Subgroups — #10713247

Order 768.

274 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2108	C₂	1	1	yes	maximal, characteristic
2	C₂×#4506	C₂	1	1	yes	maximal, characteristic
3	C₂×#4504	C₂	1	1	yes	maximal, characteristic
4	#69894	C₂	1	2	yes	maximal
5	#67817	C₂	1	2	yes	maximal
6	#13171	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×#405	C₂²	1	1	yes	characteristic
10	C₂×#819	C₂²	1	2	yes	—
11	C₂×#818	C₂²	1	2	yes	—
12	#4995	—	2	2	no	—
13	C₃×#346	—	2	1	no	—
14	C₃×C₄×C₁₆	—	2	1	no	—
15	#4554	—	2	2	no	—
16	#4506	C₄	1	4	yes	complemented
17	#4504	—	2	1	no	—
18	#4504	—	2	1	no	—
19	C₂×#368	—	3	1	no	—
20	C₂×#366	—	3	1	no	—
21	#2108	D₃	1	1	yes	characteristic
22	#1952	—	3	2	no	—
23	#1387	—	3	2	no	—
24	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₃×#117	D₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₃	—	2	2	no	—
28	C₂×#208	—	2	2	no	—
29	C₃×M₃₂	—	2	1	no	—
30	C₃×M₃₂	Q₈	1	2	yes	—
31	C₂×C₃×C₁₆	—	2	1	no	—
32	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
34	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
35	C₂×C₃×C₁₆	Q₈	1	2	yes	—
36	C₃×#97	—	2	1	no	—
37	C₃×C₄×C₈	—	2	1	no	—
38	#819	—	2	2	no	—
39	#819	C₂×C₄	1	4	yes	—
40	#818	—	2	2	no	—
41	#818	—	2	2	no	—
42	#808	—	4	2	no	—
43	#807	—	4	2	no	—
44	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
46	C₂×#108	—	3	2	no	—
47	#428	—	6	2	no	—
48	C₂×#107	—	3	2	no	—
49	#405	C₂×D₃	1	1	yes	characteristic
50	#368	—	3	4	no	—
51	#366	—	6	1	no	—
52	#366	—	6	1	no	—
53	#346	—	2	1	no	—
54	C₄×C₁₆	—	2	1	no	—
55	#343	—	6	2	no	—
56	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₃	—	4	2	no	—
58	C₃×M₁₆	—	2	1	no	—
59	C₃×M₁₆	Q₁₆	1	2	yes	—
60	C₂×C₃×C₈	—	2	1	no	—
61	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃×C₈	#41	1	1	yes	characteristic
66	C₂×C₃×C₈	#41	1	1	yes	characteristic
67	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
68	C₃×C₄⋊C₄	D₈	1	1	yes	characteristic
69	C₃×C₂²⋊C₄	—	2	1	no	—
70	C₃×C₄²	—	2	1	no	—
71	C₄⋊Dic₃	—	4	2	no	—
72	C₄⋊Dic₃	—	4	2	no	—
73	#208	—	2	4	no	—
74	#208	—	4	2	no	—
75	C₂×C₃⋊C₈	—	4	2	no	—
76	C₃×C₁₆	—	2	2	no	—
77	C₃×C₁₆	C₄⋊C₄	1	2	yes	—
78	C₃×C₁₆	C₄⋊C₄	1	2	yes	—
79	C₂×M₁₆	—	6	2	no	—
80	C₂×M₁₆	D₁₂	1	1	yes	characteristic
81	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
82	#117	D₁₂	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	6	2	no	—
84	M₃₂	—	2	1	no	—
85	M₃₂	Dic₆	1	2	yes	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	Dic₆	1	2	yes	—
88	C₂×C₁₆	C₄×D₃	1	2	yes	—
89	C₂×C₁₆	#67	1	1	yes	characteristic
90	C₂×C₁₆	#67	1	1	yes	characteristic
91	#108	—	3	4	no	—
92	#108	—	6	2	no	—
93	#107	—	6	2	no	—
94	#107	—	6	2	no	—
95	#107	—	12	2	no	—
96	#106	—	12	2	no	—
97	#97	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₂³×C₃	#135	1	1	yes	characteristic
100	C₂×C₃×C₄	—	2	1	no	—
101	C₂×C₃×C₄	—	2	1	no	—
102	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
103	C₂×C₃×C₄	#121	1	1	yes	characteristic
104	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₃×C₄	#135	1	1	yes	characteristic
108	C₂×Dic₃	—	8	2	no	—
109	C₂×Dic₃	—	16	2	no	—
110	C₃×C₈	—	2	2	no	—
