Subgroups — #10804045

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79044	C₂	1	2	yes	maximal
2	#78442	C₂	1	1	yes	maximal, characteristic
3	C₃×#1848	C₂	1	1	yes	maximal, characteristic
4	#70705	C₂	1	4	yes	maximal
5	#70703	C₂	1	4	yes	maximal
6	#70514	C₂	1	2	yes	maximal
7	C₄×#815	C₂	1	1	yes	maximal, characteristic
8	#21734	—	3	1	no	maximal, Sylow
9	#5470	C₂²	1	2	yes	—
10	C₂×C₃×#108	C₂²	1	4	yes	—
11	C₃×#423	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#5122	C₂²	1	4	yes	—
14	#5119	C₂²	1	4	yes	—
15	#5106	C₂²	1	4	yes	—
16	C₂×#815	C₂²	1	2	yes	—
17	C₂×#815	C₂²	1	2	yes	—
18	C₄²×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#4831	—	2	4	no	—
21	#4830	—	2	4	no	—
22	#4683	C₄	1	4	yes	complemented
23	#4682	C₄	1	2	yes	complemented
24	#4680	C₄	1	2	yes	complemented
25	#4555	C₂²	1	8	yes	—
26	#3160	—	3	1	no	—
27	#2960	—	3	2	no	—
28	#1994	—	3	2	no	—
29	#1946	—	3	4	no	—
30	#1848	D₃	1	1	yes	characteristic
31	#1773	—	3	4	no	—
32	C₄×#97	—	3	1	no	—
33	C₂×C₃×M₁₆	C₂³	1	4	yes	—
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
36	#926	C₂³	1	4	yes	—
37	#924	—	2	4	no	—
38	C₂×C₄×Dic₃	—	2	2	no	—
39	C₂×C₄×Dic₃	C₂³	1	2	yes	—
40	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₃⋊C₈	C₂³	1	1	yes	characteristic
42	#870	C₂×C₄	1	4	yes	—
43	#869	C₂×C₄	1	4	yes	—
44	C₄×Dic₆	—	2	4	no	—
45	C₃×#108	—	2	8	no	—
46	C₃×C₄⋊C₈	—	2	4	no	—
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
50	#818	C₂×C₄	1	8	yes	—
51	#817	C₂×C₄	1	8	yes	—
52	#815	—	2	4	no	—
53	#815	—	2	4	no	—
54	#815	Q₈	1	8	yes	complemented
55	Dic₃⋊C₈	—	2	8	no	—
56	C₈×Dic₃	—	2	4	no	—
57	C₄×C₃⋊C₈	D₄	1	4	yes	—
58	#533	—	3	2	no	—
59	#500	—	3	2	no	—
60	#499	—	3	4	no	—
61	#498	—	3	2	no	—
62	#446	—	6	4	no	—
63	C₈×Q₈	—	6	4	no	—
64	C₂×#108	C₂×D₃	1	4	yes	—
65	#427	—	3	4	no	—
66	#424	—	3	4	no	—
67	#423	C₂×D₃	1	2	yes	—
68	#405	—	3	4	no	—
69	C₂×#97	—	3	2	no	—
70	C₂×#97	—	3	2	no	—
71	C₂×C₄×C₈	—	3	1	no	—
72	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₄³	—	3	1	no	—
74	#339	—	3	8	no	—
75	C₂²×C₃×C₄	#41	1	2	yes	—
76	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₃	—	2	2	no	—
78	C₂×Dic₆	—	2	4	no	—
79	C₃×M₁₆	—	2	8	no	—
80	C₂×C₃×C₈	—	2	4	no	—
81	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
84	C₂×C₃×C₈	#41	1	4	yes	—
85	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₄²	C₂×D₄	1	2	yes	—
88	#217	—	2	4	no	—
89	C₄⋊Dic₃	—	2	4	no	—
90	C₄⋊Dic₃	C₂²×C₄	1	8	yes	—
91	Dic₃⋊C₄	—	2	8	no	—
92	C₄×Dic₃	—	2	4	no	—
93	C₄×Dic₃	—	2	2	no	—
94	C₄×Dic₃	—	4	4	no	—
95	C₄×Dic₃	C₂×Q₈	1	4	yes	—
96	C₂×C₃⋊C₈	—	2	2	no	—
97	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
98	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₃⋊C₈	C₂×Q₈	1	4	yes	—
100	C₂×M₁₆	C₂²×D₃	1	4	yes	—
101	C₂²×C₈	—	3	1	no	—
102	C₂²×C₈	—	3	1	no	—
103	C₂²×C₈	C₂²×D₃	1	2	yes	—
104	C₄⋊Q₈	—	3	4	no	—
105	#125	—	3	4	no	—
106	C₂²⋊Q₈	—	6	4	no	—
107	C₄×Q₈	—	6	4	no	—
108	#117	—	3	4	no	—
109	C₂×C₄²	—	3	2	no	—
110	C₂×C₄²	—	6	2	no	—
111	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
112	#108	—	2	8	no	—
113	#107	—	3	8	no	—
114	#106	—	3	8	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	6	8	no	—
