Subgroups — #149412

Order 512.

446 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	C₂×#2161	C₂	1	1	yes	maximal, characteristic
2	#12561	C₂	1	1	yes	maximal, characteristic
3	#12561	C₂	1	1	yes	maximal, characteristic
4	#12559	C₂	1	1	yes	maximal, characteristic
5	C₂×#1375	C₂	1	1	yes	maximal, characteristic
6	#10865	C₂	1	1	yes	maximal, characteristic
7	#10860	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	C₂²×#108	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂²×#97	C₂²	1	1	yes	characteristic
15	#2161	—	2	2	no	—
16	#2161	C₄	1	4	yes	complemented
17	C₂×#368	—	2	2	no	—
18	C₂×#341	—	2	1	no	—
19	C₂×#339	—	2	1	no	—
20	#1729	—	2	1	no	—
21	#1728	—	2	1	no	—
22	#1382	—	2	2	no	—
23	#1382	—	2	2	no	—
24	#1381	—	2	2	no	—
25	#1375	—	2	2	no	—
26	C₂²×M₁₆	—	2	1	no	—
27	C₂²×M₁₆	—	2	1	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	1	yes	characteristic
31	C₂×#117	—	2	1	no	—
32	C₂²×C₄²	D₄	1	1	yes	characteristic
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	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	C₂×C₄	1	2	yes	—
44	C₂×M₃₂	D₄	1	2	yes	—
45	C₂²×C₁₆	D₄	1	2	yes	—
46	C₂²×C₁₆	D₄	1	2	yes	—
47	C₂×#108	—	2	1	no	—
48	C₂×#108	—	2	1	no	—
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	—	2	1	no	—
58	C₂×#108	—	2	1	no	—
59	C₂×#108	—	2	1	no	—
60	C₂×#108	—	2	1	no	—
61	C₂×#108	—	2	1	no	—
62	C₂×#108	C₂×C₄	1	2	yes	—
63	C₂×#108	C₂×C₄	1	2	yes	—
64	C₂×#108	Q₈	1	2	yes	—
65	#427	—	2	1	no	—
66	#427	—	2	1	no	—
67	#427	—	2	1	no	—
68	#427	—	2	1	no	—
69	C₂×#107	Q₈	1	2	yes	—
70	C₂×#106	—	2	1	no	—
71	C₂×#97	—	2	1	no	—
72	C₂×#97	—	2	1	no	—
73	C₂×#97	—	2	1	no	—
74	C₂×#97	—	2	1	no	—
75	C₂×#97	—	2	1	no	—
76	C₂×#97	—	2	1	no	—
77	#368	—	2	8	no	—
78	#341	—	4	2	no	—
79	#339	—	4	2	no	—
80	#336	—	4	2	no	—
81	#336	—	4	2	no	—
82	C₂³×C₄	—	2	1	no	—
83	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	4	1	no	—
88	C₂×M₁₆	—	4	1	no	—
89	C₂×M₁₆	—	4	1	no	—
90	C₂×M₁₆	—	4	1	no	—
91	C₂×M₁₆	—	4	1	no	—
92	C₂×M₁₆	—	4	1	no	—
93	C₂×M₁₆	—	4	1	no	—
94	C₂×M₁₆	—	4	1	no	—
95	C₂×M₁₆	—	4	1	no	—
96	C₂×M₁₆	—	4	1	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	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	4	1	no	—
109	C₂²×C₈	—	4	1	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	D₈	1	2	yes	—
112	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	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×Q₈	1	1	yes	characteristic
117	C₂²×C₈	C₂×Q₈	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	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	C₂²×C₈	#41	1	1	yes	characteristic
126	#117	—	4	1	no	—
127	#117	—	4	1	no	—
128	#117	—	4	1	no	—
129	#117	—	4	1	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₂²⋊C₄	—	4	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	—	2	1	no	—
135	C₂×C₄²	—	2	1	no	—
136	C₂×C₄²	—	2	1	no	—
137	C₂×C₄²	—	4	1	no	—
138	C₂×C₄²	D₈	1	2	yes	—
139	M₃₂	—	2	2	no	—
140	M₃₂	—	2	2	no	—
141	M₃₂	—	2	2	no	—
142	M₃₂	—	2	2	no	—
143	M₃₂	—	4	1	no	—
144	M₃₂	—	4	1	no	—
145	M₃₂	—	4	1	no	—
146	M₃₂	—	4	1	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
153	#108	—	2	2	no	—
154	#108	—	2	2	no	—
155	#108	—	2	2	no	—
156	#108	—	2	2	no	—
157	#108	—	2	2	no	—
158	#108	—	2	2	no	—
159	#108	—	4	1	no	—
160	#108	—	4	1	no	—
161	#108	—	4	1	no	—
162	#108	—	4	1	no	—
163	#108	—	4	1	no	—
164	#108	—	4	1	no	—
165	#107	—	2	2	no	—
166	#107	—	2	2	no	—
167	#106	—	4	1	no	—
168	#106	—	4	1	no	—
169	#97	—	4	1	no	—
170	#97	—	4	1	no	—
171	#97	—	4	1	no	—
172	#97	—	4	1	no	—
173	C₂⁴	#137	1	1	yes	characteristic
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	8	2	no	—
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	#121	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
183	C₂²×C₄	#123	1	1	yes	characteristic
184	C₂²×C₄	#123	1	1	yes	characteristic
185	C₂²×C₄	#127	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
188	C₂²×C₄	#135	1	1	yes	characteristic
189	C₂²×C₄	#135	1	1	yes	characteristic
190	C₂²×C₄	#137	1	1	yes	characteristic
191	C₂²×C₄	#137	1	1	yes	characteristic
