Finite Group Databasebeta

Subgroups — #140809

Order 512.

300 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 94 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	#12013	C₂	1	1	yes	maximal, characteristic, complemented
3	#11633	C₂	1	2	yes	maximal, complemented
4	#11492	C₂	1	2	yes	maximal, complemented
5	#11410	C₂	1	1	yes	maximal, characteristic, complemented
6	#3169	C₄	1	1	yes	characteristic
7	#2931	C₂²	1	2	yes	—
8	#2163	C₂²	1	2	yes	—
9	#2162	C₂²	1	1	yes	characteristic
10	#2162	C₂²	1	1	yes	characteristic
11	#1589	—	2	1	no	—
12	#1589	—	2	1	no	—
13	C₈×D₈	—	2	1	no	—
14	C₈×D₈	—	2	1	no	—
15	#1567	C₄	1	1	yes	characteristic
16	#1492	C₄	1	2	yes	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1370	—	2	1	no	—
19	#1367	—	2	1	no	—
20	#1330	—	2	4	no	—
21	#1315	—	2	4	no	—
22	#1312	—	2	1	no	—
23	C₈×C₁₆	—	2	1	no	—
24	C₂×#131	—	2	2	no	—
25	#532	C₂×C₄	1	2	yes	—
26	#500	C₈	1	1	yes	characteristic
27	#493	C₈	1	1	yes	characteristic
28	#486	C₈	1	2	yes	—
29	#466	—	2	1	no	—
30	#466	—	2	1	no	—
31	C₄×D₈	—	2	1	no	—
32	C₄×D₈	—	2	1	no	—
33	#436	—	2	2	no	—
34	#436	—	2	2	no	—
35	C₈×D₄	—	2	2	no	—
36	C₈×D₄	—	2	2	no	—
37	#423	C₂×C₄	1	2	yes	—
38	#408	—	2	2	no	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊C₁₆	—	2	2	no	—
43	C₄⋊C₁₆	—	2	1	no	—
44	C₄⋊C₁₆	D₄	1	2	yes	—
45	C₂²⋊C₁₆	—	2	1	no	—
46	C₂²⋊C₁₆	—	2	2	no	—
47	C₂²⋊C₁₆	D₄	1	2	yes	—
48	#346	—	2	2	no	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	—	2	1	no	—
51	#335	—	2	1	no	—
52	#335	—	2	1	no	—
53	D₄⋊C₈	—	2	2	no	—
54	D₄⋊C₈	—	2	2	no	—
55	C₈²	D₄	1	1	yes	characteristic
56	C₈²	D₄	1	1	yes	characteristic
57	C₈²	D₄	1	1	yes	characteristic
58	C₈²	D₄	1	1	yes	characteristic
59	C₂×#41	—	2	2	no	—
60	C₂×D₈	—	2	1	no	—
61	C₂×D₈	—	2	1	no	—
62	#131	—	8	4	no	—
63	C₂×M₁₆	—	2	2	no	—
64	C₂×M₁₆	—	4	2	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	4	2	no	—
67	C₂²×C₈	M₁₆	1	1	yes	characteristic
68	C₂²×C₈	M₁₆	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	#41	1	1	yes	characteristic
71	C₄⋊Q₈	C₂×C₈	1	2	yes	—
72	#127	C₂×C₈	1	2	yes	—
73	#124	—	2	2	no	—
74	#121	—	2	2	no	—
75	#121	—	2	2	no	—
76	C₄×D₄	—	2	2	no	—
77	C₄×D₄	—	2	2	no	—
78	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	#107	—	2	1	no	—
83	#107	—	2	1	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	—	2	2	no	—
88	C₂²⋊C₈	—	4	2	no	—
89	C₂²⋊C₈	—	4	2	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₈	C₂×C₈	1	1	yes	characteristic
96	C₄×C₈	C₂×C₈	1	1	yes	characteristic
97	C₄×C₈	C₂×D₄	1	1	yes	characteristic
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×C₈	C₂×D₄	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	#41	—	8	2	no	—
102	#41	—	8	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
111	C₂²×C₄	#131	1	1	yes	characteristic
112	D₈	—	4	2	no	—
113	D₈	—	4	2	no	—
114	M₁₆	—	8	4	no	—
115	M₁₆	—	8	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	8	4	no	—
125	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×C₈	#121	1	1	yes	characteristic
130	C₂×C₈	#121	1	1	yes	characteristic
131	C₂×C₈	#124	1	1	yes	characteristic
132	C₂×C₈	#127	1	1	yes	characteristic
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₄²	C₄×D₄	1	1	yes	characteristic
139	C₄²	C₂²×C₈	1	1	yes	characteristic
140	C₄²	#131	1	1	yes	characteristic
141	C₁₆	—	4	2	no	—
142	C₁₆	—	4	2	no	—
143	C₁₆	—	8	2	no	—
144	C₂³	—	4	2	no	—
145	C₂³	#436	1	1	yes	characteristic
146	Q₈	—	8	2	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	8	2	no	—
150	D₄	—	8	2	no	—
151	D₄	—	8	4	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	#390	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₄	#408	1	1	yes	characteristic
163	C₂×C₄	C₈×D₄	1	1	yes	characteristic
164	C₂×C₄	C₈×D₄	1	1	yes	characteristic
165	C₂×C₄	#436	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	4	1	no	—
173	C₈	—	8	2	no	—
174	C₈	#350	1	2	yes	—
175	C₈	#350	1	2	yes	—
176	C₈	#472	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	8	2	no	—
181	C₂²	—	8	2	no	—
182	C₂²	—	8	2	no	—
183	C₂²	#1954	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	2	no	—
189	C₄	#1976	1	1	yes	characteristic
190	C₄	#1978	1	1	yes	characteristic
191	C₄	C₂×#350	1	1	yes	characteristic
192	C₄	C₂×#350	1	1	yes	characteristic
193	C₂	—	4	1	no	—
194	C₂	—	16	2	no	—
195	C₂	#9668	1	1	yes	characteristic
196	C₂	#13191	1	1	yes	characteristic
197	C₂	#13197	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	#3169	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ 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. #12013	C₂	semidirect
3. #11633	C₂	semidirect
4. #11492	C₂	semidirect
5. #11410	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
#12013	2
#11633	2
#11492	2
#11410	2