Finite Group Databasebeta

Subgroups — #141403

Order 512.

366 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₂×#1709	C₂	1	1	yes	maximal, characteristic
2	#11962	C₂	1	2	yes	maximal, complemented
3	#11961	C₂	1	2	yes	maximal
4	#11410	C₂	1	1	yes	maximal, characteristic, complemented
5	#8824	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#494	C₂²	1	1	yes	characteristic
7	C₂×#493	C₂²	1	1	yes	characteristic
8	C₂×#486	—	2	1	no	—
9	C₂×#486	—	2	1	no	—
10	#2162	C₂²	1	2	yes	—
11	#1837	—	2	1	no	—
12	#1709	—	2	1	no	—
13	#1709	—	2	1	no	—
14	#1709	C₄	1	2	yes	—
15	#1709	C₄	1	2	yes	—
16	#1567	C₂²	1	2	yes	—
17	#1566	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1362	—	2	4	no	—
20	#1361	—	2	4	no	—
21	#1312	—	2	1	no	—
22	C₈×C₁₆	—	2	1	no	—
23	C₂²×Q₁₆	—	2	2	no	—
24	C₂²×D₈	—	2	2	no	—
25	C₂×C₄⋊Q₈	—	2	1	no	—
26	C₂×#127	—	2	1	no	—
27	#494	—	2	1	no	—
28	#494	—	2	2	no	—
29	#494	C₂×C₄	1	2	yes	—
30	#493	—	2	1	no	—
31	#493	—	2	2	no	—
32	#493	C₂×C₄	1	2	yes	—
33	#486	—	2	2	no	—
34	#486	—	2	2	no	—
35	#486	—	2	2	no	—
36	#486	—	2	2	no	—
37	#486	—	4	2	no	—
38	#486	—	4	2	no	—
39	#423	—	2	2	no	—
40	C₂×#102	—	4	2	no	—
41	C₂×#102	—	4	2	no	—
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	C₄⋊C₁₆	D₄	1	4	yes	—
46	C₂²⋊C₁₆	—	2	2	no	—
47	C₄×C₁₆	—	2	2	no	—
48	#341	—	4	2	no	—
49	#335	—	2	2	no	—
50	#335	—	2	2	no	—
51	#334	—	4	2	no	—
52	C₈²	C₂×C₄	1	1	yes	characteristic
53	C₈²	C₂×C₄	1	1	yes	characteristic
54	C₈²	D₄	1	1	yes	characteristic
55	C₈²	D₄	1	1	yes	characteristic
56	C₂²×Q₈	—	4	1	no	—
57	C₂²×D₄	—	4	1	no	—
58	C₂²×D₄	—	4	1	no	—
59	C₂²×D₄	—	4	1	no	—
60	C₂×Q₁₆	—	2	4	no	—
61	C₂×Q₁₆	—	4	2	no	—
62	C₂×Q₁₆	—	8	4	no	—
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	2	no	—
65	C₂×D₈	—	8	4	no	—
66	C₂×M₁₆	—	4	2	no	—
67	C₂²×C₈	D₈	1	2	yes	—
68	C₂²×C₈	SD₁₆	1	2	yes	—
69	C₂²×C₈	#41	1	2	yes	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	—	4	1	no	—
72	C₄⋊Q₈	—	4	1	no	—
73	C₄⋊Q₈	—	4	1	no	—
74	#127	—	2	2	no	—
75	#127	—	4	1	no	—
76	#127	—	4	1	no	—
77	#127	—	4	1	no	—
78	C₂×C₄⋊C₄	—	4	1	no	—
79	C₂×C₄⋊C₄	—	4	1	no	—
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	—	2	4	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	#102	—	4	4	no	—
85	#102	—	4	4	no	—
86	#102	—	4	4	no	—
87	#102	—	4	4	no	—
88	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₈	D₈	1	2	yes	—
93	C₄×C₈	SD₁₆	1	2	yes	—
94	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₄×C₈	C₂×D₄	1	1	yes	characteristic
96	C₄×C₈	C₂×D₄	1	2	yes	—
97	C₂⁴	—	8	1	no	—
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	8	4	no	—
100	C₂×Q₈	—	8	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂×D₄	—	8	4	no	—
105	C₂×D₄	—	8	4	no	—
106	C₂×D₄	—	8	1	no	—
107	C₂×D₄	—	8	1	no	—
108	C₂×D₄	—	8	1	no	—
109	C₂×D₄	—	8	4	no	—
110	C₂²×C₄	—	8	1	no	—
111	C₂²×C₄	#124	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
113	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
114	Q₁₆	—	8	4	no	—
115	Q₁₆	—	8	4	no	—
116	D₈	—	8	4	no	—
117	D₈	—	8	4	no	—
118	M₁₆	—	8	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	8	2	no	—
123	C₂×C₈	#102	1	2	yes	—
124	C₂×C₈	#102	1	2	yes	—
125	C₂×C₈	#102	1	2	yes	—
126	C₂×C₈	#102	1	2	yes	—
127	C₂×C₈	C₄×D₄	1	2	yes	—
128	C₂×C₈	#121	1	2	yes	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	8	1	no	—
132	C₄⋊C₄	—	8	1	no	—
133	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₄²	#127	1	1	yes	characteristic
135	C₄²	C₂×D₈	1	1	yes	characteristic
136	C₄²	C₂×SD₁₆	1	1	yes	characteristic
137	C₁₆	—	4	4	no	—
138	C₂³	—	8	2	no	—
139	C₂³	—	16	2	no	—
140	C₂³	—	16	2	no	—
141	C₂³	—	16	2	no	—
142	C₂³	#486	1	1	yes	characteristic
143	Q₈	—	8	4	no	—
144	Q₈	—	16	2	no	—
145	D₄	—	8	4	no	—
146	D₄	—	8	4	no	—
147	D₄	—	8	4	no	—
148	D₄	—	16	2	no	—
149	D₄	—	16	2	no	—
150	D₄	—	16	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	#390	1	1	yes	characteristic
156	C₂×C₄	C₂×#102	1	1	yes	characteristic
157	C₂×C₄	C₂×#102	1	1	yes	characteristic
158	C₂×C₄	#486	1	1	yes	characteristic
159	C₂×C₄	#492	1	1	yes	characteristic
160	C₂×C₄	#493	1	1	yes	characteristic
161	C₂×C₄	#508	1	4	yes	—
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	16	2	no	—
168	C₈	#472	1	4	yes	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	16	2	no	—
172	C₂²	—	16	2	no	—
173	C₂²	—	16	2	no	—
174	C₂²	—	32	4	no	—
175	C₂²	#1957	1	1	yes	characteristic
176	C₂²	#2242	1	2	yes	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	16	2	no	—
180	C₄	#1978	1	2	yes	—
181	C₄	#2138	1	2	yes	—
182	C₄	#2148	1	2	yes	—
183	C₄	#2249	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	32	2	no	—
186	C₂	#9773	1	1	yes	characteristic
187	C₂	#13204	1	1	yes	characteristic
188	C₂	#13214	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	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. #11962	C₂	semidirect
4. #11410	C₂	semidirect
5. #8824	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1709	2
#11962	2
#11961	2
#11410	2
#8824	2