Finite Group Databasebeta

Subgroups — #122889

Order 512.

304 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 82 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	#19603	C₂	1	1	yes	maximal, characteristic
2	#12013	C₂	1	1	yes	maximal, characteristic, complemented
3	#12004	C₂	1	1	yes	maximal, characteristic, complemented
4	#7097	C₂	1	2	yes	maximal, complemented
5	#7096	C₂	1	2	yes	maximal, complemented
6	#3169	—	2	1	no	—
7	#3169	C₄	1	2	yes	—
8	#3147	—	2	1	no	—
9	#2163	—	2	1	no	—
10	#2150	—	2	1	no	—
11	#1713	C₂²	1	1	yes	characteristic
12	#1712	C₂²	1	1	yes	characteristic
13	#1709	—	2	1	no	—
14	#1706	—	2	1	no	—
15	#1706	C₄	1	2	yes	—
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	#1370	—	2	1	no	—
18	#1370	C₂²	1	2	yes	—
19	#1370	C₂²	1	2	yes	—
20	#1367	—	2	1	no	—
21	C₂×#135	—	4	1	no	—
22	C₂×#135	—	4	1	no	—
23	#532	—	2	2	no	—
24	C₂×M₃₂	—	4	1	no	—
25	#500	—	2	1	no	—
26	#500	C₂×C₄	1	2	yes	—
27	#495	—	4	1	no	—
28	#495	—	4	1	no	—
29	#494	—	2	1	no	—
30	#493	—	2	1	no	—
31	#493	—	2	1	no	—
32	#493	C₂×C₄	1	2	yes	—
33	#492	—	2	2	no	—
34	#486	—	2	2	no	—
35	#486	—	2	2	no	—
36	#466	—	4	1	no	—
37	#466	—	4	1	no	—
38	#466	—	4	2	no	—
39	C₄×D₈	—	4	2	no	—
40	C₄×D₈	—	4	1	no	—
41	C₄×D₈	—	4	1	no	—
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	2	no	—
46	C₄⋊C₁₆	—	2	2	no	—
47	C₄⋊C₁₆	—	4	1	no	—
48	C₂²⋊C₁₆	—	4	1	no	—
49	#346	—	4	1	no	—
50	C₈²	C₂×C₄	1	1	yes	characteristic
51	C₈²	D₄	1	1	yes	characteristic
52	C₈²	D₄	1	1	yes	characteristic
53	C₈²	C₂³	1	1	yes	characteristic
54	C₂×#41	—	4	2	no	—
55	#135	—	8	4	no	—
56	#135	—	8	4	no	—
57	C₂×Q₁₆	—	4	1	no	—
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×SD₁₆	—	4	2	no	—
60	C₂×SD₁₆	—	4	2	no	—
61	C₂×D₈	—	4	2	no	—
62	C₂×D₈	—	4	1	no	—
63	C₂×D₈	—	4	1	no	—
64	C₂×D₈	—	4	1	no	—
65	C₂×D₈	—	4	1	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	D₈	1	1	yes	characteristic
68	C₂²×C₈	D₈	1	1	yes	characteristic
69	C₂²×C₈	SD₁₆	1	1	yes	characteristic
70	C₂²×C₈	SD₁₆	1	1	yes	characteristic
71	C₄⋊Q₈	—	2	2	no	—
72	#127	—	2	2	no	—
73	#124	—	4	2	no	—
74	#121	—	4	2	no	—
75	#121	—	4	2	no	—
76	C₄×D₄	—	4	2	no	—
77	C₄×D₄	—	4	2	no	—
78	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
79	M₃₂	—	8	2	no	—
80	C₂×C₁₆	—	4	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	#107	—	4	1	no	—
83	#107	—	4	1	no	—
84	#107	—	4	2	no	—
85	#102	—	4	4	no	—
86	#102	—	4	2	no	—
87	#102	—	4	2	no	—
88	C₄×C₈	—	2	1	no	—
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	1	yes	characteristic
93	C₄×C₈	D₈	1	1	yes	characteristic
94	C₄×C₈	SD₁₆	1	1	yes	characteristic
95	C₄×C₈	SD₁₆	1	1	yes	characteristic
96	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₄×C₈	C₂×D₄	1	1	yes	characteristic
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	#41	—	8	4	no	—
100	#41	—	8	4	no	—
101	C₂×Q₈	—	4	2	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	8	2	no	—
106	C₂²×C₄	—	8	2	no	—
107	C₂²×C₄	#100	1	1	yes	characteristic
108	C₂²×C₄	#102	1	1	yes	characteristic
109	C₂²×C₄	#102	1	1	yes	characteristic
110	Q₁₆	—	8	2	no	—
111	Q₁₆	—	8	2	no	—
112	SD₁₆	—	8	4	no	—
113	SD₁₆	—	8	4	no	—
114	D₈	—	8	4	no	—
115	D₈	—	8	2	no	—
116	D₈	—	8	2	no	—
117	D₈	—	8	1	no	—
118	D₈	—	8	2	no	—
119	D₈	—	8	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	#102	1	1	yes	characteristic
129	C₂×C₈	#102	1	1	yes	characteristic
130	C₂×C₈	#102	1	1	yes	characteristic
131	C₂×C₈	#102	1	1	yes	characteristic
132	C₂×C₈	C₂×D₈	1	1	yes	characteristic
133	C₂×C₈	C₂×D₈	1	1	yes	characteristic
134	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₂²⋊C₄	—	8	2	no	—
139	C₂²⋊C₄	—	8	2	no	—
140	C₄²	#100	1	1	yes	characteristic
141	C₄²	#102	1	1	yes	characteristic
142	C₄²	#102	1	1	yes	characteristic
143	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₁₆	—	8	2	no	—
145	C₁₆	—	8	2	no	—
146	C₂³	—	8	2	no	—
147	C₂³	#331	1	1	yes	characteristic
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	D₄	—	8	4	no	—
151	D₄	—	8	2	no	—
152	D₄	—	8	2	no	—
153	D₄	—	8	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	16	4	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	16	4	no	—
163	C₂×C₄	#331	1	1	yes	characteristic
164	C₂×C₄	#331	1	1	yes	characteristic
165	C₂×C₄	#331	1	1	yes	characteristic
166	C₂×C₄	C₂×#100	1	1	yes	characteristic
167	C₂×C₄	C₂×#102	1	1	yes	characteristic
168	C₂×C₄	C₂×#102	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	4	1	no	—
176	C₈	#361	1	2	yes	—
177	C₈	#509	1	1	yes	characteristic
178	C₈	#509	1	2	yes	—
179	C₈	#510	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	16	2	no	—
184	C₂²	—	16	2	no	—
185	C₂²	—	16	2	no	—
186	C₂²	C₂×#331	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	16	2	no	—
192	C₄	#2139	1	1	yes	characteristic
193	C₄	#2140	1	1	yes	characteristic
194	C₄	C₂×#361	1	1	yes	characteristic
195	C₄	#2148	1	1	yes	characteristic
196	C₂	—	4	1	no	—
197	C₂	—	32	2	no	—
198	C₂	#11920	1	1	yes	characteristic
199	C₂	#11930	1	1	yes	characteristic
200	C₂	#11931	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #12013	C₂	semidirect
3. #12004	C₂	semidirect
4. #7097	C₂	semidirect
5. #7096	C₂	semidirect

Maximal subgroups

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

subgroup	index
#19603	2
#12013	2
#12004	2
#7097	2
#7096	2