Finite Group Databasebeta

Subgroups — #10709893

Order 768.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2241	C₂	1	1	yes	maximal, characteristic
2	#71382	C₂	1	1	yes	maximal, characteristic
3	#71361	C₂	1	1	yes	maximal, characteristic
4	#71310	C₂	1	1	yes	maximal, characteristic
5	#71298	C₂	1	1	yes	maximal, characteristic
6	#69065	C₂	1	1	yes	maximal, characteristic
7	#67498	C₂	1	1	yes	maximal, characteristic
8	#12514	—	3	1	no	maximal, Sylow
9	C₃×#500	C₂²	1	1	yes	characteristic
10	C₃×#494	C₂²	1	1	yes	characteristic
11	#5093	—	2	1	no	—
12	#5079	—	2	1	no	—
13	#5078	—	2	1	no	—
14	#5078	—	2	1	no	—
15	#4699	C₂²	1	1	yes	characteristic
16	#4682	C₂²	1	1	yes	characteristic
17	C₃×#358	—	2	2	no	—
18	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#4564	—	2	2	no	—
20	#4503	—	2	2	no	—
21	#4492	—	2	2	no	—
22	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₈×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#2241	—	3	1	no	—
25	#2241	—	3	1	no	—
26	#2241	—	3	1	no	—
27	#2241	D₃	1	1	yes	characteristic
28	#1713	—	3	1	no	—
29	#1710	—	3	1	no	—
30	#1311	—	3	1	no	—
31	C₂×C₃×Q₁₆	—	2	1	no	—
32	C₂×C₃×Q₁₆	—	2	1	no	—
33	C₃×C₄⋊Q₈	—	2	1	no	—
34	C₃×C₄⋊Q₈	—	2	1	no	—
35	C₂×Dic₁₂	—	2	1	no	—
36	C₂×Dic₁₂	—	2	1	no	—
37	#869	—	2	1	no	—
38	#869	—	2	1	no	—
39	C₂×C₃×C₁₆	D₄	1	2	yes	—
40	C₃×#107	—	2	1	no	—
41	C₃×#107	—	2	1	no	—
42	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
43	#835	—	4	2	no	—
44	#818	—	2	1	no	—
45	#818	—	2	1	no	—
46	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
47	#810	—	4	2	no	—
48	#807	—	4	2	no	—
49	#807	—	4	2	no	—
50	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
52	#500	—	3	1	no	—
53	#500	—	6	1	no	—
54	#500	—	6	1	no	—
55	#500	C₂×D₃	1	1	yes	characteristic
56	#494	—	3	1	no	—
57	#494	C₂×D₃	1	1	yes	characteristic
58	#487	—	6	1	no	—
59	#487	—	6	1	no	—
60	#358	—	2	2	no	—
61	#358	—	6	2	no	—
62	#358	—	6	2	no	—
63	#358	—	6	2	no	—
64	C₄×C₁₆	—	3	1	no	—
65	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₈²	—	3	1	no	—
67	C₂×C₃×Q₈	—	4	1	no	—
68	C₂×C₃×Q₈	—	4	1	no	—
69	C₂×Dic₆	—	4	1	no	—
70	C₂×Dic₆	—	4	1	no	—
71	C₃×Q₁₆	—	4	2	no	—
72	C₃×Q₁₆	—	4	2	no	—
73	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄⋊C₄	—	4	1	no	—
76	C₃×C₄⋊C₄	—	4	1	no	—
77	C₃×C₄⋊C₄	—	4	1	no	—
78	C₃×C₄⋊C₄	—	4	1	no	—
79	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄⋊Dic₃	—	4	1	no	—
81	C₄⋊Dic₃	—	4	1	no	—
82	C₄⋊Dic₃	—	4	1	no	—
83	C₄⋊Dic₃	—	4	1	no	—
84	C₂×C₃⋊C₈	D₈	1	2	yes	—
85	C₂×C₃⋊C₈	D₈	1	2	yes	—
86	Dic₁₂	—	4	2	no	—
87	Dic₁₂	—	4	2	no	—
88	C₃×C₁₆	—	2	2	no	—
89	C₃⋊C₁₆	—	2	2	no	—
90	C₂×Q₁₆	—	2	1	no	—
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	6	1	no	—
93	C₂×Q₁₆	—	6	1	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	6	1	no	—
97	C₄⋊Q₈	—	6	1	no	—
98	C₂×C₁₆	—	3	2	no	—
