Subgroups — #146972

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#364	C₂	1	1	yes	maximal, characteristic
2	C₂²×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#1379	C₂	1	2	yes	maximal
5	#12545	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×#108	C₄	1	1	yes	characteristic
10	C₂×#427	C₄	1	1	yes	characteristic
11	C₂×#405	C₂²	1	2	yes	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#364	—	2	2	no	—
14	C₂×#364	—	2	2	no	—
15	C₂×#364	C₄	1	4	yes	complemented
16	C₂×#346	—	2	2	no	—
17	C₂×#346	—	2	2	no	—
18	C₂×#346	C₄	1	4	yes	complemented
19	#1772	—	2	2	no	—
20	#1772	—	2	2	no	—
21	C₂×#339	C₄	1	2	yes	—
22	#1379	—	2	4	no	—
23	#1374	—	2	4	no	—
24	C₂²×M₁₆	C₂×C₄	1	2	yes	—
25	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₈	C₂³	1	1	yes	characteristic
27	C₂×#117	C₂×C₄	1	2	yes	—
28	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×M₃₂	—	2	2	no	—
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	C₂×C₄	1	4	yes	—
35	C₂²×C₁₆	—	2	1	no	—
36	C₂²×C₁₆	—	2	1	no	—
37	C₂²×C₁₆	—	2	1	no	—
38	C₂²×C₁₆	—	2	1	no	—
39	C₂²×C₁₆	—	2	1	no	—
40	C₂²×C₁₆	—	2	1	no	—
41	C₂²×C₁₆	—	2	1	no	—
42	C₂²×C₁₆	—	2	1	no	—
43	C₂²×C₁₆	—	2	2	no	—
44	C₂²×C₁₆	C₂×C₄	1	2	yes	—
45	C₂²×C₁₆	C₂×C₄	1	2	yes	—
46	C₂²×C₁₆	D₄	1	2	yes	—
47	C₂²×C₁₆	D₄	1	2	yes	—
48	C₂²×C₁₆	Q₈	1	2	yes	—
49	C₂²×C₁₆	Q₈	1	2	yes	—
50	C₂×#108	—	2	2	no	—
51	C₂×#108	—	2	2	no	—
52	C₂×#108	C₈	1	4	yes	—
53	#427	—	2	2	no	—
54	#427	—	2	2	no	—
55	C₂×#107	C₈	1	2	yes	—
56	C₂×#106	C₈	1	2	yes	—
57	#405	—	2	2	no	—
58	#405	—	2	2	no	—
59	#405	—	2	2	no	—
60	#405	—	2	2	no	—
61	C₂×#97	—	2	2	no	—
62	C₂×C₄×C₈	—	2	2	no	—
63	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
64	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
65	C₂×C₄×C₈	D₄	1	2	yes	—
66	C₂×C₄×C₈	D₄	1	2	yes	—
67	C₂×C₄×C₈	D₄	1	2	yes	—
68	C₂×C₄×C₈	Q₈	1	2	yes	—
69	#364	—	2	8	no	—
70	#346	—	2	8	no	—
71	#339	—	2	4	no	—
72	C₂³×C₄	—	2	1	no	—
73	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	2	2	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	C₂×C₈	1	4	yes	—
80	C₂²×C₈	—	2	2	no	—
81	C₂²×C₈	C₄²	1	2	yes	—
82	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₈	C₄⋊C₄	1	2	yes	—
84	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₈	C₂×C₈	1	2	yes	—
87	C₂²×C₈	M₁₆	1	2	yes	—
88	C₂²×C₈	M₁₆	1	2	yes	—
89	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₈	#41	1	1	yes	characteristic
100	C₂²×C₈	#41	1	1	yes	characteristic
101	C₂²×C₈	#41	1	1	yes	characteristic
102	C₂²×C₈	#41	1	1	yes	characteristic
103	#117	—	2	2	no	—
104	#117	—	2	2	no	—
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
108	C₂×C₂²⋊C₄	—	2	2	no	—
109	C₂×C₄²	—	2	1	no	—
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	—	2	1	no	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₄²	1	2	yes	—
114	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
115	C₂×C₄²	C₄⋊C₄	1	2	yes	—
116	M₃₂	—	2	4	no	—
117	M₃₂	—	2	4	no	—
118	M₃₂	—	4	2	no	—
119	M₃₂	—	4	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	—	4	2	no	—
134	C₂×C₁₆	—	4	2	no	—
135	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
136	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
137	#108	—	2	8	no	—
138	#107	—	2	4	no	—
139	#106	—	2	4	no	—
140	#97	—	4	2	no	—
141	#97	—	4	2	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	—	4	2	no	—
144	C₄×C₈	C₂²⋊C₄	1	4	yes	—
145	C₄×C₈	C₂²⋊C₄	1	4	yes	—
146	C₄×C₈	C₄⋊C₄	1	4	yes	—
147	C₄×C₈	C₄⋊C₄	1	4	yes	—
148	C₂⁴	#131	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
158	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
159	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
160	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
164	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
165	C₂²×C₄	#131	1	1	yes	characteristic
166	C₂²×C₄	#131	1	1	yes	characteristic
167	C₂²×C₄	#131	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	2	4	no	—
170	M₁₆	—	4	2	no	—
171	M₁₆	—	4	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	#95	1	4	yes	—
