Subgroups — #149132

Order 512.

421 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12567	C₂	1	1	yes	maximal, characteristic
2	#12567	C₂	1	1	yes	maximal, characteristic
3	C₂×#1384	C₂	1	1	yes	maximal, characteristic
4	C₂×#1384	C₂	1	1	yes	maximal, characteristic
5	#12551	C₂	1	1	yes	maximal, characteristic
6	#10871	C₂	1	1	yes	maximal, characteristic
7	#10871	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	#1740	—	2	1	no	—
16	#1740	—	2	1	no	—
17	#1733	—	2	1	no	—
18	#1733	—	2	1	no	—
19	#1386	—	2	2	no	—
20	#1386	C₄	1	4	yes	complemented
21	#1384	—	2	2	no	—
22	#1384	—	2	2	no	—
23	#1377	—	2	2	no	—
24	C₂²×M₁₆	D₄	1	1	yes	characteristic
25	C₂³×C₈	C₂³	1	1	yes	characteristic
26	C₂×#117	—	2	1	no	—
27	C₂×#117	—	2	1	no	—
28	C₂×#117	—	2	1	no	—
29	C₂×#117	—	2	1	no	—
30	C₂×#117	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	1	no	—
32	C₂×M₃₂	—	2	1	no	—
33	C₂×M₃₂	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
42	C₂×M₃₂	D₄	1	2	yes	—
43	C₂²×C₁₆	—	2	1	no	—
44	C₂²×C₁₆	—	2	1	no	—
45	#427	—	2	1	no	—
46	#427	—	2	1	no	—
47	#427	—	2	1	no	—
48	#427	—	2	1	no	—
49	#427	—	2	1	no	—
50	#427	—	2	1	no	—
51	#427	—	2	1	no	—
52	#427	—	2	1	no	—
53	#427	—	2	1	no	—
54	#427	—	2	1	no	—
55	#427	—	2	1	no	—
56	#427	—	2	1	no	—
57	#427	—	2	1	no	—
58	#427	—	2	1	no	—
59	#427	C₂×C₄	1	2	yes	—
60	#427	D₄	1	2	yes	—
61	C₂×#107	—	2	1	no	—
62	C₂×#107	—	2	1	no	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	—	2	1	no	—
65	C₂×#106	—	2	1	no	—
66	C₂×#106	—	2	1	no	—
67	C₂×#106	C₂×C₄	1	2	yes	—
68	C₂×#106	Q₈	1	2	yes	—
69	#405	—	2	1	no	—
70	#405	—	2	1	no	—
71	#405	—	2	1	no	—
72	#405	—	2	1	no	—
73	C₂×#97	—	2	1	no	—
74	C₂×C₄×C₈	—	2	1	no	—
75	#342	—	4	2	no	—
76	#342	—	4	2	no	—
77	#339	—	4	2	no	—
78	#339	—	4	2	no	—
79	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×M₁₆	—	2	1	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂×M₁₆	—	2	1	no	—
83	C₂×M₁₆	—	2	1	no	—
84	C₂×M₁₆	—	2	1	no	—
85	C₂×M₁₆	SD₁₆	1	2	yes	—
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₂²×C₈	#41	1	1	yes	characteristic
92	C₂²×C₈	#41	1	1	yes	characteristic
93	C₂²×C₈	#41	1	1	yes	characteristic
94	C₂²×C₈	#41	1	1	yes	characteristic
95	C₂²×C₈	#41	1	1	yes	characteristic
96	C₂²×C₈	#41	1	1	yes	characteristic
97	C₂²×C₈	#41	1	1	yes	characteristic
98	C₂²×C₈	#41	1	1	yes	characteristic
99	C₂²×C₈	#41	1	1	yes	characteristic
100	C₂²×C₈	#41	1	1	yes	characteristic
101	#117	—	2	1	no	—
102	#117	—	2	1	no	—
103	#117	—	2	1	no	—
104	#117	—	2	1	no	—
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	#117	—	4	1	no	—
108	#117	—	4	1	no	—
109	#117	—	4	1	no	—
110	#117	—	4	1	no	—
111	#117	—	4	1	no	—
112	#117	—	4	1	no	—
113	#117	—	4	1	no	—
114	#117	—	4	1	no	—
115	#117	—	4	1	no	—
116	#117	—	4	1	no	—
117	#117	—	4	1	no	—
118	#117	—	4	1	no	—
119	#117	—	4	1	no	—
120	#117	—	4	1	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	1	no	—
125	C₂×C₄⋊C₄	—	4	1	no	—
126	C₂×C₂²⋊C₄	—	4	1	no	—
127	C₂×C₂²⋊C₄	—	4	1	no	—
128	C₂×C₂²⋊C₄	—	4	1	no	—
