	4	no	—
152	C₄⋊C₄	—	6	2	no	—
153	C₄⋊C₄	—	6	2	no	—
154	C₄⋊C₄	—	6	2	no	—
155	C₄²	—	12	2	no	—
156	C₄²	—	12	2	no	—
157	C₄²	C₂×Dic₆	1	1	yes	characteristic
158	C₄²	C₂×Dic₆	1	1	yes	characteristic
159	C₄²	#235	1	1	yes	characteristic
160	C₄²	C₂×#67	1	1	yes	characteristic
161	C₂²×C₃	#389	1	1	yes	characteristic
162	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
163	C₂²×C₃	#436	1	1	yes	characteristic
164	C₂²×C₃	C₄×SD₁₆	1	2	yes	—
165	C₂²×C₃	#443	1	2	yes	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
169	C₃×C₄	#424	1	1	yes	characteristic
170	C₃×C₄	C₈×Q₈	1	1	yes	characteristic
171	C₃×C₄	#446	1	1	yes	characteristic
172	C₃×C₄	#465	1	2	yes	—
173	C₃×C₄	#498	1	1	yes	characteristic
174	C₃×C₄	#499	1	1	yes	characteristic
175	Dic₃	—	4	4	no	—
176	Dic₃	—	8	2	no	—
177	C₂³	C₄×D₁₂	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	6	4	no	—
180	C₂×C₄	—	12	2	no	—
181	C₂×C₄	—	12	2	no	—
182	C₂×C₄	—	12	2	no	—
183	C₂×C₄	—	12	2	no	—
184	C₂×C₄	—	12	2	no	—
185	C₂×C₄	C₄×Dic₆	1	1	yes	characteristic
186	C₂×C₄	#869	1	1	yes	characteristic
187	C₂×C₄	#870	1	1	yes	characteristic
188	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
189	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
190	C₂×C₄	#901	1	1	yes	characteristic
191	C₂×C₄	#901	1	1	yes	characteristic
192	C₂×C₄	C₂×#204	1	2	yes	—
193	C₂×C₄	#904	1	2	yes	—
194	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
195	C₂×C₄	#924	1	1	yes	characteristic
196	C₂×C₄	#930	1	1	yes	characteristic
197	C₈	—	2	4	no	—
198	C₈	—	2	2	no	—
199	C₈	—	2	4	no	—
200	C₈	—	24	2	no	—
201	C₈	Dic₃⋊C₈	1	4	yes	complemented
202	C₂×C₃	C₈×SD₁₆	1	2	yes	—
203	C₂×C₃	#1590	1	2	yes	—
204	C₂×C₃	#1939	1	1	yes	characteristic
205	C₂×C₃	#1941	1	1	yes	characteristic
206	C₂×C₃	#1945	1	1	yes	characteristic
207	C₂²	C₈×D₁₂	1	1	yes	characteristic
208	C₂²	#4688	1	1	yes	characteristic
209	C₂²	C₄×#204	1	2	yes	—
210	C₂²	#4700	1	2	yes	—
211	C₂²	#4928	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	12	4	no	—
215	C₄	—	24	2	no	—
216	C₄	C₈×Dic₆	1	1	yes	characteristic
217	C₄	#4679	1	1	yes	characteristic
218	C₄	#4680	1	1	yes	characteristic
219	C₄	#4683	1	1	yes	characteristic
220	C₄	C₂×Dic₃⋊C₈	1	1	yes	characteristic
221	C₄	#5102	1	1	yes	characteristic
222	C₄	#5114	1	2	yes	—
223	C₃	#9593	1	1	yes	characteristic, complemented, Sylow
224	C₂	C₈×#204	1	2	yes	—
225	C₂	#68055	1	2	yes	—
226	C₂	#69769	1	1	yes	characteristic
227	C₂	#69786	1	1	yes	characteristic
228	C₂	#69794	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₈²	C₂
socle, prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₈²	D₃
hypocenter	C₃	#9593
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
41. #817	C₈	semidirect
130. C₃×C₈	C₄⋊C₈	semidirect
201. C₈	Dic₃⋊C₈	semidirect
223. C₃	#9593	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈²	2
#69786	2
C₄×#817	2
C₂×#4457	2
#67655	2
#9593	3