Subgroups — #147049

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 355 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×#1388	C₂	1	2	yes	maximal
5	#12561	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	1	no	—
16	C₂×#364	—	2	1	no	—
17	C₂×#364	C₄	1	4	yes	complemented
18	#1848	—	2	2	no	—
19	#1848	C₄	1	4	yes	complemented
20	#1772	—	2	1	no	—
21	#1772	—	2	1	no	—
22	#1772	C₄	1	2	yes	—
23	#1772	C₄	1	2	yes	—
24	C₂×#339	—	2	1	no	—
25	#1388	—	2	4	no	—
26	#1382	—	2	4	no	—
27	C₂²×M₁₆	—	2	1	no	—
28	C₂²×M₁₆	—	2	2	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	2	no	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	—	2	2	no	—
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	—	2	2	no	—
38	C₂×M₃₂	C₂×C₄	1	4	yes	—
39	C₂²×C₁₆	—	2	2	no	—
40	C₂×#108	—	2	1	no	—
41	C₂×#108	—	2	1	no	—
42	C₂×#108	—	2	1	no	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	—	2	4	no	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	—	4	1	no	—
50	C₂×#108	—	4	1	no	—
51	C₂×#108	—	4	2	no	—
52	C₂×#108	C₂×C₄	1	4	yes	—
53	C₂×#108	C₂×C₄	1	2	yes	—
54	C₂×#108	D₄	1	2	yes	—
55	C₂×#108	Q₈	1	2	yes	—
56	#427	—	2	1	no	—
57	#427	—	2	1	no	—
58	#427	C₂×C₄	1	2	yes	—
59	#427	D₄	1	2	yes	—
60	C₂×#107	Q₈	1	2	yes	—
61	C₂×#106	—	2	1	no	—
62	#423	—	2	4	no	—
63	#423	—	4	2	no	—
64	C₂×C₄⋊C₈	—	4	2	no	—
65	#405	—	2	2	no	—
66	#405	—	2	2	no	—
67	#405	—	4	1	no	—
68	#405	—	4	1	no	—
69	C₂×#97	—	4	1	no	—
70	C₂×C₄×C₈	—	4	1	no	—
71	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
72	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
73	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈	D₄	1	2	yes	—
76	C₂×C₄×C₈	D₄	1	1	yes	characteristic
77	C₂×C₄×C₈	D₄	1	1	yes	characteristic
78	C₂×C₄×C₈	Q₈	1	2	yes	—
79	#364	—	2	4	no	—
80	#364	—	2	4	no	—
81	#339	—	2	2	no	—
82	#339	—	2	2	no	—
83	C₂³×C₄	—	2	1	no	—
84	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	2	4	no	—
89	C₂×M₁₆	—	4	1	no	—
90	C₂×M₁₆	—	4	1	no	—
91	C₂×M₁₆	—	4	1	no	—
92	C₂×M₁₆	—	4	2	no	—
93	C₂×M₁₆	—	4	2	no	—
94	C₂×M₁₆	—	4	2	no	—
95	C₂×M₁₆	—	4	2	no	—
96	C₂×M₁₆	—	4	2	no	—
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	—	4	2	no	—
99	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
102	C₂²×C₈	C₄⋊C₄	1	2	yes	—
103	C₂²×C₈	D₈	1	1	yes	characteristic
104	C₂²×C₈	D₈	1	1	yes	characteristic
105	C₂²×C₈	SD₁₆	1	1	yes	characteristic
106	C₂²×C₈	SD₁₆	1	1	yes	characteristic
107	C₂²×C₈	SD₁₆	1	2	yes	—
108	C₂²×C₈	Q₁₆	1	2	yes	—
109	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₄	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×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	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	#117	—	2	2	no	—
124	#117	—	2	2	no	—
125	#117	—	4	1	no	—
126	#117	—	4	1	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₂²⋊C₄	—	4	1	no	—
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	—	4	1	no	—
133	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
136	C₂×C₄²	C₄⋊C₄	1	2	yes	—
137	M₃₂	—	2	4	no	—
138	M₃₂	—	2	4	no	—
139	M₃₂	—	4	2	no	—
140	M₃₂	—	4	2	no	—
141	C₂×C₁₆	—	2	4	no	—
142	C₂×C₁₆	—	4	2	no	—
143	C₂×C₁₆	—	4	2	no	—
144	#108	—	2	2	no	—
145	#108	—	2	4	no	—
146	#108	—	2	4	no	—
147	#108	—	2	2	no	—
148	#108	—	2	2	no	—
149	#108	—	4	4	no	—
150	#108	—	4	4	no	—
151	#108	—	4	2	no	—
152	#108	—	4	2	no	—
153	#108	C₄⋊C₄	1	4	yes	—
154	#107	—	2	2	no	—
155	#107	C₄⋊C₄	1	4	yes	—
156	#106	—	2	2	no	—
157	#106	—	2	2	no	—
158	C₄⋊C₈	—	4	8	no	—
159	#97	—	4	2	no	—
160	#97	—	4	2	no	—
161	C₄×C₈	—	4	2	no	—
162	C₄×C₈	—	4	2	no	—
163	C₄×C₈	C₄²	1	2	yes	—
164	C₄×C₈	C₄²	1	2	yes	—
165	C₄×C₈	C₂²⋊C₄	1	2	yes	—
166	C₄×C₈	C₂²⋊C₄	1	2	yes	—
167	C₄×C₈	C₄⋊C₄	1	2	yes	—
168	C₄×C₈	C₄⋊C₄	1	2	yes	—
169	C₄×C₈	C₄⋊C₄	1	2	yes	—
170	C₄×C₈	C₄⋊C₄	1	2	yes	—
171	C₂⁴	#137	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	#121	1	1	yes	characteristic
182	C₂²×C₄	#121	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
187	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
188	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
189	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂²×C₄	#137	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	4	no	—
198	M₁₆	—	8	2	no	—
199	M₁₆	—	8	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	4	no	—
204	C₂×C₈	—	8	2	no	—
205	C₂×C₈	—	8	2	no	—
206	C₂×C₈	#95	1	2	yes	—