99	C₂×C₁₆	#67	1	2	yes	—
100	#107	—	2	1	no	—
101	#107	—	2	1	no	—
102	#107	—	6	1	no	—
103	#107	—	6	1	no	—
104	#107	—	12	2	no	—
105	#107	—	12	2	no	—
106	#103	—	12	2	no	—
107	#103	—	12	2	no	—
108	C₄×C₈	—	3	1	no	—
109	C₄×C₈	—	3	1	no	—
110	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₃×Q₈	—	8	1	no	—
112	C₃×Q₈	—	8	1	no	—
113	C₃×Q₈	—	8	2	no	—
114	C₂×C₃×C₄	—	8	1	no	—
115	C₂×C₃×C₄	—	8	1	no	—
116	C₂×C₃×C₄	#127	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂×Dic₃	—	8	1	no	—
120	C₂×Dic₃	—	8	1	no	—
121	Dic₆	—	8	1	no	—
122	Dic₆	—	8	1	no	—
123	Dic₆	—	8	2	no	—
124	C₃×C₈	#135	1	2	yes	—
125	C₃×C₈	#135	1	2	yes	—
126	C₃⋊C₈	—	2	2	no	—
127	C₃⋊C₈	—	2	2	no	—
128	C₂×Q₈	—	4	1	no	—
129	C₂×Q₈	—	4	1	no	—
130	C₂×Q₈	—	12	1	no	—
131	C₂×Q₈	—	12	1	no	—
132	Q₁₆	—	4	2	no	—
133	Q₁₆	—	4	2	no	—
134	Q₁₆	—	12	2	no	—
135	Q₁₆	—	12	2	no	—
136	C₂×C₈	—	3	2	no	—
137	C₂×C₈	—	3	2	no	—
138	C₂×C₈	D₃×D₄	1	1	yes	characteristic
139	C₂×C₈	C₂×#67	1	1	yes	characteristic
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	12	1	no	—
145	C₄⋊C₄	—	12	1	no	—
146	C₄⋊C₄	—	12	1	no	—
147	C₄⋊C₄	—	12	1	no	—
148	C₄²	D₃×D₄	1	1	yes	characteristic
149	C₁₆	—	2	2	no	—
150	C₁₆	—	6	2	no	—
151	C₂²×C₃	#493	1	1	yes	characteristic
152	C₃×C₄	—	16	1	no	—
153	C₃×C₄	—	16	1	no	—
154	C₃×C₄	#495	1	1	yes	characteristic
155	C₃×C₄	#495	1	1	yes	characteristic
156	C₃×C₄	#510	1	2	yes	—
157	C₃×C₄	#510	1	2	yes	—
158	Dic₃	—	16	1	no	—
159	Dic₃	—	16	1	no	—
160	Q₈	—	8	1	no	—
161	Q₈	—	8	2	no	—
162	Q₈	—	8	1	no	—
163	Q₈	—	24	1	no	—
164	Q₈	—	24	2	no	—
165	Q₈	—	24	1	no	—
166	C₂×C₄	—	8	1	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	—	24	1	no	—
169	C₂×C₄	—	24	1	no	—
170	C₂×C₄	D₃×D₈	1	1	yes	characteristic
171	C₂×C₄	D₃×D₈	1	1	yes	characteristic
172	C₂×C₄	#940	1	1	yes	characteristic
173	C₈	—	6	2	no	—
174	C₈	—	6	2	no	—
175	C₈	#916	1	2	yes	—
176	C₈	#950	1	2	yes	—
177	C₂×C₃	#1709	1	1	yes	characteristic
178	C₂×C₃	#2251	1	1	yes	characteristic
179	C₂×C₃	#2251	1	1	yes	characteristic
180	C₂²	#5151	1	1	yes	characteristic
181	C₄	—	16	1	no	—
182	C₄	—	16	1	no	—
183	C₄	—	48	1	no	—
184	C₄	—	48	1	no	—
185	C₄	#4915	1	2	yes	—
186	C₄	#4918	1	2	yes	—
187	C₄	#5059	1	1	yes	characteristic
188	C₄	#5168	1	1	yes	characteristic
189	C₃	#12514	1	1	yes	characteristic, complemented, Sylow
190	C₂	#71182	1	1	yes	characteristic
191	C₂	#71482	1	1	yes	characteristic
192	C₂	#71501	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#5151
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#2241	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	#2241	D₃
hypocenter	C₃	#12514
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
189. C₃	#12514	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2241	2
#71382	2
#71361	2
#71310	2
#71298	2
#69065	2
#67498	2
#12514	3