Subgroups — #149265

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×#1952	C₂	1	1	yes	maximal, characteristic
3	#21156	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	1	yes	maximal, characteristic
5	C₂×#1382	C₂	1	1	yes	maximal, characteristic
6	#12557	C₂	1	1	yes	maximal, characteristic
7	C₂×#1379	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂×#428	—	2	1	no	—
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂²×#107	C₂²	1	1	yes	characteristic
15	C₂×#405	C₂²	1	1	yes	characteristic
16	#2317	—	2	1	no	—
17	C₂×#367	—	2	2	no	—
18	C₂×#367	—	2	2	no	—
19	C₂×#367	—	2	2	no	—
20	#1952	—	2	2	no	—
21	#1952	—	2	2	no	—
22	#1944	—	2	2	no	—
23	#1942	—	2	2	no	—
24	C₂×#343	—	2	1	no	—
25	#1736	—	2	1	no	—
26	#1387	—	2	2	no	—
27	#1382	C₄	1	4	yes	complemented
28	#1380	—	2	2	no	—
29	#1379	—	2	2	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	D₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	—	2	1	no	—
34	C₂×#117	D₄	1	1	yes	characteristic
35	C₂²×C₄⋊C₄	—	2	1	no	—
36	C₂²×C₄⋊C₄	—	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	Q₈	1	2	yes	—
43	C₂²×C₁₆	—	2	1	no	—
44	C₂²×C₁₆	—	2	1	no	—
45	C₂²×C₁₆	—	2	1	no	—
46	C₂²×C₁₆	—	2	1	no	—
47	C₂²×C₁₆	—	2	1	no	—
48	C₂²×C₁₆	C₂×C₄	1	2	yes	—
49	C₂²×C₁₆	D₄	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	C₂×C₄	1	2	yes	—
55	C₂×#108	D₄	1	2	yes	—
56	C₂×#108	Q₈	1	2	yes	—
57	#428	—	2	4	no	—
58	#428	—	4	2	no	—
59	#427	—	2	1	no	—
60	#427	—	2	1	no	—
61	#427	C₂×C₄	1	2	yes	—
62	#427	Q₈	1	2	yes	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	—	2	1	no	—
65	C₂×#107	—	2	1	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	—	2	1	no	—
68	C₂×#107	—	2	1	no	—
69	C₂×#107	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	C₂×#107	—	2	1	no	—
72	C₂×#107	—	2	1	no	—
73	C₂×#107	—	4	1	no	—
74	C₂×#107	D₄	1	2	yes	—
75	C₂×#107	D₄	1	2	yes	—
76	C₂×#107	D₄	1	2	yes	—
77	C₂×#106	—	2	1	no	—
78	C₂×#106	—	4	1	no	—
79	#405	—	2	1	no	—
80	#405	—	2	1	no	—
81	#405	—	2	1	no	—
82	#405	—	2	1	no	—
83	C₂×#97	—	2	1	no	—
84	C₂×C₄×C₈	—	2	1	no	—
85	#389	—	2	4	no	—
86	#385	—	4	2	no	—
87	#384	—	4	1	no	—
88	#382	—	4	1	no	—
89	#367	—	2	8	no	—
90	#343	—	2	4	no	—
91	#340	—	2	4	no	—
92	C₂³×C₄	—	4	1	no	—
93	C₂³×C₄	—	4	1	no	—
94	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	Q₁₆	1	2	yes	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	4	1	no	—
109	C₂²×C₈	C₂²×C₄	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×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×Q₈	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	1	no	—
125	#117	—	2	1	no	—
126	#117	—	2	2	no	—
127	#117	—	2	1	no	—
128	#117	—	2	1	no	—
129	#117	—	4	1	no	—
130	#117	—	4	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	4	no	—
145	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
146	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	—	2	1	no	—
148	C₂×C₂²⋊C₄	—	4	1	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	4	1	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	—	4	1	no	—
154	M₃₂	—	4	1	no	—
155	C₂×C₁₆	—	2	2	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	—	2	2	no	—
162	C₂×C₁₆	—	4	1	no	—
163	C₂×C₁₆	—	4	1	no	—
164	C₂×C₁₆	C₄⋊C₄	1	4	yes	complemented
165	#108	—	2	2	no	—
166	#108	—	2	2	no	—
167	#108	—	2	2	no	—
168	#108	C₄⋊C₄	1	4	yes	complemented
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#107	—	2	2	no	—
172	#107	—	2	2	no	—
173	#107	—	2	2	no	—
174	#107	—	4	4	no	—
175	#107	—	4	1	no	—
176	#107	—	4	1	no	—
177	#107	—	4	1	no	—
178	#107	—	4	1	no	—
179	#107	C₄⋊C₄	1	4	yes	—
180	#106	—	2	2	no	—
181	#106	—	2	2	no	—
182	#106	—	4	4	no	—
183	#97	—	4	1	no	—
184	#97	—	4	1	no	—
185	C₄×C₈	—	4	1	no	—
186	C₄×C₈	—	4	1	no	—
187	#95	—	4	4	no	—
188	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	8	1	no	—
195	C₂²×C₄	—	8	1	no	—
196	C₂²×C₄	—	8	1	no	—
197	C₂²×C₄	—	8	1	no	—
198	C₂²×C₄	—	16	2	no	—
199	C₂²×C₄	—	16	2	no	—
200	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
201	C₂²×C₄	#120	1	1	yes	characteristic
202	C₂²×C₄	#120	1	1	yes	characteristic
203	C₂²×C₄	#121	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
209	C₂²×C₄	#135	1	1	yes	characteristic
210	C₂²×C₄	#135	1	1	yes	characteristic
211	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
212	C₂²×C₄	#137	1	1	yes	characteristic
