Finite Group Databasebeta

Subgroups — #153350

Order 512.

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

normal subgroups only

This group has 201 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	#13628	C₂	1	1	yes	maximal, characteristic, complemented
3	#13620	C₂	1	1	yes	maximal, characteristic
4	C₂×#1417	C₂	1	1	yes	maximal, characteristic
5	#13243	C₂	1	1	yes	maximal, characteristic
6	#13173	C₂	1	1	yes	maximal, characteristic, complemented
7	#7473	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	C₂×#368	C₂²	1	1	yes	characteristic
13	#2158	C₂²	1	1	yes	characteristic
14	#2145	—	2	1	no	—
15	#2108	C₂²	1	1	yes	characteristic
16	#1976	—	2	1	no	—
17	#1953	—	2	1	no	—
18	#1426	—	2	2	no	—
19	C₄⋊C₃₂	—	2	2	no	—
20	#1417	—	2	2	no	—
21	#1417	—	2	2	no	—
22	#1388	—	2	1	no	—
23	C₂×#135	—	2	1	no	—
24	#508	—	2	1	no	—
25	#508	—	2	1	no	—
26	#508	—	2	1	no	—
27	#508	—	2	1	no	—
28	C₂×Q₃₂	Q₈	1	1	yes	characteristic
29	C₂×SD₃₂	—	2	1	no	—
30	C₂×D₁₆	Q₈	1	1	yes	characteristic
31	C₂×M₃₂	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	#430	—	4	1	no	—
34	C₂×#108	—	2	1	no	—
35	C₂×#108	—	2	1	no	—
36	#427	—	2	1	no	—
37	#421	—	4	1	no	—
38	#410	—	4	1	no	—
39	#405	D₄	1	1	yes	characteristic
40	M₆₄	—	2	1	no	—
41	M₆₄	—	2	1	no	—
42	C₂×C₃₂	—	2	1	no	—
43	C₂×C₃₂	—	2	1	no	—
44	C₂×C₃₂	—	2	1	no	—
45	C₂×C₃₂	D₄	1	2	yes	—
46	#368	—	2	1	no	—
47	#368	—	2	1	no	—
48	#368	—	2	1	no	—
49	#368	D₄	1	2	yes	—
50	#367	—	2	1	no	—
51	#366	—	2	1	no	—
52	#360	—	4	2	no	—
53	#346	—	2	1	no	—
54	C₄×C₁₆	—	2	1	no	—
55	#344	—	4	1	no	—
56	C₂×#41	—	4	1	no	—
57	#135	—	4	1	no	—
58	#135	—	4	1	no	—
59	#135	—	4	1	no	—
60	#135	—	4	1	no	—
61	C₂×Q₁₆	—	2	1	no	—
62	C₂×SD₁₆	—	4	1	no	—
63	C₂×D₈	—	2	1	no	—
64	C₂×M₁₆	—	4	1	no	—
65	C₂×M₁₆	—	4	1	no	—
66	C₂×M₁₆	D₈	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	#117	—	4	1	no	—
69	#117	D₈	1	1	yes	characteristic
70	Q₃₂	—	2	1	no	—
71	Q₃₂	—	2	1	no	—
72	SD₃₂	—	4	1	no	—
73	SD₃₂	—	4	1	no	—
74	D₁₆	—	2	1	no	—
75	D₁₆	—	2	1	no	—
76	M₃₂	—	2	1	no	—
77	M₃₂	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
80	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₁₆	#41	1	1	yes	characteristic
83	C₂×C₁₆	#41	1	1	yes	characteristic
84	C₂×C₁₆	#41	1	1	yes	characteristic
85	#108	—	2	2	no	—
86	#108	—	4	1	no	—
87	#108	—	4	1	no	—
88	#108	—	4	1	no	—
89	#108	—	8	2	no	—
90	#107	—	4	1	no	—
91	#106	—	4	1	no	—
92	#104	—	8	2	no	—
93	#99	—	8	2	no	—
94	#97	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₃₂	—	2	2	no	—
97	C₃₂	—	4	1	no	—
98	#41	—	8	1	no	—
99	#41	—	8	1	no	—
100	#41	—	8	1	no	—
101	#41	—	8	1	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂×D₄	—	4	1	no	—
104	C₂×D₄	—	8	1	no	—
105	C₂²×C₄	—	8	1	no	—
106	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
107	Q₁₆	—	4	1	no	—
108	Q₁₆	—	4	1	no	—
109	SD₁₆	—	8	1	no	—
110	SD₁₆	—	8	1	no	—
111	D₈	—	4	1	no	—
112	D₈	—	4	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	4	2	no	—
116	M₁₆	—	8	1	no	—
117	M₁₆	—	8	1	no	—
118	M₁₆	—	8	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	8	1	no	—
121	C₂×C₈	—	8	1	no	—
122	C₂×C₈	#121	1	1	yes	characteristic
123	C₂×C₈	#121	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₈	#135	1	1	yes	characteristic
126	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₈	#137	1	1	yes	characteristic
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	8	1	no	—
130	C₂²⋊C₄	—	8	1	no	—
131	C₂²⋊C₄	SD₃₂	1	2	yes	—
132	C₄²	—	2	1	no	—
133	C₄²	—	8	1	no	—
134	C₁₆	—	4	1	no	—
135	C₁₆	#121	1	1	yes	characteristic
136	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
137	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
138	C₁₆	#126	1	1	yes	characteristic
139	C₂³	—	8	1	no	—
140	C₂³	C₂×SD₃₂	1	1	yes	characteristic
141	Q₈	—	8	1	no	—
142	Q₈	—	8	1	no	—
143	D₄	—	8	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	16	1	no	—
146	D₄	—	16	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	16	1	no	—
150	C₂×C₄	—	16	1	no	—
151	C₂×C₄	—	16	1	no	—
152	C₂×C₄	#459	1	1	yes	characteristic
153	C₂×C₄	#466	1	1	yes	characteristic
154	C₂×C₄	#474	1	1	yes	characteristic
155	C₂×C₄	#508	1	1	yes	characteristic
156	C₂×C₄	#509	1	1	yes	characteristic
157	C₂×C₄	#509	1	1	yes	characteristic
158	C₈	—	4	1	no	—
159	C₈	—	8	2	no	—
160	C₈	—	16	1	no	—
161	C₈	#396	1	1	yes	characteristic
162	C₈	#464	1	1	yes	characteristic
163	C₈	#468	1	1	yes	characteristic
164	C₈	#478	1	1	yes	characteristic
165	C₂²	—	4	1	no	—
166	C₂²	—	16	1	no	—
167	C₂²	—	32	1	no	—
168	C₂²	#2207	1	1	yes	characteristic
169	C₂²	#2217	1	1	yes	characteristic
170	C₂²	#2226	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	16	1	no	—
173	C₄	—	32	1	no	—
174	C₄	#2067	1	1	yes	characteristic
175	C₄	#2210	1	1	yes	characteristic
176	C₄	#2218	1	1	yes	characteristic
177	C₄	#2228	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	—	32	1	no	—
182	C₂	#13476	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13476
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. #13628	C₂	semidirect
6. #13173	C₂	semidirect
7. #7473	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
#13628	2
#13620	2
C₂×#1417	2
#13243	2
#13173	2
#7473	2