Subgroups — #10693796

Order 768.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#334	C₂	1	1	yes	maximal, characteristic
2	#70420	C₂	1	2	yes	maximal
3	C₄×#819	C₂	1	1	yes	maximal, characteristic
4	#69724	C₂	1	1	yes	maximal, characteristic
5	#67706	C₂	1	2	yes	maximal
6	#9607	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	C₄	1	1	yes	characteristic
10	C₂×#819	C₄	1	1	yes	characteristic
11	#4925	C₂²	1	1	yes	characteristic
12	#4925	C₂²	1	1	yes	characteristic
13	C₄×#208	C₂²	1	2	yes	—
14	C₃×#334	—	2	1	no	—
15	C₃×#334	—	2	1	no	—
16	#4552	C₄	1	2	yes	—
17	#4480	—	2	2	no	—
18	#4479	—	2	2	no	—
19	#4455	—	2	1	no	—
20	C₈×C₃⋊C₈	—	2	1	no	—
21	C₄×#108	—	3	1	no	—
22	#1565	—	3	2	no	—
23	C₂×#334	D₃	1	1	yes	characteristic
24	#1450	—	3	1	no	—
25	#1283	—	3	2	no	—
26	C₂²×C₃×C₈	—	2	2	no	—
27	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×#208	C₂×C₄	1	2	yes	—
31	C₂×#208	C₂×C₄	1	2	yes	—
32	C₃×C₄⋊C₈	—	2	2	no	—
33	C₃×C₄⋊C₈	—	2	2	no	—
34	C₃×C₄×C₈	D₄	1	1	yes	characteristic
35	C₃×C₄×C₈	D₄	1	1	yes	characteristic
36	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
37	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
38	#830	—	2	1	no	—
39	#830	Q₈	1	2	yes	—
40	#819	—	2	2	no	—
41	#819	C₈	1	4	yes	complemented
42	C₄⋊(C₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
44	#803	—	2	2	no	—
45	C₄×C₃⋊C₈	—	2	2	no	—
46	C₄×C₃⋊C₈	—	2	1	no	—
47	C₄×C₃⋊C₈	—	2	1	no	—
48	#431	—	3	1	no	—
49	#431	—	3	1	no	—
50	C₂×#108	—	3	1	no	—
51	C₂×#108	—	3	1	no	—
52	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
53	C₄×M₁₆	—	3	2	no	—
54	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	#341	—	3	2	no	—
56	#335	—	6	2	no	—
57	#334	—	2	1	no	—
58	#334	—	2	1	no	—
59	#334	—	6	2	no	—
60	#322	—	6	1	no	—
61	C₈²	—	6	1	no	—
62	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₃×C₄	#41	1	1	yes	characteristic
65	C₂×C₃×C₈	—	2	1	no	—
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	4	2	no	—
68	C₂×C₃×C₈	—	8	2	no	—
69	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
74	C₂×C₃×C₈	M₁₆	1	2	yes	—
75	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
78	C₃×C₄²	#41	1	1	yes	characteristic
79	#208	—	2	2	no	—
80	#208	—	2	2	no	—
81	#208	—	2	2	no	—
82	#208	C₂×C₈	1	4	yes	—
83	C₂×C₃⋊C₈	—	2	2	no	—
84	C₂×C₃⋊C₈	—	2	2	no	—
85	C₂×C₃⋊C₈	—	2	2	no	—
86	C₂×C₃⋊C₈	—	2	2	no	—
87	C₂×M₁₆	—	3	2	no	—
88	C₂×M₁₆	—	3	2	no	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
91	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
92	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
93	#108	—	3	4	no	—
94	#108	—	6	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	3	2	no	—
98	C₄⋊C₈	—	6	1	no	—
99	C₂²⋊C₈	—	3	2	no	—
100	C₂²⋊C₈	—	6	1	no	—
101	#97	—	6	2	no	—
102	C₄×C₈	—	6	2	no	—
103	C₄×C₈	—	6	1	no	—
104	C₄×C₈	—	6	1	no	—
105	C₄×C₈	Dic₆	1	1	yes	characteristic
106	C₄×C₈	Dic₆	1	1	yes	characteristic
107	C₄×C₈	#67	1	1	yes	characteristic
108	C₄×C₈	#67	1	1	yes	characteristic
109	C₂³×C₃	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	4	1	no	—
112	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	#121	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₃×C₄	#125	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₃×C₄	#131	1	1	yes	characteristic
121	C₂×C₃×C₄	#131	1	1	yes	characteristic
122	C₃×C₈	—	2	1	no	—