207	C₂×C₈	#95	1	2	yes	—
208	C₂×C₈	#102	1	1	yes	characteristic
209	C₂×C₈	#102	1	1	yes	characteristic
210	C₂×C₈	#102	1	1	yes	characteristic
211	C₂×C₈	#102	1	1	yes	characteristic
212	C₂×C₈	#102	1	2	yes	—
213	C₂×C₈	#102	1	2	yes	—
214	C₂×C₈	#103	1	2	yes	—
215	C₂×C₈	#103	1	2	yes	—
216	C₂×C₈	#103	1	2	yes	—
217	C₂×C₈	#103	1	2	yes	—
218	C₂×C₈	#106	1	2	yes	—
219	C₂×C₈	#106	1	2	yes	—
220	C₂×C₈	#107	1	2	yes	—
221	C₂×C₈	#107	1	2	yes	—
222	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
223	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄⋊C₄	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	C₄×D₄	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×Q₈	1	1	yes	characteristic
235	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
236	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
237	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
238	C₂×C₈	#121	1	1	yes	characteristic
239	C₂×C₈	#121	1	1	yes	characteristic
240	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₈	#124	1	1	yes	characteristic
243	C₂×C₈	#124	1	1	yes	characteristic
244	C₂×C₈	#125	1	1	yes	characteristic
245	C₂×C₈	#125	1	1	yes	characteristic
246	C₂×C₈	#127	1	1	yes	characteristic
247	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
248	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
249	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	#95	1	2	yes	—
261	C₄²	#95	1	2	yes	—
262	C₁₆	—	4	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	#417	1	1	yes	characteristic
265	C₂³	#419	1	1	yes	characteristic
266	C₂³	#428	1	1	yes	characteristic
267	C₂³	#461	1	1	yes	characteristic
268	C₂³	#462	1	1	yes	characteristic
269	C₂³	#474	1	1	yes	characteristic
270	C₂³	#475	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	C₂×#95	1	1	yes	characteristic
280	C₂×C₄	#380	1	1	yes	characteristic
281	C₂×C₄	#380	1	1	yes	characteristic
282	C₂×C₄	#389	1	1	yes	characteristic
283	C₂×C₄	#389	1	1	yes	characteristic
284	C₂×C₄	#390	1	1	yes	characteristic
285	C₂×C₄	#391	1	1	yes	characteristic
286	C₂×C₄	C₂×#102	1	1	yes	characteristic
287	C₂×C₄	C₂×#102	1	1	yes	characteristic
288	C₂×C₄	C₂×#103	1	1	yes	characteristic
289	C₂×C₄	C₂×#103	1	1	yes	characteristic
290	C₂×C₄	#417	1	1	yes	characteristic
291	C₂×C₄	#418	1	1	yes	characteristic
292	C₂×C₄	C₂×#106	1	1	yes	characteristic
293	C₂×C₄	C₂×#107	1	1	yes	characteristic
294	C₂×C₄	#428	1	1	yes	characteristic
295	C₂×C₄	#459	1	1	yes	characteristic
296	C₂×C₄	#459	1	1	yes	characteristic
297	C₂×C₄	#460	1	1	yes	characteristic
298	C₂×C₄	#460	1	1	yes	characteristic
299	C₂×C₄	#461	1	1	yes	characteristic
300	C₂×C₄	#462	1	1	yes	characteristic
301	C₂×C₄	#474	1	1	yes	characteristic
302	C₂×C₄	#475	1	1	yes	characteristic
303	C₂×C₄	#476	1	1	yes	characteristic
304	C₂×C₄	#476	1	1	yes	characteristic
305	C₂×C₄	#477	1	1	yes	characteristic
306	C₂×C₄	#477	1	1	yes	characteristic
307	C₈	—	4	4	no	—
308	C₈	—	8	4	no	—
309	C₈	#340	1	2	yes	—
310	C₈	#340	1	2	yes	—
311	C₈	#340	1	2	yes	—
312	C₈	#340	1	2	yes	—
313	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
314	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
315	C₈	#382	1	1	yes	characteristic
316	C₈	#384	1	1	yes	characteristic
317	C₈	#389	1	1	yes	characteristic
318	C₈	#389	1	1	yes	characteristic
319	C₈	#390	1	1	yes	characteristic
320	C₈	#391	1	1	yes	characteristic
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	#1736	1	1	yes	characteristic
329	C₂²	#1802	1	1	yes	characteristic
330	C₂²	#1803	1	1	yes	characteristic
331	C₂²	#1947	1	1	yes	characteristic
332	C₂²	#1948	1	1	yes	characteristic
333	C₂²	#1981	1	1	yes	characteristic
334	C₂²	#1988	1	1	yes	characteristic
335	C₄	—	2	4	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	4	no	—
338	C₄	C₄⋊#95	1	1	yes	characteristic
339	C₄	C₂×#340	1	1	yes	characteristic
340	C₄	#1802	1	1	yes	characteristic
341	C₄	#1803	1	1	yes	characteristic
342	C₄	#1947	1	1	yes	characteristic
343	C₄	#1948	1	1	yes	characteristic
344	C₄	#1982	1	1	yes	characteristic
345	C₄	#1987	1	1	yes	characteristic
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	—	2	1	no	—
352	C₂	#10849	1	1	yes	characteristic
353	C₂	#12822	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1802
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
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×#1388	2
#12561	2