Subgroups — #147308

Order 512.

509 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2155	C₂	1	1	yes	maximal, characteristic
2	#20446	C₂	1	1	yes	maximal, characteristic
3	C₂×#1389	C₂	1	1	yes	maximal, characteristic
4	C₂×#1388	C₂	1	1	yes	maximal, characteristic
5	#12567	C₂	1	1	yes	maximal, characteristic
6	C₂×#1378	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×#427	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	#2873	—	2	1	no	—
15	C₂×#405	C₂²	1	1	yes	characteristic
16	#2155	—	2	2	no	—
17	#2155	C₄	1	4	yes	complemented
18	C₂×#365	—	2	2	no	—
19	#1773	—	2	2	no	—
20	#1772	—	2	2	no	—
21	C₂×#344	—	2	1	no	—
22	#1733	—	2	1	no	—
23	#1733	—	2	1	no	—
24	#1389	C₄	1	4	yes	complemented
25	#1388	—	2	2	no	—
26	#1386	—	2	2	no	—
27	#1378	C₄	1	4	yes	complemented
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	—	2	1	no	—
34	C₂×#117	D₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
45	C₂×M₃₂	C₂×C₄	1	2	yes	—
46	C₂×M₃₂	C₂×C₄	1	2	yes	—
47	C₂²×C₁₆	—	2	1	no	—
48	C₂²×C₁₆	—	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	C₂×C₄	1	2	yes	—
58	C₂×#108	C₂×C₄	1	2	yes	—
59	C₂×#108	D₄	1	2	yes	—
60	C₂×#108	Q₈	1	2	yes	—
61	#427	—	2	1	no	—
62	#427	—	2	1	no	—
63	#427	—	2	1	no	—
64	#427	—	2	1	no	—
65	#427	—	2	1	no	—
66	#427	C₂×C₄	1	2	yes	—
67	#427	C₂×C₄	1	2	yes	—
68	#427	D₄	1	2	yes	—
69	C₂×#107	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	C₂×#106	—	2	1	no	—
72	C₂×#106	D₄	1	2	yes	—
73	#405	—	2	1	no	—
74	#405	—	2	1	no	—
75	#405	—	2	4	no	—
76	#405	—	4	2	no	—
77	#405	C₂×C₄	1	2	yes	—
78	#405	Q₈	1	2	yes	—
79	C₄×M₁₆	—	4	2	no	—
80	C₂×#97	—	2	1	no	—
81	C₂×C₄×C₈	—	2	1	no	—
82	#365	—	2	8	no	—
83	#344	—	4	2	no	—
84	#339	—	2	4	no	—
85	#339	—	4	2	no	—
86	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	2	1	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	2	2	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	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
103	C₂×M₁₆	SD₁₆	1	2	yes	—
104	C₂×M₁₆	SD₁₆	1	2	yes	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	—	4	1	no	—
108	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Q₈	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	C₂²×C₈	#41	1	1	yes	characteristic
122	#117	—	2	1	no	—
123	#117	—	2	1	no	—
124	#117	—	2	2	no	—
125	#117	—	2	2	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	—	4	1	no	—
132	#117	—	4	1	no	—
133	#117	C₄⋊C₄	1	2	yes	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₂²⋊C₄	—	4	1	no	—
139	C₂×C₂²⋊C₄	—	4	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	—	4	1	no	—
142	C₂×C₄²	—	4	1	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	—	4	1	no	—
149	M₃₂	—	4	1	no	—
150	M₃₂	C₄²	1	4	yes	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	4	1	no	—
156	C₂×C₁₆	—	4	1	no	—
157	#108	—	2	2	no	—
158	#108	—	2	2	no	—
159	#108	—	2	2	no	—
160	#108	—	2	2	no	—
161	#108	—	2	2	no	—
162	#108	—	4	1	no	—
163	#108	—	4	1	no	—
164	#108	C₄⋊C₄	1	4	yes	—
165	#107	—	2	2	no	—
166	#107	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	—	4	1	no	—
169	#106	—	2	2	no	—
170	#106	—	2	2	no	—
171	#106	—	4	1	no	—
172	#106	C₂²⋊C₄	1	4	yes	—
173	#97	—	2	2	no	—
174	#97	—	2	2	no	—
175	#97	—	4	4	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	4	4	no	—
179	C₂⁴	#135	1	1	yes	characteristic
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	8	1	no	—
183	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
186	C₂²×C₄	#120	1	1	yes	characteristic
187	C₂²×C₄	#121	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
189	C₂²×C₄	#123	1	1	yes	characteristic
190	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
191	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
192	C₂²×C₄	#135	1	1	yes	characteristic
193	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂²×C₄	#137	1	1	yes	characteristic
196	C₂²×C₄	#137	1	1	yes	characteristic
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	8	1	no	—
207	M₁₆	—	8	1	no	—
208	M₁₆	#106	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	8	1	no	—
217	C₂×C₈	—	8	1	no	—
