Finite Group Databasebeta

Subgroups — #10709919

Order 768.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2248	C₂	1	1	yes	maximal, characteristic
2	#71312	C₂	1	1	yes	maximal, characteristic, complemented
3	#71298	C₂	1	1	yes	maximal, characteristic, complemented
4	#71291	C₂	1	1	yes	maximal, characteristic
5	#71275	C₂	1	1	yes	maximal, characteristic
6	#69066	C₂	1	1	yes	maximal, characteristic, complemented
7	#67498	C₂	1	1	yes	maximal, characteristic, complemented
8	#12520	—	3	1	no	maximal, Sylow
9	C₂×C₃×SD₃₂	—	2	2	no	—
10	C₃×#494	C₂²	1	1	yes	characteristic
11	C₃×#493	C₂²	1	1	yes	characteristic
12	C₂×#829	—	2	2	no	—
13	C₂×#827	—	2	2	no	—
14	#5078	—	2	2	no	—
15	#5070	—	2	2	no	—
16	C₂×Dic₂₄	—	2	2	no	—
17	#4699	C₂²	1	1	yes	characteristic
18	#4699	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₈×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#2248	—	3	1	no	—
23	#2248	D₃	1	1	yes	characteristic
24	#2247	—	3	1	no	—
25	#2247	—	3	1	no	—
26	#1713	—	3	1	no	—
27	#1709	—	3	1	no	—
28	#1311	—	3	1	no	—
29	C₂×C₃×Q₁₆	—	2	1	no	—
30	C₂×C₃×Q₁₆	—	2	1	no	—
31	C₂×C₃×D₈	—	2	1	no	—
32	C₂×C₃×D₈	—	2	1	no	—
33	C₃×C₄⋊Q₈	—	2	1	no	—
34	C₃×#127	—	2	1	no	—
35	C₂×Dic₁₂	—	2	1	no	—
36	C₂×Dic₁₂	—	2	1	no	—
37	C₂×Dic₁₂	—	2	1	no	—
38	C₂×Dic₁₂	—	2	1	no	—
39	#869	—	2	1	no	—
40	#869	—	2	1	no	—
41	C₃×SD₃₂	—	4	4	no	—
42	C₂×C₃×C₁₆	D₄	1	2	yes	—
43	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
44	#832	—	4	4	no	—
45	#829	—	4	4	no	—
46	#827	—	4	4	no	—
47	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
48	#807	—	4	4	no	—
49	C₄×C₃⋊C₈	D₄	1	2	yes	—
50	Dic₂₄	—	4	4	no	—
51	C₂×Q₃₂	—	6	2	no	—
52	C₂×Q₃₂	—	6	2	no	—
53	C₂×SD₃₂	—	2	2	no	—
54	C₂×SD₃₂	—	6	2	no	—
55	#500	—	6	2	no	—
56	#494	—	3	1	no	—
57	#494	—	3	1	no	—
58	#494	C₂×D₃	1	1	yes	characteristic
59	#493	C₂×D₃	1	1	yes	characteristic
60	#486	—	6	2	no	—
61	C₄×C₁₆	—	3	1	no	—
62	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
63	C₈²	—	3	1	no	—
64	C₂×C₃×Q₈	—	4	1	no	—
65	C₂×C₃×D₄	—	4	1	no	—
66	C₂×C₃×D₄	—	4	2	no	—
67	C₂×Dic₆	—	4	1	no	—
68	C₂×Dic₆	—	4	1	no	—
69	C₃×Q₁₆	—	4	2	no	—
70	C₃×Q₁₆	—	4	2	no	—
71	C₃×D₈	—	4	2	no	—
72	C₃×D₈	—	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	2	no	—
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₄⋊Dic₃	—	4	2	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₂×C₃⋊C₈	D₈	1	4	yes	—
80	Dic₁₂	—	4	2	no	—
81	Dic₁₂	—	4	2	no	—
82	Dic₁₂	—	4	2	no	—
83	Dic₁₂	—	4	2	no	—
84	C₃×C₁₆	D₈	1	4	yes	—
85	C₃⋊C₁₆	D₈	1	4	yes	—
86	C₂×Q₁₆	—	2	1	no	—
87	C₂×Q₁₆	—	2	1	no	—
88	C₂×Q₁₆	—	6	1	no	—
89	C₂×Q₁₆	—	6	1	no	—
90	C₂×Q₁₆	—	6	1	no	—
91	C₂×Q₁₆	—	6	1	no	—
92	C₂×D₈	—	2	1	no	—
93	C₂×D₈	—	2	1	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	6	1	no	—
96	C₄⋊Q₈	—	6	1	no	—
97	#127	—	2	1	no	—
98	Q₃₂	—	12	4	no	—
99	Q₃₂	—	12	4	no	—
100	SD₃₂	—	4	4	no	—
101	SD₃₂	—	12	4	no	—
