Subgroups — #146849

Order 512.

470 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2110	C₂	1	1	yes	maximal, characteristic
2	#12554	C₂	1	1	yes	maximal, characteristic
3	C₂×#1379	C₂	1	1	yes	maximal, characteristic
4	C₂×#1379	C₂	1	1	yes	maximal, characteristic
5	C₂×#1378	C₂	1	1	yes	maximal, characteristic
6	#12545	C₂	1	1	yes	maximal, characteristic
7	#10711	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×M₃₂	C₂²	1	1	yes	characteristic
11	C₂³×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×#405	C₂²	1	1	yes	characteristic
13	C₂×#405	C₂²	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	#2110	—	2	2	no	—
16	#2110	C₄	1	4	yes	complemented
17	C₂×#347	—	2	2	no	—
18	C₂×#344	C₄	1	1	yes	characteristic
19	C₂×#344	C₄	1	1	yes	characteristic
20	#1742	C₄	1	1	yes	characteristic
21	#1740	C₄	1	1	yes	characteristic
22	#1379	—	2	2	no	—
23	#1379	—	2	2	no	—
24	#1379	C₄	1	4	yes	complemented
25	#1378	—	2	2	no	—
26	#1374	—	2	2	no	—
27	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	1	yes	characteristic
31	C₂×#117	C₂×C₄	1	1	yes	characteristic
32	C₂×#117	C₂×C₄	1	1	yes	characteristic
33	C₂×#117	C₂×C₄	1	1	yes	characteristic
34	C₂×M₃₂	—	2	1	no	—
35	C₂×M₃₂	—	2	1	no	—
36	C₂×M₃₂	—	2	1	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂×M₃₂	—	2	1	no	—
45	C₂×M₃₂	—	2	1	no	—
46	C₂×M₃₂	C₂×C₄	1	2	yes	—
47	C₂×M₃₂	C₂×C₄	1	2	yes	—
48	C₂×M₃₂	C₂×C₄	1	2	yes	—
49	C₂×M₃₂	D₄	1	2	yes	—
50	C₂×M₃₂	D₄	1	2	yes	—
51	C₂×M₃₂	Q₈	1	2	yes	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	—	2	1	no	—
54	C₂²×C₁₆	—	2	1	no	—
55	C₂²×C₁₆	—	2	1	no	—
56	C₂²×C₁₆	—	2	1	no	—
57	C₂²×C₁₆	—	2	1	no	—
58	C₂²×C₁₆	—	2	1	no	—
59	C₂²×C₁₆	—	2	1	no	—
60	C₂²×C₁₆	C₂×C₄	1	2	yes	—
61	C₂²×C₁₆	D₄	1	2	yes	—
62	#405	—	2	1	no	—
63	#405	—	2	1	no	—
64	#405	—	2	1	no	—
65	#405	—	2	1	no	—
66	#405	—	2	1	no	—
67	#405	—	2	1	no	—
68	#405	—	2	1	no	—
69	#405	—	2	1	no	—
70	#405	—	2	1	no	—
71	#405	—	2	1	no	—
72	#405	C₂×C₄	1	2	yes	—
73	#405	C₂×C₄	1	2	yes	—
74	C₂×#97	—	2	1	no	—
75	C₂×#97	—	2	1	no	—
76	C₂×#97	—	2	1	no	—
77	C₂×C₄×C₈	—	2	1	no	—
78	C₂×C₄×C₈	—	2	1	no	—
79	C₂×C₄×C₈	—	2	1	no	—
80	#347	—	2	8	no	—
81	#344	—	2	2	no	—
82	#344	—	2	2	no	—
83	#343	—	2	2	no	—
84	#342	—	2	2	no	—
85	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×M₁₆	—	2	1	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	2	1	no	—
90	C₂×M₁₆	—	2	1	no	—
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	C₄²	1	2	yes	—
102	C₂×M₁₆	M₁₆	1	2	yes	—
103	C₂×M₁₆	M₁₆	1	2	yes	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	C₂²×C₈	#41	1	1	yes	characteristic
117	C₂²×C₈	#41	1	1	yes	characteristic
118	C₂²×C₈	#41	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	C₂²×C₈	#41	1	1	yes	characteristic
121	#117	—	2	1	no	—
122	#117	—	2	1	no	—
123	#117	—	2	1	no	—
124	#117	—	2	1	no	—
125	#117	—	2	1	no	—
126	#117	—	2	1	no	—
127	#117	—	2	1	no	—
128	#117	—	2	1	no	—
129	#117	—	2	1	no	—
130	#117	—	2	1	no	—
131	#117	—	2	1	no	—
132	#117	C₄²	1	2	yes	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	M₁₆	1	2	yes	—
136	C₂×C₂²⋊C₄	—	2	1	no	—
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₂²⋊C₄	M₁₆	1	2	yes	—
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	—	2	1	no	—
142	M₃₂	—	2	2	no	—
143	M₃₂	—	2	2	no	—
144	M₃₂	—	2	2	no	—
145	M₃₂	—	2	2	no	—
146	M₃₂	—	2	2	no	—
147	M₃₂	—	2	2	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	4	1	no	—
152	M₃₂	—	4	1	no	—
153	M₃₂	C₂²⋊C₄	1	4	yes	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	—	4	1	no	—
164	C₂×C₁₆	—	4	1	no	—
165	C₂×C₁₆	—	4	1	no	—
166	C₂×C₁₆	—	4	1	no	—
167	C₂×C₁₆	—	4	1	no	—
168	C₂×C₁₆	—	4	1	no	—
169	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
170	#97	—	2	2	no	—
171	#97	—	2	2	no	—
172	#97	—	4	1	no	—
173	#97	—	4	1	no	—
174	#97	—	4	1	no	—
175	#97	—	4	1	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	4	1	no	—
179	C₄×C₈	—	4	1	no	—
180	C₄×C₈	—	4	1	no	—
181	C₄×C₈	—	4	1	no	—
182	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
187	C₂²×C₄	#117	1	1	yes	characteristic
188	C₂²×C₄	#117	1	1	yes	characteristic
189	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
