Finite Group Databasebeta

Subgroups — #150153

Order 512.

217 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#12024	C₂	1	1	yes	maximal, characteristic, complemented
4	#7384	C₂	1	2	yes	maximal, complemented
5	#7383	C₂	1	2	yes	maximal, complemented
6	#3168	—	2	1	no	—
7	C₂²×C₃₂	—	2	1	no	—
8	#2253	C₄	1	1	yes	characteristic
9	#2249	C₄	1	1	yes	characteristic
10	#2242	—	2	1	no	—
11	#2217	—	2	1	no	—
12	#2216	C₄	1	1	yes	characteristic
13	#2215	C₄	1	1	yes	characteristic
14	C₄×Q₃₂	C₂²	1	1	yes	characteristic
15	C₄×SD₃₂	—	2	1	no	—
16	C₄×D₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	#1568	—	2	1	no	—
19	C₄×C₃₂	—	2	1	no	—
20	C₄×C₃₂	C₂²	1	2	yes	—
21	#1372	—	2	1	no	—
22	#1371	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	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₆	D₄	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	#432	—	4	1	no	—
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃₂	—	2	4	no	—
41	C₂×C₃₂	—	2	2	no	—
42	#367	—	2	1	no	—
43	#366	C₂×C₄	1	1	yes	characteristic
44	#366	C₂×C₄	1	1	yes	characteristic
45	#358	—	2	1	no	—
46	#358	—	2	1	no	—
47	#357	—	2	1	no	—
48	#357	—	2	1	no	—
49	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
50	C₄×C₁₆	D₄	1	1	yes	characteristic
51	C₄×C₁₆	D₄	1	1	yes	characteristic
52	C₄×C₁₆	C₂³	1	1	yes	characteristic
53	#335	—	2	2	no	—
54	#334	—	4	1	no	—
55	C₂×Q₁₆	—	2	1	no	—
56	C₂×SD₁₆	—	4	1	no	—
57	C₂×D₈	—	2	1	no	—
58	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈	SD₁₆	1	1	yes	characteristic
60	#126	—	8	1	no	—
61	#125	—	4	1	no	—
62	#124	—	4	1	no	—
63	#123	—	8	1	no	—
64	C₂²⋊Q₈	—	4	1	no	—
65	#121	—	4	1	no	—
66	C₄×Q₈	—	4	1	no	—
67	C₄×D₄	—	4	1	no	—
68	C₄×D₄	—	8	1	no	—
69	#117	—	8	1	no	—
70	C₂×C₄²	D₈	1	1	yes	characteristic
71	Q₃₂	—	2	2	no	—
72	SD₃₂	—	4	2	no	—
73	D₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	2	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	#106	—	4	1	no	—
83	C₄⋊C₈	—	4	2	no	—
84	#103	—	4	1	no	—
85	#103	—	4	1	no	—
86	#102	—	4	1	no	—
87	#102	—	4	1	no	—
88	C₂²⋊C₈	—	8	1	no	—
89	#97	—	8	1	no	—
90	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₈	D₈	1	1	yes	characteristic
92	C₄×C₈	SD₁₆	1	1	yes	characteristic
93	C₄×C₈	C₂×D₄	1	1	yes	characteristic
94	C₃₂	—	2	4	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂×D₄	—	8	1	no	—
98	C₂²×C₄	—	8	1	no	—
99	C₂²×C₄	#102	1	1	yes	characteristic
100	C₂²×C₄	D₁₆	1	1	yes	characteristic
101	C₂²×C₄	SD₃₂	1	1	yes	characteristic
102	Q₁₆	—	4	1	no	—
103	Q₁₆	—	4	1	no	—
104	SD₁₆	—	8	2	no	—
105	D₈	—	4	1	no	—
106	D₈	—	4	1	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	8	2	no	—
111	C₂×C₈	#102	1	1	yes	characteristic
112	C₂×C₈	#102	1	1	yes	characteristic
113	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	8	1	no	—
119	C₄⋊C₄	—	8	1	no	—
120	C₄⋊C₄	—	8	1	no	—
121	C₂²⋊C₄	—	8	1	no	—
122	C₂²⋊C₄	—	8	1	no	—
123	C₂²⋊C₄	—	8	1	no	—
124	C₂²⋊C₄	—	8	1	no	—
125	C₂²⋊C₄	—	8	1	no	—
126	C₄²	—	8	1	no	—
127	C₄²	#102	1	1	yes	characteristic
128	C₄²	D₁₆	1	1	yes	characteristic
129	C₄²	SD₃₂	1	1	yes	characteristic
130	C₄²	C₂×D₈	1	1	yes	characteristic
131	C₁₆	—	2	1	no	—
132	C₁₆	—	2	1	no	—
133	C₁₆	#104	1	2	yes	—
134	C₁₆	#131	1	2	yes	—
135	C₂³	—	8	1	no	—
136	C₂³	#357	1	1	yes	characteristic
137	Q₈	—	8	1	no	—
138	Q₈	—	8	1	no	—
139	D₄	—	8	1	no	—
140	D₄	—	8	1	no	—
141	D₄	—	16	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	16	2	no	—
150	C₂×C₄	#357	1	1	yes	characteristic
151	C₂×C₄	#357	1	1	yes	characteristic
152	C₂×C₄	#357	1	1	yes	characteristic
153	C₂×C₄	C₂×#102	1	1	yes	characteristic
154	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
155	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	16	2	no	—
159	C₈	#359	1	2	yes	—
160	C₈	#408	1	1	yes	characteristic
161	C₈	C₂×#104	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	16	1	no	—
165	C₂²	—	16	1	no	—
166	C₂²	—	16	1	no	—
167	C₂²	#1417	1	2	yes	—
168	C₂²	C₂×#357	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	16	1	no	—
174	C₄	—	16	1	no	—
175	C₄	—	16	1	no	—
176	C₄	#1492	1	1	yes	characteristic
177	C₄	C₂×#359	1	1	yes	characteristic
178	C₄	#2262	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	32	1	no	—
181	C₂	#11671	1	1	yes	characteristic
182	C₂	#13615	1	1	yes	characteristic
183	C₂	C₂×#1417	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	#2215	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. C₂×C₄×C₃₂	C₂	semidirect
3. #12024	C₂	semidirect
4. #7384	C₂	semidirect
5. #7383	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33735	2
C₂×C₄×C₃₂	2
#12024	2
#7384	2
#7383	2