Subgroups — #483577

Order 512.

437 conjugacy classes of subgroups.

The subgroup lattice has 241 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#12061	C₂	1	2	yes	maximal, complemented
4	#12060	C₂	1	4	yes	maximal, complemented
5	#12059	C₂	1	2	yes	maximal, complemented
6	#11996	C₂	1	1	yes	maximal, characteristic
7	#11607	C₂	1	2	yes	maximal
8	#11402	C₂	1	1	yes	maximal, characteristic
9	C₂×C₈×Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₁₆	C₄	1	1	yes	characteristic
11	#2183	—	2	4	no	—
12	#2183	C₂²	1	4	yes	—
13	Q₈×C₁₆	C₂²	1	4	yes	—
14	#2150	—	2	2	no	—
15	#2150	C₂²	1	2	yes	—
16	C₄×M₃₂	—	2	2	no	—
17	C₄×M₃₂	C₂²	1	1	yes	characteristic
18	C₄×M₃₂	C₂²	1	1	yes	characteristic
19	C₈×Q₁₆	C₄	1	2	yes	—
20	C₈×Q₁₆	C₄	1	1	yes	characteristic
21	C₈×Q₁₆	C₄	1	1	yes	characteristic
22	C₈×Q₁₆	C₂²	1	1	yes	characteristic
23	C₈×Q₁₆	C₂²	1	2	yes	—
24	C₈×Q₁₆	C₂²	1	1	yes	characteristic
25	C₂×#335	C₄	1	1	yes	characteristic
26	C₂×Q₈⋊C₈	C₄	1	2	yes	—
27	C₂×C₈²	C₂²	1	1	yes	characteristic
28	#1367	C₂²	1	2	yes	—
29	#1367	C₂²	1	2	yes	—
30	Q₈⋊C₁₆	C₂²	1	4	yes	—
31	Q₈⋊C₁₆	C₂²	1	4	yes	—
32	#1312	C₂²	1	2	yes	—
33	#1312	C₂²	1	2	yes	—
34	C₂²×Q₁₆	C₈	1	1	yes	characteristic
35	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	D₄	1	2	yes	—
39	C₈×Q₈	—	2	2	no	—
40	C₈×Q₈	—	2	2	no	—
41	C₈×Q₈	C₂×C₄	1	2	yes	—
42	C₈×Q₈	C₂×C₄	1	2	yes	—
43	C₈×Q₈	C₂³	1	2	yes	—
44	C₈×Q₈	C₂³	1	2	yes	—
45	C₄×Q₁₆	C₈	1	2	yes	—
46	C₄×Q₁₆	C₈	1	1	yes	characteristic
47	C₄×Q₁₆	C₈	1	1	yes	characteristic
48	C₄×Q₁₆	C₂×C₄	1	2	yes	—
49	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×#107	C₈	1	1	yes	characteristic
52	C₂×C₄⋊C₈	—	2	2	no	—
53	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₂×#103	C₈	1	2	yes	—
55	C₂×C₄×C₈	—	2	2	no	—
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	C₄⋊C₁₆	—	2	4	no	—
60	C₄⋊C₁₆	—	2	4	no	—
61	C₄⋊C₁₆	—	2	2	no	—
62	C₄⋊C₁₆	C₂³	1	4	yes	—
63	#346	—	2	4	no	—
64	#346	—	2	1	no	—
65	#346	D₄	1	2	yes	—
66	C₄×C₁₆	—	2	4	no	—
67	C₄×C₁₆	C₂³	1	2	yes	—
68	C₄×C₁₆	C₂³	1	2	yes	—
69	#335	C₂×C₄	1	1	yes	characteristic
70	#335	C₂×C₄	1	2	yes	—
71	#335	C₂×C₄	1	1	yes	characteristic
72	Q₈⋊C₈	C₂×C₄	1	2	yes	—
73	Q₈⋊C₈	C₂×C₄	1	2	yes	—
74	Q₈⋊C₈	C₂×C₄	1	2	yes	—
75	Q₈⋊C₈	C₂×C₄	1	2	yes	—
76	C₈²	C₂×C₄	1	1	yes	characteristic
77	C₈²	C₂×C₄	1	1	yes	characteristic
78	C₈²	C₂³	1	1	yes	characteristic
79	C₈²	C₂³	1	1	yes	characteristic
80	C₂²×Q₈	C₂×C₈	1	2	yes	—
81	C₂×Q₁₆	—	2	4	no	—
82	C₂×Q₁₆	C₂×C₈	1	2	yes	—
83	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
84	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	#41	1	1	yes	characteristic
91	C₄×Q₈	—	2	2	no	—
92	C₄×Q₈	—	2	2	no	—
93	C₄×Q₈	C₂×C₈	1	2	yes	—
94	C₄×Q₈	C₂×C₈	1	2	yes	—
95	C₄×Q₈	C₂²×C₄	1	2	yes	—
96	C₄×Q₈	C₂²×C₄	1	2	yes	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
99	C₂×C₄²	—	2	2	no	—
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	4	2	no	—
103	M₃₂	—	4	4	no	—
