Subgroups — #147050

Order 512.

596 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	C₂²×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#1389	C₂	1	2	yes	maximal
5	#12559	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×#108	—	2	1	no	—
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂²×#108	C₂²	1	1	yes	characteristic
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂×#423	—	2	1	no	—
12	C₂×#405	—	2	1	no	—
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×#364	—	2	2	no	—
15	C₂×#364	—	2	2	no	—
16	C₂×#364	C₄	1	2	yes	complemented
17	C₂×#364	C₄	1	2	yes	complemented
18	#1848	—	2	2	no	—
19	#1848	C₄	1	4	yes	complemented
20	#1772	—	2	2	no	—
21	#1772	C₄	1	4	yes	—
22	C₂×#339	—	2	1	no	—
23	#1389	—	2	4	no	—
24	#1381	—	2	4	no	—
25	C₂²×M₁₆	—	2	1	no	—
26	C₂²×M₁₆	—	2	2	no	—
27	C₂³×C₈	D₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	—	2	1	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
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	C₂²×C₁₆	—	2	2	no	—
38	C₂×#108	—	2	1	no	—
39	C₂×#108	—	2	1	no	—
40	C₂×#108	—	2	1	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	4	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	—	4	1	no	—
48	C₂×#108	—	4	1	no	—
49	C₂×#108	—	4	2	no	—
50	C₂×#108	C₂×C₄	1	4	yes	—
51	C₂×#108	C₂×C₄	1	2	yes	—
52	C₂×#108	D₄	1	2	yes	—
53	C₂×#108	Q₈	1	2	yes	—
54	#427	—	2	1	no	—
55	#427	—	2	1	no	—
56	#427	C₂×C₄	1	2	yes	—
57	#427	D₄	1	2	yes	—
58	C₂×#107	—	2	1	no	—
59	C₂×#106	Q₈	1	2	yes	—
60	#423	—	2	4	no	—
61	#423	—	4	2	no	—
62	C₂×C₄⋊C₈	—	4	2	no	—
63	#405	—	2	4	no	—
64	#405	—	4	2	no	—
65	C₂×#97	—	4	1	no	—
66	C₂×C₄×C₈	—	4	1	no	—
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
70	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
71	C₂×C₄×C₈	D₄	1	2	yes	—
72	C₂×C₄×C₈	D₄	1	2	yes	—
73	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
75	#364	—	2	4	no	—
76	#364	—	2	4	no	—
77	#339	—	2	4	no	—
78	C₂³×C₄	—	2	1	no	—
79	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×M₁₆	—	2	2	no	—
81	C₂×M₁₆	—	2	2	no	—
82	C₂×M₁₆	—	2	2	no	—
83	C₂×M₁₆	—	2	4	no	—
84	C₂×M₁₆	—	4	1	no	—
85	C₂×M₁₆	—	4	1	no	—
86	C₂×M₁₆	—	4	1	no	—
87	C₂×M₁₆	—	4	2	no	—
88	C₂×M₁₆	—	4	2	no	—
89	C₂×M₁₆	—	4	2	no	—
90	C₂×M₁₆	—	4	2	no	—
91	C₂×M₁₆	—	4	2	no	—
92	C₂²×C₈	—	4	1	no	—
93	C₂²×C₈	—	4	2	no	—
94	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
95	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
96	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	D₈	1	1	yes	characteristic
99	C₂²×C₈	D₈	1	1	yes	characteristic
100	C₂²×C₈	SD₁₆	1	2	yes	—
101	C₂²×C₈	SD₁₆	1	2	yes	—
102	C₂²×C₈	Q₁₆	1	1	yes	characteristic
103	C₂²×C₈	Q₁₆	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
113	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	C₂²×C₈	#41	1	1	yes	characteristic
117	C₂²×C₈	#41	1	1	yes	characteristic
118	#117	—	2	2	no	—
119	#117	—	2	2	no	—
120	#117	—	4	1	no	—
121	#117	—	4	1	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₂²⋊C₄	—	4	1	no	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	4	1	no	—
