Subgroups — #149244

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 364 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	C₂×#1946	C₂	1	1	yes	maximal, characteristic
3	#21154	C₂	1	1	yes	maximal, characteristic
4	C₂×#1384	C₂	1	1	yes	maximal, characteristic
5	C₂×#1384	C₂	1	1	yes	maximal, characteristic
6	#12563	C₂	1	1	yes	maximal, characteristic
7	C₂×#1375	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	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×#424	—	2	1	no	—
15	C₂²×#97	C₂²	1	1	yes	characteristic
16	#2296	—	2	1	no	—
17	#2155	—	2	2	no	—
18	#2155	—	2	2	no	—
19	C₂×#365	—	2	2	no	—
20	#1946	—	2	2	no	—
21	#1946	—	2	2	no	—
22	#1944	—	2	2	no	—
23	#1943	—	2	2	no	—
24	#1735	—	2	1	no	—
25	C₂×#339	—	2	1	no	—
26	#1384	—	2	2	no	—
27	#1384	C₄	1	4	yes	complemented
28	#1383	—	2	2	no	—
29	#1375	—	2	2	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂³×C₈	D₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	—	2	1	no	—
34	C₂×#117	—	2	1	no	—
35	C₂×#117	—	2	1	no	—
36	C₂²×C₄²	D₄	1	1	yes	characteristic
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	—	2	1	no	—
44	C₂×M₃₂	—	2	1	no	—
45	C₂×M₃₂	—	2	1	no	—
46	C₂×M₃₂	—	2	1	no	—
47	C₂×M₃₂	C₂×C₄	1	2	yes	—
48	C₂×M₃₂	D₄	1	2	yes	—
49	C₂²×C₁₆	Q₈	1	2	yes	—
50	C₂²×C₁₆	Q₈	1	2	yes	—
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	D₄	1	2	yes	—
57	#427	—	2	1	no	—
58	#427	—	2	1	no	—
59	#427	—	2	1	no	—
60	#427	—	2	1	no	—
61	#427	—	2	1	no	—
62	#427	—	2	1	no	—
63	#427	—	2	1	no	—
64	#427	—	2	1	no	—
65	#427	C₂×C₄	1	2	yes	—
66	#427	C₂×C₄	1	2	yes	—
67	#427	D₄	1	2	yes	—
68	#427	Q₈	1	2	yes	—
69	C₂×#107	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	C₂×#107	—	2	1	no	—
72	C₂×#106	D₄	1	2	yes	—
73	C₂×#106	D₄	1	2	yes	—
74	C₂×#106	Q₈	1	2	yes	—
75	#424	—	2	4	no	—
76	#424	—	4	2	no	—
77	C₂×C₄⋊C₈	—	4	2	no	—
78	C₂×#97	—	2	1	no	—
79	C₂×#97	—	2	1	no	—
80	C₂×#97	—	2	1	no	—
81	C₂×#97	—	2	1	no	—
82	C₂×#97	—	2	1	no	—
83	C₂×#97	—	2	1	no	—
84	#389	—	2	4	no	—
85	#385	—	4	2	no	—
86	C₄×C₄⋊C₄	—	4	2	no	—
87	#365	—	2	8	no	—
88	#340	—	2	4	no	—
89	#339	—	2	4	no	—
90	C₂³×C₄	—	2	1	no	—
91	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×M₁₆	—	2	2	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	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	4	1	no	—
105	C₂²×C₈	SD₁₆	1	2	yes	—
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×D₄	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	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	#117	—	2	2	no	—
121	#117	—	2	2	no	—
122	#117	—	2	2	no	—
123	#117	—	2	2	no	—
124	#117	—	4	1	no	—
125	#117	—	4	1	no	—
126	#117	—	4	1	no	—
127	#117	—	4	1	no	—
128	#117	—	4	1	no	—
129	#117	—	4	1	no	—
130	#117	—	4	1	no	—
131	#117	—	4	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	4	4	no	—
136	C₂×C₂²⋊C₄	—	4	1	no	—
137	C₂×C₂²⋊C₄	—	4	1	no	—
138	C₂×C₂²⋊C₄	—	4	1	no	—
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	4	1	no	—
145	C₂×C₄²	—	4	1	no	—
146	C₂×C₄²	—	4	1	no	—
147	C₂×C₄²	SD₁₆	1	2	yes	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	4	1	no	—
153	M₃₂	—	4	1	no	—
154	M₃₂	—	4	1	no	—
155	M₃₂	—	4	1	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	C₄⋊C₄	1	4	yes	complemented
162	#108	—	2	2	no	—
163	#108	—	2	2	no	—
164	#108	—	4	1	no	—
165	#108	—	4	1	no	—
166	#107	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	—	2	2	no	—
169	#107	—	2	2	no	—
170	#107	—	4	1	no	—
171	#107	—	4	1	no	—
172	#106	—	2	2	no	—
173	#106	—	2	2	no	—
174	#106	—	2	2	no	—
175	#106	—	2	2	no	—
176	#106	C₄⋊C₄	1	4	yes	—
177	#106	C₄⋊C₄	1	4	yes	complemented
178	C₄⋊C₈	—	4	8	no	—
179	#97	—	4	1	no	—
180	#97	—	4	1	no	—
181	#97	—	4	1	no	—
182	#97	—	4	1	no	—
183	#95	—	4	4	no	—
184	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	8	2	no	—
192	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
193	C₂²×C₄	#120	1	1	yes	characteristic
194	C₂²×C₄	#120	1	1	yes	characteristic
195	C₂²×C₄	#121	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
201	C₂²×C₄	#135	1	1	yes	characteristic
202	C₂²×C₄	#135	1	1	yes	characteristic
203	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂²×C₄	#137	1	1	yes	characteristic
205	C₂²×C₄	#137	1	1	yes	characteristic
