Finite Group Databasebeta

Subgroups — #150140

Order 512.

239 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₄×#508	C₂	1	1	yes	maximal, characteristic
2	#13640	C₂	1	1	yes	maximal, characteristic, complemented
3	#12024	C₂	1	1	yes	maximal, characteristic, complemented
4	#7386	C₂	1	2	yes	maximal, complemented
5	#7385	C₂	1	2	yes	maximal, complemented
6	C₂×#508	C₄	1	1	yes	characteristic
7	C₄×#135	—	2	1	no	—
8	C₄×Q₃₂	C₄	1	1	yes	characteristic
9	C₄×Q₃₂	C₂²	1	1	yes	characteristic
10	C₄×SD₃₂	—	2	1	no	—
11	C₄×SD₃₂	—	2	1	no	—
12	C₄×D₁₆	C₄	1	1	yes	characteristic
13	C₄×D₁₆	C₂²	1	1	yes	characteristic
14	#2158	C₄	1	1	yes	characteristic
15	#2138	—	2	1	no	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1568	—	2	1	no	—
18	C₄⋊C₃₂	—	2	1	no	—
19	C₂²⋊C₃₂	—	2	1	no	—
20	#1397	C₂²	1	2	yes	—
21	#1372	C₂²	1	2	yes	—
22	#1371	—	2	1	no	—
23	C₂×#135	—	2	1	no	—
24	C₄×#41	—	4	1	no	—
25	#508	—	2	2	no	—
26	#508	—	2	2	no	—
27	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
28	C₂×SD₃₂	—	2	1	no	—
29	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	—	2	1	no	—
34	C₄×SD₁₆	—	4	1	no	—
35	C₄×SD₁₆	—	4	1	no	—
36	C₄×D₈	—	2	1	no	—
37	C₄×D₈	—	2	1	no	—
38	#432	—	4	1	no	—
39	#427	—	2	1	no	—
40	#416	—	4	1	no	—
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₃₂	—	2	2	no	—
43	#367	—	2	1	no	—
44	#366	C₂×C₄	1	1	yes	characteristic
45	#366	C₂×C₄	1	1	yes	characteristic
46	#358	—	2	1	no	—
47	#358	—	2	1	no	—
48	#357	—	2	1	no	—
49	#357	—	2	1	no	—
50	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₄×C₁₆	D₄	1	1	yes	characteristic
52	C₄×C₁₆	D₄	1	1	yes	characteristic
53	C₄×C₁₆	C₂³	1	1	yes	characteristic
54	#335	—	2	2	no	—
55	#334	—	4	1	no	—
56	C₂×#41	—	4	1	no	—
57	#135	—	2	4	no	—
58	#135	—	4	2	no	—
59	C₂×Q₁₆	—	2	1	no	—
60	C₂×SD₁₆	—	4	1	no	—
61	C₂×D₈	—	2	1	no	—
62	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	SD₁₆	1	1	yes	characteristic
64	C₄×Q₈	—	4	1	no	—
65	C₄×Q₈	—	4	1	no	—
66	C₄×D₄	—	4	1	no	—
67	C₄×D₄	—	4	1	no	—
68	C₄×D₄	—	8	1	no	—
69	C₄×D₄	—	8	1	no	—
70	#117	—	4	1	no	—
71	#117	—	8	1	no	—
72	C₂×C₄²	—	8	1	no	—
73	C₂×C₄²	D₈	1	1	yes	characteristic
74	Q₃₂	—	2	1	no	—
75	Q₃₂	M₁₆	1	2	yes	—
76	SD₃₂	—	2	4	no	—
77	D₁₆	—	2	1	no	—
78	D₁₆	M₁₆	1	2	yes	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
86	#107	—	2	1	no	—
87	#107	—	2	1	no	—
88	#106	—	4	1	no	—
89	C₄⋊C₈	—	4	2	no	—
90	#103	—	4	1	no	—
91	#103	—	4	1	no	—
92	#102	—	4	1	no	—
93	#102	—	4	1	no	—
94	C₂²⋊C₈	—	8	1	no	—
95	#97	—	8	1	no	—
96	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₈	D₈	1	1	yes	characteristic
98	C₄×C₈	SD₁₆	1	1	yes	characteristic
99	C₄×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃₂	—	4	2	no	—
101	#41	—	4	4	no	—
102	#41	—	8	2	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×D₄	—	4	1	no	—
105	C₂×D₄	—	8	1	no	—
106	C₂²×C₄	—	8	1	no	—
107	C₂²×C₄	—	8	1	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	#102	1	1	yes	characteristic
110	C₂²×C₄	D₁₆	1	1	yes	characteristic
111	C₂²×C₄	SD₃₂	1	1	yes	characteristic
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	SD₁₆	—	4	4	no	—
115	D₈	—	2	2	no	—
116	D₈	—	4	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	8	2	no	—
121	C₂×C₈	#102	1	1	yes	characteristic
122	C₂×C₈	#102	1	1	yes	characteristic
123	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	8	1	no	—
128	C₄⋊C₄	—	8	1	no	—
129	C₂²⋊C₄	—	8	1	no	—
130	C₂²⋊C₄	—	8	1	no	—
131	C₂²⋊C₄	—	8	1	no	—
132	C₄²	—	8	1	no	—
133	C₄²	—	8	1	no	—
134	C₄²	—	8	1	no	—
135	C₄²	#102	1	1	yes	characteristic
136	C₄²	D₁₆	1	1	yes	characteristic
137	C₄²	SD₃₂	1	1	yes	characteristic
138	C₄²	C₂×D₈	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	1	no	—
141	C₁₆	—	2	1	no	—
142	C₁₆	C₂×M₁₆	1	1	yes	characteristic
143	C₁₆	C₂×M₁₆	1	1	yes	characteristic
144	C₂³	—	8	1	no	—
145	C₂³	#357	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	Q₈	—	8	1	no	—
148	D₄	—	4	2	no	—
149	D₄	—	8	4	no	—
150	D₄	—	8	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	8	1	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	—	16	2	no	—
160	C₂×C₄	—	16	2	no	—
161	C₂×C₄	#357	1	1	yes	characteristic
162	C₂×C₄	#357	1	1	yes	characteristic
163	C₂×C₄	#357	1	1	yes	characteristic
164	C₂×C₄	C₂×#102	1	1	yes	characteristic
165	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
166	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	16	2	no	—
171	C₈	#407	1	1	yes	characteristic
172	C₈	#421	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	8	2	no	—
177	C₂²	—	16	1	no	—
178	C₂²	—	16	1	no	—
179	C₂²	C₂×#357	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	8	2	no	—
186	C₄	—	16	1	no	—
187	C₄	—	16	1	no	—
188	C₄	#1486	1	1	yes	characteristic
189	C₄	#2143	1	1	yes	characteristic
190	C₂	—	4	1	no	—
191	C₂	—	16	2	no	—
192	C₂	#11657	1	1	yes	characteristic
193	C₂	#13622	1	1	yes	characteristic
194	C₂	#13625	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13640	C₂	semidirect
3. #12024	C₂	semidirect
4. #7386	C₂	semidirect
5. #7385	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#508	2
#13640	2
#12024	2
#7386	2
#7385	2