Subgroups — #147000

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	#20797	C₂	1	1	yes	maximal, characteristic
4	C₂×#1389	C₂	1	4	yes	maximal
5	C₂²×M₃₂	C₂²	1	2	yes	—
6	C₂²×#108	—	2	1	no	—
7	C₂²×#108	C₂²	1	2	yes	—
8	C₂×#427	—	2	1	no	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#423	—	2	1	no	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#2285	—	2	1	no	—
13	C₄×M₃₂	—	2	1	no	—
14	C₄×M₃₂	—	2	1	no	—
15	C₄×M₃₂	C₄	1	2	yes	complemented
16	C₄×M₃₂	C₄	1	2	yes	complemented
17	C₂×#346	—	2	1	no	—
18	C₂×C₄×C₁₆	—	2	1	no	—
19	#1848	—	2	1	no	—
20	#1848	—	2	1	no	—
21	#1848	C₄	1	2	yes	complemented
22	#1848	C₄	1	2	yes	complemented
23	#1846	—	2	1	no	—
24	#1846	C₄	1	2	yes	—
25	#1845	—	2	1	no	—
26	#1844	C₄	1	2	yes	—
27	#1389	—	2	8	no	—
28	C₂²×M₁₆	—	2	2	no	—
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	1	yes	characteristic
31	C₂×#117	—	2	2	no	—
32	C₂²×C₄²	D₄	1	1	yes	characteristic
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×M₃₂	C₂×C₄	1	4	yes	—
37	C₂×M₃₂	D₄	1	4	yes	—
38	C₂×M₃₂	Q₈	1	4	yes	—
39	C₂²×C₁₆	—	2	2	no	—
40	C₂×#108	—	2	4	no	—
41	C₂×#108	—	2	4	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	4	2	no	—
46	C₂×#108	—	4	2	no	—
47	C₂×#108	C₂×C₄	1	4	yes	—
48	#427	—	2	4	no	—
49	#427	—	2	4	no	—
50	#427	—	4	2	no	—
51	#427	—	4	2	no	—
52	C₂×#107	—	2	2	no	—
53	C₂×#107	—	2	2	no	—
54	C₂×#106	—	2	2	no	—
55	C₂×#106	C₂×C₄	1	4	yes	—
56	#423	—	2	2	no	—
57	#423	—	2	2	no	—
58	#423	—	4	2	no	—
59	C₂×C₄⋊C₈	—	4	2	no	—
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
72	#384	—	2	2	no	—
73	#382	—	2	2	no	—
74	#380	—	4	2	no	—
75	C₄×C₄⋊C₄	—	4	2	no	—
76	#346	—	2	2	no	—
77	#346	—	2	2	no	—
78	C₄×C₁₆	—	2	2	no	—
79	C₄×C₁₆	—	2	2	no	—
80	C₂³×C₄	SD₁₆	1	2	yes	—
81	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×M₁₆	—	2	4	no	—
83	C₂×M₁₆	—	4	2	no	—
84	C₂×M₁₆	—	4	2	no	—
85	C₂×M₁₆	—	4	2	no	—
86	C₂×M₁₆	—	4	2	no	—
87	C₂×M₁₆	—	4	2	no	—
88	C₂²×C₈	—	4	2	no	—
89	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	D₈	1	2	yes	—
96	C₂²×C₈	SD₁₆	1	2	yes	—
97	C₂²×C₈	SD₁₆	1	2	yes	—
98	C₂²×C₈	Q₁₆	1	2	yes	—
99	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₈	C₂×D₄	1	2	yes	—
105	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₈	C₂×Q₈	1	2	yes	—
107	C₂²×C₈	#41	1	2	yes	—
108	C₂²×C₈	#41	1	2	yes	—
109	#117	—	4	2	no	—
110	#117	—	4	2	no	—
111	#117	—	4	2	no	—
112	#117	—	4	2	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	4	4	no	—
115	C₂×C₂²⋊C₄	—	4	2	no	—
116	C₂×C₄²	—	4	2	no	—
117	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄²	D₈	1	2	yes	—
124	C₂×C₄²	SD₁₆	1	2	yes	—
125	C₂×C₄²	Q₁₆	1	2	yes	—
126	M₃₂	—	2	4	no	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	C₄⋊C₄	1	8	yes	complemented