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	8	1	no	—
209	M₁₆	—	8	1	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	8	1	no	—
219	C₂×C₈	—	8	1	no	—
220	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄⋊C₄	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×Q₈	1	1	yes	characteristic
226	C₂×C₈	#120	1	1	yes	characteristic
227	C₂×C₈	#121	1	1	yes	characteristic
228	C₂×C₈	#121	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	C₂²⋊Q₈	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	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₈	#123	1	1	yes	characteristic
237	C₂×C₈	#123	1	1	yes	characteristic
238	C₂×C₈	#123	1	1	yes	characteristic
239	C₂×C₈	#125	1	1	yes	characteristic
240	C₂×C₈	#125	1	1	yes	characteristic
241	C₂×C₈	#125	1	1	yes	characteristic
242	C₂×C₈	#127	1	1	yes	characteristic
243	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
245	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
246	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
247	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	8	8	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	8	1	no	—
260	C₂²⋊C₄	—	8	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	8	1	no	—
270	C₄²	—	8	1	no	—
271	C₁₆	—	2	4	no	—
272	C₁₆	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	#442	1	1	yes	characteristic
275	C₂³	#449	1	1	yes	characteristic
276	C₂³	#450	1	1	yes	characteristic
277	C₂³	#468	1	1	yes	characteristic
278	C₂³	#476	1	1	yes	characteristic
279	C₂³	#478	1	1	yes	characteristic
280	C₂³	#501	1	1	yes	characteristic
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	#385	1	1	yes	characteristic
292	C₂×C₄	#385	1	1	yes	characteristic
293	C₂×C₄	#389	1	1	yes	characteristic
294	C₂×C₄	#392	1	1	yes	characteristic
295	C₂×C₄	#395	1	1	yes	characteristic
296	C₂×C₄	#395	1	1	yes	characteristic
297	C₂×C₄	#398	1	1	yes	characteristic
298	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
299	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
300	C₂×C₄	#441	1	1	yes	characteristic
301	C₂×C₄	#448	1	1	yes	characteristic
302	C₂×C₄	#448	1	1	yes	characteristic
303	C₂×C₄	#449	1	1	yes	characteristic
304	C₂×C₄	#450	1	1	yes	characteristic
305	C₂×C₄	#452	1	1	yes	characteristic
306	C₂×C₄	#452	1	1	yes	characteristic
307	C₂×C₄	#465	1	1	yes	characteristic
308	C₂×C₄	#465	1	1	yes	characteristic
309	C₂×C₄	#468	1	1	yes	characteristic
310	C₂×C₄	#475	1	1	yes	characteristic
311	C₂×C₄	#475	1	1	yes	characteristic
312	C₂×C₄	#476	1	1	yes	characteristic
313	C₂×C₄	#478	1	1	yes	characteristic
314	C₂×C₄	#479	1	1	yes	characteristic
315	C₂×C₄	#479	1	1	yes	characteristic
316	C₂×C₄	#498	1	1	yes	characteristic
317	C₂×C₄	#499	1	1	yes	characteristic
318	C₂×C₄	#501	1	1	yes	characteristic
319	C₈	—	4	2	no	—
320	C₈	—	8	2	no	—
321	C₈	#384	1	1	yes	characteristic
322	C₈	#385	1	1	yes	characteristic
323	C₈	#389	1	1	yes	characteristic
324	C₈	#389	1	1	yes	characteristic
325	C₈	#395	1	1	yes	characteristic
326	C₈	#398	1	1	yes	characteristic
327	C₈	#398	1	1	yes	characteristic
328	C₈	#399	1	1	yes	characteristic
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	#1864	1	1	yes	characteristic
337	C₂²	#1864	1	1	yes	characteristic
338	C₂²	#1943	1	1	yes	characteristic
339	C₂²	#2000	1	1	yes	characteristic
340	C₂²	#2058	1	1	yes	characteristic
341	C₂²	#2059	1	1	yes	characteristic
342	C₂²	#2088	1	1	yes	characteristic
343	C₄	—	8	2	no	—
344	C₄	—	8	4	no	—
345	C₄	—	8	4	no	—
346	C₄	—	16	2	no	—
347	C₄	#1444	1	1	yes	characteristic
348	C₄	#1865	1	1	yes	characteristic
349	C₄	#1865	1	1	yes	characteristic
350	C₄	#1941	1	1	yes	characteristic
351	C₄	#2000	1	1	yes	characteristic
352	C₄	#2058	1	1	yes	characteristic
353	C₄	#2059	1	1	yes	characteristic
354	C₄	#2088	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	#11327	1	1	yes	characteristic
362	C₂	#13459	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#498
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2058
prime radical	C₂⁴	AGL(1,ℤ/8)
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
27. #1384	C₄	semidirect
161. C₂×C₁₆	C₄⋊C₄	semidirect
177. #106	C₄⋊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
C₂×#1946	2
#21154	2
C₂×#1384	2
#12563	2
C₂×#1375	2