Finite Group Databasebeta

Subgroups — #151972

Order 512.

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

normal subgroups only

This group has 288 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×#2179	C₂	1	1	yes	maximal, characteristic
2	C₂×#1418	C₂	1	1	yes	maximal, characteristic
3	#13616	C₂	1	1	yes	maximal, characteristic
4	C₂×#1398	C₂	1	1	yes	maximal, characteristic, complemented
5	#12674	C₂	1	1	yes	maximal, characteristic, complemented
6	#7478	C₂	1	1	yes	maximal, characteristic, complemented
7	#7474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508	C₂²	1	1	yes	characteristic
9	C₂×#443	—	2	1	no	—
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×M₆₄	C₂²	1	1	yes	characteristic
12	#2179	—	2	2	no	—
13	#2179	C₄	1	4	yes	complemented
14	C₂×#368	C₂²	1	1	yes	characteristic
15	C₂×#367	C₂²	1	1	yes	characteristic
16	C₂×#366	C₂²	1	1	yes	characteristic
17	C₂×#359	—	2	1	no	—
18	C₂×#346	C₂²	1	1	yes	characteristic
19	C₄×#107	—	2	1	no	—
20	#1418	C₄	1	4	yes	complemented
21	#1416	C₄	1	2	yes	complemented
22	#1415	C₄	1	2	yes	complemented
23	#1398	—	2	2	no	—
24	#1380	—	2	1	no	—
25	C₂×#135	—	2	1	no	—
26	C₂×#131	—	4	1	no	—
27	#508	—	2	1	no	—
28	#508	—	2	1	no	—
29	#508	C₂×C₄	1	2	yes	—
30	#508	C₂×C₄	1	2	yes	—
31	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
32	C₂×SD₃₂	—	2	1	no	—
33	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	#443	—	2	4	no	—
37	#443	—	4	2	no	—
38	C₂×#108	—	2	1	no	—
39	C₂×#107	—	2	1	no	—
40	C₂×#107	—	2	1	no	—
41	C₂×#104	—	4	1	no	—
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₄×C₄⋊C₄	—	4	1	no	—
44	M₆₄	—	2	1	no	—
45	M₆₄	—	2	1	no	—
46	M₆₄	—	2	1	no	—
47	M₆₄	C₂×C₄	1	2	yes	—
48	C₂×C₃₂	—	2	1	no	—
49	C₂×C₃₂	C₂×C₄	1	2	yes	—
50	#368	—	2	1	no	—
51	#368	C₂×C₄	1	2	yes	—
52	#367	—	2	1	no	—
53	#367	C₂×C₄	1	2	yes	—
54	#366	—	2	1	no	—
55	#366	C₂×C₄	1	2	yes	—
56	#359	—	2	4	no	—
57	#346	C₂×C₄	1	2	yes	—
58	#346	D₄	1	2	yes	—
59	#340	—	4	1	no	—
60	C₂×#41	—	4	1	no	—
61	#135	—	2	2	no	—
62	#135	—	2	2	no	—
63	#135	—	4	1	no	—
64	#135	—	4	1	no	—
65	C₂×Q₁₆	—	2	1	no	—
66	C₂×SD₁₆	—	4	1	no	—
67	C₂×D₈	—	2	1	no	—
68	#131	—	4	4	no	—
69	#131	—	8	2	no	—
70	C₂×M₁₆	—	4	1	no	—
71	C₂×M₁₆	—	8	1	no	—
72	C₂²×C₈	—	8	1	no	—
73	C₂²×C₈	D₈	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄²	—	8	1	no	—
78	C₂×C₄²	D₈	1	1	yes	characteristic
79	Q₃₂	—	2	1	no	—
80	Q₃₂	C₄²	1	2	yes	complemented
81	SD₃₂	—	2	2	no	—
82	SD₃₂	—	2	2	no	—
83	D₁₆	—	2	1	no	—
84	D₁₆	C₄²	1	2	yes	complemented
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
88	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
92	C₂×C₁₆	#41	1	1	yes	characteristic
93	C₂×C₁₆	#41	1	1	yes	characteristic
94	#108	—	2	2	no	—
95	#108	—	4	1	no	—
96	#107	—	2	2	no	—
97	#107	—	2	2	no	—
98	#107	—	4	1	no	—
99	#107	—	4	1	no	—
100	#104	—	4	4	no	—
101	C₄×C₈	C₂²⋊C₄	1	2	yes	—
102	C₄×C₈	SD₁₆	1	2	yes	—
103	#95	—	8	1	no	—
104	C₃₂	—	2	2	no	—
105	C₃₂	—	2	2	no	—
