Finite Group Databasebeta

Subgroups — #150882

Order 512.

245 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 68 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	#33608	C₂	1	1	yes	maximal, characteristic
2	#13556	C₂	1	1	yes	maximal, characteristic
3	#12840	C₂	1	1	yes	maximal, characteristic
4	#7425	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₃₂	C₄	1	1	yes	characteristic
6	C₂×#504	C₂²	1	1	yes	characteristic
7	#3059	—	2	1	no	—
8	C₂×M₆₄	—	2	1	no	—
9	#2219	C₄	1	2	yes	—
10	#2219	C₂²	1	2	yes	—
11	#2216	—	2	1	no	—
12	#2216	—	2	1	no	—
13	C₂×#368	—	2	1	no	—
14	#2158	C₄	1	1	yes	characteristic
15	#2141	—	2	1	no	—
16	#1813	—	2	1	no	—
17	#1400	—	2	1	no	—
18	#1400	C₂²	1	2	yes	—
19	#1385	C₂²	1	2	yes	—
20	C₂²×Q₁₆	—	2	1	no	—
21	C₂×#131	D₄	1	1	yes	characteristic
22	#536	—	4	1	no	—
23	C₂×Q₃₂	—	2	1	no	—
24	C₂×Q₃₂	—	4	1	no	—
25	C₂×Q₃₂	—	4	1	no	—
26	C₂×Q₃₂	C₂×C₄	1	2	yes	—
27	#504	—	4	1	no	—
28	#504	—	4	1	no	—
29	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	D₄	1	1	yes	characteristic
31	C₂×M₃₂	C₂³	1	1	yes	characteristic
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₁₆	D₄	1	1	yes	characteristic
35	#470	—	4	1	no	—
36	#470	—	4	1	no	—
37	#467	—	2	2	no	—
38	#467	—	2	2	no	—
39	C₂×#108	—	2	2	no	—
40	#427	—	2	1	no	—
41	#419	—	4	1	no	—
42	#413	—	4	1	no	—
43	M₆₄	—	2	2	no	—
44	M₆₄	—	4	1	no	—
45	C₂×C₃₂	—	4	1	no	—
46	#368	—	4	1	no	—
47	#368	—	4	1	no	—
48	#367	C₂×C₄	1	2	yes	—
49	#366	—	2	1	no	—
50	#358	—	2	2	no	—
51	#358	—	2	2	no	—
52	#338	—	4	1	no	—
53	C₂×#41	D₈	1	1	yes	characteristic
54	C₂²×Q₈	—	4	1	no	—
55	C₂×Q₁₆	—	2	2	no	—
56	C₂×Q₁₆	—	4	2	no	—
57	C₂×Q₁₆	—	4	2	no	—
58	C₂×Q₁₆	—	4	1	no	—
59	#131	—	4	1	no	—
60	#131	—	4	1	no	—
61	C₂×M₁₆	—	4	2	no	—
62	C₂×M₁₆	D₈	1	1	yes	characteristic
63	C₂×M₁₆	SD₁₆	1	1	yes	characteristic
64	C₂×M₁₆	SD₁₆	1	1	yes	characteristic
65	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₄⋊Q₈	—	8	1	no	—
69	#124	—	8	1	no	—
70	#123	—	8	1	no	—
71	#123	—	8	1	no	—
72	C₂²⋊Q₈	—	4	2	no	—
73	C₂²⋊Q₈	—	4	2	no	—
74	#117	—	4	1	no	—
75	Q₃₂	—	4	2	no	—
76	Q₃₂	—	4	1	no	—
77	Q₃₂	—	4	1	no	—
78	M₃₂	—	4	1	no	—
79	M₃₂	—	4	1	no	—
80	M₃₂	—	4	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
91	#108	—	4	2	no	—
92	#108	—	4	2	no	—
93	#107	—	2	2	no	—
94	#106	—	4	1	no	—
95	#103	—	4	2	no	—
96	#103	—	4	2	no	—
97	#101	—	8	1	no	—
98	#100	—	8	1	no	—
99	C₃₂	—	4	2	no	—
100	#41	—	4	1	no	—
101	#41	—	4	1	no	—
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	8	2	no	—
104	C₂×Q₈	—	8	2	no	—
105	C₂×Q₈	—	8	1	no	—
106	C₂×Q₈	D₁₆	1	1	yes	characteristic
107	C₂×D₄	D₁₆	1	1	yes	characteristic
108	C₂×D₄	SD₃₂	1	1	yes	characteristic
109	C₂×D₄	SD₃₂	1	1	yes	characteristic
110	C₂²×C₄	—	8	1	no	—
111	C₂²×C₄	#102	1	1	yes	characteristic
112	C₂²×C₄	#102	1	1	yes	characteristic
113	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	1	no	—
119	M₁₆	—	4	1	no	—
120	M₁₆	—	8	2	no	—
121	M₁₆	—	8	2	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	8	2	no	—
129	C₂×C₈	#102	1	1	yes	characteristic
130	C₂×C₈	#102	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	8	2	no	—
135	C₄⋊C₄	—	8	2	no	—
136	C₂²⋊C₄	—	8	1	no	—
137	C₂²⋊C₄	—	8	2	no	—
138	C₂²⋊C₄	—	8	2	no	—
139	C₄²	—	8	1	no	—
140	C₁₆	—	2	1	no	—
141	C₁₆	—	2	1	no	—
142	C₁₆	—	2	1	no	—
143	C₁₆	#99	1	1	yes	characteristic
144	C₁₆	#99	1	1	yes	characteristic
145	C₂³	#357	1	1	yes	characteristic
146	C₂³	#357	1	1	yes	characteristic
147	C₂³	C₂×SD₃₂	1	1	yes	characteristic
148	Q₈	—	4	1	no	—
149	Q₈	—	8	2	no	—
150	Q₈	—	8	2	no	—
151	Q₈	—	16	2	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	4	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	16	2	no	—
164	C₂×C₄	#357	1	1	yes	characteristic
165	C₂×C₄	#357	1	1	yes	characteristic
166	C₂×C₄	C₂×#102	1	1	yes	characteristic
167	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	16	2	no	—
172	C₈	C₂×#99	1	1	yes	characteristic
173	C₈	#421	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	C₂×#357	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	16	2	no	—
185	C₄	—	16	2	no	—
186	C₄	#1519	1	1	yes	characteristic
187	C₄	#2145	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	4	1	no	—
190	C₂	—	4	1	no	—
191	C₂	—	4	1	no	—
192	C₂	#11886	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#11886
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7425	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33608	2
#13556	2
#12840	2
#7425	2