Subgroups — #10711535

Order 768.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₃⋊C₁₆	C₂	1	2	yes	maximal
2	#69749	C₂	1	1	yes	maximal, characteristic
3	C₃×#1374	C₂	1	1	yes	maximal, characteristic
4	#67804	C₂	1	2	yes	maximal
5	#67657	C₂	1	1	yes	maximal, characteristic
6	#12880	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#815	C₄	1	1	yes	characteristic
10	C₂×#815	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
14	#4932	C₄	1	1	yes	characteristic
15	C₂×#803	C₄	1	1	yes	characteristic
16	Dic₃⋊C₁₆	—	2	4	no	—
17	C₂×C₄⋊C₁₆	—	3	2	no	—
18	#1726	—	3	1	no	—
19	#1374	—	3	2	no	—
20	#1374	—	3	1	no	—
21	#1374	D₃	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₁₆	—	2	2	no	—
30	C₂×C₃×C₁₆	—	2	2	no	—
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₃×C₄×C₈	—	2	2	no	—
33	C₃⋊#95	C₈	1	2	yes	—
34	C₃⋊#95	C₈	1	2	yes	—
35	#815	—	2	2	no	—
36	#815	—	2	2	no	—
37	C₈×Dic₃	—	2	2	no	—
38	C₈×Dic₃	D₄	1	2	yes	—
39	C₈×Dic₃	Q₈	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	#803	—	2	2	no	—
44	C₂²×C₁₆	—	3	2	no	—
45	C₂²×C₁₆	C₂×D₃	1	2	yes	—
46	C₂×#97	—	3	1	no	—
47	C₂×#97	—	3	1	no	—
48	C₂×#97	—	3	1	no	—
49	C₂×C₄×C₈	—	3	1	no	—
50	C₂×C₄×C₈	—	3	1	no	—
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	#376	—	3	1	no	—
53	C₄⋊C₁₆	—	6	4	no	—
54	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
55	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×Dic₃	C₂×C₈	1	2	yes	—
59	C₂×C₃×C₈	—	2	1	no	—
60	C₂×C₃×C₈	—	2	1	no	—
61	C₂×C₃×C₈	M₁₆	1	2	yes	—
62	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₈	#41	1	2	yes	—
65	C₂×C₃×C₈	#41	1	2	yes	—
66	C₃×C₄²	—	2	1	no	—
67	C₃×C₄²	—	2	1	no	—
68	C₄×Dic₃	—	2	1	no	—
69	C₄×Dic₃	—	2	1	no	—
70	C₄×Dic₃	C₄⋊C₄	1	2	yes	—
71	C₄×Dic₃	M₁₆	1	2	yes	—
72	C₂×C₃⋊C₈	—	2	1	no	—
73	C₂×C₃⋊C₈	—	2	1	no	—
74	C₂×C₃⋊C₈	—	2	1	no	—
75	C₂×C₃⋊C₈	—	2	1	no	—
76	C₂×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
77	C₂×C₃⋊C₈	M₁₆	1	2	yes	—
78	C₃×C₁₆	—	4	2	no	—
79	C₃⋊C₁₆	—	4	2	no	—
80	C₂²×C₈	—	3	1	no	—
81	C₂²×C₈	—	3	1	no	—
82	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₄²	—	3	1	no	—
85	C₂×C₄²	—	3	1	no	—
86	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	6	2	no	—
91	C₂×C₁₆	—	6	2	no	—
92	C₂×C₁₆	—	6	2	no	—
93	#97	—	6	2	no	—
94	#97	—	6	2	no	—
95	#97	—	6	2	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	3	2	no	—
98	C₄×C₈	—	3	2	no	—
99	C₄×C₈	—	6	2	no	—
100	#95	—	3	2	no	—
101	#95	—	3	2	no	—
102	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	#117	1	1	yes	characteristic
108	C₂×C₃×C₄	#117	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
112	C₂×Dic₃	—	2	2	no	—
113	C₂×Dic₃	—	2	2	no	—
114	C₂×Dic₃	—	2	1	no	—
115	C₂×Dic₃	—	2	1	no	—
116	C₂×Dic₃	—	2	2	no	—
117	C₂×Dic₃	—	2	2	no	—
118	C₂×Dic₃	C₄⋊C₈	1	4	yes	—
119	C₃×C₈	—	2	2	no	—
120	C₃×C₈	#124	1	2	yes	—