104	C₂×C₁₆	—	2	4	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₂×D₄	1	4	yes	—
107	#107	C₂×C₈	1	2	yes	—
108	#107	C₂×C₈	1	1	yes	characteristic
109	#107	C₂×C₈	1	1	yes	characteristic
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	C₂²×C₄	1	2	yes	—
116	C₄⋊C₈	C₂²×C₄	1	2	yes	—
117	#103	C₂×C₈	1	2	yes	—
118	#103	C₂×C₈	1	2	yes	—
119	#103	C₂×C₈	1	2	yes	—
120	#103	C₂×C₈	1	2	yes	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	C₂×C₈	1	1	yes	characteristic
127	C₄×C₈	C₂×C₈	1	1	yes	characteristic
128	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
129	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₄×C₈	C₂×D₄	1	1	yes	characteristic
134	C₄×C₈	#41	1	1	yes	characteristic
135	C₄×C₈	C₂⁴	1	1	yes	characteristic
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	C₂²×C₈	1	2	yes	—
140	C₂×Q₈	C₂²×C₈	1	2	yes	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
145	C₂²×C₄	#131	1	1	yes	characteristic
146	Q₁₆	—	2	4	no	—
147	Q₁₆	—	2	2	no	—
148	Q₁₆	M₃₂	1	4	yes	complemented
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	4	no	—
158	C₂×C₈	C₄×D₄	1	1	yes	characteristic
159	C₂×C₈	C₄×D₄	1	1	yes	characteristic
160	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₈	#137	1	4	yes	—
165	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₈	C₂×#41	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	C₂²×C₈	1	2	yes	—
173	C₄⋊C₄	C₂²×C₈	1	2	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₄×D₄	1	1	yes	characteristic
179	C₄²	C₂²×C₈	1	1	yes	characteristic
180	C₄²	#131	1	1	yes	characteristic
181	C₄²	C₂³×C₄	1	1	yes	characteristic
182	C₁₆	—	2	4	no	—
183	C₁₆	—	4	2	no	—
184	C₁₆	—	4	4	no	—
185	C₂³	C₈×D₄	1	1	yes	characteristic
186	Q₈	—	2	2	no	—
187	Q₈	—	2	2	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	4	4	no	—
190	Q₈	C₂×M₃₂	1	4	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	C₈×D₄	1	1	yes	characteristic
201	C₂×C₄	C₈×D₄	1	1	yes	characteristic
202	C₂×C₄	C₈×D₄	1	1	yes	characteristic
203	C₂×C₄	#441	1	2	yes	—
204	C₂×C₄	#444	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×#131	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	4	1	no	—
215	C₈	—	4	2	no	—
216	C₈	—	4	2	no	—
217	C₈	#441	1	2	yes	—
218	C₈	C₂×M₃₂	1	2	yes	—
219	C₈	C₂×#137	1	2	yes	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	#1579	1	2	yes	—
223	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	4	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	#1579	1	2	yes	—
232	C₄	C₂×#441	1	1	yes	characteristic
233	C₄	C₂×#444	1	1	yes	characteristic
234	C₄	C₂²×M₃₂	1	1	yes	characteristic
235	C₄	#3407	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	C₂×#1579	1	1	yes	characteristic
238	C₂	D₄×M₃₂	1	1	yes	characteristic
239	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12061	C₂	semidirect
4. #12060	C₂	semidirect
5. #12059	C₂	semidirect
148. Q₁₆	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₃₂	2
C₂×C₈×Q₁₆	2
#12061	2
#12060	2
#12059	2
#11996	2
#11607	2
#11402	2