Subgroups — #93657

Order 512.

228 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12898	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	#8805	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
7	#1842	—	2	1	no	—
8	#1831	—	2	1	no	—
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#335	C₄	1	2	yes	—
11	C₂×#334	—	2	1	no	—
12	C₂×#334	—	2	1	no	—
13	C₂×#334	C₄	1	2	yes	—
14	C₂²×C₁₆	—	2	2	no	—
15	C₂×C₄⋊C₈	—	2	2	no	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×C₄⋊C₈	—	4	1	no	—
18	C₂×C₄⋊C₈	—	4	1	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	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
25	C₄³	D₄	1	1	yes	characteristic
26	C₄⋊C₁₆	—	4	2	no	—
27	C₄⋊C₁₆	—	4	1	no	—
28	C₄⋊C₁₆	—	4	1	no	—
29	C₄⋊C₁₆	—	4	1	no	—
30	C₄⋊C₁₆	—	4	1	no	—
31	#336	—	4	1	no	—
32	#336	—	4	1	no	—
33	#335	—	4	2	no	—
34	#334	—	4	2	no	—
35	#334	—	4	1	no	—
36	#334	—	4	1	no	—
37	#334	—	4	1	no	—
38	#334	—	4	1	no	—
39	C₂²×C₈	—	4	2	no	—
40	C₂²×C₈	—	4	2	no	—
41	C₂²×C₈	C₄²	1	1	yes	characteristic
42	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄²	D₈	1	1	yes	characteristic
49	C₂×C₄²	SD₁₆	1	1	yes	characteristic
50	C₂×C₄²	SD₁₆	1	1	yes	characteristic
51	C₂×C₄²	Q₁₆	1	1	yes	characteristic
52	C₂×C₁₆	—	4	2	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	—	4	2	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄⋊C₈	—	4	2	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	—	8	2	no	—
60	C₄⋊C₈	—	8	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	4	1	no	—
68	C₄×C₈	D₈	1	1	yes	characteristic
69	C₄×C₈	D₈	1	1	yes	characteristic
70	C₄×C₈	SD₁₆	1	1	yes	characteristic
71	C₄×C₈	SD₁₆	1	1	yes	characteristic
72	C₄×C₈	SD₁₆	1	1	yes	characteristic
73	C₄×C₈	SD₁₆	1	1	yes	characteristic
74	C₄×C₈	Q₁₆	1	1	yes	characteristic
75	C₄×C₈	Q₁₆	1	1	yes	characteristic
76	C₂²×C₄	#95	1	1	yes	characteristic
77	C₂²×C₄	#102	1	1	yes	characteristic
78	C₂²×C₄	#102	1	1	yes	characteristic
79	C₂²×C₄	#103	1	1	yes	characteristic
80	C₂²×C₄	#103	1	1	yes	characteristic
81	C₂²×C₄	#106	1	1	yes	characteristic
82	C₂²×C₄	#107	1	1	yes	characteristic
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	8	2	no	—
91	C₂×C₈	—	8	2	no	—
92	C₂×C₈	—	8	2	no	—
93	C₂×C₈	—	8	2	no	—
94	C₂×C₈	—	8	2	no	—
95	C₂×C₈	—	8	2	no	—
96	C₂×C₈	#100	1	1	yes	characteristic
97	C₂×C₈	#101	1	1	yes	characteristic
98	C₂×C₈	#102	1	1	yes	characteristic
99	C₂×C₈	#102	1	1	yes	characteristic
100	C₂×C₈	#103	1	1	yes	characteristic
101	C₂×C₈	#103	1	1	yes	characteristic
102	C₂×C₈	#106	1	1	yes	characteristic
103	C₂×C₈	#106	1	1	yes	characteristic
104	C₂×C₈	#107	1	1	yes	characteristic
105	C₂×C₈	#107	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	4	1	no	—
112	C₄²	#100	1	1	yes	characteristic
113	C₄²	#100	1	1	yes	characteristic
114	C₄²	#101	1	1	yes	characteristic
115	C₄²	#101	1	1	yes	characteristic
116	C₄²	#102	1	1	yes	characteristic
117	C₄²	#102	1	1	yes	characteristic
118	C₄²	#103	1	1	yes	characteristic
119	C₄²	#103	1	1	yes	characteristic
120	C₁₆	—	8	2	no	—
121	C₂³	#340	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	#331	1	1	yes	characteristic
134	C₂×C₄	#331	1	1	yes	characteristic
135	C₂×C₄	#332	1	1	yes	characteristic
136	C₂×C₄	#332	1	1	yes	characteristic
137	C₂×C₄	#332	1	1	yes	characteristic
138	C₂×C₄	#332	1	1	yes	characteristic
139	C₂×C₄	#333	1	1	yes	characteristic
140	C₂×C₄	#333	1	1	yes	characteristic
141	C₂×C₄	#338	1	1	yes	characteristic
142	C₂×C₄	#340	1	1	yes	characteristic
143	C₂×C₄	#340	1	1	yes	characteristic
144	C₂×C₄	#343	1	1	yes	characteristic
145	C₂×C₄	#343	1	1	yes	characteristic
146	C₂×C₄	#343	1	1	yes	characteristic
147	C₂×C₄	#359	1	2	yes	—
148	C₂×C₄	#360	1	2	yes	—
149	C₂×C₄	#365	1	2	yes	—
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	4	1	no	—
155	C₈	—	16	2	no	—
156	C₈	—	16	2	no	—
157	C₈	#331	1	1	yes	characteristic
158	C₈	#332	1	1	yes	characteristic
159	C₈	#332	1	1	yes	characteristic
160	C₈	#333	1	1	yes	characteristic
161	C₂²	#1292	1	1	yes	characteristic
162	C₂²	#1296	1	1	yes	characteristic
163	C₂²	#1296	1	1	yes	characteristic
164	C₂²	#1382	1	1	yes	characteristic
165	C₂²	#1383	1	1	yes	characteristic
166	C₂²	#1386	1	1	yes	characteristic
167	C₂²	#1388	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	#1292	1	1	yes	characteristic
177	C₄	#1292	1	1	yes	characteristic
178	C₄	#1296	1	1	yes	characteristic
179	C₄	#1309	1	1	yes	characteristic
180	C₄	#1360	1	2	yes	—
181	C₄	#1363	1	2	yes	—
182	C₂	#7050	1	1	yes	characteristic
183	C₂	#7243	1	1	yes	characteristic
184	C₂	#7244	1	1	yes	characteristic
185	C₂	#7275	1	1	yes	characteristic
186	C₂	#7277	1	1	yes	characteristic
187	C₂	#7278	1	1	yes	characteristic
188	C₂	#7280	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#340
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₄²×C₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12898	2
#8807	2
#8805	2