Finite Group Databasebeta

Subgroups — #152000

Order 512.

202 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₂×#1426	C₂	1	1	yes	maximal, characteristic
2	#13634	C₂	1	1	yes	maximal, characteristic
3	C₂×#1398	C₂	1	1	yes	maximal, characteristic
4	#12674	C₂	1	1	yes	maximal, characteristic
5	#12671	C₂	1	1	yes	maximal, characteristic
6	#7478	C₂	1	1	yes	maximal, characteristic
7	#7477	C₂	1	1	yes	maximal, characteristic
8	C₂×M₆₄	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×#368	C₂²	1	1	yes	characteristic
11	#2158	C₂²	1	1	yes	characteristic
12	C₂×#367	C₂²	1	1	yes	characteristic
13	C₂×#366	C₂²	1	1	yes	characteristic
14	C₂×#346	C₂²	1	1	yes	characteristic
15	C₄×#107	—	2	1	no	—
16	#1772	—	2	1	no	—
17	#1426	C₄	1	4	yes	complemented
18	#1424	C₄	1	2	yes	complemented
19	#1423	C₄	1	2	yes	complemented
20	#1398	C₄	1	4	yes	complemented
21	#1382	—	2	1	no	—
22	#1380	—	2	1	no	—
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂³	1	1	yes	characteristic
25	C₂×#108	—	2	1	no	—
26	#427	—	2	1	no	—
27	C₂×#107	—	2	1	no	—
28	C₂×#107	—	2	1	no	—
29	#405	—	4	1	no	—
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄	—	4	1	no	—
32	M₆₄	—	2	1	no	—
33	M₆₄	C₂×C₄	1	2	yes	—
34	M₆₄	C₂×C₄	1	2	yes	—
35	M₆₄	C₂×C₄	1	2	yes	—
36	C₂×C₃₂	—	2	1	no	—
37	C₂×C₃₂	C₂×C₄	1	2	yes	—
38	#368	—	2	1	no	—
39	#368	C₂×C₄	1	2	yes	—
40	#367	—	2	1	no	—
41	#367	—	2	1	no	—
42	#367	C₂×C₄	1	2	yes	—
43	#366	—	2	1	no	—
44	#366	C₂×C₄	1	2	yes	—
45	#366	C₂×C₄	1	1	yes	characteristic
46	#366	C₂×C₄	1	1	yes	characteristic
47	#346	C₂×C₄	1	2	yes	—
48	#346	Q₈	1	2	yes	—
49	#340	—	4	1	no	—
50	#339	—	4	1	no	—
51	C₂×M₁₆	—	4	1	no	—
52	C₂²×C₈	D₈	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	#117	—	4	1	no	—
55	C₂×C₄⋊C₄	—	4	1	no	—
56	C₂×C₄⋊C₄	—	4	1	no	—
57	C₂×C₄²	—	8	1	no	—
58	C₂×C₄²	D₈	1	1	yes	characteristic
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
62	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₁₆	#41	1	1	yes	characteristic
67	C₂×C₁₆	#41	1	1	yes	characteristic
68	#108	—	2	2	no	—
69	#108	—	4	1	no	—
70	#107	—	2	2	no	—
71	#107	—	2	1	no	—
72	#107	—	2	1	no	—
73	#107	—	2	2	no	—
74	#107	—	4	1	no	—
75	#107	—	4	1	no	—
76	#106	—	4	1	no	—
77	#97	—	8	1	no	—
78	C₄×C₈	—	8	1	no	—
79	C₄×C₈	C₄⋊C₄	1	2	yes	—
80	C₄×C₈	Q₁₆	1	2	yes	—
81	#95	—	8	1	no	—
82	C₃₂	—	2	2	no	—
83	C₃₂	C₄²	1	4	yes	complemented
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	8	1	no	—
86	C₂²×C₄	—	8	1	no	—
87	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
88	M₁₆	—	4	2	no	—
89	M₁₆	—	8	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	8	1	no	—
93	C₂×C₈	#107	1	2	yes	—
94	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₈	C₄×D₄	1	1	yes	characteristic
96	C₂×C₈	C₄×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₈	#135	1	1	yes	characteristic
99	C₂×C₈	#135	1	1	yes	characteristic
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	8	1	no	—
105	C₄⋊C₄	—	8	1	no	—
106	C₂²⋊C₄	—	8	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	8	1	no	—
109	C₄²	—	16	2	no	—
110	C₄²	#107	1	2	yes	—
111	C₁₆	—	2	2	no	—
112	C₁₆	C₂×C₄²	1	1	yes	characteristic
113	C₁₆	#117	1	1	yes	characteristic
114	C₁₆	C₄×Q₈	1	1	yes	characteristic
115	C₁₆	C₄×Q₈	1	1	yes	characteristic
116	C₂³	#509	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	2	no	—
124	C₂×C₄	—	16	1	no	—
125	C₂×C₄	—	16	1	no	—
126	C₂×C₄	—	16	1	no	—
127	C₂×C₄	—	16	1	no	—
128	C₂×C₄	C₂×#107	1	1	yes	characteristic
129	C₂×C₄	C₄×D₈	1	1	yes	characteristic
130	C₂×C₄	C₄×D₈	1	1	yes	characteristic
131	C₂×C₄	#509	1	1	yes	characteristic
132	C₂×C₄	#510	1	1	yes	characteristic
133	C₂×C₄	#510	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	8	2	no	—
136	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₈	#427	1	1	yes	characteristic
138	C₈	C₄×Q₁₆	1	1	yes	characteristic
139	C₈	C₄×Q₁₆	1	1	yes	characteristic
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#2159	1	1	yes	characteristic
143	C₂²	#2175	1	1	yes	characteristic
144	C₂²	#2177	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	16	2	no	—
147	C₄	—	16	1	no	—
148	C₄	—	16	1	no	—
149	C₄	—	16	2	no	—
150	C₄	C₄×#107	1	1	yes	characteristic
151	C₄	#2159	1	1	yes	characteristic
152	C₄	#2175	1	1	yes	characteristic
153	C₄	#2176	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#12674	1	1	yes	characteristic
157	C₂	#13657	1	1	yes	characteristic
158	C₂	#13658	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ 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
17. #1426	C₄	semidirect
18. #1424	C₄	semidirect
19. #1423	C₄	semidirect
20. #1398	C₄	semidirect
83. C₃₂	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1426	2
#13634	2
C₂×#1398	2
#12674	2
#12671	2
#7478	2
#7477	2