Finite Group Databasebeta

Subgroups — #140408

Order 512.

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

normal subgroups only

This group has 196 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	#12881	C₂	1	1	yes	maximal, characteristic
2	#12660	C₂	1	1	yes	maximal, characteristic
3	#12656	C₂	1	1	yes	maximal, characteristic
4	#12011	C₂	1	1	yes	maximal, characteristic
5	#12002	C₂	1	1	yes	maximal, characteristic
6	#7162	C₂	1	1	yes	maximal, characteristic
7	#7162	C₂	1	1	yes	maximal, characteristic
8	C₂×#364	C₄	1	1	yes	characteristic
9	C₂×#364	C₄	1	1	yes	characteristic
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	C₂×#364	C₂²	1	1	yes	characteristic
14	C₂×#347	—	2	1	no	—
15	C₄×M₃₂	C₂²	1	1	yes	characteristic
16	C₄×M₃₂	C₂²	1	1	yes	characteristic
17	#1726	C₂²	1	1	yes	characteristic
18	#1377	—	2	1	no	—
19	#1377	—	2	1	no	—
20	#1376	C₄	1	1	yes	characteristic
21	#1376	C₄	1	1	yes	characteristic
22	#1375	—	2	1	no	—
23	#1373	—	2	2	no	—
24	#1369	—	2	2	no	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
31	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
33	C₂×#97	D₄	1	1	yes	characteristic
34	C₂×#97	D₄	1	1	yes	characteristic
35	C₂×#97	Q₈	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#376	Q₈	1	1	yes	characteristic
40	#364	—	2	2	no	—
41	#364	—	2	2	no	—
42	#364	—	2	1	no	—
43	#364	—	2	1	no	—
44	#364	—	2	1	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	—	2	1	no	—
48	#364	D₄	1	2	yes	—
49	#364	Q₈	1	2	yes	—
50	#347	—	4	2	no	—
51	#346	—	2	1	no	—
52	#346	—	2	1	no	—
53	C₄×C₁₆	—	2	1	no	—
54	C₄×C₁₆	—	2	1	no	—
55	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
62	M₃₂	—	2	1	no	—
63	M₃₂	—	2	1	no	—
64	M₃₂	—	2	2	no	—
65	M₃₂	—	2	2	no	—
66	M₃₂	—	2	1	no	—
67	M₃₂	—	2	1	no	—
68	M₃₂	—	2	1	no	—
69	M₃₂	—	2	1	no	—
70	M₃₂	—	4	1	no	—
71	M₃₂	—	4	1	no	—
72	M₃₂	—	4	1	no	—
73	M₃₂	—	4	1	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	—	4	1	no	—
76	M₃₂	M₁₆	1	2	yes	—
77	M₃₂	M₁₆	1	2	yes	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	4	1	no	—
83	C₂×C₁₆	—	4	1	no	—
84	C₂×C₁₆	—	4	1	no	—
85	C₂×C₁₆	—	4	1	no	—
86	#97	—	2	2	no	—
87	#97	—	2	1	no	—
88	#97	—	2	1	no	—
89	#97	—	2	1	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	C₂×D₄	1	1	yes	characteristic
96	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₄×C₈	#41	1	1	yes	characteristic
98	C₄×C₈	#41	1	1	yes	characteristic
99	#95	—	4	1	no	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	#108	1	2	yes	—
110	C₂×C₈	C₄×D₄	1	1	yes	characteristic
111	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₈	#121	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₈	#131	1	1	yes	characteristic
117	C₂×C₈	#131	1	1	yes	characteristic
118	C₂×C₈	#135	1	1	yes	characteristic
119	C₂×C₈	#135	1	1	yes	characteristic
120	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₈	#137	1	1	yes	characteristic
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	#125	1	1	yes	characteristic
128	C₄²	C₄⋊Q₈	1	1	yes	characteristic
129	C₁₆	—	2	2	no	—
130	C₁₆	—	2	2	no	—
131	C₁₆	—	4	1	no	—
132	C₁₆	—	4	1	no	—
133	C₁₆	—	4	2	no	—
134	C₁₆	—	4	2	no	—
135	C₁₆	—	8	1	no	—
136	C₁₆	—	8	1	no	—
137	C₂³	#430	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	#389	1	1	yes	characteristic
146	C₂×C₄	#423	1	1	yes	characteristic
147	C₂×C₄	#424	1	1	yes	characteristic
148	C₂×C₄	#427	1	1	yes	characteristic
149	C₂×C₄	#428	1	1	yes	characteristic
150	C₂×C₄	C₂×#108	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	4	1	no	—
153	C₈	#435	1	1	yes	characteristic
154	C₈	#435	1	1	yes	characteristic
155	C₈	#446	1	1	yes	characteristic
156	C₈	#446	1	1	yes	characteristic
157	C₈	#463	1	1	yes	characteristic
158	C₈	#464	1	1	yes	characteristic
159	C₈	#478	1	1	yes	characteristic
160	C₈	#479	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	#1569	1	1	yes	characteristic
163	C₂²	#1572	1	1	yes	characteristic
164	C₂²	#1950	1	1	yes	characteristic
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	8	1	no	—
168	C₄	—	8	1	no	—
169	C₄	#1568	1	1	yes	characteristic
170	C₄	#1574	1	1	yes	characteristic
171	C₄	#1940	1	1	yes	characteristic
172	C₄	#1949	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#9606	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1949
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9606
prime radical	C₂³	#430
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12881	2
#12660	2
#12656	2
#12011	2
#12002	2
#7162	2