Subgroups — #134397

Order 512.

366 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#446	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	#12653	C₂	1	1	yes	maximal, characteristic
4	C₂×#1338	C₂	1	2	yes	maximal
5	#7170	C₂	1	2	yes	maximal
6	C₂×#446	C₄	1	1	yes	characteristic
7	C₂×#446	C₂²	1	1	yes	characteristic
8	C₄²×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#346	C₂²	1	2	yes	—
12	#1771	—	2	1	no	—
13	#1771	—	2	1	no	—
14	#1770	C₄	1	1	yes	characteristic
15	#1770	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₈	—	2	1	no	—
17	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
18	C₄×#97	—	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	#1338	—	2	4	no	—
24	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
25	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
26	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	—	2	2	no	—
28	C₂²×C₁₆	—	2	2	no	—
29	#446	—	2	2	no	—
30	#446	—	2	2	no	—
31	#446	—	2	1	no	—
32	#446	—	2	1	no	—
33	#446	D₄	1	2	yes	—
34	#446	Q₈	1	2	yes	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×#97	—	2	1	no	—
40	C₂×#97	—	2	1	no	—
41	C₂×#97	—	2	1	no	—
42	C₂×#97	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₄⋊C₄	—	2	1	no	—
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
54	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
55	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄³	—	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	#346	—	2	2	no	—
62	#346	—	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	—	4	2	no	—
76	C₄×Q₈	—	4	2	no	—
77	C₄×Q₈	C₄⋊C₄	1	2	yes	—
78	C₄×Q₈	M₁₆	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	—	2	1	no	—
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	—	2	1	no	—
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
95	C₂×C₄²	M₁₆	1	1	yes	characteristic
96	C₂×C₄²	M₁₆	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	—	4	2	no	—
100	C₂×C₁₆	—	4	2	no	—
101	C₂×C₁₆	—	4	2	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	—	4	2	no	—
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
110	C₄⋊C₈	M₁₆	1	2	yes	—
111	#97	—	2	4	no	—
112	#97	—	2	2	no	—
113	#97	—	2	2	no	—
114	#97	—	4	2	no	—
115	#97	—	4	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	4	1	no	—
122	C₄×C₈	—	4	2	no	—
123	C₄×C₈	C₂×D₄	1	1	yes	characteristic
124	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
125	C₄×C₈	#41	1	1	yes	characteristic
126	C₄×C₈	#41	1	1	yes	characteristic
127	#95	—	2	1	no	—
128	#95	—	2	1	no	—
129	#95	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	C₄⋊C₈	1	2	yes	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	—	4	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	#104	1	2	yes	—
159	C₂×C₈	#121	1	1	yes	characteristic
160	C₂×C₈	#121	1	1	yes	characteristic
161	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₈	#135	1	2	yes	—
164	C₂×C₈	#137	1	2	yes	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	4	1	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	4	1	no	—
189	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₄²	#117	1	1	yes	characteristic
191	C₄²	C₂×M₁₆	1	1	yes	characteristic
192	C₄²	C₂×M₁₆	1	1	yes	characteristic
193	C₁₆	—	4	4	no	—
194	C₁₆	—	8	2	no	—
195	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
196	Q₈	—	4	2	no	—
197	Q₈	—	4	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	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	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	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	#349	1	2	yes	—
224	C₂×C₄	#349	1	2	yes	—
225	C₂×C₄	#350	1	2	yes	—
226	C₂×C₄	#380	1	1	yes	characteristic
227	C₂×C₄	#407	1	1	yes	characteristic
228	C₂×C₄	#416	1	1	yes	characteristic
229	C₂×C₄	#419	1	1	yes	characteristic
230	C₂×C₄	C₂×#104	1	1	yes	characteristic
231	C₂×C₄	#421	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
233	C₂×C₄	#423	1	1	yes	characteristic
234	C₂×C₄	#431	1	1	yes	characteristic
235	C₂×C₄	#432	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	1	no	—
241	C₈	—	4	2	no	—
242	C₈	#464	1	2	yes	—
243	C₈	#479	1	2	yes	—
244	C₂²	#1476	1	1	yes	characteristic
245	C₂²	#1479	1	1	yes	characteristic
246	C₂²	#1797	1	1	yes	characteristic
247	C₂²	C₂×#349	1	1	yes	characteristic
248	C₂²	C₂×#349	1	1	yes	characteristic
249	C₂²	C₂×#350	1	1	yes	characteristic
250	C₂²	#2117	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	1	no	—
262	C₄	—	8	1	no	—
263	C₄	#1488	1	1	yes	characteristic
264	C₄	#1491	1	1	yes	characteristic
265	C₄	#1805	1	1	yes	characteristic
266	C₄	#1806	1	1	yes	characteristic
267	C₂	#8566	1	1	yes	characteristic
268	C₂	#11617	1	1	yes	characteristic
269	C₂	#11619	1	1	yes	characteristic
270	C₂	#11642	1	1	yes	characteristic
271	C₂	#11644	1	1	yes	characteristic
272	C₂	#12785	1	1	yes	characteristic
273	C₂	#12787	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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
C₄×#446	2
#12879	2
#12653	2
C₂×#1338	2
#7170	2