Subgroups — #10713298

Order 768.

294 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×#1772	C₂	1	1	yes	maximal, characteristic
2	#70649	C₂	1	2	yes	maximal
3	#69766	C₂	1	1	yes	maximal, characteristic
4	C₂×#4463	C₂	1	1	yes	maximal, characteristic
5	#67732	C₂	1	2	yes	maximal
6	#13180	—	3	1	no	maximal, Sylow
7	C₂×C₃×#108	C₂²	1	1	yes	characteristic
8	C₃×#427	C₂²	1	1	yes	characteristic
9	C₃×#405	—	2	1	no	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#818	C₂²	1	2	yes	—
12	C₂×#817	—	2	1	no	—
13	C₂×#812	C₂²	1	2	yes	—
14	#4936	—	2	1	no	—
15	C₃×#339	—	2	1	no	—
16	#4490	—	2	2	no	—
17	#4489	—	2	2	no	—
18	#4463	C₄	1	4	yes	complemented
19	#2159	—	3	2	no	—
20	C₂×#364	—	3	1	no	—
21	#1846	—	3	1	no	—
22	#1772	D₃	1	1	yes	characteristic
23	#1387	—	3	2	no	—
24	C₂×C₃×M₁₆	—	2	1	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
27	C₃×#117	—	2	1	no	—
28	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	2	no	—
30	C₃×#108	—	2	1	no	—
31	C₃×#108	Q₈	1	2	yes	—
32	C₃×#107	D₄	1	2	yes	—
33	C₃×#106	—	2	1	no	—
34	C₃×#97	—	4	1	no	—
35	C₃×C₄×C₈	—	4	1	no	—
36	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₃×C₄×C₈	Q₈	1	2	yes	—
38	C₃⋊#95	—	4	2	no	—
39	#818	—	2	2	no	—
40	#818	—	2	2	no	—
41	#817	—	4	1	no	—
42	#817	—	4	1	no	—
43	#812	—	2	2	no	—
44	#812	C₂×C₄	1	4	yes	—
45	C₂×C₃⋊C₁₆	—	2	2	no	—
46	C₂×M₃₂	—	3	2	no	—
47	C₂×#108	C₂×D₃	1	1	yes	characteristic
48	#427	C₂×D₃	1	1	yes	characteristic
49	C₂×#107	—	3	2	no	—
50	C₂×#106	—	6	1	no	—
51	#405	—	2	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#382	—	6	1	no	—
54	#367	—	6	2	no	—
55	#366	—	6	2	no	—
56	#364	—	3	4	no	—
57	#339	—	2	1	no	—
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₃	—	4	2	no	—
61	C₃×M₁₆	—	2	2	no	—
62	C₃×M₁₆	—	4	1	no	—
63	C₂×C₃×C₈	—	4	1	no	—
64	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
65	C₂×C₃×C₈	D₈	1	1	yes	characteristic
66	C₂×C₃×C₈	D₈	1	1	yes	characteristic
67	C₂×C₃×C₈	Q₁₆	1	2	yes	—
68	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₈	#41	1	1	yes	characteristic
73	C₂×C₃×C₈	#41	1	1	yes	characteristic
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₂²⋊C₄	—	4	1	no	—
76	C₃×C₄²	—	2	1	no	—
77	C₃×C₄²	—	4	1	no	—
78	C₃×C₄²	C₄⋊C₄	1	2	yes	—
79	C₄⋊Dic₃	—	4	2	no	—
80	C₄⋊Dic₃	—	4	2	no	—
81	C₃⋊C₁₆	—	2	4	no	—
82	C₂×M₁₆	—	2	1	no	—
83	C₂²×C₈	#67	1	1	yes	characteristic
84	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
85	#117	—	2	1	no	—
86	C₂×C₄⋊C₄	—	6	2	no	—
87	C₂×C₄²	#67	1	1	yes	characteristic
88	M₃₂	—	3	4	no	—
89	M₃₂	—	6	2	no	—
90	C₂×C₁₆	—	6	2	no	—
91	#108	—	2	1	no	—
92	#108	Dic₆	1	2	yes	—
93	#107	—	6	2	no	—
94	#107	—	6	2	no	—
95	#107	D₁₂	1	2	yes	—
96	#106	—	2	1	no	—
97	#106	—	12	1	no	—
98	#106	—	12	1	no	—
99	#97	—	4	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	Dic₆	1	2	yes	—
102	C₄×C₈	C₄×D₃	1	2	yes	—
103	#95	—	12	2	no	—
104	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	4	2	no	—
107	C₂×C₃×C₄	—	4	2	no	—
108	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	#121	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
