Subgroups — #10804058

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3162	C₂	1	1	yes	maximal, characteristic
2	#83328	C₂	1	2	yes	maximal
3	#71840	C₂	1	2	yes	maximal
4	#70663	C₂	1	4	yes	maximal
5	#70659	C₂	1	4	yes	maximal
6	#69842	C₂	1	1	yes	maximal, characteristic
7	#69804	C₂	1	1	yes	maximal, characteristic
8	#21735	—	3	1	no	maximal, Sylow
9	C₃×#533	C₂²	1	2	yes	—
10	C₃×#501	C₄	1	8	yes	complemented
11	C₃×#427	C₂²	1	4	yes	—
12	C₂×C₃×#97	C₂²	1	1	yes	characteristic
13	C₂×#819	C₂²	1	4	yes	—
14	C₂×#815	C₂²	1	2	yes	—
15	C₂×C₈×Dic₃	C₂²	1	2	yes	—
16	#5024	—	2	4	no	—
17	Q₈×C₃⋊C₈	—	2	4	no	—
18	#4999	C₂²	1	4	yes	—
19	#4997	C₂²	1	4	yes	—
20	#4932	C₂²	1	1	yes	characteristic
21	#4924	C₂²	1	2	yes	—
22	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	#4532	C₂²	1	8	yes	—
24	#3162	D₃	1	1	yes	characteristic
25	#2960	—	3	2	no	—
26	#1995	—	3	2	no	—
27	#1946	—	3	4	no	—
28	#1849	—	3	1	no	—
29	#1772	—	3	4	no	—
30	#1726	—	3	1	no	—
31	C₂²×C₃×C₈	C₂³	1	2	yes	—
32	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
33	C₃×#125	C₂×C₄	1	4	yes	—
34	C₃×C₂²⋊Q₈	—	2	4	no	—
35	C₃×C₄×Q₈	—	2	4	no	—
36	C₃×#117	C₂³	1	4	yes	—
37	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₃	C₂³	1	2	yes	—
39	C₂×#208	C₂³	1	4	yes	—
40	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₃×#107	C₂×C₄	1	8	yes	—
43	C₃×#106	C₂×C₄	1	8	yes	—
44	C₃×#97	C₂×C₄	1	4	yes	—
45	C₃⋊#95	—	2	2	no	—
46	#819	—	2	8	no	—
47	#815	—	2	4	no	—
48	C₈×Dic₃	Q₈	1	8	yes	complemented
49	C₄⋊(C₃⋊C₈)	—	2	4	no	—
50	C₄⋊(C₃⋊C₈)	—	2	8	no	—
51	#803	—	2	4	no	—
52	C₄×C₃⋊C₈	—	2	4	no	—
53	C₄×C₃⋊C₈	D₄	1	2	yes	—
54	C₄×C₃⋊C₈	D₄	1	2	yes	—
55	#533	C₂×D₃	1	2	yes	—
56	#501	Dic₃	1	8	yes	complemented
57	#446	—	6	4	no	—
58	C₈×Q₈	—	6	4	no	—
59	C₂×#108	—	3	4	no	—
60	#427	C₂×D₃	1	4	yes	—
61	#424	—	3	4	no	—
62	#423	—	3	2	no	—
63	#405	—	3	4	no	—
64	C₂×#97	—	3	2	no	—
65	C₂×#97	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄×C₈	—	3	2	no	—
67	C₂×C₄×C₈	—	3	1	no	—
68	#376	—	3	1	no	—
69	#339	—	3	8	no	—
70	C₂×C₃×Q₈	—	2	4	no	—
71	C₂²×C₃×C₄	#41	1	2	yes	—
72	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₃	—	2	2	no	—
74	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
75	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
76	C₂×C₃×C₈	#41	1	4	yes	—
77	C₃×C₄⋊C₄	—	2	8	no	—
78	C₃×C₄⋊C₄	—	2	4	no	—
79	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
80	C₃×C₂²⋊C₄	—	2	4	no	—
81	C₃×C₄²	—	2	4	no	—
82	C₃×C₄²	C₂²×C₄	1	2	yes	—
83	C₃×C₄²	C₂×D₄	1	2	yes	—
84	C₄×Dic₃	—	2	2	no	—
85	C₄×Dic₃	C₂×Q₈	1	4	yes	—
86	#208	—	2	8	no	—
87	C₂×C₃⋊C₈	—	2	2	no	—
88	C₂×C₃⋊C₈	—	2	4	no	—
89	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃⋊C₈	C₂×Q₈	1	4	yes	—
93	C₂×M₁₆	—	3	4	no	—
94	C₂²×C₈	—	3	1	no	—
95	C₂²×C₈	—	3	1	no	—
96	C₂²×C₈	C₂²×D₃	1	2	yes	—
97	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
98	#125	C₂×Dic₃	1	4	yes	—
99	C₂²⋊Q₈	—	2	4	no	—
100	C₄×Q₈	—	2	4	no	—
101	#117	C₂²×D₃	1	4	yes	—
102	C₂×C₄²	—	3	2	no	—
103	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
104	#108	—	6	8	no	—
105	#107	C₂×Dic₃	1	8	yes	—