111	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
113	C₃×C₈	C₂²⋊Q₈	1	1	yes	characteristic
114	C₃×C₈	#125	1	1	yes	characteristic
115	C₃⋊C₈	—	4	4	no	—
116	C₂²×C₄	—	12	2	no	—
117	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
118	M₁₆	—	2	1	no	—
119	M₁₆	—	6	4	no	—
120	M₁₆	—	12	2	no	—
121	M₁₆	Dic₁₂	1	2	yes	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	12	2	no	—
124	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
125	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
126	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
127	C₂×C₈	#235	1	1	yes	characteristic
128	C₂×C₈	#235	1	1	yes	characteristic
129	C₂×C₈	C₂×#67	1	1	yes	characteristic
130	C₄⋊C₄	—	12	2	no	—
131	C₄⋊C₄	—	12	2	no	—
132	C₄⋊C₄	D₂₄	1	1	yes	characteristic
133	C₄⋊C₄	D₂₄	1	1	yes	characteristic
134	C₂²⋊C₄	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₁₆	—	2	2	no	—
137	C₁₆	Dic₃⋊C₄	1	2	yes	—
138	C₁₆	Dic₃⋊C₄	1	2	yes	—
139	C₂²×C₃	—	2	1	no	—
140	C₂²×C₃	—	2	1	no	—
141	C₂²×C₃	C₄×Q₁₆	1	1	yes	characteristic
142	C₂²×C₃	#466	1	1	yes	characteristic
143	C₂²×C₃	#499	1	1	yes	characteristic
144	C₃×C₄	—	4	1	no	—
145	C₃×C₄	—	4	1	no	—
146	C₃×C₄	#389	1	1	yes	characteristic
147	C₃×C₄	C₄×D₈	1	1	yes	characteristic
148	C₃×C₄	#467	1	1	yes	characteristic
149	C₃×C₄	#500	1	1	yes	characteristic
150	Dic₃	—	16	2	no	—
151	C₂³	#904	1	1	yes	characteristic
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	24	2	no	—
155	C₂×C₄	—	48	2	no	—
156	C₂×C₄	#869	1	1	yes	characteristic
157	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
158	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
159	C₂×C₄	#904	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
161	C₂×C₄	#930	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	12	4	no	—
164	C₈	C₄×Dic₆	1	1	yes	characteristic
165	C₈	#870	1	1	yes	characteristic
166	C₈	C₂×Dic₃⋊C₄	1	1	yes	characteristic
167	C₈	#924	1	1	yes	characteristic
168	C₂×C₃	—	2	1	no	—
169	C₂×C₃	—	2	1	no	—
170	C₂×C₃	#1942	1	1	yes	characteristic
171	C₂×C₃	#2221	1	1	yes	characteristic
172	C₂×C₃	#2222	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#4683	1	1	yes	characteristic
176	C₂²	C₄×Dic₁₂	1	1	yes	characteristic
177	C₂²	#5115	1	1	yes	characteristic
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	48	2	no	—
181	C₄	#4682	1	1	yes	characteristic
182	C₄	C₄×D₂₄	1	1	yes	characteristic
183	C₄	#4928	1	1	yes	characteristic
184	C₄	#5116	1	1	yes	characteristic
185	C₃	#13171	1	1	yes	characteristic, complemented, Sylow
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#69770	1	1	yes	characteristic
189	C₂	#71454	1	1	yes	characteristic
190	C₂	#71456	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#5115
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#2108	C₂
socle	C₂²×C₃	#466
prime radical	C₂³×C₃	#135
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#2108	D₃
hypocenter	C₃	#13171
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
16. #4506	C₄	semidirect
185. C₃	#13171	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2108	2
C₂×#4506	2
C₂×#4504	2
#69894	2
#67817	2
#13171	3