Finite Group Databasebeta

Subgroups — #153410

Order 512.

199 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 66 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	#13638	C₂	1	1	yes	maximal, characteristic, complemented
2	#13633	C₂	1	1	yes	maximal, characteristic, complemented
3	#13622	C₂	1	1	yes	maximal, characteristic
4	C₂×#1417	C₂	1	1	yes	maximal, characteristic
5	#13243	C₂	1	1	yes	maximal, characteristic
6	#12911	C₂	1	1	yes	maximal, characteristic, complemented
7	#7470	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	1	yes	characteristic
12	#2158	C₂²	1	1	yes	characteristic
13	#2158	C₂²	1	1	yes	characteristic
14	#2155	—	2	1	no	—
15	#2145	—	2	1	no	—
16	#2108	C₂²	1	1	yes	characteristic
17	#1976	—	2	1	no	—
18	#1833	—	2	1	no	—
19	#1426	—	2	2	no	—
20	#1424	—	2	1	no	—
21	#1423	—	2	1	no	—
22	#1418	—	2	2	no	—
23	#1417	—	2	1	no	—
24	#1417	—	2	1	no	—
25	C₂×#135	—	2	1	no	—
26	#508	—	2	1	no	—
27	#508	—	2	1	no	—
28	#508	—	2	1	no	—
29	#508	—	2	1	no	—
30	C₂×Q₃₂	Q₈	1	1	yes	characteristic
31	C₂×SD₃₂	—	2	1	no	—
32	C₂×D₁₆	Q₈	1	1	yes	characteristic
33	C₂×M₃₂	D₄	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	1	yes	characteristic
35	C₂×#108	—	2	1	no	—
36	#427	—	2	1	no	—
37	#427	—	2	1	no	—
38	#421	—	4	1	no	—
39	#410	—	4	1	no	—
40	#405	D₄	1	1	yes	characteristic
41	M₆₄	—	2	1	no	—
42	M₆₄	D₄	1	2	yes	—
43	C₂×C₃₂	—	2	1	no	—
44	C₂×C₃₂	—	2	1	no	—
45	C₂×C₃₂	—	2	1	no	—
46	C₂×C₃₂	D₄	1	1	yes	characteristic
47	C₂×C₃₂	D₄	1	1	yes	characteristic
48	#368	—	2	1	no	—
49	#368	—	2	1	no	—
50	#367	—	2	1	no	—
51	#367	—	2	1	no	—
52	#366	—	2	1	no	—
53	#366	—	2	1	no	—
54	#365	—	4	2	no	—
55	#360	—	4	2	no	—
56	#346	—	2	1	no	—
57	C₄×C₁₆	—	2	1	no	—
58	#344	—	4	1	no	—
59	H(3,ℤ/4)	—	4	1	no	—
60	C₂×#41	—	4	1	no	—
61	#135	—	4	1	no	—
62	#135	—	4	1	no	—
63	#135	—	4	1	no	—
64	#135	—	4	1	no	—
65	C₂×Q₁₆	—	2	1	no	—
66	C₂×SD₁₆	—	4	1	no	—
67	C₂×D₈	—	2	1	no	—
68	C₂×M₁₆	—	4	1	no	—
69	C₂×M₁₆	D₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	#117	—	4	1	no	—
72	#117	—	4	1	no	—
73	#117	D₈	1	1	yes	characteristic
74	Q₃₂	—	2	1	no	—
75	Q₃₂	—	2	1	no	—
76	SD₃₂	—	4	1	no	—
77	SD₃₂	—	4	1	no	—
78	D₁₆	—	2	1	no	—
79	D₁₆	—	2	1	no	—
80	M₃₂	—	2	1	no	—
81	M₃₂	D₈	1	2	yes	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
84	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
85	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₁₆	#41	1	1	yes	characteristic
87	C₂×C₁₆	#41	1	1	yes	characteristic
88	C₂×C₁₆	#41	1	1	yes	characteristic
89	#108	—	4	1	no	—
90	#108	—	4	1	no	—
91	#107	—	4	1	no	—
92	#107	—	4	1	no	—
93	#106	—	4	1	no	—
94	#106	—	4	1	no	—
95	#104	—	8	2	no	—
96	#99	—	8	2	no	—
97	#97	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₃₂	—	2	1	no	—
100	C₃₂	—	2	1	no	—
101	C₃₂	—	2	2	no	—