213	C₂²×C₄	#137	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄⋊C₄	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×Q₈	1	1	yes	characteristic
234	C₂×C₈	#120	1	1	yes	characteristic
235	C₂×C₈	#121	1	1	yes	characteristic
236	C₂×C₈	#121	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊Q₈	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	C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	#123	1	1	yes	characteristic
245	C₂×C₈	#123	1	1	yes	characteristic
246	C₂×C₈	#123	1	1	yes	characteristic
247	C₂×C₈	#125	1	1	yes	characteristic
248	C₂×C₈	#125	1	1	yes	characteristic
249	C₂×C₈	#125	1	1	yes	characteristic
250	C₂×C₈	#127	1	1	yes	characteristic
251	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
252	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
253	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
254	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
255	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	8	1	no	—
267	C₄⋊C₄	—	8	1	no	—
268	C₄⋊C₄	—	8	1	no	—
269	C₄⋊C₄	—	8	1	no	—
270	C₄⋊C₄	—	16	4	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	4	2	no	—
279	C₁₆	—	2	4	no	—
280	C₁₆	—	4	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	2	2	no	—
283	C₂³	#440	1	1	yes	characteristic
284	C₂³	#447	1	1	yes	characteristic
285	C₂³	#449	1	1	yes	characteristic
286	C₂³	#469	1	1	yes	characteristic
287	C₂³	#477	1	1	yes	characteristic
288	C₂³	#478	1	1	yes	characteristic
289	C₂³	#501	1	1	yes	characteristic
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	16	2	no	—
301	C₂×C₄	—	16	2	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	—	16	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	#385	1	1	yes	characteristic
306	C₂×C₄	#385	1	1	yes	characteristic
307	C₂×C₄	#389	1	1	yes	characteristic
308	C₂×C₄	#392	1	1	yes	characteristic
309	C₂×C₄	#395	1	1	yes	characteristic
310	C₂×C₄	#395	1	1	yes	characteristic
311	C₂×C₄	#398	1	1	yes	characteristic
312	C₂×C₄	C₄×D₈	1	1	yes	characteristic
313	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
314	C₂×C₄	#440	1	1	yes	characteristic
315	C₂×C₄	#447	1	1	yes	characteristic
316	C₂×C₄	#449	1	1	yes	characteristic
317	C₂×C₄	#451	1	1	yes	characteristic
318	C₂×C₄	#451	1	1	yes	characteristic
319	C₂×C₄	#452	1	1	yes	characteristic
320	C₂×C₄	#452	1	1	yes	characteristic
321	C₂×C₄	#466	1	1	yes	characteristic
322	C₂×C₄	#467	1	1	yes	characteristic
323	C₂×C₄	#470	1	1	yes	characteristic
324	C₂×C₄	#474	1	1	yes	characteristic
325	C₂×C₄	#474	1	1	yes	characteristic
326	C₂×C₄	#477	1	1	yes	characteristic
327	C₂×C₄	#478	1	1	yes	characteristic
328	C₂×C₄	#479	1	1	yes	characteristic
329	C₂×C₄	#479	1	1	yes	characteristic
330	C₂×C₄	#499	1	1	yes	characteristic
331	C₂×C₄	#500	1	1	yes	characteristic
332	C₂×C₄	#501	1	1	yes	characteristic
333	C₈	—	4	4	no	—
334	C₈	—	4	2	no	—
335	C₈	#384	1	1	yes	characteristic
336	C₈	#385	1	1	yes	characteristic
337	C₈	#389	1	1	yes	characteristic
338	C₈	#389	1	1	yes	characteristic
339	C₈	#395	1	1	yes	characteristic
340	C₈	#398	1	1	yes	characteristic
341	C₈	#398	1	1	yes	characteristic
342	C₈	#399	1	1	yes	characteristic
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	#1862	1	1	yes	characteristic
352	C₂²	#1863	1	1	yes	characteristic
353	C₂²	#1944	1	1	yes	characteristic
354	C₂²	#1999	1	1	yes	characteristic
355	C₂²	#2057	1	1	yes	characteristic
356	C₂²	#2059	1	1	yes	characteristic
357	C₂²	#2087	1	1	yes	characteristic
358	C₂²	#2193	1	2	yes	—
359	C₂²	#2194	1	2	yes	—
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	4	no	—
363	C₄	—	16	2	no	—
364	C₄	—	16	2	no	—
365	C₄	#1444	1	1	yes	characteristic
366	C₄	#1862	1	1	yes	characteristic
367	C₄	#1863	1	1	yes	characteristic
368	C₄	#1942	1	1	yes	characteristic
369	C₄	#2001	1	1	yes	characteristic
370	C₄	#2057	1	1	yes	characteristic
371	C₄	#2059	1	1	yes	characteristic
372	C₄	#2086	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	#11328	1	1	yes	characteristic
378	C₂	#12946	1	2	yes	—
379	C₂	#13320	1	1	yes	characteristic
380	C₂	#13321	1	1	yes	characteristic
381	C₂	#13456	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
27. #1382	C₄	semidirect
164. C₂×C₁₆	C₄⋊C₄	semidirect
168. #108	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#367	2
C₂×#1952	2
#21156	2
C₂×#1387	2
C₂×#1382	2
#12557	2
C₂×#1379	2