129	C₂×C₂²⋊C₄	—	4	1	no	—
130	C₂×C₂²⋊C₄	D₈	1	2	yes	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	—	4	1	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	—	4	1	no	—
135	C₂×C₄²	—	4	1	no	—
136	M₃₂	—	2	2	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	2	no	—
139	M₃₂	—	2	2	no	—
140	M₃₂	—	4	1	no	—
141	M₃₂	—	4	1	no	—
142	M₃₂	—	4	1	no	—
143	M₃₂	—	4	1	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	4	1	no	—
147	C₂×C₁₆	—	4	1	no	—
148	C₂×C₁₆	—	4	1	no	—
149	C₂×C₁₆	—	4	1	no	—
150	#107	—	2	2	no	—
151	#107	—	2	2	no	—
152	#107	—	4	1	no	—
153	#107	—	4	1	no	—
154	#107	—	4	1	no	—
155	#107	—	4	1	no	—
156	#107	—	4	1	no	—
157	#107	—	4	1	no	—
158	#106	—	2	2	no	—
159	#106	—	2	2	no	—
160	#106	—	2	2	no	—
161	#106	—	2	2	no	—
162	#106	—	2	2	no	—
163	#106	—	2	2	no	—
164	#106	—	4	1	no	—
165	#106	—	4	1	no	—
166	#97	—	4	1	no	—
167	#97	—	4	1	no	—
168	C₄×C₈	—	4	1	no	—
169	C₄×C₈	—	4	1	no	—
170	C₂⁴	C₂×D₈	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	8	1	no	—
175	C₂²×C₄	—	8	1	no	—
176	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
177	C₂²×C₄	#121	1	1	yes	characteristic
178	C₂²×C₄	#121	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
180	C₂²×C₄	#123	1	1	yes	characteristic
181	C₂²×C₄	#123	1	1	yes	characteristic
182	C₂²×C₄	#124	1	1	yes	characteristic
183	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
184	C₂²×C₄	#135	1	1	yes	characteristic
185	C₂²×C₄	#135	1	1	yes	characteristic
186	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂²×C₄	#137	1	1	yes	characteristic
188	C₂²×C₄	#137	1	1	yes	characteristic
189	C₂²×C₄	#137	1	1	yes	characteristic
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	#117	1	1	yes	characteristic
198	C₂×C₈	#117	1	1	yes	characteristic
199	C₂×C₈	#117	1	1	yes	characteristic
200	C₂×C₈	C₄×D₄	1	1	yes	characteristic
201	C₂×C₈	C₄×D₄	1	1	yes	characteristic
202	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
203	C₂×C₈	#121	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂×C₈	#123	1	1	yes	characteristic
207	C₂×C₈	#123	1	1	yes	characteristic
208	C₂×C₈	#123	1	1	yes	characteristic
209	C₂×C₈	#123	1	1	yes	characteristic
210	C₂×C₈	#124	1	1	yes	characteristic
211	C₂×C₈	#124	1	1	yes	characteristic
212	C₂×C₈	#125	1	1	yes	characteristic
213	C₂×C₈	#125	1	1	yes	characteristic
214	C₂×C₈	#125	1	1	yes	characteristic
215	C₂×C₈	#126	1	1	yes	characteristic
216	C₂×C₈	#126	1	1	yes	characteristic
217	C₂×C₈	#126	1	1	yes	characteristic
218	C₂×C₈	#126	1	1	yes	characteristic
219	C₂×C₈	#126	1	1	yes	characteristic
220	C₂×C₈	#126	1	1	yes	characteristic
221	C₂×C₈	#126	1	1	yes	characteristic
222	C₂×C₈	#126	1	1	yes	characteristic
223	C₂×C₈	#126	1	1	yes	characteristic
224	C₂×C₈	#126	1	1	yes	characteristic
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	8	1	no	—
234	C₄⋊C₄	—	8	1	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	8	1	no	—
240	C₂²⋊C₄	—	8	1	no	—
241	C₂²⋊C₄	—	8	1	no	—
242	C₂²⋊C₄	—	8	1	no	—
243	C₂²⋊C₄	—	8	1	no	—
