Subgroups — #10804057

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	#78479	C₂	1	1	yes	maximal, characteristic
2	#78445	C₂	1	2	yes	maximal
3	C₃×#1726	C₂	1	1	yes	maximal, characteristic
4	#69844	C₂	1	4	yes	maximal
5	#69842	C₂	1	1	yes	maximal, characteristic
6	#69779	C₂	1	4	yes	maximal
7	#69441	C₂	1	2	yes	maximal
8	#21735	—	3	1	no	maximal, Sylow
9	#5470	C₂²	1	2	yes	—
10	C₂×C₃×#97	C₂²	1	2	yes	—
11	C₂×C₃×#97	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×#376	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	#4702	C₄	1	8	yes	complemented
21	#4679	—	2	4	no	—
22	C₈×Dic₆	—	2	4	no	—
23	#4523	C₂²	1	8	yes	—
24	#3162	—	3	1	no	—
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	D₃	1	1	yes	characteristic
31	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂³	1	2	yes	—
33	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄²	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×#208	C₂³	1	4	yes	—
39	#870	C₂×C₄	1	4	yes	—
40	#869	C₂×C₄	1	4	yes	—
41	C₄×Dic₆	—	2	4	no	—
42	C₃×#97	—	2	4	no	—
43	C₃×#97	C₂×C₄	1	4	yes	—
44	C₃×C₄×C₈	D₄	1	2	yes	—
45	C₃×C₄×C₈	D₄	1	2	yes	—
46	C₃×C₄×C₈	Q₈	1	8	yes	complemented
47	C₃×#95	—	2	2	no	—
48	#819	—	2	8	no	—
49	#818	C₂×C₄	1	8	yes	—
50	#817	C₂×C₄	1	8	yes	—
51	#815	—	2	4	no	—
52	Dic₃⋊C₈	—	2	8	no	—
53	C₈×Dic₃	—	2	4	no	—
54	C₄⋊(C₃⋊C₈)	—	2	4	no	—
55	#533	—	3	2	no	—
56	#501	—	3	8	no	—
57	#446	—	6	4	no	—
58	C₈×Q₈	—	6	4	no	—
59	C₂×#108	—	3	4	no	—
60	#427	—	3	4	no	—
61	#424	—	3	4	no	—
62	#423	—	3	2	no	—
63	#405	—	3	4	no	—
64	C₂×#97	C₂×D₃	1	2	yes	—
65	C₂×#97	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
67	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
68	#376	C₂×D₃	1	1	yes	characteristic
69	#339	—	3	8	no	—
70	C₂²×C₃×C₄	—	2	2	no	—
71	C₂²×C₃×C₄	#41	1	2	yes	—
72	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₆	—	2	4	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
76	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
80	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
81	C₂×C₃×C₈	#41	1	4	yes	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	C₂²×C₄	1	2	yes	—
84	C₃×C₄²	C₂×D₄	1	2	yes	—
85	C₃×C₄²	C₂×Q₈	1	4	yes	—
86	#217	—	2	4	no	—
87	C₄⋊Dic₃	—	2	4	no	—
88	C₄⋊Dic₃	C₂²×C₄	1	8	yes	—
89	Dic₃⋊C₄	—	2	8	no	—
90	C₄×Dic₃	—	2	4	no	—
91	#208	—	2	8	no	—
92	C₂×C₃⋊C₈	—	2	4	no	—
93	C₂×M₁₆	—	3	4	no	—
94	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₃	1	2	yes	—
97	C₄⋊Q₈	—	3	4	no	—
98	#125	—	3	4	no	—
99	C₂²⋊Q₈	—	6	4	no	—
100	C₄×Q₈	—	6	4	no	—
101	#117	—	3	4	no	—
102	C₂×C₄²	C₂²×D₃	1	2	yes	—
103	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
104	#108	—	6	8	no	—
105	#107	—	3	8	no	—
106	#106	—	3	8	no	—
107	C₄⋊C₈	—	6	8	no	—
108	C₄⋊C₈	—	6	4	no	—
109	#97	—	2	4	no	—
110	#97	—	6	4	no	—
111	#97	C₄×D₃	1	4	yes	—
112	C₄×C₈	—	6	4	no	—
113	C₄×C₈	Dic₆	1	8	yes	complemented
114	C₄×C₈	D₁₂	1	2	yes	—
115	C₄×C₈	D₁₂	1	2	yes	—
116	#95	—	2	2	no	—
117	C₂³×C₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	—	4	4	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	8	no	—
127	Dic₆	—	4	8	no	—
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	C₄×Q₈	1	8	yes	—
130	C₃×C₈	C₄⋊Q₈	1	2	yes	—
131	C₃×C₈	C₄⋊Q₈	1	2	yes	—
132	C₃⋊C₈	—	4	4	no	—
133	C₂×Q₈	—	6	4	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	#235	1	2	yes	—
136	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
137	M₁₆	—	6	8	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	6	4	no	—
140	C₂×C₈	C₂×Dic₆	1	4	yes	—
141	C₂×C₈	C₂×Dic₆	1	4	yes	—
142	C₂×C₈	C₂×C₄×D₃	1	4	yes	—
143	C₂×C₈	C₂×D₁₂	1	2	yes	—
144	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
145	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
146	C₂×C₈	#235	1	4	yes	—
147	C₄⋊C₄	—	3	8	no	—
148	C₄⋊C₄	—	6	8	no	—
149	C₄⋊C₄	—	6	4	no	—
150	C₂²⋊C₄	—	6	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	6	4	no	—
153	C₄²	C₂×Dic₆	1	4	yes	—
154	C₄²	C₂×C₄×D₃	1	2	yes	—
155	C₄²	C₂×D₁₂	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	4	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	8	no	—
165	C₂³	C₂×#235	1	1	yes	characteristic
166	Q₈	—	12	8	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	6	8	no	—
171	C₂×C₄	C₄×D₁₂	1	4	yes	—
172	C₂×C₄	C₂²×Dic₆	1	2	yes	—
173	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
175	C₂×C₄	C₂×#235	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	12	4	no	—
178	C₈	C₄×Dic₆	1	8	yes	—
179	C₈	#869	1	2	yes	—
180	C₈	#869	1	2	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	#5470	1	2	yes	—
186	C₂²	C₂×C₄×D₁₂	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	4	4	no	—
189	C₄	—	12	8	no	—
190	C₄	C₂×C₄×Dic₆	1	2	yes	—
191	C₄	C₂×#869	1	1	yes	characteristic
192	C₄	#5477	1	1	yes	characteristic
193	C₃	#21735	1	1	yes	characteristic, complemented, Sylow
194	C₂	—	2	2	no	—
195	C₂	C₄×#869	1	1	yes	characteristic
196	C₂	#78579	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₃×#1726	C₂
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₂×#41
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	#1726	D₃
hypocenter	C₃	#21735
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
20. #4702	C₄	semidirect
46. C₃×C₄×C₈	Q₈	semidirect
113. C₄×C₈	Dic₆	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
#78479	2
#78445	2
C₃×#1726	2
#69844	2
#69842	2
#69779	2
#69441	2
#21735	3