194	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
195	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
196	C₂²×C₄	#131	1	1	yes	characteristic
197	C₂²×C₄	#131	1	1	yes	characteristic
198	C₂²×C₄	#131	1	1	yes	characteristic
199	C₂²×C₄	#131	1	1	yes	characteristic
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₈	#117	1	1	yes	characteristic
224	C₂×C₈	#117	1	1	yes	characteristic
225	C₂×C₈	#117	1	1	yes	characteristic
226	C₂×C₈	#117	1	1	yes	characteristic
227	C₂×C₈	#117	1	1	yes	characteristic
228	C₂×C₈	#117	1	1	yes	characteristic
229	C₂×C₈	C₄×D₄	1	1	yes	characteristic
230	C₂×C₈	C₄×D₄	1	1	yes	characteristic
231	C₂×C₈	C₄×D₄	1	1	yes	characteristic
232	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
233	C₂×C₈	#121	1	1	yes	characteristic
234	C₂×C₈	#121	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₈	#123	1	1	yes	characteristic
238	C₂×C₈	#123	1	1	yes	characteristic
239	C₂×C₈	#123	1	1	yes	characteristic
240	C₂×C₈	#123	1	1	yes	characteristic
241	C₂×C₈	#124	1	1	yes	characteristic
242	C₂×C₈	#124	1	1	yes	characteristic
243	C₂×C₈	#125	1	1	yes	characteristic
244	C₂×C₈	#125	1	1	yes	characteristic
245	C₂×C₈	#126	1	1	yes	characteristic
246	C₂×C₈	#126	1	1	yes	characteristic
247	C₂×C₈	#126	1	1	yes	characteristic
248	C₂×C₈	#126	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₁₆	—	2	4	no	—
268	C₁₆	—	2	4	no	—
269	C₁₆	—	4	2	no	—
270	C₁₆	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₄×M₁₆	1	1	yes	characteristic
273	C₂³	#432	1	1	yes	characteristic
274	C₂³	#432	1	1	yes	characteristic
275	C₂³	#435	1	1	yes	characteristic
276	C₂³	#435	1	1	yes	characteristic
277	C₂³	#436	1	1	yes	characteristic
278	C₂³	#446	1	1	yes	characteristic
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	#376	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
288	C₂×C₄	#387	1	1	yes	characteristic
289	C₂×C₄	#387	1	1	yes	characteristic
290	C₂×C₄	#388	1	1	yes	characteristic
291	C₂×C₄	#388	1	1	yes	characteristic
292	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
293	C₂×C₄	#405	1	1	yes	characteristic
294	C₂×C₄	#405	1	1	yes	characteristic
295	C₂×C₄	#432	1	1	yes	characteristic
296	C₂×C₄	#432	1	1	yes	characteristic
297	C₂×C₄	#433	1	1	yes	characteristic
298	C₂×C₄	#433	1	1	yes	characteristic
299	C₂×C₄	#433	1	1	yes	characteristic
300	C₂×C₄	#433	1	1	yes	characteristic
301	C₂×C₄	#435	1	1	yes	characteristic
302	C₂×C₄	#435	1	1	yes	characteristic
303	C₂×C₄	#435	1	1	yes	characteristic
304	C₂×C₄	#435	1	1	yes	characteristic
305	C₂×C₄	#435	1	1	yes	characteristic
306	C₂×C₄	#435	1	1	yes	characteristic
307	C₂×C₄	#436	1	1	yes	characteristic
308	C₂×C₄	#436	1	1	yes	characteristic
309	C₂×C₄	#436	1	1	yes	characteristic
310	C₂×C₄	#446	1	1	yes	characteristic
311	C₂×C₄	#446	1	1	yes	characteristic
312	C₂×C₄	#446	1	1	yes	characteristic
313	C₈	—	4	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	2	no	—
316	C₈	#380	1	1	yes	characteristic
317	C₈	#381	1	1	yes	characteristic
318	C₈	#382	1	1	yes	characteristic
319	C₈	#383	1	1	yes	characteristic
320	C₈	#387	1	1	yes	characteristic
321	C₈	#387	1	1	yes	characteristic
322	C₈	#388	1	1	yes	characteristic
323	C₈	#388	1	1	yes	characteristic
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	#1745	1	1	yes	characteristic
332	C₂²	#1753	1	1	yes	characteristic
333	C₂²	#1771	1	1	yes	characteristic
334	C₂²	#1917	1	1	yes	characteristic
335	C₂²	#1917	1	1	yes	characteristic
336	C₂²	#1924	1	1	yes	characteristic
337	C₂²	#1924	1	1	yes	characteristic
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	#1434	1	1	yes	characteristic
342	C₄	#1746	1	1	yes	characteristic
343	C₄	#1753	1	1	yes	characteristic
344	C₄	#1771	1	1	yes	characteristic
345	C₄	#1917	1	1	yes	characteristic
346	C₄	#1917	1	1	yes	characteristic
347	C₄	#1924	1	1	yes	characteristic
348	C₄	#1924	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	#10798	1	1	yes	characteristic
356	C₂	#13126	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #2110	C₄	semidirect
24. #1379	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2110	2
#12554	2
C₂×#1379	2
C₂×#1378	2
#12545	2
#10711	2