123	C₃×C₈	—	2	1	no	—
124	C₃×C₈	—	8	2	no	—
125	C₃×C₈	C₄⋊C₈	1	2	yes	—
126	C₃×C₈	C₄⋊C₈	1	2	yes	—
127	C₃⋊C₈	—	2	4	no	—
128	C₃⋊C₈	—	4	2	no	—
129	C₃⋊C₈	—	4	2	no	—
130	C₃⋊C₈	—	4	2	no	—
131	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
132	C₂²×C₄	#237	1	1	yes	characteristic
133	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
134	M₁₆	—	3	4	no	—
135	M₁₆	—	6	2	no	—
136	M₁₆	—	6	2	no	—
137	M₁₆	—	6	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	6	2	no	—
142	C₂×C₈	—	6	2	no	—
143	C₂×C₈	—	6	2	no	—
144	C₂×C₈	—	6	2	no	—
145	C₂×C₈	—	8	2	no	—
146	C₂×C₈	D₃×C₈	1	2	yes	—
147	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
148	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
149	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
150	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
151	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
152	C₄²	C₂×Dic₆	1	1	yes	characteristic
153	C₄²	D₃×D₄	1	1	yes	characteristic
154	C₄²	#239	1	1	yes	characteristic
155	C₄²	C₂×#67	1	1	yes	characteristic
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	#389	1	1	yes	characteristic
159	C₂²×C₃	C₈×Q₈	1	1	yes	characteristic
160	C₂²×C₃	#446	1	1	yes	characteristic
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	1	no	—
164	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₃×C₄	#424	1	1	yes	characteristic
166	C₃×C₄	C₈×D₄	1	1	yes	characteristic
167	C₃×C₄	#436	1	1	yes	characteristic
168	C₃×C₄	#471	1	2	yes	—
169	C₂³	#887	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	#888	1	1	yes	characteristic
173	C₂×C₄	#890	1	1	yes	characteristic
174	C₂×C₄	#893	1	1	yes	characteristic
175	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
176	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
177	C₂×C₄	#907	1	1	yes	characteristic
178	C₂×C₄	#907	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
180	C₂×C₄	#938	1	1	yes	characteristic
181	C₂×C₄	#946	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	6	4	no	—
185	C₈	—	8	2	no	—
186	C₈	—	12	2	no	—
187	C₈	—	12	2	no	—
188	C₈	—	12	2	no	—
189	C₈	Dic₃⋊C₈	1	2	yes	—
190	C₈	Dic₃⋊C₈	1	2	yes	—
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	#1939	1	1	yes	characteristic
194	C₂×C₃	#1951	1	1	yes	characteristic
195	C₂×C₃	#1953	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	#4830	1	1	yes	characteristic
199	C₂²	#4831	1	1	yes	characteristic
200	C₂²	#4972	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	#4834	1	1	yes	characteristic
205	C₄	#4838	1	1	yes	characteristic
206	C₄	C₂×Dic₃⋊C₈	1	1	yes	characteristic
207	C₄	#5119	1	1	yes	characteristic
208	C₄	#5177	1	2	yes	—
209	C₃	#9607	1	1	yes	characteristic, complemented, Sylow
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#70468	1	1	yes	characteristic
213	C₂	#70572	1	1	yes	characteristic
214	C₂	#70706	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#938
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#334	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₄×Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#334	D₃
hypocenter	C₃	#9607
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
41. #819	C₈	semidirect
209. C₃	#9607	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#334	2
#70420	2
C₄×#819	2
#69724	2
#67706	2
#9607	3