Finite Group Databasebeta

Subgroups — #72322766

Order 1024.

267 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 119 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	#153425	C₂	1	1	yes	maximal, characteristic
3	#152409	C₂	1	1	yes	maximal, characteristic
4	#94770	C₂	1	1	yes	maximal, characteristic
5	#94767	C₂	1	1	yes	maximal, characteristic
6	#94765	C₂	1	1	yes	maximal, characteristic
7	#94764	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	C₄	1	2	yes	complemented
13	#13732	C₄	1	2	yes	complemented
14	#13638	—	2	1	no	—
15	C₂×#1425	C₂²	1	1	yes	characteristic
16	#13634	—	2	1	no	—
17	#13633	C₂²	1	1	yes	characteristic
18	#13633	C₂²	1	1	yes	characteristic
19	C₂×#1424	C₂²	1	1	yes	characteristic
20	#13549	C₂²	1	1	yes	characteristic
21	#12911	—	2	1	no	—
22	#7511	—	2	2	no	—
23	#3164	—	4	1	no	—
24	C₂×M₆₄	D₄	1	1	yes	characteristic
25	C₂²×C₃₂	C₂³	1	1	yes	characteristic
26	#2255	—	4	2	no	—
27	#2253	—	2	2	no	—
28	#2252	—	2	2	no	—
29	C₂×#368	—	2	1	no	—
30	#2159	—	4	1	no	—
31	#2158	—	2	1	no	—
32	#2158	—	2	1	no	—
33	C₂×#366	—	2	1	no	—
34	#2155	—	4	1	no	—
35	#2108	D₄	1	1	yes	characteristic
36	#1833	—	4	1	no	—
37	M₁₂₈	—	2	1	no	—
38	M₁₂₈	—	2	1	no	—
39	M₁₂₈	—	2	1	no	—
40	M₁₂₈	Q₈	1	2	yes	complemented
41	C₂×C₆₄	—	2	1	no	—
42	C₂×C₆₄	Q₈	1	2	yes	complemented
43	#1426	—	4	2	no	—
44	#1425	—	2	1	no	—
45	#1425	—	2	1	no	—
46	#1424	—	2	2	no	—
47	#1424	—	2	1	no	—
48	#1424	—	2	1	no	—
49	#1424	C₂×C₄	1	2	yes	—
50	#1424	C₂×C₄	1	1	yes	characteristic
51	#1424	C₂×C₄	1	1	yes	characteristic
52	#1423	—	2	2	no	—
53	#1423	—	2	1	no	—
54	#1423	—	2	1	no	—
55	#1397	D₄	1	2	yes	—
56	C₄×C₃₂	C₂×C₄	1	2	yes	—
57	#544	—	8	1	no	—
58	C₂×M₃₂	D₈	1	1	yes	characteristic
59	C₂²×C₁₆	C₂×D₄	1	1	yes	characteristic
60	#501	—	8	2	no	—
61	#500	—	4	2	no	—
62	#499	—	4	2	no	—
63	C₂×#108	—	4	1	no	—
64	#428	—	8	1	no	—
65	#427	—	4	1	no	—
66	#427	—	4	1	no	—
67	C₂×#107	—	4	1	no	—
68	#405	D₈	1	1	yes	characteristic
69	M₆₄	SD₁₆	1	2	yes	—
70	M₆₄	Q₁₆	1	2	yes	—
71	C₂×C₃₂	C₂²⋊C₄	1	2	yes	—
72	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃₂	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃₂	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃₂	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃₂	#41	1	1	yes	characteristic
77	C₂×C₃₂	#41	1	1	yes	characteristic
78	#368	—	4	1	no	—
79	#368	—	4	1	no	—
80	#367	—	4	2	no	—
81	#367	—	4	1	no	—
82	#367	—	4	1	no	—
83	#366	—	2	2	no	—
84	#366	—	2	1	no	—
85	#366	—	2	1	no	—
86	#366	—	4	2	no	—
87	#366	—	4	1	no	—
88	#366	—	4	1	no	—
89	#365	—	8	2	no	—
90	#346	SD₁₆	1	2	yes	—
91	C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
92	#344	—	8	1	no	—
93	H(3,ℤ/4)	—	8	1	no	—
94	C₆₄	—	2	2	no	—
95	C₆₄	—	2	2	no	—
96	C₂×M₁₆	—	8	1	no	—
97	C₂×M₁₆	D₁₆	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₈	1	1	yes	characteristic
99	C₄⋊Q₈	—	8	2	no	—
100	C₄⋊Q₈	—	16	1	no	—
101	#125	—	8	2	no	—
102	#125	—	16	1	no	—
103	C₂²⋊Q₈	—	16	2	no	—
104	#117	—	8	1	no	—
105	#117	—	8	1	no	—
106	#117	D₁₆	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	8	1	no	—
108	M₃₂	SD₃₂	1	2	yes	—
109	M₃₂	Q₃₂	1	2	yes	—