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	—	2	4	no	—
133	#108	—	2	8	no	—
134	#108	—	4	4	no	—
135	#108	—	4	4	no	—
136	#108	—	4	4	no	—
137	#107	—	4	4	no	—
138	#107	—	4	4	no	—
139	#106	—	2	8	no	—
140	#106	—	4	4	no	—
141	C₄⋊C₈	—	4	4	no	—
142	C₄⋊C₈	—	4	4	no	—
143	C₄×C₈	C₄²	1	1	yes	characteristic
144	C₄×C₈	C₄²	1	1	yes	characteristic
145	C₄×C₈	C₄²	1	1	yes	characteristic
146	C₄×C₈	C₄²	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
159	#95	—	4	4	no	—
160	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	#102	1	2	yes	—
164	C₂²×C₄	#102	1	2	yes	—
165	C₂²×C₄	#103	1	2	yes	—
166	C₂²×C₄	#103	1	2	yes	—
167	C₂²×C₄	#106	1	2	yes	—
168	C₂²×C₄	#106	1	2	yes	—
169	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	#124	1	1	yes	characteristic
173	C₂²×C₄	#124	1	1	yes	characteristic
174	C₂²×C₄	#127	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
178	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
183	M₁₆	—	4	4	no	—
184	M₁₆	—	4	4	no	—
185	M₁₆	—	8	2	no	—
186	M₁₆	—	8	2	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	8	2	no	—
189	C₂×C₈	—	8	2	no	—
190	C₂×C₈	#95	1	1	yes	characteristic
191	C₂×C₈	#95	1	1	yes	characteristic
192	C₂×C₈	#95	1	1	yes	characteristic
193	C₂×C₈	#95	1	1	yes	characteristic
194	C₂×C₈	#102	1	2	yes	—
195	C₂×C₈	#102	1	2	yes	—
196	C₂×C₈	#102	1	2	yes	—
197	C₂×C₈	#102	1	2	yes	—
198	C₂×C₈	#103	1	2	yes	—
199	C₂×C₈	#103	1	2	yes	—
200	C₂×C₈	#103	1	2	yes	—
201	C₂×C₈	#103	1	2	yes	—
202	C₂×C₈	#106	1	2	yes	—
203	C₂×C₈	#106	1	2	yes	—
204	C₂×C₈	#107	1	2	yes	—
205	C₂×C₈	#107	1	2	yes	—
206	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
209	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₈	#117	1	2	yes	—
212	C₂×C₈	C₄×D₄	1	2	yes	—
213	C₂×C₈	C₄×D₄	1	2	yes	—
214	C₂×C₈	C₄×Q₈	1	2	yes	—
215	C₂×C₈	C₄×Q₈	1	2	yes	—
216	C₂×C₈	#121	1	2	yes	—
217	C₂×C₈	#121	1	2	yes	—
218	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
219	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
220	C₂×C₈	#125	1	2	yes	—
221	C₂×C₈	C₄⋊Q₈	1	2	yes	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	8	8	no	—
225	C₂²⋊C₄	—	8	2	no	—
226	C₂²⋊C₄	—	8	2	no	—
227	C₄²	—	8	2	no	—
228	C₄²	—	8	2	no	—
229	C₄²	#95	1	1	yes	characteristic
230	C₄²	#95	1	1	yes	characteristic
231	C₄²	#95	1	1	yes	characteristic
232	C₄²	#95	1	1	yes	characteristic
233	C₄²	#102	1	2	yes	—
234	C₄²	#102	1	2	yes	—
235	C₄²	#103	1	2	yes	—
236	C₄²	#103	1	2	yes	—
237	C₄²	#107	1	2	yes	—
238	C₄²	#107	1	2	yes	—
239	C₁₆	—	2	8	no	—
240	C₂³	—	4	2	no	—
241	C₂³	C₂×#102	1	1	yes	characteristic
242	C₂³	C₂×#103	1	1	yes	characteristic
243	C₂³	C₂×#106	1	1	yes	characteristic
244	C₂³	#486	1	1	yes	characteristic
