Finite Group Databasebeta

Subgroups — #150787

Order 512.

219 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 72 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	#33735	C₂	1	1	yes	maximal, characteristic
2	#13639	C₂	1	1	yes	maximal, characteristic, complemented
3	#12889	C₂	1	1	yes	maximal, characteristic, complemented
4	#7419	C₂	1	2	yes	maximal, complemented
5	#7418	C₂	1	2	yes	maximal
6	#3168	—	2	1	no	—
7	#2253	C₄	1	1	yes	characteristic, complemented
8	#2249	C₄	1	1	yes	characteristic, complemented
9	#2242	—	2	1	no	—
10	#2217	—	2	1	no	—
11	#2216	C₂²	1	1	yes	characteristic
12	#2215	C₂²	1	1	yes	characteristic
13	C₄×Q₃₂	C₄	1	1	yes	characteristic, complemented
14	C₄×SD₃₂	—	2	1	no	—
15	C₄×D₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1846	—	2	1	no	—
18	C₄⋊C₃₂	—	2	1	no	—
19	C₂²⋊C₃₂	C₂²	1	2	yes	—
20	C₄×C₃₂	—	2	1	no	—
21	#1380	—	2	1	no	—
22	#1380	C₂²	1	2	yes	—
23	#529	—	4	1	no	—
24	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
25	C₂×SD₃₂	—	2	1	no	—
26	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂³	1	1	yes	characteristic
29	#499	—	2	1	no	—
30	#495	—	2	1	no	—
31	#488	—	4	1	no	—
32	#467	—	2	1	no	—
33	#466	—	2	1	no	—
34	#465	—	4	1	no	—
35	C₄×Q₁₆	—	2	1	no	—
36	C₄×SD₁₆	—	4	1	no	—
37	C₄×D₈	—	2	1	no	—
38	C₂×#107	—	2	2	no	—
39	C₂×#106	—	4	1	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	#382	—	4	1	no	—
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	C₂×C₄	1	1	yes	characteristic
52	C₄×C₁₆	D₄	1	1	yes	characteristic
53	C₄×C₁₆	D₄	1	1	yes	characteristic
54	C₂×Q₁₆	—	2	1	no	—
55	C₂×SD₁₆	—	4	1	no	—
56	C₂×D₈	—	2	1	no	—
57	C₂²×C₈	D₈	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	#126	—	8	1	no	—
60	#125	—	4	1	no	—
61	#124	—	4	1	no	—
62	#123	—	8	1	no	—
63	C₂²⋊Q₈	—	4	1	no	—
64	#121	—	4	1	no	—
65	C₄×Q₈	—	4	1	no	—
66	C₄×D₄	—	4	1	no	—
67	C₄×D₄	—	8	1	no	—
68	#117	—	8	1	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄²	D₈	1	1	yes	characteristic
71	Q₃₂	—	4	1	no	—
72	SD₃₂	—	4	2	no	—
73	D₁₆	—	4	1	no	—
74	C₂×C₁₆	—	4	1	no	—
75	C₂×C₁₆	—	4	1	no	—
76	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
80	#107	—	2	1	no	—
81	#107	—	2	1	no	—
82	#107	—	4	2	no	—
83	#107	—	4	2	no	—
84	#106	—	4	1	no	—
85	#106	—	8	2	no	—
86	#103	—	4	1	no	—
87	#103	—	4	1	no	—
88	#102	—	4	1	no	—
89	#102	—	4	1	no	—
90	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₈	SD₁₆	1	1	yes	characteristic
93	C₄×C₈	SD₁₆	1	1	yes	characteristic
94	#95	—	8	1	no	—
95	#95	—	8	1	no	—
96	C₃₂	—	4	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂×D₄	—	4	1	no	—
99	C₂×D₄	—	8	1	no	—
100	C₂²×C₄	—	8	1	no	—
101	C₂²×C₄	—	8	2	no	—
102	C₂²×C₄	D₁₆	1	1	yes	characteristic
103	C₂²×C₄	D₁₆	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
105	Q₁₆	—	4	1	no	—
106	Q₁₆	—	4	1	no	—
107	SD₁₆	—	8	2	no	—
108	D₈	—	4	1	no	—
109	D₈	—	4	1	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	#102	1	1	yes	characteristic
114	C₂×C₈	#102	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	8	2	no	—
120	C₄⋊C₄	—	8	2	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	8	1	no	—
123	C₄⋊C₄	—	8	1	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₂²⋊C₄	—	8	1	no	—
126	C₂²⋊C₄	—	8	1	no	—
127	C₂²⋊C₄	—	8	1	no	—
128	C₂²⋊C₄	—	8	1	no	—
129	C₂²⋊C₄	—	8	1	no	—
130	C₄²	—	8	1	no	—
131	C₄²	#102	1	1	yes	characteristic
132	C₄²	#102	1	1	yes	characteristic
133	C₄²	SD₃₂	1	1	yes	characteristic
134	C₄²	SD₃₂	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	2	1	no	—
137	C₁₆	—	2	1	no	—
138	C₁₆	—	2	1	no	—
139	C₂³	—	8	1	no	—
140	C₂³	C₂×D₁₆	1	1	yes	characteristic
141	Q₈	—	8	1	no	—
142	Q₈	—	8	1	no	—
143	D₄	—	8	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	16	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	—	16	2	no	—
156	C₂×C₄	—	16	2	no	—
157	C₂×C₄	#357	1	1	yes	characteristic
158	C₂×C₄	#357	1	1	yes	characteristic
159	C₂×C₄	#357	1	1	yes	characteristic
160	C₂×C₄	#357	1	1	yes	characteristic
161	C₂×C₄	C₂×#102	1	1	yes	characteristic
162	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	#360	1	2	yes	—
166	C₈	#410	1	1	yes	characteristic
167	C₈	#421	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	16	1	no	—
171	C₂²	—	16	1	no	—
172	C₂²	—	16	1	no	—
173	C₂²	C₂×#357	1	1	yes	characteristic
174	C₂²	#2262	1	2	yes	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	16	1	no	—
181	C₄	—	16	1	no	—
182	C₄	—	16	1	no	—
183	C₄	—	32	2	no	—
184	C₄	#1517	1	1	yes	characteristic
185	C₄	C₂×#360	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	32	1	no	—
188	C₂	#11856	1	1	yes	characteristic
189	C₂	#13615	1	1	yes	characteristic
190	C₂	#13622	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13639	C₂	semidirect
3. #12889	C₂	semidirect
4. #7419	C₂	semidirect
7. #2253	C₄	semidirect
8. #2249	C₄	semidirect
13. C₄×Q₃₂	C₄	semidirect
15. C₄×D₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33735	2
#13639	2
#12889	2
#7419	2
#7418	2