Finite Group Databasebeta

Subgroups — #142103

Order 512.

356 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 146 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	#33313	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#7275	C₂	1	4	yes	maximal
5	C₂×#500	C₄	1	2	yes	—
6	C₂×#498	—	2	1	no	—
7	#2862	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	—	2	1	no	—
10	#1846	—	2	1	no	—
11	#1846	—	2	1	no	—
12	#1846	C₄	1	2	yes	—
13	#1846	C₂²	1	2	yes	—
14	#1842	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#334	—	2	1	no	—
17	C₂×#334	C₂²	1	2	yes	—
18	#1374	—	2	1	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂²×C₁₆	—	2	2	no	—
21	#500	—	4	4	no	—
22	#498	—	4	2	no	—
23	#498	—	4	2	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×#107	C₂×C₄	1	4	yes	—
26	C₂×#106	—	2	2	no	—
27	C₂×#106	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	—	4	2	no	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	#391	—	4	2	no	—
37	#391	—	4	2	no	—
38	#382	—	2	2	no	—
39	#382	—	2	2	no	—
40	C₄³	D₄	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	Q₈	1	4	yes	complemented
43	C₄×C₁₆	—	4	2	no	—
44	#334	—	2	4	no	—
45	#334	—	4	2	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₄⋊Q₈	—	4	4	no	—
53	C₄⋊Q₈	—	4	4	no	—
54	C₂×C₄⋊C₄	—	2	4	no	—
55	C₂×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	D₈	1	2	yes	—
59	C₂×C₄²	SD₁₆	1	2	yes	—
60	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
61	C₂×C₁₆	—	2	4	no	—
62	C₂×C₁₆	—	4	2	no	—
63	C₂×C₁₆	—	4	2	no	—
64	#107	—	4	4	no	—
65	#107	—	4	4	no	—
66	#106	—	4	4	no	—
67	#106	—	4	2	no	—
68	#106	—	4	2	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	8	4	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	D₈	1	2	yes	—
76	C₄×C₈	SD₁₆	1	2	yes	—
77	C₄×C₈	SD₁₆	1	2	yes	—
78	C₄×C₈	Q₁₆	1	2	yes	—
79	C₄×C₈	C₂×Q₈	1	2	yes	—
80	C₄×C₈	#41	1	2	yes	—
81	#95	—	4	4	no	—
82	#95	—	4	2	no	—
83	#95	—	4	2	no	—
84	C₂×Q₈	—	8	4	no	—
85	C₂×Q₈	—	8	4	no	—
86	C₂×Q₈	—	8	4	no	—
87	C₂²×C₄	—	4	4	no	—
88	C₂²×C₄	#102	1	2	yes	—
89	C₂²×C₄	#102	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
92	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	8	4	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	#101	1	2	yes	—
100	C₂×C₈	#102	1	2	yes	—
101	C₂×C₈	#103	1	2	yes	—
102	C₂×C₈	#103	1	2	yes	—
103	C₂×C₈	C₄×Q₈	1	2	yes	—
104	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₈	C₂×D₈	1	1	yes	characteristic
106	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₈	#137	1	2	yes	—
108	C₄⋊C₄	—	4	4	no	—
109	C₄⋊C₄	—	4	4	no	—
110	C₄⋊C₄	—	8	4	no	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	#100	1	2	yes	—
116	C₄²	#102	1	2	yes	—
117	C₄²	#124	1	1	yes	characteristic
118	C₄²	C₄⋊Q₈	1	1	yes	characteristic
119	C₄²	C₂×SD₁₆	1	1	yes	characteristic
120	C₄²	C₂×Q₁₆	1	1	yes	characteristic
121	C₁₆	—	4	4	no	—
122	C₂³	C₂×#102	1	1	yes	characteristic
123	Q₈	—	16	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	#331	1	4	yes	—
133	C₂×C₄	#359	1	4	yes	—
134	C₂×C₄	#360	1	4	yes	—
135	C₂×C₄	#391	1	1	yes	characteristic
136	C₂×C₄	C₂×#100	1	1	yes	characteristic
137	C₂×C₄	C₂×#101	1	1	yes	characteristic
138	C₂×C₄	C₂×#102	1	1	yes	characteristic
139	C₂×C₄	C₂×#103	1	1	yes	characteristic
140	C₂×C₄	#419	1	1	yes	characteristic
141	C₂×C₄	#486	1	1	yes	characteristic
142	C₂×C₄	#487	1	1	yes	characteristic
143	C₂×C₄	#498	1	1	yes	characteristic
144	C₂×C₄	#500	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	16	2	no	—
148	C₈	#332	1	4	yes	—
149	C₈	#474	1	2	yes	—
150	C₈	#475	1	2	yes	—
151	C₂²	C₂×#331	1	1	yes	characteristic
152	C₂²	#1546	1	1	yes	characteristic
153	C₂²	#1990	1	1	yes	characteristic
154	C₂²	C₂×#359	1	2	yes	—
155	C₂²	C₂×#360	1	2	yes	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	2	no	—
161	C₄	—	16	4	no	—
162	C₄	#1360	1	8	yes	—
163	C₄	C₂×#332	1	1	yes	characteristic
164	C₄	#1548	1	1	yes	characteristic
165	C₄	#1988	1	1	yes	characteristic
166	C₄	#1997	1	1	yes	characteristic
167	C₂	#9898	1	1	yes	characteristic
168	C₂	C₂×#1360	1	2	yes	—
169	C₂	#11968	1	2	yes	—
170	C₂	#13285	1	1	yes	characteristic
171	C₂	#13287	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×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
42. C₄⋊C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33313	2
C₄×C₄⋊C₁₆	2
#8805	2
#7275	2