Subgroups — #134396

Order 512.

360 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20516	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	#7167	C₂	1	2	yes	maximal
6	C₂×#446	C₄	1	1	yes	characteristic
7	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
8	#2295	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	#1771	—	2	1	no	—
13	#1770	C₄	1	1	yes	characteristic
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#1768	—	2	1	no	—
17	#1767	C₄	1	1	yes	characteristic
18	#1726	—	2	1	no	—
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	#1374	—	2	1	no	—
21	#1374	—	2	1	no	—
22	#1374	—	2	1	no	—
23	Q₈⋊C₁₆	—	2	4	no	—
24	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	—	2	2	no	—
26	C₂²×C₁₆	—	2	2	no	—
27	#446	—	2	2	no	—
28	#446	—	2	2	no	—
29	C₈×Q₈	—	2	1	no	—
30	C₈×Q₈	—	2	1	no	—
31	C₈×Q₈	D₄	1	2	yes	—
32	C₈×Q₈	Q₈	1	2	yes	—
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#97	—	2	1	no	—
38	C₂×#97	—	2	1	no	—
39	C₂×#97	—	2	1	no	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	#391	C₈	1	1	yes	characteristic
49	#391	C₈	1	1	yes	characteristic
50	#389	C₈	1	1	yes	characteristic
51	#389	C₈	1	1	yes	characteristic
52	#387	—	2	1	no	—
53	#387	—	2	1	no	—
54	C₄×C₄⋊C₄	—	2	1	no	—
55	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	#376	—	2	1	no	—
57	C₄³	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₁₆	—	2	4	no	—
59	C₄⋊C₁₆	—	2	2	no	—
60	C₄⋊C₁₆	—	2	2	no	—
61	C₄×C₁₆	—	2	2	no	—
62	C₄×C₁₆	—	2	2	no	—
63	#336	—	2	1	no	—
64	#336	—	2	1	no	—
65	#336	—	2	1	no	—
66	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₄×Q₈	—	2	1	no	—
74	C₄×Q₈	—	2	1	no	—
75	C₄×Q₈	C₄⋊C₄	1	2	yes	—
76	C₄×Q₈	M₁₆	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄²	M₁₆	1	1	yes	characteristic
90	C₂×C₄²	M₁₆	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₁₆	—	2	4	no	—
93	C₂×C₁₆	—	4	2	no	—
94	C₂×C₁₆	—	4	2	no	—
95	C₂×C₁₆	—	4	2	no	—
96	C₂×C₁₆	—	4	2	no	—
97	C₂×C₁₆	—	4	2	no	—
98	C₄⋊C₈	—	2	4	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	1	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
104	C₄⋊C₈	M₁₆	1	2	yes	—
105	#97	—	2	4	no	—
106	#97	—	4	2	no	—
107	#97	—	4	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	—	4	2	no	—
117	C₄×C₈	C₂×D₄	1	1	yes	characteristic
118	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
119	C₄×C₈	#41	1	1	yes	characteristic
120	C₄×C₈	#41	1	1	yes	characteristic
121	#95	—	2	1	no	—
122	#95	—	2	1	no	—
123	#95	—	2	1	no	—
124	#95	—	2	1	no	—
125	#95	—	2	1	no	—
126	#95	—	2	1	no	—
127	#95	—	2	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	C₄⋊C₈	1	2	yes	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
142	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	#104	1	2	yes	—
157	C₂×C₈	#121	1	1	yes	characteristic
158	C₂×C₈	#121	1	1	yes	characteristic
159	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₈	#135	1	2	yes	—
162	C₂×C₈	#137	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	4	1	no	—
181	C₄²	—	4	1	no	—
182	C₄²	—	4	1	no	—
183	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₄²	#117	1	1	yes	characteristic
185	C₄²	C₂×M₁₆	1	1	yes	characteristic
186	C₄²	C₂×M₁₆	1	1	yes	characteristic
187	C₁₆	—	4	4	no	—
188	C₁₆	—	8	2	no	—
189	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
190	Q₈	—	2	4	no	—
191	Q₈	—	4	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	#350	1	2	yes	—
218	C₂×C₄	#380	1	1	yes	characteristic
219	C₂×C₄	#407	1	1	yes	characteristic
220	C₂×C₄	#416	1	1	yes	characteristic
221	C₂×C₄	#419	1	1	yes	characteristic
222	C₂×C₄	C₂×#104	1	1	yes	characteristic
223	C₂×C₄	#421	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
225	C₂×C₄	#423	1	1	yes	characteristic
226	C₂×C₄	#431	1	1	yes	characteristic
227	C₂×C₄	#432	1	1	yes	characteristic
228	C₂×C₄	#504	1	2	yes	—
229	C₂×C₄	#504	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	—	4	2	no	—
234	C₈	—	4	1	no	—
235	C₈	—	4	2	no	—
236	C₈	#464	1	2	yes	—
237	C₈	#479	1	2	yes	—
238	C₂²	#1476	1	1	yes	characteristic
239	C₂²	#1479	1	1	yes	characteristic
240	C₂²	#1797	1	1	yes	characteristic
241	C₂²	#2113	1	1	yes	characteristic
242	C₂²	#2113	1	1	yes	characteristic
243	C₂²	C₂×#350	1	1	yes	characteristic
244	C₂²	#2117	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	8	1	no	—
254	C₄	—	8	1	no	—
255	C₄	#1488	1	1	yes	characteristic
256	C₄	#1491	1	1	yes	characteristic
257	C₄	#1805	1	1	yes	characteristic
258	C₄	#1806	1	1	yes	characteristic
259	C₄	#2151	1	2	yes	—
260	C₄	#2164	1	2	yes	—
261	C₂	#8566	1	1	yes	characteristic
262	C₂	#11614	1	1	yes	characteristic
263	C₂	#11616	1	1	yes	characteristic
264	C₂	#11638	1	1	yes	characteristic
265	C₂	#11640	1	1	yes	characteristic
266	C₂	#12784	1	1	yes	characteristic
267	C₂	#12787	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#20516	2
#12879	2
C₄×C₄⋊C₁₆	2
C₂×Q₈⋊C₁₆	2
#7167	2