113	C₂×C₃×C₄	#137	1	1	yes	characteristic
114	C₂×Dic₃	—	8	4	no	—
115	C₂×Dic₃	—	8	2	no	—
116	C₃×C₈	—	4	2	no	—
117	C₃×C₈	#107	1	2	yes	—
118	C₃×C₈	#107	1	2	yes	—
119	C₃×C₈	C₄×D₄	1	1	yes	characteristic
120	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
121	C₃×C₈	#125	1	1	yes	characteristic
122	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	12	2	no	—
125	C₂²×C₄	C₂×#67	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	—	4	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
130	C₂×C₈	#213	1	1	yes	characteristic
131	C₂×C₈	#213	1	1	yes	characteristic
132	C₂×C₈	#216	1	2	yes	—
133	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
134	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
137	C₂×C₈	#235	1	1	yes	characteristic
138	C₂×C₈	#235	1	1	yes	characteristic
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	12	2	no	—
141	C₄⋊C₄	—	12	2	no	—
142	C₂²⋊C₄	—	4	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	4	1	no	—
145	C₄²	Dic₃⋊C₄	1	2	yes	—
146	C₁₆	—	6	4	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	#428	1	1	yes	characteristic
149	C₂²×C₃	#459	1	1	yes	characteristic
150	C₂²×C₃	#477	1	1	yes	characteristic
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	—	8	2	no	—
153	C₃×C₄	#389	1	1	yes	characteristic
154	C₃×C₄	C₂×#107	1	1	yes	characteristic
155	C₃×C₄	#462	1	1	yes	characteristic
156	C₃×C₄	#474	1	1	yes	characteristic
157	Dic₃	—	16	2	no	—
158	C₂³	#949	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	24	4	no	—
163	C₂×C₄	—	24	2	no	—
164	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
165	C₂×C₄	#924	1	1	yes	characteristic
166	C₂×C₄	#930	1	1	yes	characteristic
167	C₂×C₄	C₂×#213	1	1	yes	characteristic
168	C₂×C₄	C₂×#216	1	1	yes	characteristic
169	C₂×C₄	#951	1	1	yes	characteristic
170	C₈	—	4	2	no	—
171	C₈	#807	1	2	yes	—
172	C₈	#807	1	2	yes	—
173	C₈	C₄×Dic₆	1	1	yes	characteristic
174	C₈	#869	1	1	yes	characteristic
175	C₈	#870	1	1	yes	characteristic
176	C₈	C₄×D₁₂	1	1	yes	characteristic
177	C₂×C₃	—	2	1	no	—
178	C₂×C₃	—	2	1	no	—
179	C₂×C₃	#1947	1	1	yes	characteristic
180	C₂×C₃	#2212	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#4995	1	1	yes	characteristic
183	C₂²	#5015	1	1	yes	characteristic
184	C₂²	#5021	1	1	yes	characteristic
185	C₄	—	4	2	no	—
186	C₄	—	8	2	no	—
187	C₄	—	48	2	no	—
188	C₄	#4928	1	1	yes	characteristic
189	C₄	C₂×#807	1	1	yes	characteristic
190	C₄	#5007	1	1	yes	characteristic
191	C₄	#5031	1	1	yes	characteristic
192	C₃	#13180	1	1	yes	characteristic, complemented, Sylow
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#69466	1	1	yes	characteristic
196	C₂	#70925	1	2	yes	—

	subgroup	G/H
center	C₂²	#5015
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1772	C₂
socle	C₂²×C₃	#459
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1772	D₃
hypocenter	C₃	#13180
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
18. #4463	C₄	semidirect
192. C₃	#13180	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1772	2
#70649	2
#69766	2
C₂×#4463	2
#67732	2
#13180	3