102	#41	—	8	1	no	—
103	#41	—	8	1	no	—
104	#41	—	8	1	no	—
105	#41	—	8	1	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂×D₄	—	4	1	no	—
108	C₂×D₄	—	8	1	no	—
109	C₂²×C₄	—	8	1	no	—
110	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
111	Q₁₆	—	4	1	no	—
112	Q₁₆	—	4	1	no	—
113	SD₁₆	—	8	1	no	—
114	SD₁₆	—	8	1	no	—
115	D₈	—	4	1	no	—
116	D₈	—	4	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	2	1	no	—
119	M₁₆	—	8	1	no	—
120	M₁₆	—	8	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	8	1	no	—
123	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₈	#123	1	1	yes	characteristic
125	C₂×C₈	#127	1	1	yes	characteristic
126	C₂×C₈	C₂×D₈	1	1	yes	characteristic
127	C₂×C₈	#137	1	1	yes	characteristic
128	C₂×C₈	#137	1	1	yes	characteristic
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	8	1	no	—
131	C₄⋊C₄	—	8	1	no	—
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	8	1	no	—
134	C₂²⋊C₄	—	8	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	8	1	no	—
137	C₄²	—	8	1	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
140	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
141	C₁₆	#123	1	1	yes	characteristic
142	C₁₆	#123	1	1	yes	characteristic
143	C₂³	—	8	1	no	—
144	C₂³	#509	1	1	yes	characteristic
145	Q₈	—	8	1	no	—
146	Q₈	—	8	1	no	—
147	D₄	—	8	1	no	—
148	D₄	—	8	1	no	—
149	D₄	—	16	1	no	—
150	D₄	—	16	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	16	1	no	—
154	C₂×C₄	—	16	1	no	—
155	C₂×C₄	—	16	1	no	—
156	C₂×C₄	—	16	1	no	—
157	C₂×C₄	#474	1	1	yes	characteristic
158	C₂×C₄	#480	1	1	yes	characteristic
159	C₂×C₄	#493	1	1	yes	characteristic
160	C₂×C₄	#508	1	1	yes	characteristic
161	C₂×C₄	#508	1	1	yes	characteristic
162	C₂×C₄	#509	1	1	yes	characteristic
163	C₈	—	4	1	no	—
164	C₈	—	16	1	no	—
165	C₈	#399	1	1	yes	characteristic
166	C₈	#474	1	1	yes	characteristic
167	C₈	#480	1	1	yes	characteristic
168	C₈	#497	1	1	yes	characteristic
169	C₂²	—	4	1	no	—
170	C₂²	—	16	1	no	—
171	C₂²	—	32	1	no	—
172	C₂²	#2228	1	1	yes	characteristic
173	C₂²	#2234	1	1	yes	characteristic
174	C₂²	#2250	1	1	yes	characteristic
175	C₄	—	8	1	no	—
176	C₄	—	16	1	no	—
177	C₄	—	32	1	no	—
178	C₄	—	32	1	no	—
179	C₄	#2093	1	1	yes	characteristic
180	C₄	#2228	1	1	yes	characteristic
181	C₄	#2234	1	1	yes	characteristic
182	C₄	#2249	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	—	32	1	no	—
187	C₂	#13528	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2234
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13528
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #13638	C₂	semidirect
2. #13633	C₂	semidirect
6. #12911	C₂	semidirect
7. #7470	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13638	2
#13633	2
#13622	2
C₂×#1417	2
#13243	2
#12911	2
#7470	2