Finite Group Databasebeta

Subgroups — #146261

Order 512.

283 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 80 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	#13213	C₂	1	1	yes	maximal, characteristic
2	#13206	C₂	1	1	yes	maximal, characteristic
3	#12892	C₂	1	1	yes	maximal, characteristic, complemented
4	#12887	C₂	1	1	yes	maximal, characteristic, complemented
5	#9773	C₂	1	1	yes	maximal, characteristic
6	#8790	C₂	1	1	yes	maximal, characteristic, complemented
7	#7140	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#493	C₂²	1	1	yes	characteristic
9	C₂×#361	—	2	2	no	—
10	C₂×#357	—	2	2	no	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×#346	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	1	yes	characteristic
14	#1848	C₂²	1	1	yes	characteristic
15	#1846	C₂²	1	1	yes	characteristic
16	#1545	—	2	2	no	—
17	#1450	C₂²	1	1	yes	characteristic
18	#1383	—	2	2	no	—
19	#1381	—	2	2	no	—
20	#1286	—	2	2	no	—
21	C₂²×D₈	—	2	1	no	—
22	C₂²×D₈	—	2	1	no	—
23	C₂×#127	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	C₂²×C₁₆	D₄	1	2	yes	—
26	#493	—	2	1	no	—
27	#493	—	2	1	no	—
28	#493	—	2	1	no	—
29	#493	Q₈	1	2	yes	—
30	#431	D₄	1	2	yes	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#108	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	#423	—	2	1	no	—
38	#423	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#382	—	2	1	no	—
41	#361	—	4	4	no	—
42	#357	—	4	4	no	—
43	#346	—	2	1	no	—
44	#346	—	2	1	no	—
45	#346	—	2	1	no	—
46	C₄×C₁₆	—	2	1	no	—
47	#331	—	4	4	no	—
48	#322	—	2	1	no	—
49	C₈²	—	2	1	no	—
50	C₂²×D₄	—	4	1	no	—
51	C₂²×D₄	—	4	2	no	—
52	C₂×D₈	—	2	2	no	—
53	C₂×D₈	—	2	2	no	—
54	C₂×D₈	—	4	1	no	—
55	C₂×D₈	—	4	1	no	—
56	C₂×D₈	—	4	2	no	—
57	C₂×D₈	—	4	2	no	—
58	C₂×D₈	—	4	2	no	—
59	C₂×D₈	—	4	2	no	—
60	C₂×M₁₆	—	4	1	no	—
61	C₂×M₁₆	—	4	1	no	—
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
64	#127	—	2	2	no	—
65	#127	—	4	1	no	—
66	#127	—	4	2	no	—
67	C₂×C₄⋊C₄	—	4	1	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	M₃₂	—	2	2	no	—
70	M₃₂	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	#108	—	4	2	no	—
76	#108	—	4	2	no	—
77	#108	—	4	2	no	—
78	#108	—	4	2	no	—
79	#107	—	4	2	no	—
80	#106	—	4	2	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₂²⋊C₈	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	#95	—	4	2	no	—
91	C₂⁴	—	8	1	no	—
92	C₂×D₄	—	4	2	no	—
93	C₂×D₄	—	4	4	no	—
94	C₂×D₄	—	8	2	no	—
95	C₂×D₄	—	8	1	no	—
96	C₂×D₄	—	8	1	no	—
97	C₂×D₄	—	8	1	no	—
98	C₂×D₄	—	8	1	no	—
99	C₂×D₄	—	8	1	no	—
100	C₂×D₄	—	16	4	no	—
101	C₂²×C₄	—	8	1	no	—
102	C₂²×C₄	#127	1	1	yes	characteristic
103	C₂²×C₄	#135	1	2	yes	—
104	D₈	—	4	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	8	2	no	—
107	D₈	—	8	2	no	—
108	M₁₆	—	8	1	no	—
109	M₁₆	—	8	1	no	—
110	M₁₆	—	8	1	no	—
111	M₁₆	—	8	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₈	#123	1	1	yes	characteristic
118	C₂×C₈	#123	1	1	yes	characteristic
119	C₂×C₈	#135	1	2	yes	—
120	C₂×C₈	#135	1	2	yes	—
121	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
123	C₄⋊C₄	—	8	1	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
126	C₄²	#123	1	1	yes	characteristic
127	C₄²	AGL(1,ℤ/8)	1	2	yes	—
128	C₁₆	—	4	2	no	—
129	C₁₆	—	4	2	no	—
130	C₂³	—	8	2	no	—
131	C₂³	—	16	4	no	—
132	C₂³	—	16	1	no	—
133	C₂³	—	16	1	no	—
134	C₂³	#495	1	1	yes	characteristic
135	D₄	—	8	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	16	4	no	—
138	D₄	—	16	4	no	—
139	D₄	—	16	1	no	—
140	D₄	—	16	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	16	2	no	—
143	C₂×C₄	—	16	1	no	—
144	C₂×C₄	#399	1	1	yes	characteristic
145	C₂×C₄	#478	1	2	yes	—
146	C₂×C₄	#482	1	2	yes	—
147	C₂×C₄	#495	1	1	yes	characteristic
148	C₂×C₄	#495	1	1	yes	characteristic
149	C₂×C₄	#496	1	1	yes	characteristic
150	C₂×C₄	#496	1	1	yes	characteristic
151	C₂×C₄	#496	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	—	16	1	no	—
154	C₈	—	16	1	no	—
155	C₈	#478	1	2	yes	—
156	C₈	#478	1	2	yes	—
157	C₈	#482	1	2	yes	—
158	C₈	#482	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	—	16	2	no	—
161	C₂²	—	16	4	no	—
162	C₂²	—	32	2	no	—
163	C₂²	—	32	2	no	—
164	C₂²	#1711	1	1	yes	characteristic
165	C₂²	#1717	1	1	yes	characteristic
166	C₂²	#2092	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	32	1	no	—
170	C₄	#1717	1	1	yes	characteristic
171	C₄	#1717	1	1	yes	characteristic
172	C₄	#2092	1	1	yes	characteristic
173	C₄	#2092	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	32	2	no	—
177	C₂	#10678	1	1	yes	characteristic
178	C₂	#13523	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
3. #12892	C₂	semidirect
4. #12887	C₂	semidirect
6. #8790	C₂	semidirect
7. #7140	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13213	2
#13206	2
#12892	2
#12887	2
#9773	2
#8790	2
#7140	2