Finite Group Databasebeta

Subgroups — #135143

Order 512.

299 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 90 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	#17956	C₂	1	1	yes	maximal, characteristic
2	#12004	C₂	1	1	yes	maximal, characteristic, complemented
3	#11986	C₂	1	2	yes	maximal
4	#8823	C₂	1	1	yes	maximal, characteristic, complemented
5	#7245	C₂	1	2	yes	maximal, complemented
6	#3169	C₂²	1	1	yes	characteristic
7	#2931	—	2	1	no	—
8	#2150	C₂²	1	2	yes	—
9	#1848	—	2	1	no	—
10	#1589	—	2	1	no	—
11	#1589	C₄	1	2	yes	—
12	C₈×D₈	—	2	1	no	—
13	C₈×D₈	C₄	1	2	yes	—
14	#1567	—	2	1	no	—
15	#1567	C₂²	1	2	yes	—
16	#1567	C₂²	1	1	yes	characteristic
17	#1492	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1370	—	2	1	no	—
20	#1370	—	2	1	no	—
21	#1365	—	2	2	no	—
22	#1361	—	2	2	no	—
23	C₂×#131	—	4	1	no	—
24	#532	—	2	1	no	—
25	C₂×M₃₂	—	2	2	no	—
26	#500	D₄	1	1	yes	characteristic
27	#493	D₄	1	1	yes	characteristic
28	#486	—	2	1	no	—
29	#466	—	2	1	no	—
30	#466	Q₈	1	2	yes	—
31	C₄×D₈	—	2	1	no	—
32	C₄×D₈	C₂×C₄	1	2	yes	—
33	#436	—	2	2	no	—
34	#436	—	4	1	no	—
35	C₈×D₄	—	2	2	no	—
36	C₈×D₄	—	4	1	no	—
37	C₂×#108	—	4	2	no	—
38	#423	—	2	1	no	—
39	#423	—	2	2	no	—
40	#408	—	4	1	no	—
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₄⋊C₁₆	—	2	2	no	—
45	C₄⋊C₁₆	—	2	2	no	—
46	#335	—	2	1	no	—
47	#335	—	2	2	no	—
48	#335	—	2	2	no	—
49	#335	—	4	1	no	—
50	#335	—	4	1	no	—
51	#335	C₂×C₄	1	2	yes	—
52	D₄⋊C₈	—	2	2	no	—
53	D₄⋊C₈	—	2	2	no	—
54	C₈²	C₂×C₄	1	1	yes	characteristic
55	C₈²	C₂×C₄	1	1	yes	characteristic
56	C₈²	D₄	1	1	yes	characteristic
57	C₈²	D₄	1	1	yes	characteristic
58	C₂×#41	—	4	1	no	—
59	C₂×D₈	—	2	1	no	—
60	C₂×D₈	C₄⋊C₄	1	2	yes	—
61	#131	—	8	4	no	—
62	C₂×M₁₆	—	4	1	no	—
63	C₂×M₁₆	—	4	2	no	—
64	C₂×M₁₆	—	4	2	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	4	2	no	—
67	C₂²×C₈	SD₁₆	1	1	yes	characteristic
68	C₂²×C₈	SD₁₆	1	1	yes	characteristic
69	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₈	#41	1	1	yes	characteristic
71	C₄⋊Q₈	—	2	1	no	—
72	#127	—	2	1	no	—
73	#124	—	4	1	no	—
74	#121	—	2	2	no	—
75	#121	—	4	1	no	—
76	C₄×D₄	—	2	2	no	—
77	C₄×D₄	—	4	1	no	—
78	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
79	M₃₂	—	8	2	no	—
80	C₂×C₁₆	—	4	2	no	—
81	#108	—	8	4	no	—
82	#107	—	2	1	no	—
83	#107	C₄⋊C₄	1	2	yes	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄⋊C₈	—	4	1	no	—
87	C₄⋊C₈	—	4	2	no	—
88	#102	—	2	2	no	—
89	#102	—	2	2	no	—
90	C₂²⋊C₈	—	4	2	no	—
91	C₂²⋊C₈	—	4	2	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₈	D₈	1	1	yes	characteristic
98	C₄×C₈	D₈	1	1	yes	characteristic
99	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	#41	—	8	2	no	—
104	#41	—	8	2	no	—
105	C₂×Q₈	—	4	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂×D₄	—	4	1	no	—
109	C₂×D₄	—	4	1	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
114	D₈	—	4	2	no	—
115	D₈	—	4	1	no	—
116	D₈	—	4	1	no	—
117	M₁₆	—	8	4	no	—
118	M₁₆	—	8	2	no	—
119	M₁₆	—	8	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	8	4	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	#102	1	1	yes	characteristic
135	C₂×C₈	#102	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₈	#117	1	1	yes	characteristic
138	C₂×C₈	#121	1	1	yes	characteristic
139	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₄²	#121	1	1	yes	characteristic
146	C₄²	C₂×D₈	1	1	yes	characteristic
147	C₄²	#137	1	1	yes	characteristic
148	C₁₆	—	8	2	no	—
149	C₂³	—	4	2	no	—
150	C₂³	#476	1	1	yes	characteristic
151	Q₈	—	8	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	4	no	—
156	D₄	—	8	1	no	—
157	D₄	—	8	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	#380	1	1	yes	characteristic
168	C₂×C₄	C₂×#102	1	1	yes	characteristic
169	C₂×C₄	#417	1	1	yes	characteristic
170	C₂×C₄	#459	1	1	yes	characteristic
171	C₂×C₄	#461	1	1	yes	characteristic
172	C₂×C₄	#474	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	8	2	no	—
182	C₈	—	16	2	no	—
183	C₈	#361	1	2	yes	—
184	C₈	#472	1	1	yes	characteristic
185	C₈	#473	1	1	yes	characteristic
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	2	no	—
191	C₂²	—	8	2	no	—
192	C₂²	#1802	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	2	no	—
197	C₄	#1813	1	1	yes	characteristic
198	C₄	#1816	1	1	yes	characteristic
199	C₄	C₂×#361	1	1	yes	characteristic
200	C₄	#2148	1	1	yes	characteristic
201	C₄	#2213	1	1	yes	characteristic
202	C₄	#2214	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	16	2	no	—
205	C₂	#8684	1	1	yes	characteristic
206	C₂	#12818	1	1	yes	characteristic
207	C₂	#12821	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #12004	C₂	semidirect
4. #8823	C₂	semidirect
5. #7245	C₂	semidirect

Maximal subgroups

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

subgroup	index
#17956	2
#12004	2
#11986	2
#8823	2
#7245	2