176	C₂×C₈	C₄×C₈	1	2	yes	—
177	C₂×C₈	C₄×C₈	1	2	yes	—
178	C₂×C₈	#97	1	4	yes	—
179	C₂×C₈	C₂²⋊C₈	1	2	yes	—
180	C₂×C₈	C₂²⋊C₈	1	2	yes	—
181	C₂×C₈	C₂²⋊C₈	1	4	yes	—
182	C₂×C₈	C₄⋊C₈	1	2	yes	—
183	C₂×C₈	C₄⋊C₈	1	2	yes	—
184	C₂×C₈	C₄⋊C₈	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	#117	1	1	yes	characteristic
192	C₂×C₈	#117	1	1	yes	characteristic
193	C₂×C₈	C₄×D₄	1	1	yes	characteristic
194	C₂×C₈	C₄×D₄	1	1	yes	characteristic
195	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
196	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
197	C₂×C₈	#121	1	1	yes	characteristic
198	C₂×C₈	#121	1	1	yes	characteristic
199	C₂×C₈	#121	1	1	yes	characteristic
200	C₂×C₈	#121	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
205	C₂×C₈	#124	1	1	yes	characteristic
206	C₂×C₈	#124	1	1	yes	characteristic
207	C₂×C₈	#125	1	1	yes	characteristic
208	C₂×C₈	#125	1	1	yes	characteristic
209	C₂×C₈	#127	1	1	yes	characteristic
210	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
211	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
212	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	#95	1	4	yes	—
223	C₁₆	—	2	4	no	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	4	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	#405	1	1	yes	characteristic
228	C₂³	#408	1	1	yes	characteristic
229	C₂³	#424	1	1	yes	characteristic
230	C₂³	C₈×D₄	1	1	yes	characteristic
231	C₂³	#436	1	1	yes	characteristic
232	C₂³	C₈×Q₈	1	1	yes	characteristic
233	C₂³	#446	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂×#95	1	1	yes	characteristic
241	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₄	#389	1	1	yes	characteristic
244	C₂×C₄	#389	1	1	yes	characteristic
245	C₂×C₄	#390	1	1	yes	characteristic
246	C₂×C₄	#391	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
248	C₂×C₄	C₂×#97	1	1	yes	characteristic
249	C₂×C₄	#405	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
252	C₂×C₄	#408	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
255	C₂×C₄	#424	1	1	yes	characteristic
256	C₂×C₄	C₈×D₄	1	1	yes	characteristic
257	C₂×C₄	C₈×D₄	1	1	yes	characteristic
258	C₂×C₄	C₈×D₄	1	1	yes	characteristic
259	C₂×C₄	#436	1	1	yes	characteristic
260	C₂×C₄	#436	1	1	yes	characteristic
261	C₂×C₄	#436	1	1	yes	characteristic
262	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
263	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
264	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
265	C₂×C₄	#446	1	1	yes	characteristic
266	C₂×C₄	#446	1	1	yes	characteristic
267	C₂×C₄	#446	1	1	yes	characteristic
268	C₈	—	4	4	no	—
269	C₈	#336	1	4	yes	—
270	C₈	#336	1	4	yes	—
271	C₈	#380	1	1	yes	characteristic
272	C₈	#380	1	1	yes	characteristic
273	C₈	#382	1	1	yes	characteristic
274	C₈	#384	1	1	yes	characteristic
275	C₈	#389	1	1	yes	characteristic
276	C₈	#389	1	1	yes	characteristic
277	C₈	#390	1	1	yes	characteristic
278	C₈	#391	1	1	yes	characteristic
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	#1729	1	1	yes	characteristic
286	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
287	C₂²	#1770	1	1	yes	characteristic
288	C₂²	#1939	1	1	yes	characteristic
289	C₂²	#1939	1	1	yes	characteristic
290	C₂²	#1954	1	1	yes	characteristic
291	C₂²	#1984	1	1	yes	characteristic
292	C₂²	#2171	1	4	yes	—
293	C₄	—	4	4	no	—
294	C₄	—	8	4	no	—
295	C₄	C₄⋊#95	1	1	yes	characteristic
296	C₄	C₂×#336	1	1	yes	characteristic
297	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
298	C₄	#1770	1	1	yes	characteristic
299	C₄	#1939	1	1	yes	characteristic
300	C₄	#1939	1	1	yes	characteristic
301	C₄	#1954	1	1	yes	characteristic
302	C₄	#1984	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	#10834	1	1	yes	characteristic
308	C₂	#12665	1	2	yes	—
309	C₂	#13162	1	2	yes	—
310	C₂	#13196	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. C₂×#364	C₄	semidirect
18. C₂×#346	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#364	2
C₂²×#346	2
C₂×#1772	2
C₂×#1379	2
#12545	2