Subgroups — #128539

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#12653	C₂	1	1	yes	maximal, characteristic
3	C₄×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
8	C₄²×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₂×#346	—	2	1	no	—
11	C₂×#346	—	2	1	no	—
12	C₂×#346	—	2	1	no	—
13	C₂×C₄×C₁₆	C₂²	1	2	yes	—
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×C₄⋊C₄	—	2	1	no	—
16	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
17	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₈	—	2	1	no	—
19	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
20	C₄²×C₈	—	2	1	no	—
21	C₄²×C₈	C₂²	1	1	yes	characteristic
22	#1374	—	2	1	no	—
23	Q₈⋊C₁₆	C₄	1	8	yes	complemented
24	C₂×C₄×Q₈	C₈	1	2	yes	—
25	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	—	2	2	no	—
27	C₂²×C₁₆	—	2	2	no	—
28	C₈×Q₈	—	2	2	no	—
29	C₈×Q₈	—	2	2	no	—
30	C₈×Q₈	C₈	1	4	yes	—
31	C₈×Q₈	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	—	2	1	no	—
37	C₂×C₄×C₈	—	2	1	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₄×C₄⋊C₄	—	2	1	no	—
48	C₄×C₄⋊C₄	—	2	2	no	—
49	C₄×C₄⋊C₄	C₈	1	2	yes	—
50	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₄³	—	2	1	no	—
52	C₄³	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₁₆	—	2	2	no	—
54	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
55	#346	—	2	4	no	—
56	#346	—	2	4	no	—
57	#346	—	4	2	no	—
58	C₄×C₁₆	—	2	2	no	—
59	C₄×C₁₆	C₂×C₄	1	4	yes	—
60	#336	—	2	1	no	—
61	#336	—	2	1	no	—
62	#336	—	2	1	no	—
63	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₄×Q₈	—	2	4	no	—
71	C₄×Q₈	—	2	4	no	—
72	C₄×Q₈	—	2	1	no	—
73	C₄×Q₈	—	2	1	no	—
74	C₄×Q₈	C₄²	1	2	yes	—
75	C₄×Q₈	C₂×C₈	1	2	yes	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₂×C₄²	—	2	2	no	—
81	C₂×C₄²	—	2	1	no	—
82	C₂×C₄²	—	2	2	no	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂×C₈	1	2	yes	—
87	C₂×C₄²	M₁₆	1	2	yes	—
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	—	2	4	no	—
92	C₂×C₁₆	—	2	4	no	—
93	C₂×C₁₆	—	4	2	no	—
94	C₂×C₁₆	—	4	2	no	—
95	C₄⋊C₈	—	2	4	no	—
96	C₄⋊C₈	—	2	4	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	C₄²	1	2	yes	—
100	C₄⋊C₈	C₂×C₈	1	2	yes	—
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	2	4	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	4	2	no	—
107	C₄×C₈	—	4	2	no	—
108	C₄×C₈	—	4	2	no	—
109	C₄×C₈	C₂×C₈	1	2	yes	—
110	C₄×C₈	M₁₆	1	2	yes	—
111	C₄×C₈	SD₁₆	1	2	yes	—
112	C₄×C₈	Q₁₆	1	2	yes	—
113	C₄×C₈	C₂²×C₄	1	2	yes	—
114	C₄×C₈	#41	1	2	yes	—
115	#95	—	2	2	no	—
116	#95	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	C₄×C₈	1	2	yes	—
120	C₂×Q₈	M₃₂	1	4	yes	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
125	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
126	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	C₂²⋊C₈	1	2	yes	—
140	C₂×C₈	C₂²⋊C₈	1	2	yes	—
141	C₂×C₈	#103	1	2	yes	—
142	C₂×C₈	#104	1	2	yes	—
143	C₂×C₈	C₄×D₄	1	2	yes	—
144	C₂×C₈	C₄×D₄	1	2	yes	—
145	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
147	C₂×C₈	#135	1	2	yes	—
148	C₄⋊C₄	—	2	8	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	C₄×C₈	1	2	yes	—
154	C₄⋊C₄	M₃₂	1	4	yes	—
155	C₄²	—	2	8	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	—	4	4	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#103	1	2	yes	—
167	C₄²	#104	1	2	yes	—
168	C₄²	C₂×C₄²	1	1	yes	characteristic
169	C₄²	#117	1	1	yes	characteristic
170	C₄²	C₂²×C₈	1	1	yes	characteristic
171	C₄²	C₂×M₁₆	1	1	yes	characteristic
172	C₁₆	—	2	8	no	—
173	C₁₆	—	4	4	no	—
174	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	Q₈	—	2	2	no	—
176	Q₈	—	2	4	no	—
177	Q₈	#346	1	4	yes	complemented
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	Q₈⋊C₈	1	4	yes	—
192	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
194	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
195	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×#103	1	1	yes	characteristic
197	C₂×C₄	#416	1	1	yes	characteristic
198	C₂×C₄	C₂×#104	1	1	yes	characteristic
199	C₂×C₄	C₂×#104	1	1	yes	characteristic
200	C₂×C₄	#431	1	1	yes	characteristic
201	C₂×C₄	#431	1	1	yes	characteristic
202	C₂×C₄	C₂×M₃₂	1	2	yes	—
203	C₂×C₄	C₂×M₃₂	1	2	yes	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	4	2	no	—
207	C₈	—	4	2	no	—
208	C₈	Q₈⋊C₈	1	4	yes	—
209	C₈	C₄×SD₁₆	1	2	yes	—
210	C₈	C₄×Q₁₆	1	2	yes	—
211	C₂²	C₂×Q₈⋊C₈	1	1	yes	characteristic
212	C₂²	#1476	1	1	yes	characteristic
213	C₂²	C₄×C₂²⋊C₈	1	1	yes	characteristic
214	C₂²	#2112	1	2	yes	—
215	C₂²	#2117	1	2	yes	—
216	C₄	—	2	4	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	4	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	4	no	—
223	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
224	C₄	#1476	1	1	yes	characteristic
225	C₄	C₄×#104	1	1	yes	characteristic
226	C₄	C₄×#103	1	1	yes	characteristic
227	C₄	C₂×#346	1	1	yes	characteristic
228	C₄	C₄×M₃₂	1	1	yes	characteristic
229	C₄	#2163	1	2	yes	—
230	C₂	C₄×Q₈⋊C₈	1	1	yes	characteristic
231	C₂	#11607	1	2	yes	—
232	C₂	#11615	1	2	yes	—
233	C₂	#12599	1	1	yes	characteristic
234	C₂	#12604	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ 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
23. Q₈⋊C₁₆	C₄	semidirect
177. Q₈	#346	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Q₈	2
#12653	2
C₄×#346	2
C₂×Q₈⋊C₁₆	2
#7167	2