Subgroups — #149064

Order 512.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21156	C₂	1	1	yes	maximal, characteristic
2	C₂×#1387	C₂	1	1	yes	maximal, characteristic
3	C₂×#1382	C₂	1	1	yes	maximal, characteristic
4	#12557	C₂	1	1	yes	maximal, characteristic
5	C₂×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂×#1379	C₂	1	1	yes	maximal, characteristic
7	#10870	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×#427	C₂²	1	1	yes	characteristic
12	C₂²×#107	C₂²	1	1	yes	characteristic
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	#2317	—	2	1	no	—
16	#1944	—	2	2	no	—
17	#1942	—	2	1	no	—
18	#1942	—	2	1	no	—
19	C₂×#343	—	2	1	no	—
20	#1736	—	2	1	no	—
21	#1736	—	2	1	no	—
22	#1387	—	2	2	no	—
23	#1382	C₄	1	4	yes	complemented
24	#1380	—	2	2	no	—
25	#1380	—	2	1	no	—
26	#1380	—	2	1	no	—
27	#1379	—	2	2	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	D₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	1	yes	characteristic
31	C₂×#117	—	2	1	no	—
32	C₂×#117	D₄	1	1	yes	characteristic
33	C₂²×C₄⋊C₄	—	2	1	no	—
34	C₂²×C₄⋊C₄	—	2	1	no	—
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	D₄	1	2	yes	—
41	C₂²×C₁₆	—	2	1	no	—
42	C₂²×C₁₆	—	2	1	no	—
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	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	C₂×C₄	1	2	yes	—
55	#427	—	2	1	no	—
56	#427	—	2	1	no	—
57	#427	C₂×C₄	1	2	yes	—
58	#427	Q₈	1	2	yes	—
59	C₂×#107	—	2	1	no	—
60	C₂×#107	—	2	1	no	—
61	C₂×#107	—	2	1	no	—
62	C₂×#107	—	2	1	no	—
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	D₄	1	2	yes	—
70	C₂×#107	D₄	1	1	yes	characteristic
71	C₂×#107	D₄	1	1	yes	characteristic
72	C₂×#107	Q₈	1	2	yes	—
73	C₂×#106	—	2	1	no	—
74	#405	—	2	1	no	—
75	#405	—	2	1	no	—
76	#405	—	2	1	no	—
77	#405	—	2	1	no	—
78	C₂×#97	—	2	1	no	—
79	C₂×C₄×C₈	—	2	1	no	—
80	#389	—	2	2	no	—
81	#389	—	2	2	no	—
82	#385	—	4	2	no	—
83	#384	—	4	1	no	—
84	#382	—	4	1	no	—
85	#343	—	4	2	no	—
86	#340	—	2	2	no	—
87	#340	—	2	2	no	—
88	#340	—	4	1	no	—
89	#340	—	4	1	no	—
90	C₂³×C₄	—	4	1	no	—
91	C₂³×C₄	—	4	1	no	—
92	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	SD₁₆	1	2	yes	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	C₂²×C₄	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	#41	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	C₂²×C₈	#41	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	C₂²×C₈	#41	1	1	yes	characteristic
121	#117	—	2	2	no	—
122	#117	—	2	1	no	—
123	#117	—	2	2	no	—
124	#117	—	2	1	no	—
125	#117	—	2	1	no	—
126	#117	—	2	1	no	—
127	#117	—	4	1	no	—
128	#117	—	4	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	4	1	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	2	no	—
144	C₂×C₄⋊C₄	—	4	2	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	—	4	1	no	—
160	C₂×C₁₆	—	4	1	no	—
161	C₂×C₁₆	—	4	1	no	—
162	C₂×C₁₆	—	4	1	no	—
163	C₂×C₁₆	—	4	1	no	—
164	C₂×C₁₆	—	4	1	no	—
165	#108	—	2	2	no	—
166	#108	—	2	2	no	—
167	#108	—	4	1	no	—
168	#108	—	4	1	no	—
169	#107	—	2	1	no	—
170	#107	—	2	1	no	—
171	#107	—	2	2	no	—
172	#107	—	2	2	no	—
173	#107	—	2	2	no	—
174	#107	—	2	2	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	2	yes	—
180	#107	C₄⋊C₄	1	2	yes	—
181	#106	—	2	2	no	—
182	#106	—	2	2	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	2	no	—
188	#95	—	4	2	no	—
189	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	8	1	no	—
197	C₂²×C₄	—	8	1	no	—
198	C₂²×C₄	—	8	1	no	—
199	C₂²×C₄	—	8	1	no	—
200	C₂²×C₄	—	16	1	no	—
201	C₂²×C₄	—	16	1	no	—
202	C₂²×C₄	—	16	1	no	—
203	C₂²×C₄	—	16	1	no	—
204	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
205	C₂²×C₄	#120	1	1	yes	characteristic
206	C₂²×C₄	#121	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
209	C₂²×C₄	#123	1	1	yes	characteristic
210	C₂²×C₄	#124	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
213	C₂²×C₄	#135	1	1	yes	characteristic
214	C₂²×C₄	#135	1	1	yes	characteristic
215	C₂²×C₄	#137	1	1	yes	characteristic