245	C₂³	#487	1	1	yes	characteristic
246	C₂³	#492	1	1	yes	characteristic
247	C₂³	#498	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	#340	1	2	yes	—
257	C₂×C₄	#340	1	2	yes	—
258	C₂×C₄	#340	1	2	yes	—
259	C₂×C₄	#340	1	2	yes	—
260	C₂×C₄	C₂×#95	1	1	yes	characteristic
261	C₂×C₄	#382	1	1	yes	characteristic
262	C₂×C₄	#384	1	1	yes	characteristic
263	C₂×C₄	#390	1	1	yes	characteristic
264	C₂×C₄	#390	1	1	yes	characteristic
265	C₂×C₄	#391	1	1	yes	characteristic
266	C₂×C₄	#391	1	1	yes	characteristic
267	C₂×C₄	C₂×#102	1	1	yes	characteristic
268	C₂×C₄	C₂×#102	1	1	yes	characteristic
269	C₂×C₄	C₂×#102	1	1	yes	characteristic
270	C₂×C₄	C₂×#103	1	1	yes	characteristic
271	C₂×C₄	C₂×#103	1	1	yes	characteristic
272	C₂×C₄	C₂×#103	1	1	yes	characteristic
273	C₂×C₄	C₂×#106	1	1	yes	characteristic
274	C₂×C₄	C₂×#107	1	1	yes	characteristic
275	C₂×C₄	C₂×#107	1	1	yes	characteristic
276	C₂×C₄	#486	1	1	yes	characteristic
277	C₂×C₄	#486	1	1	yes	characteristic
278	C₂×C₄	#486	1	1	yes	characteristic
279	C₂×C₄	#487	1	1	yes	characteristic
280	C₂×C₄	#487	1	1	yes	characteristic
281	C₂×C₄	#487	1	1	yes	characteristic
282	C₂×C₄	#492	1	1	yes	characteristic
283	C₂×C₄	#493	1	1	yes	characteristic
284	C₂×C₄	#494	1	1	yes	characteristic
285	C₂×C₄	#498	1	1	yes	characteristic
286	C₂×C₄	#500	1	1	yes	characteristic
287	C₂×C₄	#500	1	1	yes	characteristic
288	C₈	—	8	4	no	—
289	C₈	#340	1	2	yes	—
290	C₈	#340	1	2	yes	—
291	C₈	#340	1	2	yes	—
292	C₈	#340	1	2	yes	—
293	C₈	C₄×C₄⋊C₄	1	2	yes	—
294	C₈	#380	1	2	yes	—
295	C₈	#389	1	2	yes	—
296	C₈	#389	1	2	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	C₂×#340	1	1	yes	characteristic
305	C₂²	#1834	1	1	yes	characteristic
306	C₂²	#1844	1	1	yes	characteristic
307	C₂²	#1956	1	1	yes	characteristic
308	C₂²	#1957	1	1	yes	characteristic
309	C₂²	#1990	1	1	yes	characteristic
310	C₂²	#1990	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	16	4	no	—
315	C₄	#1389	1	4	yes	complemented
316	C₄	C₄⋊#95	1	1	yes	characteristic
317	C₄	C₂×#340	1	1	yes	characteristic
318	C₄	#1834	1	1	yes	characteristic
319	C₄	#1845	1	1	yes	characteristic
320	C₄	#1956	1	1	yes	characteristic
321	C₄	#1957	1	1	yes	characteristic
322	C₄	#1990	1	1	yes	characteristic
323	C₄	#1990	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	#10841	1	1	yes	characteristic
331	C₂	C₂×#1389	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×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
15. C₄×M₃₂	C₄	semidirect
16. C₄×M₃₂	C₄	semidirect
21. #1848	C₄	semidirect
22. #1848	C₄	semidirect
129. M₃₂	C₄⋊C₄	semidirect
315. C₄	#1389	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×#1848	2
#20797	2
C₂×#1389	2