Finite Group Databasebeta

Subgroups — #72322053

Order 1024.

255 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 89 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	#513064	C₂	1	1	yes	maximal, characteristic
2	#153578	C₂	1	1	yes	maximal, characteristic
3	C₂×#7548	C₂	1	1	yes	maximal, characteristic
4	#152859	C₂	1	1	yes	maximal, characteristic
5	#94770	C₂	1	1	yes	maximal, characteristic
6	#94764	C₂	1	1	yes	maximal, characteristic
7	#94759	C₂	1	1	yes	maximal, characteristic
8	#33729	—	2	1	no	—
9	C₂×M₁₂₈	C₂²	1	1	yes	characteristic
10	C₂×M₁₂₈	C₂²	1	1	yes	characteristic
11	#13734	—	2	2	no	—
12	#13733	—	2	1	no	—
13	#13732	—	2	1	no	—
14	C₂×#1425	C₂²	1	1	yes	characteristic
15	C₂×#1425	C₂²	1	1	yes	characteristic
16	#13634	—	2	1	no	—
17	#13633	C₂²	1	1	yes	characteristic
18	C₂×#1424	C₂²	1	1	yes	characteristic
19	#13549	C₂²	1	1	yes	characteristic
20	#13173	—	2	1	no	—
21	#7548	—	2	2	no	—
22	#7548	—	2	1	no	—
23	#7548	—	2	1	no	—
24	#7479	—	2	1	no	—
25	#3164	—	4	1	no	—
26	C₂×M₆₄	D₄	1	1	yes	characteristic
27	C₂²×C₃₂	C₂³	1	1	yes	characteristic
28	#2255	—	4	2	no	—
29	#2253	—	2	2	no	—
30	#2252	—	2	2	no	—
31	C₂×#368	—	2	1	no	—
32	C₂×#368	—	2	1	no	—
33	#2159	—	4	1	no	—
34	#2158	—	2	1	no	—
35	C₂×#366	—	2	1	no	—
36	#2108	D₄	1	1	yes	characteristic
37	#1953	—	4	1	no	—
38	M₁₂₈	—	2	1	no	—
39	M₁₂₈	—	2	1	no	—
40	M₁₂₈	—	2	1	no	—
41	M₁₂₈	Q₈	1	2	yes	complemented
42	C₂×C₆₄	—	2	1	no	—
43	C₂×C₆₄	Q₈	1	2	yes	complemented
44	#1425	—	2	1	no	—
45	#1425	—	2	1	no	—
46	#1425	—	2	1	no	—
47	#1425	Q₈	1	2	yes	complemented
48	#1424	—	2	2	no	—
49	#1424	—	2	1	no	—
50	#1424	D₄	1	2	yes	—
51	#1424	D₄	1	1	yes	characteristic
52	#1424	D₄	1	1	yes	characteristic
53	#1423	—	2	2	no	—
54	#1423	—	2	1	no	—
55	#1397	—	2	1	no	—
56	C₄×C₃₂	—	2	1	no	—
57	#1388	—	4	1	no	—
58	#544	—	8	1	no	—
59	C₂×M₃₂	D₈	1	1	yes	characteristic
60	C₂²×C₁₆	C₂×D₄	1	1	yes	characteristic
61	#501	—	8	2	no	—
62	#500	—	4	2	no	—
63	#499	—	4	2	no	—
64	#430	—	8	1	no	—
65	C₂×#108	—	4	1	no	—
66	C₂×#108	—	4	1	no	—
67	#428	—	8	1	no	—
68	#427	—	4	1	no	—
69	C₂×#107	—	4	1	no	—
70	#405	D₈	1	1	yes	characteristic
71	M₆₄	—	2	1	no	—
72	M₆₄	Q₁₆	1	2	yes	—
73	C₂×C₃₂	—	2	1	no	—
74	C₂×C₃₂	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃₂	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃₂	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃₂	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃₂	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃₂	#41	1	1	yes	characteristic
80	#368	—	2	2	no	—
81	#368	—	4	1	no	—
82	#368	—	4	1	no	—
83	#368	—	4	1	no	—
84	#367	—	4	2	no	—
85	#367	—	4	1	no	—
86	#366	—	2	2	no	—
87	#366	—	2	1	no	—
88	#366	—	2	1	no	—
89	#366	—	4	2	no	—
90	#366	—	4	1	no	—
91	#346	—	2	1	no	—
92	C₄×C₁₆	—	2	1	no	—
93	#344	—	8	1	no	—
94	C₆₄	—	2	2	no	—
95	C₆₄	—	2	2	no	—
96	C₂×M₁₆	—	8	1	no	—
97	C₂×M₁₆	—	8	1	no	—
98	C₂×M₁₆	D₁₆	1	1	yes	characteristic
99	C₂²×C₈	C₂×D₈	1	1	yes	characteristic
100	C₄⋊Q₈	—	8	2	no	—
101	C₄⋊Q₈	—	16	1	no	—
102	#125	—	8	2	no	—
103	#125	—	16	1	no	—
104	C₂²⋊Q₈	—	16	2	no	—
105	#117	—	8	1	no	—
106	#117	D₁₆	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	8	1	no	—
108	M₃₂	—	2	1	no	—