102	C₂×C₁₆	—	3	2	no	—
103	C₂×C₁₆	#67	1	2	yes	—
104	#107	—	12	4	no	—
105	#102	—	12	4	no	—
106	C₄×C₈	—	3	2	no	—
107	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
108	C₂³×C₃	—	8	1	no	—
109	C₃×Q₈	—	8	1	no	—
110	C₃×Q₈	—	8	1	no	—
111	C₃×D₄	—	8	1	no	—
112	C₃×D₄	—	8	1	no	—
113	C₃×D₄	—	8	4	no	—
114	C₂×C₃×C₄	—	8	1	no	—
115	C₂×C₃×C₄	#127	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
117	C₂×Dic₃	—	8	1	no	—
118	C₂×Dic₃	—	8	1	no	—
119	Dic₆	—	8	1	no	—
120	Dic₆	—	8	1	no	—
121	Dic₆	—	8	1	no	—
122	Dic₆	—	8	1	no	—
123	C₃×C₈	C₂×D₈	1	2	yes	—
124	C₃×C₈	C₂×D₈	1	2	yes	—
125	C₃⋊C₈	—	2	4	no	—
126	C₂×Q₈	—	4	1	no	—
127	C₂×Q₈	—	12	1	no	—
128	C₂×Q₈	—	12	1	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂×D₄	—	4	1	no	—
131	Q₁₆	—	4	2	no	—
132	Q₁₆	—	4	2	no	—
133	Q₁₆	—	12	2	no	—
134	Q₁₆	—	12	2	no	—
135	Q₁₆	—	12	2	no	—
136	Q₁₆	—	12	2	no	—
137	D₈	—	4	2	no	—
138	D₈	—	4	2	no	—
139	C₂×C₈	—	3	4	no	—
140	C₂×C₈	D₃×D₄	1	1	yes	characteristic
141	C₂×C₈	C₂×#67	1	1	yes	characteristic
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	12	2	no	—
144	C₄⋊C₄	—	12	2	no	—
145	C₄²	D₃×D₄	1	1	yes	characteristic
146	C₁₆	—	3	4	no	—
147	C₁₆	#213	1	4	yes	—
148	C₂²×C₃	—	16	1	no	—
149	C₂²×C₃	—	16	2	no	—
150	C₂²×C₃	#493	1	1	yes	characteristic
151	C₃×C₄	—	16	1	no	—
152	C₃×C₄	#493	1	1	yes	characteristic
153	C₃×C₄	#493	1	1	yes	characteristic
154	C₃×C₄	#510	1	4	yes	—
155	Dic₃	—	16	1	no	—
156	Dic₃	—	16	1	no	—
157	C₂³	—	8	1	no	—
158	Q₈	—	8	1	no	—
159	Q₈	—	8	1	no	—
160	Q₈	—	24	1	no	—
161	Q₈	—	24	1	no	—
162	Q₈	—	24	1	no	—
163	Q₈	—	24	1	no	—
164	D₄	—	8	1	no	—
165	D₄	—	8	4	no	—
166	D₄	—	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	2	yes	—
171	C₂×C₄	#940	1	1	yes	characteristic
172	C₈	—	6	4	no	—
173	C₈	D₃×D₈	1	2	yes	—
174	C₈	C₂×#213	1	2	yes	—
175	C₂×C₃	—	32	1	no	—
176	C₂×C₃	#1712	1	1	yes	characteristic
177	C₂×C₃	#2251	1	2	yes	—
178	C₂²	—	16	2	no	—
179	C₂²	—	16	1	no	—
180	C₂²	#5151	1	1	yes	characteristic
181	C₄	—	16	1	no	—
182	C₄	—	48	1	no	—
183	C₄	—	48	1	no	—
184	C₄	#4915	1	4	yes	—
185	C₄	#5073	1	1	yes	characteristic
186	C₄	#5151	1	1	yes	characteristic
187	C₃	#12520	1	1	yes	characteristic, complemented, Sylow
188	C₂	—	32	1	no	—
189	C₂	#71276	1	1	yes	characteristic
190	C₂	#71482	1	2	yes	—

	subgroup	G/H
center	C₂²	#5151
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#2248	C₂
socle	C₂²×C₃	#493
prime radical	C₃×#493	C₂²
hypercenter	#2248	D₃
hypocenter	C₃	#12520
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2248

Complements

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

N	H ≅ G/N	type
2. #71312	C₂	semidirect
3. #71298	C₂	semidirect
6. #69066	C₂	semidirect
7. #67498	C₂	semidirect
187. C₃	#12520	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2248	2
#71312	2
#71298	2
#71291	2
#71275	2
#69066	2
#67498	2
#12520	3