Subgroups — #10804073

Order 768.

662 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	#78479	C₂	1	1	yes	maximal, characteristic
2	#78445	C₂	1	2	yes	maximal
3	C₃×#1726	C₂	1	1	yes	maximal, characteristic
4	#69842	C₂	1	1	yes	maximal, characteristic
5	#69829	C₂	1	4	yes	maximal
6	#69794	C₂	1	4	yes	maximal
7	#69441	C₂	1	2	yes	maximal
8	#21737	—	3	1	no	maximal, Sylow
9	#5470	C₂²	1	2	yes	—
10	C₂×C₃×#97	C₂²	1	1	yes	characteristic
11	C₂×C₃×#97	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
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	#1945	—	3	4	no	—
28	#1849	—	3	1	no	—
29	#1773	—	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	C₂×C₄	1	4	yes	—
43	C₃×#97	Q₈	1	8	yes	complemented
44	C₃×C₄×C₈	—	2	4	no	—
45	C₃×C₄×C₈	D₄	1	4	yes	—
46	C₃×#95	—	2	2	no	—
47	#819	—	2	8	no	—
48	#818	C₂×C₄	1	8	yes	—
49	#817	C₂×C₄	1	8	yes	—
50	#815	—	2	4	no	—
51	Dic₃⋊C₈	—	2	8	no	—
52	C₈×Dic₃	—	2	4	no	—
53	C₄⋊(C₃⋊C₈)	—	2	4	no	—
54	#533	—	3	2	no	—
55	#501	—	3	8	no	—
56	#446	—	6	4	no	—
57	C₈×Q₈	—	6	4	no	—
58	C₂×#108	—	3	4	no	—
59	#427	—	3	4	no	—
60	#424	—	3	4	no	—
61	#423	—	3	2	no	—
62	#405	—	3	4	no	—
63	C₂×#97	C₂×D₃	1	2	yes	—
64	C₂×#97	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
66	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
67	#376	C₂×D₃	1	1	yes	characteristic
68	#339	—	3	8	no	—
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	#41	1	2	yes	—
71	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₆	—	2	4	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
75	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
78	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
79	C₂×C₃×C₈	#41	1	4	yes	—
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	C₂²×C₄	1	2	yes	—
82	C₃×C₄²	C₂×D₄	1	2	yes	—
83	C₃×C₄²	C₂×Q₈	1	4	yes	—
84	#217	—	2	4	no	—
85	C₄⋊Dic₃	—	2	4	no	—
86	C₄⋊Dic₃	C₂²×C₄	1	8	yes	—
87	Dic₃⋊C₄	—	2	8	no	—
88	C₄×Dic₃	—	2	4	no	—
89	#208	—	2	8	no	—
90	C₂×C₃⋊C₈	—	2	4	no	—
91	C₂×M₁₆	—	3	4	no	—
92	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₃	1	2	yes	—
95	C₄⋊Q₈	—	3	4	no	—
96	#125	—	3	4	no	—
97	C₂²⋊Q₈	—	6	4	no	—
98	C₄×Q₈	—	6	4	no	—
99	#117	—	3	4	no	—
100	C₂×C₄²	C₂²×D₃	1	2	yes	—
101	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
102	#108	—	6	8	no	—
103	#107	—	3	8	no	—
104	#106	—	3	8	no	—
105	C₄⋊C₈	—	6	8	no	—
106	C₄⋊C₈	—	6	4	no	—
107	#97	—	6	4	no	—
108	#97	Dic₆	1	8	yes	complemented
109	#97	C₄×D₃	1	4	yes	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	—	6	4	no	—
112	C₄×C₈	D₁₂	1	4	yes	—
113	#95	—	2	2	no	—
114	C₂³×C₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	—	4	4	no	—
118	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
119	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
122	C₂×C₃×C₄	C₂×#41	1	2	yes	—
123	C₂×Dic₃	—	2	8	no	—
124	Dic₆	—	4	8	no	—
125	C₃×C₈	—	2	2	no	—
126	C₃×C₈	C₄×Q₈	1	8	yes	—
127	C₃×C₈	C₄⋊Q₈	1	4	yes	—
128	C₃⋊C₈	—	4	4	no	—
129	C₂×Q₈	—	6	4	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	#235	1	2	yes	—
132	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
133	M₁₆	—	6	8	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	6	4	no	—
136	C₂×C₈	C₂×Dic₆	1	4	yes	—
137	C₂×C₈	C₂×Dic₆	1	4	yes	—
138	C₂×C₈	C₂×C₄×D₃	1	4	yes	—
139	C₂×C₈	C₂×D₁₂	1	2	yes	—
140	C₂×C₈	C₂×D₁₂	1	2	yes	—
141	C₂×C₈	#235	1	4	yes	—
142	C₄⋊C₄	—	3	8	no	—
143	C₄⋊C₄	—	6	8	no	—
144	C₄⋊C₄	—	6	4	no	—
145	C₂²⋊C₄	—	6	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	6	4	no	—
148	C₄²	C₂×Dic₆	1	4	yes	—
149	C₄²	C₂×C₄×D₃	1	2	yes	—
150	C₄²	C₂×D₁₂	1	2	yes	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₃	#533	1	2	yes	—
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	4	4	no	—
156	C₃×C₄	C₂×C₄×Q₈	1	2	yes	—
157	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₃×C₄	#532	1	1	yes	characteristic
159	Dic₃	—	4	8	no	—
160	C₂³	C₂×#235	1	1	yes	characteristic
161	Q₈	—	12	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	6	8	no	—
166	C₂×C₄	C₄×D₁₂	1	4	yes	—
167	C₂×C₄	C₂²×Dic₆	1	2	yes	—
168	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
170	C₂×C₄	C₂×#235	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	12	4	no	—
173	C₈	C₄×Dic₆	1	8	yes	—
174	C₈	#869	1	4	yes	—
175	C₂×C₃	—	2	2	no	—
176	C₂×C₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
177	C₂×C₃	#2955	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	#5470	1	2	yes	—
180	C₂²	C₂×C₄×D₁₂	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	4	4	no	—
183	C₄	—	12	8	no	—
184	C₄	C₂×C₄×Dic₆	1	2	yes	—
185	C₄	C₂×#869	1	1	yes	characteristic
186	C₄	#5477	1	1	yes	characteristic
187	C₃	#21737	1	1	yes	characteristic, complemented, Sylow
188	C₂	—	2	2	no	—
189	C₂	C₄×#869	1	1	yes	characteristic
190	C₂	#78579	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×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	#21737
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
20. #4702	C₄	semidirect
43. C₃×#97	Q₈	semidirect
108. #97	Dic₆	semidirect
187. C₃	#21737	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
#69842	2
#69829	2
#69794	2
#69441	2
#21737	3