Finite Group Databasebeta

Subgroups — #142143

Order 512.

298 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 98 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	#33317	C₂	1	1	yes	maximal, characteristic
2	#12882	C₂	1	1	yes	maximal, characteristic
3	#12659	C₂	1	1	yes	maximal, characteristic
4	#7271	C₂	1	2	yes	maximal
5	#7271	C₂	1	2	yes	maximal
6	C₂×#501	—	2	1	no	—
7	C₂×#501	—	2	1	no	—
8	#2862	C₄	1	1	yes	characteristic
9	#2859	C₄	1	1	yes	characteristic
10	C₂×#364	C₂²	1	2	yes	—
11	C₂×#364	C₂²	1	2	yes	—
12	C₂×#347	—	2	1	no	—
13	#1847	—	2	1	no	—
14	#1847	—	2	1	no	—
15	#1846	C₄	1	1	yes	characteristic
16	#1846	C₄	1	1	yes	characteristic
17	#1846	C₂²	1	1	yes	characteristic
18	#1846	C₂²	1	1	yes	characteristic
19	C₄×#97	C₂²	1	1	yes	characteristic
20	#1377	—	2	1	no	—
21	#1377	—	2	1	no	—
22	#1375	—	2	1	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×#125	—	2	1	no	—
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	#501	—	4	4	no	—
28	#501	—	4	2	no	—
29	#501	—	4	2	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	—	2	1	no	—
32	C₂×#107	—	2	1	no	—
33	C₂×#107	—	2	1	no	—
34	C₂×#106	—	2	1	no	—
35	C₂×#106	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	C₂×#106	—	2	1	no	—
38	C₂×#97	D₄	1	1	yes	characteristic
39	C₂×#97	D₄	1	1	yes	characteristic
40	C₂×#97	D₄	1	1	yes	characteristic
41	C₂×#97	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#391	—	4	2	no	—
45	#389	—	4	2	no	—
46	#384	—	2	1	no	—
47	#382	—	2	1	no	—
48	#382	C₂×C₄	1	1	yes	characteristic
49	#382	C₂×C₄	1	1	yes	characteristic
50	#382	C₂×C₄	1	1	yes	characteristic
51	#382	C₂×C₄	1	1	yes	characteristic
52	C₄³	C₂×C₄	1	1	yes	characteristic
53	#364	—	2	2	no	—
54	#364	—	2	4	no	—
55	#364	Q₈	1	4	yes	complemented
56	#347	—	4	2	no	—
57	C₂²×Q₈	—	4	1	no	—
58	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄⋊Q₈	—	4	2	no	—
63	C₄⋊Q₈	—	4	2	no	—
64	#125	—	4	2	no	—
65	#125	—	4	2	no	—
66	C₂×C₄⋊C₄	—	2	1	no	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	—	4	2	no	—
71	C₂×C₄⋊C₄	—	4	1	no	—
72	C₂×C₄⋊C₄	—	4	2	no	—
73	C₂×C₄²	—	2	2	no	—
74	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
77	M₃₂	—	2	4	no	—
78	M₃₂	—	4	2	no	—
79	M₃₂	—	4	2	no	—
80	M₃₂	—	4	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₂×C₁₆	—	4	2	no	—
83	#107	—	4	2	no	—
84	#107	—	4	2	no	—
85	#107	—	4	1	no	—
86	#107	—	4	1	no	—
87	#107	—	4	2	no	—
88	#106	—	4	2	no	—
89	#106	—	4	2	no	—
90	#106	—	4	1	no	—
91	#106	—	4	1	no	—
92	#106	—	4	2	no	—
93	#97	—	2	2	no	—
94	#97	—	2	2	no	—
95	#97	—	2	2	no	—
96	#97	D₈	1	2	yes	—
97	#97	SD₁₆	1	2	yes	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	C₂×Q₈	1	2	yes	—
100	C₄×C₈	#41	1	2	yes	—
101	#95	—	4	1	no	—
102	#95	—	4	1	no	—
103	#95	—	4	1	no	—
104	#95	—	4	1	no	—
105	#95	—	4	2	no	—
106	C₂×Q₈	—	8	2	no	—
107	C₂×Q₈	—	8	4	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	4	1	no	—
111	C₂²×C₄	—	4	1	no	—
112	C₂²×C₄	—	4	1	no	—
113	C₂²×C₄	—	4	1	no	—
114	C₂²×C₄	#100	1	2	yes	—
115	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	#101	1	2	yes	—
121	C₂×C₈	#102	1	2	yes	—
122	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₈	#124	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₈	C₂×D₈	1	1	yes	characteristic
126	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₈	#137	1	2	yes	—
129	C₂×C₈	#137	1	2	yes	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	8	4	no	—
139	C₄⋊C₄	—	8	2	no	—
140	C₄⋊C₄	—	8	4	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	8	2	no	—
146	C₄²	#102	1	2	yes	—
147	C₄²	C₄×Q₈	1	2	yes	—
148	C₁₆	—	4	4	no	—
149	C₁₆	—	8	2	no	—
150	C₂³	C₂×#100	1	1	yes	characteristic
151	Q₈	—	16	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	#391	1	1	yes	characteristic
167	C₂×C₄	C₂×#101	1	1	yes	characteristic
168	C₂×C₄	C₂×#102	1	1	yes	characteristic
169	C₂×C₄	#418	1	1	yes	characteristic
170	C₂×C₄	#419	1	1	yes	characteristic
171	C₂×C₄	#419	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	2	2	no	—
174	C₈	#474	1	2	yes	—
175	C₈	#475	1	2	yes	—
176	C₈	#489	1	2	yes	—
177	C₈	#501	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	#1545	1	1	yes	characteristic
180	C₂²	#1558	1	1	yes	characteristic
181	C₂²	#1997	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	16	1	no	—
186	C₄	—	16	1	no	—
187	C₄	—	16	1	no	—
188	C₄	—	16	1	no	—
189	C₄	#1546	1	1	yes	characteristic
190	C₄	#1556	1	1	yes	characteristic
191	C₄	#1988	1	1	yes	characteristic
192	C₄	#1993	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#9905	1	1	yes	characteristic
196	C₂	#11994	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ 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
55. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33317	2
#12882	2
#12659	2
#7271	2