Finite Group Databasebeta

Subgroups — #326530

Order 512.

426 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 134 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	#33298	C₂	1	1	yes	maximal, characteristic
2	#21734	C₂	1	1	yes	maximal, characteristic
3	#10363	C₂	1	2	yes	maximal
4	#10363	C₂	1	2	yes	maximal
5	#9759	C₂	1	2	yes	maximal
6	#8794	C₂	1	2	yes	maximal
7	#8681	C₂	1	2	yes	maximal
8	#8679	C₂	1	2	yes	maximal
9	#8269	C₂	1	1	yes	maximal, characteristic
10	#3160	C₂²	1	1	yes	characteristic
11	#2960	C₂²	1	1	yes	characteristic
12	#2960	C₂²	1	1	yes	characteristic
13	#2862	C₂²	1	2	yes	—
14	#1994	C₂²	1	1	yes	characteristic
15	#1994	C₂²	1	1	yes	characteristic
16	#1993	—	2	2	no	—
17	#1993	—	2	2	no	—
18	#1946	—	2	2	no	—
19	#1848	C₂²	1	1	yes	characteristic
20	#1835	C₂²	1	2	yes	—
21	#1835	C₂²	1	2	yes	—
22	#1773	—	2	2	no	—
23	C₄×#97	C₂²	1	1	yes	characteristic
24	#1564	C₂²	1	2	yes	—
25	#1518	C₂²	1	4	yes	—
26	#1518	C₂²	1	4	yes	—
27	#1452	C₂²	1	2	yes	—
28	#1286	C₂²	1	4	yes	—
29	#1286	C₂²	1	4	yes	—
30	#1276	C₂²	1	2	yes	—
31	#533	C₂³	1	1	yes	characteristic
32	#533	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Q₈	—	2	2	no	—
34	#500	Q₈	1	2	yes	complemented
35	#499	—	2	2	no	—
36	#498	Q₈	1	2	yes	complemented
37	#446	—	2	2	no	—
38	#446	—	2	2	no	—
39	C₈×Q₈	—	2	2	no	—
40	C₈×Q₈	—	2	2	no	—
41	#431	C₂³	1	4	yes	—
42	C₂×#108	—	2	2	no	—
43	#427	—	2	2	no	—
44	#424	—	2	1	no	—
45	#424	—	2	1	no	—
46	#423	C₂³	1	1	yes	characteristic
47	#423	C₂³	1	1	yes	characteristic
48	#405	—	2	1	no	—
49	#405	—	2	1	no	—
50	C₂×#97	D₄	1	2	yes	—
51	C₂×#97	D₄	1	2	yes	—
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#391	—	4	4	no	—
55	#382	—	2	2	no	—
56	#382	C₂³	1	4	yes	—
57	C₄³	C₂³	1	1	yes	characteristic
58	#340	—	2	4	no	—
59	#340	—	2	2	no	—
60	#340	—	2	2	no	—
61	#339	—	2	2	no	—
62	#339	—	2	2	no	—
63	#335	—	2	2	no	—
64	#334	—	2	2	no	—
65	#328	—	4	4	no	—
66	#328	—	4	4	no	—
67	#322	—	2	4	no	—
68	C₂²×Q₈	—	4	2	no	—
69	C₂×M₁₆	—	2	1	no	—
70	C₂×M₁₆	—	2	1	no	—
71	C₂²×C₈	C₂×D₄	1	2	yes	—
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₄⋊Q₈	—	4	4	no	—
75	C₄⋊Q₈	—	4	4	no	—
76	C₄⋊Q₈	C₂×Q₈	1	2	yes	—
77	C₄⋊Q₈	C₂×Q₈	1	2	yes	—
78	#125	—	2	1	no	—
79	#125	—	2	1	no	—
80	C₂²⋊Q₈	—	2	2	no	—
81	C₂²⋊Q₈	—	2	2	no	—
82	C₄×Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	—	2	1	no	—
85	#117	—	2	1	no	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	4	2	no	—
88	C₂×C₄²	—	2	2	no	—
89	C₂×C₄²	C₂×D₄	1	2	yes	—
90	C₂×C₄²	C₂⁴	1	1	yes	characteristic
91	#108	—	4	4	no	—
92	#107	—	2	4	no	—
93	#106	—	2	4	no	—
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₂²⋊C₈	—	2	4	no	—
100	#97	—	2	4	no	—
101	#97	—	4	2	no	—
102	#97	—	4	1	no	—
103	#97	—	4	1	no	—
104	C₄×C₈	—	2	4	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₄×C₈	#41	1	2	yes	—
110	C₄×C₈	#41	1	2	yes	—
111	C₄×C₈	#41	1	2	yes	—
112	#95	—	2	4	no	—
113	#95	—	2	4	no	—
114	#95	—	4	4	no	—
115	C₂×Q₈	—	2	2	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	8	4	no	—
118	C₂×Q₈	—	8	8	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₄	H(5,𝔽₂)	1	2	yes	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₈	#124	1	1	yes	characteristic
135	C₂×C₈	#124	1	1	yes	characteristic
136	C₂×C₈	#124	1	2	yes	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	8	4	no	—
146	C₂²⋊C₄	—	4	1	no	—
147	C₂²⋊C₄	—	4	1	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	—	8	2	no	—
152	C₄²	#137	1	4	yes	—
153	C₄²	C₂²×Q₈	1	1	yes	characteristic
154	C₄²	C₂×#41	1	1	yes	characteristic
155	C₄²	C₂×#41	1	2	yes	—
156	C₂³	#535	1	1	yes	characteristic
157	Q₈	—	8	2	no	—
158	Q₈	—	8	2	no	—
159	Q₈	—	16	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	8	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄	C₂×#124	1	1	yes	characteristic
169	C₂×C₄	#554	1	2	yes	—
170	C₂×C₄	#555	1	2	yes	—
171	C₂×C₄	C₂×#137	1	2	yes	—
172	C₂×C₄	#575	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	—	2	2	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	8	4	no	—
179	C₈	—	8	1	no	—
180	C₈	—	8	1	no	—
181	C₂²	—	2	2	no	—
182	C₂²	#2563	1	1	yes	characteristic
183	C₂²	#3215	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	4	4	no	—
187	C₄	—	8	2	no	—
188	C₄	—	8	2	no	—
189	C₄	—	16	4	no	—
190	C₄	#3078	1	2	yes	—
191	C₄	#3135	1	1	yes	characteristic
192	C₄	#3139	1	1	yes	characteristic
193	C₂	—	2	2	no	—
194	C₂	#29147	1	1	yes	characteristic
195	C₂	#29155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
34. #500	Q₈	semidirect
36. #498	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33298	2
#21734	2
#10363	2
#9759	2
#8794	2
#8681	2
#8679	2
#8269	2