121	C₃×C₈	#125	1	2	yes	—
122	C₃⋊C₈	—	2	2	no	—
123	C₃⋊C₈	—	2	2	no	—
124	C₂²×C₄	—	3	2	no	—
125	C₂²×C₄	—	3	1	no	—
126	C₂²×C₄	—	3	1	no	—
127	C₂²×C₄	D₃×C₈	1	2	yes	—
128	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	3	2	no	—
132	C₂×C₈	—	3	2	no	—
133	C₂×C₈	—	6	1	no	—
134	C₂×C₈	—	6	1	no	—
135	C₂×C₈	—	6	1	no	—
136	C₂×C₈	—	6	1	no	—
137	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
138	C₂×C₈	#235	1	2	yes	—
139	C₂×C₈	D₃×D₄	1	1	yes	characteristic
140	C₂×C₈	#237	1	2	yes	—
141	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	3	2	no	—
145	C₄²	—	3	2	no	—
146	C₄²	—	6	1	no	—
147	C₄²	—	6	1	no	—
148	C₁₆	—	4	2	no	—
149	C₁₆	—	12	2	no	—
150	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₂²×C₃	#431	1	1	yes	characteristic
152	C₂²×C₃	#431	1	1	yes	characteristic
153	C₂²×C₃	#504	1	2	yes	—
154	C₂²×C₃	#504	1	2	yes	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	#382	1	1	yes	characteristic
157	C₃×C₄	#423	1	1	yes	characteristic
158	C₃×C₄	#432	1	1	yes	characteristic
159	C₃×C₄	#432	1	1	yes	characteristic
160	Dic₃	—	2	4	no	—
161	Dic₃	—	4	1	no	—
162	Dic₃	—	4	1	no	—
163	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	3	4	no	—
168	C₂×C₄	—	6	2	no	—
169	C₂×C₄	—	6	2	no	—
170	C₂×C₄	—	6	2	no	—
171	C₂×C₄	—	6	2	no	—
172	C₂×C₄	—	6	1	no	—
173	C₂×C₄	—	6	1	no	—
174	C₂×C₄	#872	1	1	yes	characteristic
175	C₂×C₄	#877	1	1	yes	characteristic
176	C₂×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
178	C₂×C₄	D₃×M₁₆	1	1	yes	characteristic
179	C₂×C₄	D₃×M₁₆	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	6	2	no	—
182	C₈	—	6	2	no	—
183	C₈	#885	1	2	yes	—
184	C₈	#889	1	2	yes	—
185	C₂×C₃	#1831	1	1	yes	characteristic
186	C₂×C₃	#2151	1	2	yes	—
187	C₂×C₃	#2164	1	1	yes	characteristic
188	C₂×C₃	#2164	1	1	yes	characteristic
189	C₂×C₃	#2164	1	2	yes	—
190	C₂²	#4685	1	1	yes	characteristic
191	C₂²	#4718	1	1	yes	characteristic
192	C₂²	D₃×C₄⋊C₈	1	1	yes	characteristic
193	C₂²	#4901	1	2	yes	—
194	C₂²	#4907	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	6	4	no	—
197	C₄	—	12	1	no	—
198	C₄	—	12	1	no	—
199	C₄	#4650	1	1	yes	characteristic
200	C₄	#4705	1	1	yes	characteristic
201	C₄	#4719	1	1	yes	characteristic
202	C₄	#4837	1	1	yes	characteristic
203	C₃	#12880	1	1	yes	characteristic, complemented, Sylow
204	C₂	#68605	1	1	yes	characteristic
205	C₂	#69053	1	1	yes	characteristic
206	C₂	#69074	1	1	yes	characteristic
207	C₂	#69094	1	2	yes	—
208	C₂	#69285	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator, socle, prime radical	C₂³×C₃	C₂²×C₈
Fitting	C₃×#1374	C₂
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1374	D₃
hypocenter	C₃	#12880
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
203. C₃	#12880	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₃⋊C₁₆	2
#69749	2
C₃×#1374	2
#67804	2
#67657	2
#12880	3