Finite Group Databasebeta

Subgroups — #152280

Order 512.

269 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 96 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×#2178	C₂	1	1	yes	maximal, characteristic
2	C₂×#1422	C₂	1	1	yes	maximal, characteristic, complemented
3	#13620	C₂	1	2	yes	maximal
4	#12909	C₂	1	1	yes	maximal, characteristic, complemented
5	#7479	C₂	1	2	yes	maximal, complemented
6	C₂×#508	C₂²	1	1	yes	characteristic
7	C₂×#443	—	2	1	no	—
8	C₂×M₆₄	C₂²	1	2	yes	—
9	#2178	—	2	1	no	—
10	#2178	—	2	1	no	—
11	#2178	C₄	1	2	yes	—
12	#2178	C₄	1	2	yes	—
13	C₂×#368	—	2	1	no	—
14	C₂×#368	C₂²	1	2	yes	—
15	C₂×#368	C₂²	1	1	yes	characteristic
16	C₂×#359	—	2	1	no	—
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	#1848	—	2	1	no	—
19	#1422	—	2	1	no	—
20	#1422	—	2	1	no	—
21	#1417	—	2	4	no	—
22	C₂×#135	—	2	1	no	—
23	C₂×#131	—	4	1	no	—
24	#508	—	2	1	no	—
25	#508	—	2	1	no	—
26	#508	C₂×C₄	1	2	yes	—
27	#508	D₄	1	2	yes	—
28	C₂×Q₃₂	Q₈	1	1	yes	characteristic
29	C₂×SD₃₂	—	2	1	no	—
30	C₂×D₁₆	Q₈	1	1	yes	characteristic
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	#443	—	2	2	no	—
34	#443	—	2	2	no	—
35	#443	—	4	1	no	—
36	#443	—	4	1	no	—
37	C₂×#108	—	2	1	no	—
38	C₂×#108	—	2	2	no	—
39	C₂×#108	—	4	1	no	—
40	#423	—	4	1	no	—
41	C₂×#104	—	4	1	no	—
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	M₆₄	—	2	2	no	—
44	M₆₄	—	2	2	no	—
45	C₂×C₃₂	—	2	2	no	—
46	#368	—	2	1	no	—
47	#368	—	2	2	no	—
48	#368	—	2	2	no	—
49	#368	—	4	2	no	—
50	#368	C₂×C₄	1	2	yes	—
51	#359	—	2	2	no	—
52	#359	—	2	2	no	—
53	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
54	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
55	C₄×C₁₆	D₄	1	1	yes	characteristic
56	C₄×C₁₆	D₄	1	1	yes	characteristic
57	C₂×#41	—	4	1	no	—
58	#135	—	2	2	no	—
59	#135	—	2	2	no	—
60	#135	—	4	1	no	—
61	#135	—	4	1	no	—
62	C₂×Q₁₆	—	2	1	no	—
63	C₂×SD₁₆	—	4	1	no	—
64	C₂×D₈	—	2	1	no	—
65	#131	—	4	2	no	—
66	#131	—	4	2	no	—
67	#131	—	8	1	no	—
68	#131	—	8	1	no	—
69	C₂×M₁₆	—	4	1	no	—
70	C₂×M₁₆	—	4	2	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×C₄²	D₈	1	1	yes	characteristic
76	Q₃₂	—	2	1	no	—
77	Q₃₂	C₄⋊C₄	1	2	yes	—
78	SD₃₂	—	2	2	no	—
79	SD₃₂	—	2	2	no	—
80	D₁₆	—	2	1	no	—
81	D₁₆	C₄⋊C₄	1	2	yes	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
88	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
89	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₁₆	#41	1	1	yes	characteristic
91	C₂×C₁₆	#41	1	1	yes	characteristic
92	#108	—	2	2	no	—
93	#108	—	4	2	no	—
94	#108	—	4	1	no	—
95	#108	—	4	2	no	—
96	#108	—	8	1	no	—
97	#108	—	8	1	no	—
98	C₄⋊C₈	—	8	2	no	—
99	#104	—	4	2	no	—
100	#104	—	4	2	no	—
101	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₈	SD₁₆	1	1	yes	characteristic
104	C₄×C₈	SD₁₆	1	1	yes	characteristic
105	C₃₂	—	4	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	C₂×D₈	1	1	yes	characteristic
116	Q₁₆	—	2	2	no	—
117	Q₁₆	—	4	1	no	—
118	SD₁₆	—	4	2	no	—
119	SD₁₆	—	4	2	no	—
120	D₈	—	2	2	no	—
121	D₈	—	4	1	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	8	1	no	—
124	M₁₆	—	8	2	no	—
125	M₁₆	—	8	2	no	—
126	M₁₆	—	8	2	no	—
127	M₁₆	—	8	2	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	—	8	2	no	—
131	C₂×C₈	—	8	2	no	—
132	C₂×C₈	#102	1	1	yes	characteristic
133	C₂×C₈	#102	1	1	yes	characteristic
134	C₂×C₈	D₁₆	1	1	yes	characteristic
135	C₂×C₈	D₁₆	1	1	yes	characteristic
136	C₂×C₈	SD₃₂	1	1	yes	characteristic
137	C₂×C₈	SD₃₂	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#121	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₈	#137	1	1	yes	characteristic
144	C₄²	—	2	1	no	—
145	C₄²	#102	1	1	yes	characteristic
146	C₄²	#102	1	1	yes	characteristic
147	C₁₆	—	2	1	no	—
148	C₁₆	—	2	1	no	—
149	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₁₆	#117	1	1	yes	characteristic
151	C₁₆	#121	1	1	yes	characteristic
152	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂³	—	8	1	no	—
154	C₂³	#510	1	1	yes	characteristic
155	Q₈	—	4	2	no	—
156	Q₈	—	8	1	no	—
157	D₄	—	4	2	no	—
158	D₄	—	8	2	no	—
159	D₄	—	8	2	no	—
160	D₄	—	8	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	C₂×#102	1	1	yes	characteristic
168	C₂×C₄	#459	1	1	yes	characteristic
169	C₂×C₄	#474	1	1	yes	characteristic
170	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
172	C₂×C₄	#509	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	16	2	no	—
175	C₈	#357	1	1	yes	characteristic
176	C₈	#357	1	1	yes	characteristic
177	C₈	#357	1	1	yes	characteristic
178	C₈	#357	1	1	yes	characteristic
179	C₈	#380	1	1	yes	characteristic
180	C₈	#417	1	1	yes	characteristic
181	C₈	#461	1	1	yes	characteristic
182	C₈	#476	1	1	yes	characteristic
183	C₂²	—	4	1	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	16	2	no	—
186	C₂²	#2139	1	1	yes	characteristic
187	C₂²	#2208	1	1	yes	characteristic
188	C₂²	#2224	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	8	2	no	—
192	C₄	#1802	1	1	yes	characteristic
193	C₄	C₂×#357	1	1	yes	characteristic
194	C₄	#2206	1	1	yes	characteristic
195	C₄	#2226	1	1	yes	characteristic
196	C₂	—	2	1	no	—
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	—	16	2	no	—
200	C₂	#12811	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12811
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
2. C₂×#1422	C₂	semidirect
4. #12909	C₂	semidirect
5. #7479	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2178	2
C₂×#1422	2
#13620	2
#12909	2
#7479	2