Finite Group Databasebeta

Subgroups — #134749

Order 512.

293 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 84 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	#33435	C₂	1	1	yes	maximal, characteristic
2	#12832	C₂	1	2	yes	maximal, complemented
3	#12809	C₂	1	1	yes	maximal, characteristic
4	#12807	C₂	1	1	yes	maximal, characteristic
5	#7227	C₂	1	2	yes	maximal, complemented
6	C₂×#504	C₄	1	1	yes	characteristic, complemented
7	C₂×#504	C₂²	1	1	yes	characteristic
8	#2931	C₂²	1	1	yes	characteristic
9	#2168	—	2	2	no	—
10	#2168	—	2	2	no	—
11	#2155	—	2	2	no	—
12	#2150	C₄	1	1	yes	characteristic, complemented
13	#2132	—	2	1	no	—
14	#2126	—	2	1	no	—
15	#2113	C₄	1	1	yes	characteristic, complemented
16	#2113	C₂²	1	1	yes	characteristic
17	C₂×#346	C₄	1	1	yes	characteristic, complemented
18	#1813	C₂²	1	2	yes	—
19	#1812	C₂²	1	2	yes	—
20	#1392	—	2	1	no	—
21	#1391	—	2	1	no	—
22	#1382	—	2	2	no	—
23	C₂×#131	C₂×C₄	1	1	yes	characteristic
24	C₂×#131	C₂³	1	1	yes	characteristic
25	#532	D₄	1	1	yes	characteristic
26	#504	—	4	2	no	—
27	#504	—	4	2	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	D₄	1	2	yes	—
32	C₂²×C₁₆	—	2	1	no	—
33	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₆	D₄	1	2	yes	—
36	#436	—	2	2	no	—
37	C₈×D₄	—	2	2	no	—
38	C₂×#108	—	2	2	no	—
39	#427	—	2	2	no	—
40	#423	Q₈	1	1	yes	characteristic
41	#413	—	2	2	no	—
42	#410	—	2	2	no	—
43	#408	D₄	1	1	yes	characteristic
44	#408	Q₈	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	#365	—	4	4	no	—
47	C₄⋊C₁₆	—	2	2	no	—
48	#356	—	4	1	no	—
49	#355	—	4	1	no	—
50	#354	—	4	1	no	—
51	#353	—	4	1	no	—
52	C₂²⋊C₁₆	—	2	2	no	—
53	C₂²⋊C₁₆	—	2	2	no	—
54	#346	—	2	1	no	—
55	#346	—	2	1	no	—
56	#338	—	2	2	no	—
57	H(3,ℤ/4)	—	2	2	no	—
58	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#41	C₂×D₄	1	1	yes	characteristic
60	#131	—	4	1	no	—
61	#131	—	4	1	no	—
62	#131	—	4	1	no	—
63	#131	—	4	1	no	—
64	C₂×M₁₆	—	2	1	no	—
65	C₂×M₁₆	—	4	2	no	—
66	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
67	C₂×M₁₆	C₂×Q₈	1	1	yes	characteristic
68	C₂×M₁₆	#41	1	2	yes	—
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	2	yes	—
73	C₄⋊Q₈	SD₁₆	1	1	yes	characteristic
74	#127	SD₁₆	1	1	yes	characteristic
75	#124	SD₁₆	1	1	yes	characteristic
76	#124	SD₁₆	1	1	yes	characteristic
77	#121	—	2	2	no	—
78	C₄×D₄	—	2	2	no	—
79	#117	—	4	2	no	—
80	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
81	M₃₂	—	2	4	no	—
82	M₃₂	—	4	1	no	—
83	M₃₂	—	4	2	no	—
84	M₃₂	—	4	1	no	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	4	2	no	—
89	C₂×C₁₆	—	4	2	no	—
90	C₂×C₁₆	—	4	1	no	—
91	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
93	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
94	#108	—	4	2	no	—
95	#108	—	4	2	no	—
96	#107	—	4	2	no	—
97	#106	—	4	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	#101	—	4	2	no	—
100	#100	—	4	2	no	—
101	#99	—	4	2	no	—
102	#99	—	4	2	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	—	2	2	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	#41	—	4	1	no	—
108	#41	—	4	1	no	—
109	#41	—	4	1	no	—
110	#41	—	4	1	no	—
111	C₂×Q₈	#106	1	1	yes	characteristic
112	C₂×Q₈	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	#106	1	1	yes	characteristic
116	C₂×D₄	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×D₄	#135	1	2	yes	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄	#121	1	2	yes	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	1	no	—
126	M₁₆	—	8	2	no	—
127	M₁₆	—	8	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	8	2	no	—
138	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#117	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	8	2	no	—
143	C₂²⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	8	2	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	8	2	no	—
148	C₄²	#106	1	1	yes	characteristic
149	C₄²	#106	1	1	yes	characteristic
150	C₁₆	—	2	1	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	4	1	no	—
153	C₁₆	—	4	1	no	—
154	C₁₆	#99	1	2	yes	—
155	C₂³	—	2	1	no	—
156	C₂³	#427	1	1	yes	characteristic
157	C₂³	#465	1	2	yes	—
158	Q₈	—	4	1	no	—
159	Q₈	—	4	1	no	—
160	D₄	—	4	1	no	—
161	D₄	—	4	2	no	—
162	D₄	—	4	1	no	—
163	D₄	—	4	2	no	—
164	D₄	—	8	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	16	2	no	—
176	C₂×C₄	#380	1	1	yes	characteristic
177	C₂×C₄	C₂×#106	1	1	yes	characteristic
178	C₂×C₄	#465	1	2	yes	—
179	C₈	—	2	1	no	—
180	C₈	—	2	2	no	—
181	C₈	—	4	1	no	—
182	C₈	—	4	1	no	—
183	C₈	—	16	2	no	—
184	C₈	C₂×#99	1	1	yes	characteristic
185	C₈	#410	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	4	1	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	1	no	—
192	C₂²	#1807	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	4	1	no	—
197	C₄	—	4	1	no	—
198	C₄	—	32	2	no	—
199	C₄	#1799	1	1	yes	characteristic
200	C₄	#1812	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	4	2	no	—
203	C₂	—	4	1	no	—
204	C₂	—	8	1	no	—
205	C₂	#8608	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1812
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8608
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #12832	C₂	semidirect
5. #7227	C₂	semidirect
6. C₂×#504	C₄	semidirect
12. #2150	C₄	semidirect
15. #2113	C₄	semidirect
17. C₂×#346	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33435	2
#12832	2
#12809	2
#12807	2
#7227	2