Subgroups — #147191

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33324	C₂	1	1	yes	maximal, characteristic
2	C₂×#1952	C₂	1	1	yes	maximal, characteristic
3	C₂×#1389	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	1	yes	maximal, characteristic
5	C₂×#1385	C₂	1	1	yes	maximal, characteristic
6	#12557	C₂	1	1	yes	maximal, characteristic
7	#10872	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂²×#108	C₂²	1	1	yes	characteristic
12	C₂×#428	—	2	1	no	—
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂²×#107	C₂²	1	1	yes	characteristic
15	C₂×#405	C₂²	1	1	yes	characteristic
16	#2108	—	2	2	no	—
17	#2108	—	2	2	no	—
18	C₄×M₃₂	—	2	2	no	—
19	#1952	—	2	2	no	—
20	#1952	C₄	1	4	yes	complemented
21	C₂×#344	—	2	1	no	—
22	C₂×#343	—	2	1	no	—
23	#1733	—	2	1	no	—
24	#1389	C₄	1	4	yes	complemented
25	#1387	—	2	2	no	—
26	#1385	C₄	1	4	yes	complemented
27	#1380	—	2	2	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂²×M₁₆	D₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	C₂×#117	—	2	1	no	—
33	C₂×#117	D₄	1	1	yes	characteristic
34	C₂²×C₄⋊C₄	—	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	C₂×C₄	1	2	yes	—
43	C₂×M₃₂	C₂×C₄	1	2	yes	—
44	C₂×M₃₂	D₄	1	2	yes	—
45	C₂×M₃₂	Q₈	1	2	yes	—
46	C₂×M₃₂	Q₈	1	2	yes	—
47	C₂²×C₁₆	—	2	1	no	—
48	C₂²×C₁₆	D₄	1	2	yes	—
49	C₂×#108	—	2	1	no	—
50	C₂×#108	—	2	1	no	—
51	C₂×#108	—	2	1	no	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	—	2	1	no	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	—	2	1	no	—
57	C₂×#108	C₂×C₄	1	2	yes	—
58	C₂×#108	C₂×C₄	1	2	yes	—
59	C₂×#108	C₂×C₄	1	2	yes	—
60	C₂×#108	C₂×C₄	1	2	yes	—
61	#428	—	2	4	no	—
62	#428	—	4	2	no	—
63	#427	—	2	1	no	—
64	#427	—	2	1	no	—
65	#427	—	2	1	no	—
66	#427	C₂×C₄	1	2	yes	—
67	C₂×#107	—	2	1	no	—
68	C₂×#107	—	2	1	no	—
69	C₂×#107	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	C₂×#107	—	2	1	no	—
72	C₂×#107	—	2	1	no	—
73	C₂×#107	—	4	1	no	—
74	C₂×#107	C₂×C₄	1	2	yes	—
75	C₂×#106	—	2	1	no	—
76	C₂×#106	—	4	1	no	—
77	#405	—	2	1	no	—
78	#405	—	2	1	no	—
79	#405	C₂×C₄	1	2	yes	—
80	#405	D₄	1	2	yes	—
81	C₂×#97	—	2	1	no	—
82	C₂×C₄×C₈	—	2	1	no	—
83	#346	—	2	4	no	—
84	C₄×C₁₆	—	2	4	no	—
85	#344	—	4	2	no	—
86	#343	—	2	4	no	—
87	#339	—	4	2	no	—
88	C₂³×C₄	—	4	1	no	—
89	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×M₁₆	—	2	1	no	—
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
106	C₂×M₁₆	SD₁₆	1	2	yes	—
107	C₂×M₁₆	Q₁₆	1	2	yes	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	4	1	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
118	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	C₂²×C₈	#41	1	1	yes	characteristic
121	C₂²×C₈	#41	1	1	yes	characteristic
122	C₂²×C₈	#41	1	1	yes	characteristic
123	C₂²×C₈	#41	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	#117	—	2	1	no	—
126	#117	—	2	1	no	—
127	#117	—	2	2	no	—
128	#117	—	4	1	no	—
129	#117	—	4	1	no	—
130	#117	—	4	1	no	—
131	#117	C₂²⋊C₄	1	2	yes	—
132	#117	D₈	1	2	yes	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
141	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	—	2	1	no	—
143	C₂×C₂²⋊C₄	—	4	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	4	1	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	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	C₄⋊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	C₄⋊C₄	1	4	yes	—
160	#108	—	2	2	no	—
161	#108	—	2	2	no	—
162	#108	—	2	2	no	—
163	#108	—	2	2	no	—
164	#108	—	2	2	no	—
165	#108	—	4	1	no	—
166	#108	—	4	1	no	—
167	#108	C₄²	1	4	yes	—
168	#107	—	2	2	no	—
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#107	—	4	1	no	—
172	#107	—	4	1	no	—
173	#107	—	4	1	no	—
174	#107	—	4	4	no	—
175	#106	—	2	2	no	—
176	#106	—	4	1	no	—
177	#106	—	4	4	no	—
178	#97	—	2	2	no	—
179	#97	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₂⁴	#135	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	8	1	no	—
187	C₂²×C₄	—	8	1	no	—
188	C₂²×C₄	—	8	1	no	—
189	C₂²×C₄	—	16	2	no	—
190	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
193	C₂²×C₄	#121	1	1	yes	characteristic
194	C₂²×C₄	#121	1	1	yes	characteristic
195	C₂²×C₄	#124	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
201	C₂²×C₄	#135	1	1	yes	characteristic
202	C₂²×C₄	#135	1	1	yes	characteristic
203	C₂²×C₄	#135	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	8	1	no	—
214	M₁₆	—	8	1	no	—
215	M₁₆	#103	1	4	yes	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	8	1	no	—
