Subgroups — #99296

Order 512.

392 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#8806	C₂	1	2	yes	maximal
2	#8805	C₂	1	2	yes	maximal
3	#8784	C₂	1	2	yes	maximal
4	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
5	#1842	C₂²	1	4	yes	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#335	—	2	4	no	—
10	C₂×#334	—	2	4	no	—
11	C₂×C₈²	C₄	1	1	yes	characteristic
12	C₂×C₈²	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₄	1	1	yes	characteristic
14	C₂×C₈²	C₄	1	1	yes	characteristic
15	#1277	—	2	4	no	—
16	C₂×C₄⋊C₈	—	2	4	no	—
17	C₂×C₄⋊C₈	—	2	4	no	—
18	C₂×C₄⋊C₈	—	2	4	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₂×C₄×C₈	D₄	1	2	yes	—
27	C₂×C₄×C₈	D₄	1	2	yes	—
28	C₂×C₄×C₈	Q₈	1	2	yes	—
29	C₂×C₄×C₈	Q₈	1	2	yes	—
30	C₂×C₄×C₈	Q₈	1	2	yes	—
31	C₄³	C₂³	1	1	yes	characteristic
32	#335	—	4	8	no	—
33	#334	—	4	8	no	—
34	C₈²	—	2	2	no	—
35	C₈²	—	2	2	no	—
36	C₈²	—	2	2	no	—
37	C₈²	C₈	1	4	yes	complemented
38	C₂²×C₈	—	4	4	no	—
39	C₂²×C₈	C₄⋊C₄	1	2	yes	—
40	C₂²×C₈	C₄⋊C₄	1	2	yes	—
41	C₂²×C₈	C₄⋊C₄	1	2	yes	—
42	C₂²×C₈	C₄⋊C₄	1	2	yes	—
43	C₂²×C₈	C₄⋊C₄	1	2	yes	—
44	C₂²×C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
52	C₄⋊C₈	—	4	8	no	—
53	C₄⋊C₈	—	4	8	no	—
54	C₄⋊C₈	—	4	8	no	—
55	C₄×C₈	—	2	4	no	—
56	C₄×C₈	—	2	4	no	—
57	C₄×C₈	—	2	4	no	—
58	C₄×C₈	—	2	1	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	4	no	—
65	C₄×C₈	C₂×C₈	1	2	yes	—
66	C₄×C₈	C₂×C₈	1	2	yes	—
67	C₄×C₈	C₂×C₈	1	2	yes	—
68	C₄×C₈	M₁₆	1	2	yes	—
69	C₄×C₈	M₁₆	1	2	yes	—
70	C₄×C₈	M₁₆	1	2	yes	—
71	C₄×C₈	D₈	1	4	yes	—
72	C₄×C₈	SD₁₆	1	8	yes	—
73	C₄×C₈	Q₁₆	1	4	yes	—
74	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	#127	1	1	yes	characteristic
78	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	8	4	no	—
90	C₂×C₈	—	8	4	no	—
91	C₂×C₈	—	8	4	no	—
92	C₂×C₈	C₄⋊C₈	1	4	yes	—
93	C₂×C₈	C₄⋊C₈	1	4	yes	—
94	C₂×C₈	C₄⋊C₈	1	4	yes	—
95	C₂×C₈	#106	1	4	yes	—
96	C₂×C₈	#106	1	4	yes	—
97	C₂×C₈	#107	1	4	yes	—
98	C₂×C₈	#107	1	4	yes	—
99	C₂×C₈	#108	1	4	yes	—
100	C₂×C₈	#108	1	4	yes	—
101	C₂×C₈	#108	1	4	yes	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	C₂²×C₈	1	1	yes	characteristic
106	C₄²	C₂×M₁₆	1	1	yes	characteristic
107	C₄²	C₂×M₁₆	1	1	yes	characteristic
108	C₄²	C₂×M₁₆	1	1	yes	characteristic
109	C₄²	C₂×D₈	1	2	yes	—
110	C₄²	C₂×SD₁₆	1	4	yes	—
111	C₄²	C₂×Q₁₆	1	2	yes	—
112	C₄²	#135	1	4	yes	—
113	C₂³	#384	1	1	yes	characteristic
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂×C₄	#423	1	1	yes	characteristic
124	C₂×C₄	#423	1	1	yes	characteristic
125	C₂×C₄	#423	1	1	yes	characteristic
126	C₂×C₄	C₂×#106	1	2	yes	—
127	C₂×C₄	C₂×#107	1	2	yes	—
128	C₂×C₄	#427	1	2	yes	—
129	C₂×C₄	C₂×#108	1	2	yes	—
130	C₂×C₄	C₂×#108	1	2	yes	—
131	C₂×C₄	C₂×#108	1	2	yes	—
132	C₂×C₄	#492	1	2	yes	—
133	C₂×C₄	#493	1	1	yes	characteristic
134	C₂×C₄	#494	1	1	yes	characteristic
135	C₂×C₄	#495	1	2	yes	—
136	C₂×C₄	#498	1	2	yes	—
137	C₂×C₄	#499	1	2	yes	—
138	C₂×C₄	#500	1	2	yes	—
139	C₈	—	2	4	no	—
140	C₈	—	2	4	no	—
141	C₈	—	2	4	no	—
142	C₈	—	2	4	no	—
143	C₈	—	16	4	no	—
144	C₈	#334	1	8	yes	complemented
145	C₈	#335	1	8	yes	complemented
146	C₂²	#1842	1	1	yes	characteristic
147	C₂²	#1844	1	1	yes	characteristic
148	C₂²	#1845	1	1	yes	characteristic
149	C₂²	#1846	1	1	yes	characteristic
150	C₂²	#1848	1	1	yes	characteristic
151	C₂²	#1848	1	1	yes	characteristic
152	C₂²	#1848	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	C₂×#334	1	2	yes	—
157	C₄	C₂×#335	1	2	yes	—
158	C₄	#1564	1	2	yes	—
159	C₄	#1566	1	2	yes	—
160	C₄	#1567	1	2	yes	—
161	C₄	#1568	1	2	yes	—
162	C₄	#1706	1	2	yes	—
163	C₄	#1707	1	2	yes	—
164	C₂	#8805	1	1	yes	characteristic
165	C₂	#8806	1	1	yes	characteristic
166	C₂	#8807	1	1	yes	characteristic
167	C₂	#8814	1	1	yes	characteristic
168	C₂	#8822	1	1	yes	characteristic
169	C₂	#8823	1	1	yes	characteristic
170	C₂	#8824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#384
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
37. C₈²	C₈	semidirect
144. C₈	#334	semidirect
145. C₈	#335	semidirect

Maximal subgroups

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

subgroup	index
#8806	2
#8805	2
#8784	2
C₄×C₈²	2