106	#106	C₂×Dic₃	1	8	yes	—
107	C₄⋊C₈	—	6	8	no	—
108	C₄⋊C₈	—	6	4	no	—
109	#97	—	6	4	no	—
110	#97	—	6	4	no	—
111	#97	C₂×Dic₃	1	4	yes	—
112	C₄×C₈	—	3	8	no	—
113	C₄×C₈	—	3	2	no	—
114	C₄×C₈	—	3	2	no	—
115	C₄×C₈	—	6	4	no	—
116	#95	—	6	2	no	—
117	C₂³×C₃	C₂×#41	1	1	yes	characteristic
118	C₃×Q₈	—	4	8	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	—	4	2	no	—
121	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
122	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
125	C₂×C₃×C₄	C₂×#41	1	2	yes	—
126	C₂×Dic₃	—	2	4	no	—
127	C₂×Dic₃	—	4	4	no	—
128	C₃×C₈	C₄×Q₈	1	8	yes	—
129	C₃⋊C₈	—	2	2	no	—
130	C₃⋊C₈	—	4	4	no	—
131	C₃⋊C₈	C₄⋊Q₈	1	2	yes	—
132	C₃⋊C₈	C₄⋊Q₈	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂²×C₄	—	6	2	no	—
135	C₂²×C₄	#237	1	2	yes	—
136	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
137	M₁₆	—	6	8	no	—
138	C₂×C₈	—	3	1	no	—
139	C₂×C₈	—	3	1	no	—
140	C₂×C₈	—	3	4	no	—
141	C₂×C₈	—	3	2	no	—
142	C₂×C₈	—	6	2	no	—
143	C₂×C₈	—	6	4	no	—
144	C₂×C₈	D₃×Q₈	1	4	yes	—
145	C₂×C₈	#239	1	4	yes	—
146	C₂×C₈	C₂²×Dic₃	1	4	yes	—
147	C₄⋊C₄	—	2	8	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	C₂²×Dic₃	1	8	yes	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	3	4	no	—
153	C₄²	—	6	2	no	—
154	C₄²	D₃×D₄	1	2	yes	—
155	C₄²	C₂²×Dic₃	1	2	yes	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₃	#533	1	2	yes	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	4	8	no	—
161	C₃×C₄	C₂×C₄×Q₈	1	2	yes	—
162	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₃×C₄	#532	1	1	yes	characteristic
164	Dic₃	—	4	4	no	—
165	C₂³	C₂×#237	1	1	yes	characteristic
166	Q₈	—	4	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	6	4	no	—
170	C₂×C₄	—	12	4	no	—
171	C₂×C₄	D₄×Dic₃	1	4	yes	—
172	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂×D₃×Q₈	1	2	yes	—
174	C₂×C₄	C₂×#239	1	2	yes	—
175	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
176	C₈	—	3	2	no	—
177	C₈	—	3	2	no	—
178	C₈	—	6	2	no	—
179	C₈	—	12	4	no	—
180	C₈	Q₈×Dic₃	1	8	yes	—
181	C₂×C₃	—	2	2	no	—
182	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
183	C₂×C₃	#2955	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	#5728	1	2	yes	—
186	C₂²	C₂×D₄×Dic₃	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	4	8	no	—
189	C₄	—	12	4	no	—
190	C₄	D₃×C₄⋊Q₈	1	1	yes	characteristic
191	C₄	#5725	1	1	yes	characteristic
192	C₄	C₂×Q₈×Dic₃	1	2	yes	—
193	C₃	#21735	1	1	yes	characteristic, complemented, Sylow
194	C₂	—	2	2	no	—
195	C₂	Dic₃×C₄⋊Q₈	1	1	yes	characteristic
196	C₂	#83999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×#3162	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#3162	D₃
hypocenter	C₃	#21735
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

Complements

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

N	H ≅ G/N	type
10. C₃×#501	C₄	semidirect
48. C₈×Dic₃	Q₈	semidirect
56. #501	Dic₃	semidirect
193. C₃	#21735	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3162	2
#83328	2
#71840	2
#70663	2
#70659	2
#69842	2
#69804	2
#21735	3