Subgroups — #10804034

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78445	C₂	1	2	yes	maximal
2	#78442	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	#69794	C₂	1	4	yes	maximal
5	#69792	C₂	1	1	yes	maximal, characteristic
6	#69779	C₂	1	4	yes	maximal
7	#69396	C₂	1	2	yes	maximal
8	#21732	—	3	1	no	maximal, Sylow
9	#5470	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
14	C₃×C₄³	C₂²	1	1	yes	characteristic
15	C₂×#819	C₂²	1	4	yes	—
16	#5106	C₂²	1	4	yes	—
17	#5102	C₂²	1	4	yes	—
18	#5101	C₂²	1	4	yes	—
19	#4924	C₂²	1	2	yes	—
20	#4683	C₄	1	4	yes	complemented
21	#4682	C₄	1	2	yes	complemented
22	#4680	C₄	1	2	yes	complemented
23	#4679	—	2	4	no	—
24	C₈×Dic₆	—	2	4	no	—
25	#4523	C₂²	1	8	yes	—
26	#3160	—	3	1	no	—
27	#2960	—	3	2	no	—
28	#1994	—	3	2	no	—
29	#1945	—	3	4	no	—
30	#1848	—	3	1	no	—
31	#1772	—	3	4	no	—
32	C₄²×C₈	D₃	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₃×C₄²	—	2	2	no	—
37	C₂×C₃×C₄²	C₂³	1	2	yes	—
38	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
39	#926	C₂³	1	4	yes	—
40	#924	—	2	4	no	—
41	C₂×#208	C₂³	1	4	yes	—
42	#870	C₂×C₄	1	4	yes	—
43	#869	C₂×C₄	1	4	yes	—
44	C₄×Dic₆	—	2	4	no	—
45	C₃×C₄×C₈	—	2	4	no	—
46	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
49	C₃×C₄×C₈	D₄	1	2	yes	—
50	C₃×C₄×C₈	D₄	1	2	yes	—
51	C₃×C₄×C₈	Q₈	1	8	yes	complemented
52	#819	—	2	8	no	—
53	#818	C₂×C₄	1	8	yes	—
54	#817	C₂×C₄	1	8	yes	—
55	#815	—	2	4	no	—
56	Dic₃⋊C₈	—	2	8	no	—
57	C₈×Dic₃	—	2	4	no	—
58	C₄⋊(C₃⋊C₈)	—	2	4	no	—
59	#533	—	3	2	no	—
60	#500	—	3	2	no	—
61	#499	—	3	4	no	—
62	#498	—	3	2	no	—
63	#446	—	6	4	no	—
64	C₈×Q₈	—	6	4	no	—
65	C₂×#108	—	3	4	no	—
66	#427	—	3	4	no	—
67	#424	—	3	4	no	—
68	#423	—	3	2	no	—
69	#405	—	3	4	no	—
70	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
71	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	C₄³	C₂×D₃	1	1	yes	characteristic
75	#339	—	3	8	no	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	#41	1	2	yes	—
78	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₆	—	2	4	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
87	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
88	C₂×C₃×C₈	#41	1	4	yes	—
89	C₃×C₄²	—	2	8	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₄²	C₂×D₄	1	2	yes	—
94	C₃×C₄²	C₂×Q₈	1	4	yes	—
95	#217	—	2	4	no	—
96	C₄⋊Dic₃	—	2	4	no	—
97	C₄⋊Dic₃	C₂²×C₄	1	8	yes	—
98	Dic₃⋊C₄	—	2	8	no	—
99	C₄×Dic₃	—	2	4	no	—
100	#208	—	2	8	no	—
101	C₂×C₃⋊C₈	—	2	4	no	—
102	C₂×M₁₆	—	3	4	no	—
103	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₃	1	2	yes	—
106	C₄⋊Q₈	—	3	4	no	—
107	#125	—	3	4	no	—
108	C₂²⋊Q₈	—	6	4	no	—
109	C₄×Q₈	—	6	4	no	—
110	#117	—	3	4	no	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₂²×D₃	1	2	yes	—
113	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
114	#108	—	6	8	no	—
115	#107	—	3	8	no	—
116	#106	—	3	8	no	—
117	C₄⋊C₈	—	6	8	no	—
118	C₄⋊C₈	—	6	4	no	—
119	#97	—	6	4	no	—
120	C₄×C₈	—	2	4	no	—
121	C₄×C₈	—	6	4	no	—
122	C₄×C₈	Dic₆	1	8	yes	complemented
123	C₄×C₈	C₄×D₃	1	1	yes	characteristic
124	C₄×C₈	C₄×D₃	1	2	yes	—
125	C₄×C₈	C₄×D₃	1	1	yes	characteristic