244	C₂²⋊C₄	—	8	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	8	1	no	—
250	C₄²	—	8	1	no	—
251	C₄²	—	8	1	no	—
252	C₄²	—	8	1	no	—
253	C₄²	—	8	1	no	—
254	C₄²	—	8	1	no	—
255	C₁₆	—	4	2	no	—
256	C₁₆	—	4	2	no	—
257	C₂³	—	8	1	no	—
258	C₂³	C₄×D₈	1	1	yes	characteristic
259	C₂³	#459	1	1	yes	characteristic
260	C₂³	#466	1	1	yes	characteristic
261	C₂³	#474	1	1	yes	characteristic
262	C₂³	#480	1	1	yes	characteristic
263	C₂³	#480	1	1	yes	characteristic
264	C₂³	#486	1	1	yes	characteristic
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	16	2	no	—
272	C₂×C₄	—	16	2	no	—
273	C₂×C₄	—	16	2	no	—
274	C₂×C₄	—	16	2	no	—
275	C₂×C₄	#387	1	1	yes	characteristic
276	C₂×C₄	#388	1	1	yes	characteristic
277	C₂×C₄	#388	1	1	yes	characteristic
278	C₂×C₄	#396	1	1	yes	characteristic
279	C₂×C₄	#397	1	1	yes	characteristic
280	C₂×C₄	#397	1	1	yes	characteristic
281	C₂×C₄	#400	1	1	yes	characteristic
282	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
283	C₂×C₄	#440	1	1	yes	characteristic
284	C₂×C₄	#441	1	1	yes	characteristic
285	C₂×C₄	#461	1	1	yes	characteristic
286	C₂×C₄	#464	1	1	yes	characteristic
287	C₂×C₄	#464	1	1	yes	characteristic
288	C₂×C₄	#465	1	1	yes	characteristic
289	C₂×C₄	#468	1	1	yes	characteristic
290	C₂×C₄	#470	1	1	yes	characteristic
291	C₂×C₄	#475	1	1	yes	characteristic
292	C₂×C₄	#478	1	1	yes	characteristic
293	C₂×C₄	#479	1	1	yes	characteristic
294	C₂×C₄	#481	1	1	yes	characteristic
295	C₂×C₄	#482	1	1	yes	characteristic
296	C₂×C₄	#483	1	1	yes	characteristic
297	C₂×C₄	#485	1	1	yes	characteristic
298	C₂×C₄	#485	1	1	yes	characteristic
299	C₂×C₄	#485	1	1	yes	characteristic
300	C₂×C₄	#488	1	1	yes	characteristic
301	C₂×C₄	#489	1	1	yes	characteristic
302	C₂×C₄	#491	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	#383	1	1	yes	characteristic
305	C₈	#387	1	1	yes	characteristic
306	C₈	#387	1	1	yes	characteristic
307	C₈	#388	1	1	yes	characteristic
308	C₈	#397	1	1	yes	characteristic
309	C₈	#400	1	1	yes	characteristic
310	C₈	#400	1	1	yes	characteristic
311	C₈	#401	1	1	yes	characteristic
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	#1918	1	1	yes	characteristic
320	C₂²	#1925	1	1	yes	characteristic
321	C₂²	#1934	1	1	yes	characteristic
322	C₂²	#2067	1	1	yes	characteristic
323	C₂²	#2071	1	1	yes	characteristic
324	C₂²	#2075	1	1	yes	characteristic
325	C₂²	#2099	1	1	yes	characteristic
326	C₄	—	8	2	no	—
327	C₄	—	16	2	no	—
328	C₄	—	16	2	no	—
329	C₄	—	16	2	no	—
330	C₄	—	16	2	no	—
331	C₄	#1443	1	1	yes	characteristic
332	C₄	#1919	1	1	yes	characteristic
333	C₄	#1927	1	1	yes	characteristic
334	C₄	#1935	1	1	yes	characteristic
335	C₄	#2066	1	1	yes	characteristic
336	C₄	#2074	1	1	yes	characteristic
337	C₄	#2077	1	1	yes	characteristic
338	C₄	#2101	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	#11315	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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. #1386	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12567	2
C₂×#1384	2
#12551	2
#10871	2