110	C₂×C₁₆	#102	1	2	yes	—
111	C₂×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₁₆	#124	1	1	yes	characteristic
113	C₂×C₁₆	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
117	#108	—	8	1	no	—
118	#108	—	8	1	no	—
119	#107	—	4	2	no	—
120	#107	—	4	1	no	—
121	#107	—	4	1	no	—
122	#107	—	8	2	no	—
123	#107	—	8	1	no	—
124	#107	—	8	1	no	—
125	#106	—	8	2	no	—
126	#106	—	8	1	no	—
127	#106	—	8	1	no	—
128	#97	SD₃₂	1	2	yes	—
129	C₄×C₈	#102	1	2	yes	—
130	C₃₂	#103	1	4	yes	—
131	C₃₂	C₄×Q₈	1	1	yes	characteristic
132	C₃₂	C₄×Q₈	1	1	yes	characteristic
133	C₃₂	C₂²⋊Q₈	1	1	yes	characteristic
134	C₃₂	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×Q₈	—	16	2	no	—
136	C₂²×C₄	—	16	1	no	—
137	C₂²×C₄	C₂×D₁₆	1	1	yes	characteristic
138	M₁₆	—	2	1	no	—
139	M₁₆	—	2	1	no	—
140	M₁₆	—	16	1	no	—
141	M₁₆	—	16	1	no	—
142	C₂×C₈	—	16	1	no	—
143	C₂×C₈	#357	1	2	yes	—
144	C₂×C₈	C₂×#102	1	1	yes	characteristic
145	C₂×C₈	#486	1	1	yes	characteristic
146	C₂×C₈	#500	1	1	yes	characteristic
147	C₂×C₈	C₂×SD₃₂	1	1	yes	characteristic
148	C₂×C₈	C₂×SD₃₂	1	1	yes	characteristic
149	C₂×C₈	C₂×Q₃₂	1	1	yes	characteristic
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	8	1	no	—
152	C₄⋊C₄	—	8	2	no	—
153	C₄⋊C₄	—	8	1	no	—
154	C₄⋊C₄	—	16	2	no	—
155	C₄⋊C₄	—	16	2	no	—
156	C₄⋊C₄	—	16	2	no	—
157	C₄⋊C₄	—	16	1	no	—
158	C₄⋊C₄	—	16	1	no	—
159	C₂²⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	16	1	no	—
161	C₂²⋊C₄	—	16	1	no	—
162	C₄²	—	16	1	no	—
163	C₄²	—	16	1	no	—
164	C₄²	#357	1	2	yes	—
165	C₁₆	#358	1	4	yes	—
166	C₁₆	#391	1	1	yes	characteristic
167	C₁₆	C₂×#103	1	1	yes	characteristic
168	C₁₆	#487	1	1	yes	characteristic
169	C₁₆	#498	1	1	yes	characteristic
170	C₂³	#2263	1	1	yes	characteristic
171	Q₈	—	64	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	16	1	no	—
174	C₂×C₄	—	16	1	no	—
175	C₂×C₄	—	16	2	no	—
176	C₂×C₄	—	32	1	no	—
177	C₂×C₄	—	64	1	no	—
178	C₂×C₄	C₂×#357	1	1	yes	characteristic
179	C₂×C₄	#2240	1	1	yes	characteristic
180	C₂×C₄	#2252	1	1	yes	characteristic
181	C₂×C₄	#2263	1	1	yes	characteristic
182	C₂×C₄	#2264	1	1	yes	characteristic
183	C₂×C₄	#2264	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	32	1	no	—
186	C₈	#1990	1	1	yes	characteristic
187	C₈	C₂×#358	1	1	yes	characteristic
188	C₈	#2241	1	1	yes	characteristic
189	C₈	#2254	1	1	yes	characteristic
190	C₂²	—	4	1	no	—
191	C₂²	#13617	1	1	yes	characteristic
192	C₂²	#13722	1	1	yes	characteristic
193	C₂²	#13734	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	32	1	no	—
196	C₄	—	32	2	no	—
197	C₄	—	32	1	no	—
198	C₄	—	64	1	no	—
199	C₄	#13282	1	1	yes	characteristic
200	C₄	#13618	1	1	yes	characteristic
201	C₄	#13722	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	#153018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#153018
commutator	C₂×C₃₂	C₂²×C₄
Frattini, cosocle	C₂²×C₃₂	C₂³
prime radical	C₂³	#2263
LCS	G ▷ C₂×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
12. #13733	C₄	semidirect
13. #13732	C₄	semidirect
40. M₁₂₈	Q₈	semidirect
42. C₂×C₆₄	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
#153425	2
#152409	2
#94770	2
#94767	2
#94765	2
#94764	2