126	C₄×C₈	D₁₂	1	2	yes	—
127	C₄×C₈	D₁₂	1	2	yes	—
128	C₂³×C₃	C₂×#41	1	1	yes	characteristic
129	C₂×C₃×C₄	—	2	8	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
133	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
137	C₂×C₃×C₄	C₂×#41	1	2	yes	—
138	C₂×Dic₃	—	2	8	no	—
139	Dic₆	—	4	8	no	—
140	C₃×C₈	—	2	2	no	—
141	C₃×C₈	C₄×Q₈	1	4	yes	—
142	C₃×C₈	C₄×Q₈	1	4	yes	—
143	C₃×C₈	C₄⋊Q₈	1	2	yes	—
144	C₃×C₈	C₄⋊Q₈	1	2	yes	—
145	C₃⋊C₈	—	4	4	no	—
146	C₂×Q₈	—	6	4	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	#235	1	2	yes	—
149	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
150	M₁₆	—	6	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	6	4	no	—
153	C₂×C₈	C₂×Dic₆	1	4	yes	—
154	C₂×C₈	C₂×Dic₆	1	4	yes	—
155	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
157	C₂×C₈	C₂×D₁₂	1	2	yes	—
158	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
159	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
160	C₂×C₈	#235	1	4	yes	—
161	C₄⋊C₄	—	3	8	no	—
162	C₄⋊C₄	—	6	8	no	—
163	C₄⋊C₄	—	6	4	no	—
164	C₂²⋊C₄	—	6	4	no	—
165	C₄²	—	2	8	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	6	4	no	—
168	C₄²	C₂×Dic₆	1	4	yes	—
169	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
170	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
171	C₄²	C₂×D₁₂	1	2	yes	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂²×C₃	#533	1	2	yes	—
175	C₃×C₄	—	2	8	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	#443	1	4	yes	—
178	C₃×C₄	C₂×C₄×Q₈	1	2	yes	—
179	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
180	C₃×C₄	#532	1	1	yes	characteristic
181	Dic₃	—	4	8	no	—
182	C₂³	C₂×#235	1	1	yes	characteristic
183	Q₈	—	12	8	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	6	8	no	—
188	C₂×C₄	C₄×D₁₂	1	2	yes	—
189	C₂×C₄	C₄×D₁₂	1	2	yes	—
190	C₂×C₄	C₂²×Dic₆	1	2	yes	—
191	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
193	C₂×C₄	C₂×#235	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	12	4	no	—
196	C₈	C₄×Dic₆	1	4	yes	—
197	C₈	C₄×Dic₆	1	4	yes	—
198	C₈	#869	1	2	yes	—
199	C₈	#869	1	2	yes	—
200	C₂×C₃	—	2	2	no	—
201	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₃	C₂×#443	1	2	yes	—
203	C₂²	—	2	2	no	—
204	C₂²	#5470	1	2	yes	—
205	C₂²	C₂×C₄×D₁₂	1	1	yes	characteristic
206	C₄	—	2	8	no	—
207	C₄	—	2	2	no	—
208	C₄	—	12	8	no	—
209	C₄	#4700	1	4	yes	—
210	C₄	C₂×C₄×Dic₆	1	2	yes	—
211	C₄	C₂×#869	1	1	yes	characteristic
212	C₄	#5477	1	1	yes	characteristic
213	C₃	#21732	1	1	yes	characteristic, complemented, Sylow
214	C₂	—	2	2	no	—
215	C₂	C₄×#869	1	1	yes	characteristic
216	C₂	C₂×#4700	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×C₄²×C₈	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₄²×C₈	D₃
hypocenter	C₃	#21732
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
20. #4683	C₄	semidirect
21. #4682	C₄	semidirect
22. #4680	C₄	semidirect
51. C₃×C₄×C₈	Q₈	semidirect
122. C₄×C₈	Dic₆	semidirect
213. C₃	#21732	semidirect

Maximal subgroups

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

subgroup	index
#78445	2
#78442	2
C₃×C₄²×C₈	2
#69794	2
#69792	2
#69779	2
#69396	2
#21732	3