109	M₃₂	Q₃₂	1	2	yes	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	#120	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₁₆	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₁₆	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₁₆	#137	1	1	yes	characteristic
117	#108	—	4	2	no	—
118	#108	—	8	1	no	—
119	#108	—	8	1	no	—
120	#108	—	8	1	no	—
121	#108	—	16	2	no	—
122	#107	—	4	2	no	—
123	#107	—	4	1	no	—
124	#107	—	4	1	no	—
125	#107	—	8	2	no	—
126	#107	—	8	1	no	—
127	#106	—	8	2	no	—
128	#106	—	8	1	no	—
129	#97	—	2	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₃₂	—	2	2	no	—
132	C₃₂	C₂²⋊Q₈	1	1	yes	characteristic
133	C₃₂	C₂²⋊Q₈	1	1	yes	characteristic
134	C₃₂	C₄⋊Q₈	1	1	yes	characteristic
135	C₃₂	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×Q₈	—	16	2	no	—
137	C₂²×C₄	—	16	1	no	—
138	C₂²×C₄	C₂×D₁₆	1	1	yes	characteristic
139	M₁₆	—	2	1	no	—
140	M₁₆	—	8	2	no	—
141	M₁₆	—	16	1	no	—
142	M₁₆	—	16	1	no	—
143	M₁₆	—	16	1	no	—
144	M₁₆	SD₆₄	1	2	yes	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	16	1	no	—
147	C₂×C₈	—	16	1	no	—
148	C₂×C₈	#447	1	1	yes	characteristic
149	C₂×C₈	#474	1	1	yes	characteristic
150	C₂×C₈	#500	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₃₂	1	1	yes	characteristic
152	C₂×C₈	#509	1	1	yes	characteristic
153	C₂×C₈	#510	1	1	yes	characteristic
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	8	1	no	—
156	C₄⋊C₄	—	8	2	no	—
157	C₄⋊C₄	—	8	1	no	—
158	C₄⋊C₄	—	16	2	no	—
159	C₄⋊C₄	—	16	2	no	—
160	C₄⋊C₄	—	16	2	no	—
161	C₄⋊C₄	—	16	1	no	—
162	C₂²⋊C₄	—	16	1	no	—
163	C₂²⋊C₄	SD₆₄	1	2	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	16	1	no	—
166	C₁₆	—	2	2	no	—
167	C₁₆	#395	1	1	yes	characteristic
168	C₁₆	#451	1	1	yes	characteristic
169	C₁₆	#477	1	1	yes	characteristic
170	C₁₆	#501	1	1	yes	characteristic
171	C₂³	C₂×SD₆₄	1	1	yes	characteristic
172	Q₈	—	64	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	16	1	no	—
175	C₂×C₄	—	16	1	no	—
176	C₂×C₄	—	16	2	no	—
177	C₂×C₄	—	64	1	no	—
178	C₂×C₄	#2184	1	1	yes	characteristic
179	C₂×C₄	#2224	1	1	yes	characteristic
180	C₂×C₄	#2252	1	1	yes	characteristic
181	C₂×C₄	C₂×SD₆₄	1	1	yes	characteristic
182	C₂×C₄	#2263	1	1	yes	characteristic
183	C₂×C₄	#2264	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	16	2	no	—
186	C₈	—	32	1	no	—
187	C₈	#2057	1	1	yes	characteristic
188	C₈	#2188	1	1	yes	characteristic
189	C₈	#2227	1	1	yes	characteristic
190	C₈	#2255	1	1	yes	characteristic
191	C₂²	—	4	1	no	—
192	C₂²	#13664	1	1	yes	characteristic
193	C₂²	#13704	1	1	yes	characteristic
194	C₂²	#13733	1	1	yes	characteristic
195	C₄	—	4	2	no	—
196	C₄	—	32	1	no	—
197	C₄	—	32	2	no	—
198	C₄	—	32	1	no	—
199	C₄	#13451	1	1	yes	characteristic
200	C₄	#13662	1	1	yes	characteristic
201	C₄	#13705	1	1	yes	characteristic
202	C₄	#13734	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#153322	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#153322
commutator, Frattini, cosocle	C₂²×C₃₂	C₂³
prime radical	C₂³	C₂×SD₆₄
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ #13549 ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
41. M₁₂₈	Q₈	semidirect
43. C₂×C₆₄	Q₈	semidirect
47. #1425	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#513064	2
#153578	2
C₂×#7548	2
#152859	2
#94770	2
#94764	2
#94759	2