Finite Group Databasebeta

Subgroups — #134970

Order 512.

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

normal subgroups only

This group has 300 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×#2170	C₂	1	1	yes	maximal, characteristic
2	#12906	C₂	1	1	yes	maximal, characteristic, complemented
3	#12821	C₂	1	2	yes	maximal
4	C₂×#1373	C₂	1	1	yes	maximal, characteristic, complemented
5	#7269	C₂	1	2	yes	maximal, complemented
6	C₂×#504	—	2	1	no	—
7	C₂×#443	C₂²	1	1	yes	characteristic
8	#2170	—	2	2	no	—
9	#2170	C₄	1	4	yes	complemented
10	C₂×#367	—	2	1	no	—
11	C₂×#367	—	2	1	no	—
12	C₂×#364	C₂²	1	2	yes	—
13	C₂×#364	C₂²	1	1	yes	characteristic
14	#2139	—	2	2	no	—
15	C₂×#350	—	2	1	no	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1845	C₂²	1	2	yes	—
18	#1387	—	2	2	no	—
19	#1373	—	2	2	no	—
20	C₂×#135	D₄	1	1	yes	characteristic
21	C₂×#131	—	2	1	no	—
22	#504	—	2	4	no	—
23	#504	—	4	2	no	—
24	C₂×M₃₂	—	2	1	no	—
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	—	4	1	no	—
27	C₂²×C₁₆	—	4	1	no	—
28	C₂²×C₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₁₆	D₄	1	1	yes	characteristic
30	#443	—	2	1	no	—
31	#443	—	2	1	no	—
32	#443	C₂×C₄	1	2	yes	—
33	#443	Q₈	1	2	yes	—
34	C₂×#108	D₄	1	1	yes	characteristic
35	C₂×#107	—	2	2	no	—
36	C₂×#107	—	2	2	no	—
37	C₂×#104	—	2	1	no	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#384	—	2	2	no	—
40	#367	—	4	2	no	—
41	#367	—	4	1	no	—
42	#367	—	4	1	no	—
43	#364	—	2	1	no	—
44	#364	—	2	2	no	—
45	#364	—	2	2	no	—
46	#364	C₂×C₄	1	2	yes	—
47	#358	—	4	2	no	—
48	#357	—	4	2	no	—
49	#350	—	2	4	no	—
50	C₄×C₁₆	C₂×C₄	1	2	yes	—
51	C₄×C₁₆	D₄	1	2	yes	—
52	C₂×#41	—	2	1	no	—
53	#135	—	2	1	no	—
54	#135	—	2	1	no	—
55	#135	C₄⋊C₄	1	2	yes	—
56	#135	Q₁₆	1	2	yes	—
57	C₂×Q₁₆	D₈	1	1	yes	characteristic
58	C₂×SD₁₆	—	2	1	no	—
59	C₂×D₈	D₈	1	1	yes	characteristic
60	#131	—	2	2	no	—
61	#131	—	2	2	no	—
62	#131	—	4	1	no	—
63	#131	—	4	1	no	—
64	C₂×M₁₆	—	2	1	no	—
65	C₂×M₁₆	—	4	1	no	—
66	C₂²×C₈	—	4	1	no	—
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄⋊C₄	—	8	2	no	—
71	C₂×C₄²	#41	1	1	yes	characteristic
72	M₃₂	—	2	2	no	—
73	M₃₂	—	4	1	no	—
74	M₃₂	—	4	4	no	—
75	M₃₂	—	4	2	no	—
76	M₃₂	—	4	2	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	4	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
83	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
84	C₂×C₁₆	D₈	1	1	yes	characteristic
85	C₂×C₁₆	D₈	1	1	yes	characteristic
86	C₂×C₁₆	SD₁₆	1	2	yes	—
87	#108	—	2	1	no	—
88	#108	C₄⋊C₄	1	2	yes	—
89	#107	—	4	2	no	—
90	#107	—	4	2	no	—
91	#107	—	4	2	no	—
92	#107	—	4	2	no	—
93	#104	—	2	2	no	—
94	#104	—	2	2	no	—
95	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₄×C₈	#41	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	#41	—	4	1	no	—
102	#41	—	4	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	4	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	—	8	2	no	—
109	C₂²×C₄	#121	1	1	yes	characteristic
110	Q₁₆	—	2	1	no	—
111	Q₁₆	#107	1	2	yes	—
112	SD₁₆	—	2	2	no	—
113	SD₁₆	—	2	2	no	—
114	D₈	—	2	1	no	—
115	D₈	#107	1	2	yes	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₈	#121	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₈	C₂×D₈	1	1	yes	characteristic
128	C₂×C₈	C₂×D₈	1	1	yes	characteristic
129	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₈	#135	1	1	yes	characteristic
132	C₂×C₈	#135	1	1	yes	characteristic
133	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₈	#137	1	1	yes	characteristic
135	C₄⋊C₄	—	8	2	no	—
136	C₄⋊C₄	—	8	2	no	—
137	C₄⋊C₄	—	16	4	no	—
138	C₄²	—	2	1	no	—
139	C₄²	#117	1	1	yes	characteristic
140	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
141	C₁₆	—	2	2	no	—
142	C₁₆	—	4	2	no	—
143	C₁₆	—	8	2	no	—
144	C₁₆	#102	1	2	yes	—
145	C₁₆	#102	1	2	yes	—
146	C₂³	—	4	1	no	—
147	C₂³	#472	1	1	yes	characteristic
148	Q₈	—	2	2	no	—
149	Q₈	—	4	1	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	1	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	16	4	no	—
161	C₂×C₄	—	16	2	no	—
162	C₂×C₄	#380	1	1	yes	characteristic
163	C₂×C₄	#459	1	1	yes	characteristic
164	C₂×C₄	#461	1	1	yes	characteristic
165	C₂×C₄	#466	1	1	yes	characteristic
166	C₂×C₄	#467	1	1	yes	characteristic
167	C₂×C₄	#473	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	C₂×#102	1	1	yes	characteristic
170	C₈	#417	1	1	yes	characteristic
171	C₈	C₂×#107	1	1	yes	characteristic
172	C₈	#427	1	1	yes	characteristic
173	C₈	#466	1	1	yes	characteristic
174	C₈	#467	1	1	yes	characteristic
175	C₈	#474	1	1	yes	characteristic
176	C₈	#476	1	1	yes	characteristic
177	C₂²	—	2	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	8	2	no	—
180	C₂²	#1667	1	1	yes	characteristic
181	C₂²	#1675	1	1	yes	characteristic
182	C₂²	#1816	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	32	2	no	—
187	C₄	#1669	1	1	yes	characteristic
188	C₄	#1673	1	1	yes	characteristic
189	C₄	#1802	1	1	yes	characteristic
190	C₄	#1808	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	8	2	no	—
194	C₂	#8644	1	1	yes	characteristic
195	C₂	#12438	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1667
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
2. #12906	C₂	semidirect
4. C₂×#1373	C₂	semidirect
5. #7269	C₂	semidirect
9. #2170	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2170	2
#12906	2
#12821	2
C₂×#1373	2
#7269	2