128	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
129	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
130	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
132	M₃₂	—	2	4	no	—
133	M₃₂	—	2	4	no	—
134	M₃₂	—	4	2	no	—
135	M₃₂	—	4	2	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	—	4	2	no	—
138	C₂×C₁₆	—	4	2	no	—
139	#108	—	2	2	no	—
140	#108	—	2	4	no	—
141	#108	—	2	4	no	—
142	#108	—	2	2	no	—
143	#108	—	2	2	no	—
144	#108	—	4	4	no	—
145	#108	—	4	4	no	—
146	#108	—	4	2	no	—
147	#108	—	4	2	no	—
148	#108	C₄⋊C₄	1	4	yes	—
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	#106	—	2	2	no	—
152	#106	C₄⋊C₄	1	4	yes	—
153	C₄⋊C₈	—	4	8	no	—
154	#97	—	4	4	no	—
155	C₄×C₈	—	4	4	no	—
156	C₄×C₈	C₄²	1	2	yes	—
157	C₄×C₈	C₄²	1	2	yes	—
158	C₄×C₈	C₂²⋊C₄	1	2	yes	—
159	C₄×C₈	C₂²⋊C₄	1	2	yes	—
160	C₄×C₈	C₄⋊C₄	1	2	yes	—
161	C₄×C₈	C₄⋊C₄	1	2	yes	—
162	C₄×C₈	C₄⋊C₄	1	2	yes	—
163	C₄×C₈	C₄⋊C₄	1	2	yes	—
164	C₂⁴	#137	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	#121	1	1	yes	characteristic
175	C₂²×C₄	#121	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₈	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	AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂²×C₄	#137	1	1	yes	characteristic
185	M₁₆	—	2	4	no	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	4	no	—
190	M₁₆	—	4	4	no	—
191	M₁₆	—	8	2	no	—
192	M₁₆	—	8	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	8	2	no	—
198	C₂×C₈	—	8	2	no	—
199	C₂×C₈	#95	1	2	yes	—
200	C₂×C₈	#95	1	2	yes	—
201	C₂×C₈	#102	1	2	yes	—
202	C₂×C₈	#102	1	2	yes	—
203	C₂×C₈	#102	1	2	yes	—
204	C₂×C₈	#102	1	2	yes	—
205	C₂×C₈	#103	1	2	yes	—
206	C₂×C₈	#103	1	2	yes	—
207	C₂×C₈	#103	1	2	yes	—
208	C₂×C₈	#103	1	2	yes	—
209	C₂×C₈	#106	1	2	yes	—
210	C₂×C₈	#106	1	2	yes	—
211	C₂×C₈	#107	1	1	yes	characteristic
212	C₂×C₈	#107	1	1	yes	characteristic
213	C₂×C₈	#107	1	1	yes	characteristic
214	C₂×C₈	#107	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
216	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄⋊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	C₄×D₄	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×Q₈	1	1	yes	characteristic
228	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
229	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
230	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
231	C₂×C₈	#121	1	1	yes	characteristic
232	C₂×C₈	#121	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₈	#124	1	1	yes	characteristic
236	C₂×C₈	#124	1	1	yes	characteristic
237	C₂×C₈	#125	1	1	yes	characteristic
238	C₂×C₈	#125	1	1	yes	characteristic
239	C₂×C₈	#127	1	1	yes	characteristic
240	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
241	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
242	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	4	2	no	—
252	C₄²	—	4	2	no	—
253	C₄²	#95	1	2	yes	—
254	C₄²	#95	1	2	yes	—
255	C₁₆	—	4	4	no	—
256	C₂³	—	4	2	no	—