192	C₂²×C₄	#137	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	M₁₆	—	8	1	no	—
200	M₁₆	—	8	1	no	—
201	M₁₆	—	8	1	no	—
202	M₁₆	—	8	1	no	—
203	M₁₆	—	8	1	no	—
204	M₁₆	—	8	1	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	8	1	no	—
216	C₂×C₈	—	8	1	no	—
217	C₂×C₈	—	8	1	no	—
218	C₂×C₈	—	8	1	no	—
219	C₂×C₈	—	8	1	no	—
220	C₂×C₈	—	8	1	no	—
221	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₈	#117	1	1	yes	characteristic
223	C₂×C₈	#117	1	1	yes	characteristic
224	C₂×C₈	C₄×D₄	1	1	yes	characteristic
225	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
226	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
227	C₂×C₈	#121	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₈	#123	1	1	yes	characteristic
233	C₂×C₈	#123	1	1	yes	characteristic
234	C₂×C₈	#123	1	1	yes	characteristic
235	C₂×C₈	#123	1	1	yes	characteristic
236	C₂×C₈	#123	1	1	yes	characteristic
237	C₂×C₈	#123	1	1	yes	characteristic
238	C₂×C₈	#124	1	1	yes	characteristic
239	C₂×C₈	#124	1	1	yes	characteristic
240	C₂×C₈	#125	1	1	yes	characteristic
241	C₂×C₈	#125	1	1	yes	characteristic
242	C₂×C₈	#125	1	1	yes	characteristic
243	C₂×C₈	#125	1	1	yes	characteristic
244	C₂×C₈	#126	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	C₄⋊Q₈	1	1	yes	characteristic
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	8	1	no	—
252	C₂²⋊C₄	—	8	1	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	8	1	no	—
258	C₄²	—	8	1	no	—
259	C₁₆	—	2	4	no	—
260	C₁₆	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	#441	1	1	yes	characteristic
263	C₂³	#470	1	1	yes	characteristic
264	C₂³	#474	1	1	yes	characteristic
265	C₂³	#479	1	1	yes	characteristic
266	C₂³	#480	1	1	yes	characteristic
267	C₂³	#485	1	1	yes	characteristic
268	C₂³	#497	1	1	yes	characteristic
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	#387	1	1	yes	characteristic
276	C₂×C₄	#387	1	1	yes	characteristic
277	C₂×C₄	#390	1	1	yes	characteristic
278	C₂×C₄	#397	1	1	yes	characteristic
279	C₂×C₄	#398	1	1	yes	characteristic
280	C₂×C₄	#399	1	1	yes	characteristic
281	C₂×C₄	#399	1	1	yes	characteristic
282	C₂×C₄	C₄×D₈	1	1	yes	characteristic
283	C₂×C₄	C₄×D₈	1	1	yes	characteristic
284	C₂×C₄	#441	1	1	yes	characteristic
285	C₂×C₄	#466	1	1	yes	characteristic
286	C₂×C₄	#466	1	1	yes	characteristic
287	C₂×C₄	#470	1	1	yes	characteristic
288	C₂×C₄	#474	1	1	yes	characteristic
289	C₂×C₄	#474	1	1	yes	characteristic
290	C₂×C₄	#474	1	1	yes	characteristic
291	C₂×C₄	#479	1	1	yes	characteristic
292	C₂×C₄	#479	1	1	yes	characteristic
293	C₂×C₄	#479	1	1	yes	characteristic
294	C₂×C₄	#480	1	1	yes	characteristic
295	C₂×C₄	#480	1	1	yes	characteristic
296	C₂×C₄	#480	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	#493	1	1	yes	characteristic
301	C₂×C₄	#495	1	1	yes	characteristic
302	C₂×C₄	#497	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	8	2	no	—
305	C₈	—	8	2	no	—
306	C₈	—	8	2	no	—
307	C₈	#381	1	1	yes	characteristic
308	C₈	#387	1	1	yes	characteristic
309	C₈	#387	1	1	yes	characteristic
310	C₈	#391	1	1	yes	characteristic
311	C₈	#397	1	1	yes	characteristic
312	C₈	#398	1	1	yes	characteristic
313	C₈	#400	1	1	yes	characteristic
314	C₈	#400	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	#1918	1	1	yes	characteristic
323	C₂²	#1918	1	1	yes	characteristic
324	C₂²	#1971	1	1	yes	characteristic
325	C₂²	#2075	1	1	yes	characteristic
326	C₂²	#2086	1	1	yes	characteristic
327	C₂²	#2093	1	1	yes	characteristic
328	C₂²	#2097	1	1	yes	characteristic
329	C₄	—	8	2	no	—
330	C₄	—	16	2	no	—
331	C₄	#1445	1	1	yes	characteristic
332	C₄	#1918	1	1	yes	characteristic
333	C₄	#1918	1	1	yes	characteristic
334	C₄	#1970	1	1	yes	characteristic
335	C₄	#2075	1	1	yes	characteristic
336	C₄	#2086	1	1	yes	characteristic
337	C₄	#2093	1	1	yes	characteristic
338	C₄	#2097	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	#11379	1	1	yes	characteristic
346	C₂	#13532	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ 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. #2161	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2161	2
#12561	2
#12559	2
C₂×#1375	2
#10865	2
#10860	2