Finite Group Databasebeta

Subgroups — #140500

Order 512.

207 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 90 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	#12033	C₂	1	1	yes	maximal, characteristic
2	#12032	C₂	1	1	yes	maximal, characteristic
3	#12012	C₂	1	1	yes	maximal, characteristic
4	#8824	C₂	1	1	yes	maximal, characteristic
5	#8282	C₂	1	1	yes	maximal, characteristic
6	#7296	C₂	1	1	yes	maximal, characteristic
7	#7296	C₂	1	1	yes	maximal, characteristic
8	#2162	C₂²	1	1	yes	characteristic
9	#2162	C₂²	1	1	yes	characteristic
10	#1837	—	2	1	no	—
11	C₄×#107	C₂²	1	1	yes	characteristic
12	#1770	—	2	1	no	—
13	#1567	C₂²	1	1	yes	characteristic
14	#1567	C₂²	1	1	yes	characteristic
15	#1566	—	2	1	no	—
16	C₂×#335	C₂²	1	1	yes	characteristic
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1372	—	2	2	no	—
19	#1371	—	2	2	no	—
20	#1370	C₄	1	2	yes	complemented
21	#1367	C₄	1	2	yes	complemented
22	#1279	—	2	1	no	—
23	C₂×#107	—	2	1	no	—
24	#423	—	2	1	no	—
25	#423	—	2	1	no	—
26	C₂×C₄⋊C₈	—	2	1	no	—
27	C₂×#97	—	4	1	no	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	C₄×C₄⋊C₄	—	2	1	no	—
32	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
33	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
34	C₂²⋊C₁₆	—	2	1	no	—
35	C₂²⋊C₁₆	—	2	1	no	—
36	C₄×C₁₆	—	2	1	no	—
37	C₄×C₁₆	—	2	1	no	—
38	#341	—	4	1	no	—
39	#341	—	4	1	no	—
40	#340	Q₈	1	2	yes	—
41	#336	—	4	1	no	—
42	#335	—	2	1	no	—
43	#335	—	2	1	no	—
44	#335	—	2	1	no	—
45	#335	—	2	1	no	—
46	#335	—	2	1	no	—
47	#335	D₄	1	2	yes	—
48	#334	—	4	2	no	—
49	C₈²	C₂×C₄	1	1	yes	characteristic
50	C₈²	C₂×C₄	1	1	yes	characteristic
51	C₈²	Q₈	1	1	yes	characteristic
52	C₈²	Q₈	1	1	yes	characteristic
53	C₂×M₁₆	—	4	1	no	—
54	C₂×M₁₆	—	4	1	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	—	4	1	no	—
57	C₂²×C₈	D₈	1	1	yes	characteristic
58	C₂²×C₈	D₈	1	1	yes	characteristic
59	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₈	#41	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄²	—	4	1	no	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	#107	—	4	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	4	1	no	—
69	C₄⋊C₈	—	4	1	no	—
70	C₄⋊C₈	—	4	1	no	—
71	C₄⋊C₈	—	4	1	no	—
72	C₄⋊C₈	—	4	1	no	—
73	#97	—	8	2	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₈	Q₁₆	1	1	yes	characteristic
80	C₄×C₈	Q₁₆	1	1	yes	characteristic
81	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₄×C₈	#41	1	1	yes	characteristic
85	#95	—	4	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
89	C₂²×C₄	#137	1	1	yes	characteristic
90	M₁₆	—	8	1	no	—
91	M₁₆	—	8	1	no	—
92	M₁₆	—	8	1	no	—
93	M₁₆	—	8	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	8	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	—	8	1	no	—
107	C₂×C₈	—	8	1	no	—
108	C₂×C₈	#107	1	1	yes	characteristic
109	C₂×C₈	#107	1	1	yes	characteristic
110	C₂×C₈	#107	1	1	yes	characteristic
111	C₂×C₈	#107	1	1	yes	characteristic
112	C₂×C₈	C₄×D₄	1	1	yes	characteristic
113	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₈	#125	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄²	—	8	1	no	—
119	C₄²	—	8	1	no	—
120	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₄²	#121	1	1	yes	characteristic
122	C₄²	C₂×Q₁₆	1	1	yes	characteristic
123	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₁₆	—	4	2	no	—
125	C₁₆	—	4	2	no	—
126	C₂³	#474	1	1	yes	characteristic
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	#389	1	1	yes	characteristic
134	C₂×C₄	C₂×#107	1	1	yes	characteristic
135	C₂×C₄	#428	1	1	yes	characteristic
136	C₂×C₄	#459	1	1	yes	characteristic
137	C₂×C₄	#462	1	1	yes	characteristic
138	C₂×C₄	#477	1	1	yes	characteristic
139	C₂×C₄	#508	1	1	yes	characteristic
140	C₂×C₄	#508	1	1	yes	characteristic
141	C₂×C₄	#509	1	1	yes	characteristic
142	C₂×C₄	#510	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	8	2	no	—
153	C₈	—	16	1	no	—
154	C₈	—	16	1	no	—
155	C₈	#444	1	2	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#1947	1	1	yes	characteristic
159	C₂²	#2228	1	1	yes	characteristic
160	C₂²	#2229	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	16	2	no	—
165	C₄	#1946	1	1	yes	characteristic
166	C₄	#1953	1	1	yes	characteristic
167	C₄	#2158	1	1	yes	characteristic
168	C₄	#2159	1	1	yes	characteristic
169	C₄	#2210	1	1	yes	characteristic
170	C₄	#2211	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#9630	1	1	yes	characteristic
174	C₂	#13178	1	1	yes	characteristic
175	C₂	#13182	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1947
prime radical	C₂³	#474
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
20. #1370	C₄	semidirect
21. #1367	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12033	2
#12032	2
#12012	2
#8824	2
#8282	2
#7296	2