Finite Group Databasebeta

Subgroups — #140455

Order 512.

216 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 92 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	#12891	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#12008	C₂	1	1	yes	maximal, characteristic
4	#9628	C₂	1	1	yes	maximal, characteristic
5	#9627	C₂	1	1	yes	maximal, characteristic
6	#7277	C₂	1	1	yes	maximal, characteristic
7	#7277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	—	2	1	no	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1848	C₂²	1	1	yes	characteristic
12	#1848	C₂²	1	1	yes	characteristic
13	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#1571	—	2	1	no	—
15	#1570	—	2	1	no	—
16	#1564	C₂²	1	1	yes	characteristic
17	#1564	C₂²	1	1	yes	characteristic
18	#1381	—	2	1	no	—
19	#1381	—	2	1	no	—
20	#1374	—	2	1	no	—
21	#1372	C₄	1	2	yes	complemented
22	#1371	C₄	1	2	yes	complemented
23	#1303	—	2	1	no	—
24	#1303	—	2	1	no	—
25	C₂²×C₁₆	—	2	1	no	—
26	C₂²×C₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₁₆	D₄	1	1	yes	characteristic
28	#431	—	2	1	no	—
29	#431	—	2	1	no	—
30	C₂×#108	—	2	1	no	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	#423	—	2	1	no	—
35	#423	—	2	1	no	—
36	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	—	2	1	no	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₄⋊C₁₆	—	2	1	no	—
41	C₄⋊C₁₆	Q₈	1	2	yes	—
42	C₄×C₁₆	—	4	2	no	—
43	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₄×C₁₆	Q₈	1	1	yes	characteristic
46	C₄×C₁₆	Q₈	1	1	yes	characteristic
47	#336	—	2	1	no	—
48	#335	—	4	2	no	—
49	#335	C₂×C₄	1	2	yes	—
50	#335	C₂×C₄	1	2	yes	—
51	#334	—	2	1	no	—
52	#334	—	2	1	no	—
53	#334	—	4	2	no	—
54	C₂×M₁₆	—	4	1	no	—
55	C₂×M₁₆	—	4	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	#41	1	1	yes	characteristic
59	C₂×C₄⋊C₄	—	2	1	no	—
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	—	2	1	no	—
65	C₂×C₁₆	—	2	1	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	4	1	no	—
68	C₂×C₁₆	—	4	1	no	—
69	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
73	#108	—	4	1	no	—
74	#108	—	4	1	no	—
75	#108	—	4	1	no	—
76	#108	—	4	1	no	—
77	#108	—	8	1	no	—
78	#108	—	8	1	no	—
79	C₄⋊C₈	—	2	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	2	1	no	—
82	C₄⋊C₈	—	4	1	no	—
83	C₄⋊C₈	—	4	1	no	—
84	C₄⋊C₈	—	4	1	no	—
85	C₄⋊C₈	—	4	1	no	—
86	C₄⋊C₈	D₈	1	2	yes	—
87	C₂²⋊C₈	—	4	1	no	—
88	C₂²⋊C₈	—	4	1	no	—
89	C₄×C₈	—	4	1	no	—
90	C₄×C₈	—	4	1	no	—
91	C₄×C₈	—	4	1	no	—
92	C₄×C₈	—	4	1	no	—
93	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₄×C₈	#41	1	1	yes	characteristic
97	#95	Q₁₆	1	2	yes	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	#121	1	1	yes	characteristic
100	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₄	#135	1	1	yes	characteristic
102	M₁₆	—	8	1	no	—
103	M₁₆	—	8	1	no	—
104	M₁₆	—	8	1	no	—
105	M₁₆	—	8	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
117	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₈	#125	1	1	yes	characteristic
119	C₂×C₈	#135	1	1	yes	characteristic
120	C₂×C₈	#135	1	1	yes	characteristic
121	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₈	#137	1	1	yes	characteristic
123	C₄⋊C₄	—	2	4	no	—
124	C₄²	—	4	1	no	—
125	C₄²	—	4	1	no	—
126	C₄²	C₄×D₄	1	1	yes	characteristic
127	C₄²	C₄⋊Q₈	1	1	yes	characteristic
128	C₄²	C₂×D₈	1	1	yes	characteristic
129	C₄²	#135	1	1	yes	characteristic
130	C₁₆	—	2	1	no	—
131	C₁₆	—	2	1	no	—
132	C₁₆	—	2	1	no	—
133	C₁₆	—	2	1	no	—
134	C₁₆	—	4	2	no	—
135	C₂³	#467	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	#389	1	1	yes	characteristic
142	C₂×C₄	C₄×D₈	1	1	yes	characteristic
143	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
144	C₂×C₄	#463	1	1	yes	characteristic
145	C₂×C₄	#464	1	1	yes	characteristic
146	C₂×C₄	#466	1	1	yes	characteristic
147	C₂×C₄	#471	1	1	yes	characteristic
148	C₂×C₄	#471	1	1	yes	characteristic
149	C₂×C₄	#499	1	1	yes	characteristic
150	C₂×C₄	#500	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	4	2	no	—
154	C₈	—	8	2	no	—
155	C₈	—	8	2	no	—
156	C₈	—	16	1	no	—
157	C₈	—	16	1	no	—
158	C₈	#427	1	1	yes	characteristic
159	C₈	#428	1	1	yes	characteristic
160	C₈	#478	1	1	yes	characteristic
161	C₈	#479	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#1683	1	1	yes	characteristic
165	C₂²	#1684	1	1	yes	characteristic
166	C₂²	#1942	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	4	no	—
171	C₄	#1681	1	1	yes	characteristic
172	C₄	#1682	1	1	yes	characteristic
173	C₄	#1949	1	1	yes	characteristic
174	C₄	#1951	1	1	yes	characteristic
175	C₄	#2178	1	2	yes	—
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#9616	1	1	yes	characteristic
179	C₂	#13169	1	1	yes	characteristic
180	C₂	#13173	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1942
prime radical	C₂³	#467
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ 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
21. #1372	C₄	semidirect
22. #1371	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12891	2
C₁₆×C₄⋊C₄	2
#12008	2
#9628	2
#9627	2
#7277	2