106	#41	—	4	2	no	—
107	#41	—	4	2	no	—
108	#41	—	8	1	no	—
109	#41	—	8	1	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	8	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	8	1	no	—
115	C₂²×C₄	—	8	1	no	—
116	C₂²×C₄	—	8	1	no	—
117	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
118	Q₁₆	—	2	2	no	—
119	Q₁₆	—	4	1	no	—
120	SD₁₆	—	4	2	no	—
121	SD₁₆	—	4	2	no	—
122	D₈	—	2	2	no	—
123	D₈	—	4	1	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	8	4	no	—
126	M₁₆	—	8	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	8	4	no	—
130	C₂×C₈	—	8	1	no	—
131	C₂×C₈	#102	1	2	yes	—
132	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₈	C₄×D₄	1	1	yes	characteristic
134	C₂×C₈	C₄×D₄	1	1	yes	characteristic
135	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₈	#135	1	1	yes	characteristic
137	C₂×C₈	#135	1	1	yes	characteristic
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	8	1	no	—
141	C₄⋊C₄	—	8	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	16	2	no	—
144	C₄²	#102	1	2	yes	—
145	C₁₆	—	2	2	no	—
146	C₁₆	C₂×C₄²	1	1	yes	characteristic
147	C₁₆	#117	1	1	yes	characteristic
148	C₁₆	C₄×D₄	1	1	yes	characteristic
149	C₁₆	C₄×D₄	1	1	yes	characteristic
150	C₂³	—	8	1	no	—
151	C₂³	#509	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	Q₈	—	8	1	no	—
154	D₄	—	4	2	no	—
155	D₄	—	8	2	no	—
156	D₄	—	8	2	no	—
157	D₄	—	8	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	16	1	no	—
167	C₂×C₄	—	16	1	no	—
168	C₂×C₄	—	16	1	no	—
169	C₂×C₄	—	16	1	no	—
170	C₂×C₄	C₂×#102	1	1	yes	characteristic
171	C₂×C₄	C₄×D₈	1	1	yes	characteristic
172	C₂×C₄	C₄×D₈	1	1	yes	characteristic
173	C₂×C₄	#509	1	1	yes	characteristic
174	C₂×C₄	#509	1	1	yes	characteristic
175	C₂×C₄	#510	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	8	2	no	—
178	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
179	C₈	#416	1	1	yes	characteristic
180	C₈	C₄×SD₁₆	1	1	yes	characteristic
181	C₈	C₄×SD₁₆	1	1	yes	characteristic
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	16	1	no	—
186	C₂²	—	16	1	no	—
187	C₂²	#2139	1	1	yes	characteristic
188	C₂²	#2177	1	1	yes	characteristic
189	C₂²	#2177	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	2	no	—
192	C₄	—	16	2	no	—
193	C₄	—	16	2	no	—
194	C₄	C₄×#102	1	1	yes	characteristic
195	C₄	#2140	1	1	yes	characteristic
196	C₄	#2175	1	1	yes	characteristic
197	C₄	#2175	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	16	2	no	—
201	C₂	#12637	1	1	yes	characteristic
202	C₂	#13657	1	1	yes	characteristic
203	C₂	#13658	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₂×#1398	C₂	semidirect
5. #12674	C₂	semidirect
6. #7478	C₂	semidirect
7. #7474	C₂	semidirect
13. #2179	C₄	semidirect
20. #1418	C₄	semidirect
21. #1416	C₄	semidirect
22. #1415	C₄	semidirect
80. Q₃₂	C₄²	semidirect
84. D₁₆	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2179	2
C₂×#1418	2
#13616	2
C₂×#1398	2
#12674	2
#7478	2
#7474	2