224	C₂×C₈	—	8	1	no	—
225	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₈	#117	1	1	yes	characteristic
229	C₂×C₈	#117	1	1	yes	characteristic
230	C₂×C₈	#117	1	1	yes	characteristic
231	C₂×C₈	C₄×D₄	1	1	yes	characteristic
232	C₂×C₈	C₄×D₄	1	1	yes	characteristic
233	C₂×C₈	C₄×D₄	1	1	yes	characteristic
234	C₂×C₈	C₄×D₄	1	1	yes	characteristic
235	C₂×C₈	C₄×D₄	1	1	yes	characteristic
236	C₂×C₈	C₄×D₄	1	1	yes	characteristic
237	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
238	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
239	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
240	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
241	C₂×C₈	#120	1	1	yes	characteristic
242	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₈	#123	1	1	yes	characteristic
247	C₂×C₈	#123	1	1	yes	characteristic
248	C₂×C₈	#123	1	1	yes	characteristic
249	C₂×C₈	#125	1	1	yes	characteristic
250	C₂×C₈	#125	1	1	yes	characteristic
251	C₂×C₈	#126	1	1	yes	characteristic
252	C₂×C₈	#126	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	8	1	no	—
259	C₄⋊C₄	—	8	1	no	—
260	C₄⋊C₄	—	8	1	no	—
261	C₄⋊C₄	#102	1	2	yes	—
262	C₄⋊C₄	#102	1	2	yes	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	8	1	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	8	1	no	—
271	C₁₆	—	2	4	no	—
272	C₁₆	—	2	4	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	#416	1	1	yes	characteristic
276	C₂³	C₄×SD₁₆	1	1	yes	characteristic
277	C₂³	C₄×Q₁₆	1	1	yes	characteristic
278	C₂³	#465	1	1	yes	characteristic
279	C₂³	#466	1	1	yes	characteristic
280	C₂³	#488	1	1	yes	characteristic
281	C₂³	#499	1	1	yes	characteristic
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	16	1	no	—
289	C₂×C₄	—	16	1	no	—
290	C₂×C₄	—	16	2	no	—
291	C₂×C₄	—	16	2	no	—
292	C₂×C₄	—	16	2	no	—
293	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
294	C₂×C₄	#386	1	1	yes	characteristic
295	C₂×C₄	#388	1	1	yes	characteristic
296	C₂×C₄	#388	1	1	yes	characteristic
297	C₂×C₄	#389	1	1	yes	characteristic
298	C₂×C₄	#389	1	1	yes	characteristic
299	C₂×C₄	#391	1	1	yes	characteristic
300	C₂×C₄	C₂×#102	1	1	yes	characteristic
301	C₂×C₄	C₂×#103	1	1	yes	characteristic
302	C₂×C₄	#416	1	1	yes	characteristic
303	C₂×C₄	C₄×D₈	1	1	yes	characteristic
304	C₂×C₄	C₄×D₈	1	1	yes	characteristic
305	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
306	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
307	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
308	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
309	C₂×C₄	#465	1	1	yes	characteristic
310	C₂×C₄	#465	1	1	yes	characteristic
311	C₂×C₄	#465	1	1	yes	characteristic
312	C₂×C₄	#466	1	1	yes	characteristic
313	C₂×C₄	#467	1	1	yes	characteristic
314	C₂×C₄	#467	1	1	yes	characteristic
315	C₂×C₄	#486	1	1	yes	characteristic
316	C₂×C₄	#487	1	1	yes	characteristic
317	C₂×C₄	#488	1	1	yes	characteristic
318	C₂×C₄	#498	1	1	yes	characteristic
319	C₂×C₄	#499	1	1	yes	characteristic
320	C₂×C₄	#500	1	1	yes	characteristic
321	C₈	—	2	4	no	—
322	C₈	—	4	4	no	—
323	C₈	—	8	2	no	—
324	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
325	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
326	C₈	#385	1	1	yes	characteristic
327	C₈	#385	1	1	yes	characteristic
328	C₈	#387	1	1	yes	characteristic
329	C₈	#387	1	1	yes	characteristic
330	C₈	#387	1	1	yes	characteristic
331	C₈	#387	1	1	yes	characteristic
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	C₄×#103	1	1	yes	characteristic
343	C₂²	#1893	1	1	yes	characteristic
344	C₂²	#1933	1	1	yes	characteristic
345	C₂²	#1934	1	1	yes	characteristic
346	C₂²	#1941	1	1	yes	characteristic
347	C₂²	#1942	1	1	yes	characteristic
348	C₂²	#1991	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	16	2	no	—
352	C₄	—	16	2	no	—
353	C₄	#1436	1	1	yes	characteristic
354	C₄	C₄×#102	1	1	yes	characteristic
355	C₄	#1892	1	1	yes	characteristic
356	C₄	#1933	1	1	yes	characteristic
357	C₄	#1934	1	1	yes	characteristic
358	C₄	#1941	1	1	yes	characteristic
359	C₄	#1942	1	1	yes	characteristic
360	C₄	#1990	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	#10880	1	1	yes	characteristic
368	C₂	#13068	1	1	yes	characteristic
369	C₂	#13069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
20. #1952	C₄	semidirect
24. #1389	C₄	semidirect
26. #1385	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33324	2
C₂×#1952	2
C₂×#1389	2
C₂×#1387	2
C₂×#1385	2
#12557	2
#10872	2