218	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₈	#117	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×D₄	1	1	yes	characteristic
226	C₂×C₈	C₄×D₄	1	1	yes	characteristic
227	C₂×C₈	C₄×D₄	1	1	yes	characteristic
228	C₂×C₈	C₄×D₄	1	1	yes	characteristic
229	C₂×C₈	C₄×D₄	1	1	yes	characteristic
230	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
231	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
232	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
233	C₂×C₈	C₄×Q₈	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	C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₈	#123	1	1	yes	characteristic
239	C₂×C₈	#123	1	1	yes	characteristic
240	C₂×C₈	#124	1	1	yes	characteristic
241	C₂×C₈	#125	1	1	yes	characteristic
242	C₂×C₈	#125	1	1	yes	characteristic
243	C₂×C₈	#126	1	1	yes	characteristic
244	C₂×C₈	#126	1	1	yes	characteristic
245	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	8	1	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	8	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	8	1	no	—
264	C₁₆	—	2	4	no	—
265	C₁₆	—	4	2	no	—
266	C₂³	—	8	1	no	—
267	C₂³	#427	1	1	yes	characteristic
268	C₂³	C₄×SD₁₆	1	1	yes	characteristic
269	C₂³	C₄×SD₁₆	1	1	yes	characteristic
270	C₂³	#452	1	1	yes	characteristic
271	C₂³	#463	1	1	yes	characteristic
272	C₂³	#479	1	1	yes	characteristic
273	C₂³	#482	1	1	yes	characteristic
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	16	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₄	#385	1	1	yes	characteristic
282	C₂×C₄	#385	1	1	yes	characteristic
283	C₂×C₄	#386	1	1	yes	characteristic
284	C₂×C₄	#387	1	1	yes	characteristic
285	C₂×C₄	#388	1	1	yes	characteristic
286	C₂×C₄	#389	1	1	yes	characteristic
287	C₂×C₄	C₂×#106	1	1	yes	characteristic
288	C₂×C₄	#428	1	1	yes	characteristic
289	C₂×C₄	#428	1	1	yes	characteristic
290	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
291	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
292	C₂×C₄	#440	1	1	yes	characteristic
293	C₂×C₄	#440	1	1	yes	characteristic
294	C₂×C₄	#441	1	1	yes	characteristic
295	C₂×C₄	#442	1	1	yes	characteristic
296	C₂×C₄	#448	1	1	yes	characteristic
297	C₂×C₄	#449	1	1	yes	characteristic
298	C₂×C₄	#450	1	1	yes	characteristic
299	C₂×C₄	#460	1	1	yes	characteristic
300	C₂×C₄	#461	1	1	yes	characteristic
301	C₂×C₄	#464	1	1	yes	characteristic
302	C₂×C₄	#475	1	1	yes	characteristic
303	C₂×C₄	#476	1	1	yes	characteristic
304	C₂×C₄	#478	1	1	yes	characteristic
305	C₂×C₄	#481	1	1	yes	characteristic
306	C₂×C₄	#483	1	1	yes	characteristic
307	C₂×C₄	#485	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	4	4	no	—
310	C₈	—	8	2	no	—
311	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
312	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
313	C₈	#387	1	1	yes	characteristic
314	C₈	#387	1	1	yes	characteristic
315	C₈	#387	1	1	yes	characteristic
316	C₈	#388	1	1	yes	characteristic
317	C₈	#389	1	1	yes	characteristic
318	C₈	#391	1	1	yes	characteristic
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	C₄×#106	1	1	yes	characteristic
327	C₂²	#1854	1	1	yes	characteristic
328	C₂²	#1865	1	1	yes	characteristic
329	C₂²	#1877	1	1	yes	characteristic
330	C₂²	#1920	1	1	yes	characteristic
331	C₂²	#1926	1	1	yes	characteristic
332	C₂²	#1949	1	1	yes	characteristic
333	C₄	—	8	2	no	—
334	C₄	—	8	4	no	—
335	C₄	—	16	2	no	—
336	C₄	#1436	1	1	yes	characteristic
337	C₄	#1776	1	1	yes	characteristic
338	C₄	#1853	1	1	yes	characteristic
339	C₄	#1864	1	1	yes	characteristic
340	C₄	#1880	1	1	yes	characteristic
341	C₄	#1919	1	1	yes	characteristic
342	C₄	#1927	1	1	yes	characteristic
343	C₄	#1948	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	#10908	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
prime radical	C₂⁴	#135
LCS	G ▷ C₂×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
17. #2155	C₄	semidirect
24. #1389	C₄	semidirect
27. #1378	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2155	2
#20446	2
C₂×#1389	2
C₂×#1388	2
#12567	2
C₂×#1378	2
#10872	2