117	#97	—	3	8	no	—
118	#97	—	6	4	no	—
119	#97	—	6	4	no	—
120	C₄×C₈	—	3	4	no	—
121	C₄×C₈	—	6	4	no	—
122	C₄×C₈	C₄×D₃	1	1	yes	characteristic
123	C₄×C₈	C₄×D₃	1	2	yes	—
124	C₄×C₈	C₄×D₃	1	1	yes	characteristic
125	C₂³×C₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
128	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
129	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
132	C₂×C₃×C₄	C₂×#41	1	2	yes	—
133	C₂×Dic₃	—	2	8	no	—
134	C₂×Dic₃	—	2	4	no	—
135	C₂×Dic₃	—	4	4	no	—
136	Dic₆	—	4	8	no	—
137	C₃×C₈	—	4	4	no	—
138	C₃×C₈	C₄×Q₈	1	4	yes	—
139	C₃×C₈	C₄×Q₈	1	4	yes	—
140	C₃⋊C₈	—	2	2	no	—
141	C₃⋊C₈	C₄⋊Q₈	1	4	yes	—
142	C₂×Q₈	—	6	4	no	—
143	C₂²×C₄	—	6	2	no	—
144	C₂²×C₄	#239	1	2	yes	—
145	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
146	M₁₆	—	2	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	3	2	no	—
149	C₂×C₈	—	3	4	no	—
150	C₂×C₈	—	3	2	no	—
151	C₂×C₈	—	6	2	no	—
152	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
154	C₂×C₈	#237	1	4	yes	—
155	C₂×C₈	D₃×Q₈	1	4	yes	—
156	C₄⋊C₄	—	3	8	no	—
157	C₄⋊C₄	—	6	8	no	—
158	C₄⋊C₄	—	6	4	no	—
159	C₂²⋊C₄	—	6	4	no	—
160	C₄²	—	3	4	no	—
161	C₄²	—	6	4	no	—
162	C₄²	—	6	2	no	—
163	C₄²	—	12	4	no	—
164	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
165	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
166	C₄²	D₃×D₄	1	2	yes	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₃	#533	1	2	yes	—
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	#444	1	4	yes	—
172	C₃×C₄	C₂×C₄×Q₈	1	2	yes	—
173	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	C₃×C₄	#532	1	1	yes	characteristic
175	Dic₃	—	4	4	no	—
176	Dic₃	—	4	8	no	—
177	C₂³	C₂×#239	1	1	yes	characteristic
178	Q₈	—	12	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	6	4	no	—
181	C₂×C₄	—	6	8	no	—
182	C₂×C₄	—	12	4	no	—
183	C₂×C₄	#893	1	2	yes	—
184	C₂×C₄	#893	1	2	yes	—
185	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
187	C₂×C₄	C₂×#237	1	2	yes	—
188	C₂×C₄	C₂×D₃×Q₈	1	2	yes	—
189	C₈	—	3	4	no	—
190	C₈	—	4	4	no	—
191	C₈	—	6	2	no	—
192	C₈	#887	1	4	yes	—
193	C₈	#887	1	4	yes	—
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
196	C₂×C₃	C₂×#444	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂×#893	1	1	yes	characteristic
199	C₂²	#5728	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	12	4	no	—
202	C₄	—	12	8	no	—
203	C₄	#4894	1	4	yes	—
204	C₄	C₂×#887	1	2	yes	—
205	C₄	#5716	1	1	yes	characteristic
206	C₄	D₃×C₄⋊Q₈	1	1	yes	characteristic
207	C₃	#21734	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	2	2	no	—
209	C₂	#74320	1	1	yes	characteristic
210	C₂	C₂×#4894	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×#1848	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#1848	D₃
hypocenter	C₃	#21734
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
22. #4683	C₄	semidirect
23. #4682	C₄	semidirect
24. #4680	C₄	semidirect
54. #815	Q₈	semidirect
207. C₃	#21734	semidirect

Maximal subgroups

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

subgroup	index
#79044	2
#78442	2
C₃×#1848	2
#70705	2
#70703	2
#70514	2
C₄×#815	2
#21734	3