257	C₂³	#417	1	1	yes	characteristic
258	C₂³	#419	1	1	yes	characteristic
259	C₂³	#428	1	1	yes	characteristic
260	C₂³	#461	1	1	yes	characteristic
261	C₂³	#462	1	1	yes	characteristic
262	C₂³	#474	1	1	yes	characteristic
263	C₂³	#475	1	1	yes	characteristic
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	C₂×#95	1	1	yes	characteristic
273	C₂×C₄	#380	1	1	yes	characteristic
274	C₂×C₄	#380	1	1	yes	characteristic
275	C₂×C₄	#389	1	1	yes	characteristic
276	C₂×C₄	#389	1	1	yes	characteristic
277	C₂×C₄	#390	1	1	yes	characteristic
278	C₂×C₄	#391	1	1	yes	characteristic
279	C₂×C₄	C₂×#102	1	1	yes	characteristic
280	C₂×C₄	C₂×#102	1	1	yes	characteristic
281	C₂×C₄	C₂×#103	1	1	yes	characteristic
282	C₂×C₄	C₂×#103	1	1	yes	characteristic
283	C₂×C₄	#417	1	1	yes	characteristic
284	C₂×C₄	#418	1	1	yes	characteristic
285	C₂×C₄	C₂×#106	1	1	yes	characteristic
286	C₂×C₄	C₂×#107	1	1	yes	characteristic
287	C₂×C₄	#428	1	1	yes	characteristic
288	C₂×C₄	#459	1	1	yes	characteristic
289	C₂×C₄	#459	1	1	yes	characteristic
290	C₂×C₄	#460	1	1	yes	characteristic
291	C₂×C₄	#460	1	1	yes	characteristic
292	C₂×C₄	#461	1	1	yes	characteristic
293	C₂×C₄	#462	1	1	yes	characteristic
294	C₂×C₄	#474	1	1	yes	characteristic
295	C₂×C₄	#475	1	1	yes	characteristic
296	C₂×C₄	#476	1	1	yes	characteristic
297	C₂×C₄	#476	1	1	yes	characteristic
298	C₂×C₄	#477	1	1	yes	characteristic
299	C₂×C₄	#477	1	1	yes	characteristic
300	C₈	—	4	4	no	—
301	C₈	—	8	4	no	—
302	C₈	#340	1	2	yes	—
303	C₈	#340	1	2	yes	—
304	C₈	#340	1	2	yes	—
305	C₈	#340	1	2	yes	—
306	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
307	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
308	C₈	#382	1	1	yes	characteristic
309	C₈	#384	1	1	yes	characteristic
310	C₈	#389	1	1	yes	characteristic
311	C₈	#389	1	1	yes	characteristic
312	C₈	#390	1	1	yes	characteristic
313	C₈	#391	1	1	yes	characteristic
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	#1736	1	1	yes	characteristic
322	C₂²	#1802	1	1	yes	characteristic
323	C₂²	#1803	1	1	yes	characteristic
324	C₂²	#1947	1	1	yes	characteristic
325	C₂²	#1948	1	1	yes	characteristic
326	C₂²	#1981	1	1	yes	characteristic
327	C₂²	#1988	1	1	yes	characteristic
328	C₄	—	2	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	8	4	no	—
331	C₄	C₄⋊#95	1	1	yes	characteristic
332	C₄	C₂×#340	1	1	yes	characteristic
333	C₄	#1802	1	1	yes	characteristic
334	C₄	#1803	1	1	yes	characteristic
335	C₄	#1947	1	1	yes	characteristic
336	C₄	#1948	1	1	yes	characteristic
337	C₄	#1982	1	1	yes	characteristic
338	C₄	#1987	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	#10849	1	1	yes	characteristic
346	C₂	#13182	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1947
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ 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
16. C₂×#364	C₄	semidirect
17. C₂×#364	C₄	semidirect
19. #1848	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#364	2
C₂×#1848	2
C₂×#1772	2
C₂×#1389	2
#12559	2