216	C₂²×C₄	#137	1	1	yes	characteristic
217	C₂²×C₄	#137	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	8	1	no	—
225	M₁₆	—	8	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	8	1	no	—
231	C₂×C₈	—	8	1	no	—
232	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
233	C₂×C₈	#117	1	1	yes	characteristic
234	C₂×C₈	#117	1	1	yes	characteristic
235	C₂×C₈	C₄×D₄	1	1	yes	characteristic
236	C₂×C₈	C₄×D₄	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	C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₈	#123	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	#124	1	1	yes	characteristic
248	C₂×C₈	#124	1	1	yes	characteristic
249	C₂×C₈	#124	1	1	yes	characteristic
250	C₂×C₈	#125	1	1	yes	characteristic
251	C₂×C₈	#125	1	1	yes	characteristic
252	C₂×C₈	#125	1	1	yes	characteristic
253	C₂×C₈	#125	1	1	yes	characteristic
254	C₂×C₈	#125	1	1	yes	characteristic
255	C₂×C₈	#126	1	1	yes	characteristic
256	C₂×C₈	#126	1	1	yes	characteristic
257	C₂×C₈	#126	1	1	yes	characteristic
258	C₂×C₈	#126	1	1	yes	characteristic
259	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	8	1	no	—
273	C₄⋊C₄	—	8	1	no	—
274	C₄⋊C₄	—	8	1	no	—
275	C₄⋊C₄	—	8	1	no	—
276	C₄⋊C₄	—	16	2	no	—
277	C₄⋊C₄	—	16	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	2	no	—
286	C₁₆	—	4	2	no	—
287	C₁₆	—	4	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	#440	1	1	yes	characteristic
293	C₂³	#447	1	1	yes	characteristic
294	C₂³	#468	1	1	yes	characteristic
295	C₂³	#476	1	1	yes	characteristic
296	C₂³	#478	1	1	yes	characteristic
297	C₂³	#482	1	1	yes	characteristic
298	C₂³	#489	1	1	yes	characteristic
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	16	2	no	—
316	C₂×C₄	—	16	2	no	—
317	C₂×C₄	—	16	2	no	—
318	C₂×C₄	—	16	1	no	—
319	C₂×C₄	—	16	1	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	#385	1	1	yes	characteristic
322	C₂×C₄	#387	1	1	yes	characteristic
323	C₂×C₄	#388	1	1	yes	characteristic
324	C₂×C₄	#393	1	1	yes	characteristic
325	C₂×C₄	#394	1	1	yes	characteristic
326	C₂×C₄	#398	1	1	yes	characteristic
327	C₂×C₄	#400	1	1	yes	characteristic
328	C₂×C₄	C₄×D₈	1	1	yes	characteristic
329	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
330	C₂×C₄	#441	1	1	yes	characteristic
331	C₂×C₄	#448	1	1	yes	characteristic
332	C₂×C₄	#452	1	1	yes	characteristic
333	C₂×C₄	#452	1	1	yes	characteristic
334	C₂×C₄	#465	1	1	yes	characteristic
335	C₂×C₄	#466	1	1	yes	characteristic
336	C₂×C₄	#470	1	1	yes	characteristic
337	C₂×C₄	#474	1	1	yes	characteristic
338	C₂×C₄	#474	1	1	yes	characteristic
339	C₂×C₄	#475	1	1	yes	characteristic
340	C₂×C₄	#479	1	1	yes	characteristic
341	C₂×C₄	#479	1	1	yes	characteristic
342	C₂×C₄	#479	1	1	yes	characteristic
343	C₂×C₄	#480	1	1	yes	characteristic
344	C₂×C₄	#483	1	1	yes	characteristic
345	C₂×C₄	#485	1	1	yes	characteristic
346	C₂×C₄	#486	1	1	yes	characteristic
347	C₂×C₄	#488	1	1	yes	characteristic
348	C₂×C₄	#491	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	2	no	—
351	C₈	#382	1	1	yes	characteristic
352	C₈	#387	1	1	yes	characteristic
353	C₈	#388	1	1	yes	characteristic
354	C₈	#389	1	1	yes	characteristic
355	C₈	#397	1	1	yes	characteristic
356	C₈	#398	1	1	yes	characteristic
357	C₈	#400	1	1	yes	characteristic
358	C₈	#400	1	1	yes	characteristic
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	#1862	1	1	yes	characteristic
372	C₂²	#1920	1	1	yes	characteristic
373	C₂²	#1935	1	1	yes	characteristic
374	C₂²	#2023	1	1	yes	characteristic
375	C₂²	#2033	1	1	yes	characteristic
376	C₂²	#2088	1	1	yes	characteristic
377	C₂²	#2101	1	1	yes	characteristic
378	C₂²	#2192	1	1	yes	characteristic
379	C₂²	#2193	1	1	yes	characteristic
380	C₂²	#2194	1	1	yes	characteristic
381	C₂²	#2195	1	1	yes	characteristic
382	C₄	—	8	1	no	—
383	C₄	—	8	1	no	—
384	C₄	—	8	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	16	2	no	—
387	C₄	—	16	2	no	—
388	C₄	#1442	1	1	yes	characteristic
389	C₄	#1865	1	1	yes	characteristic
390	C₄	#1918	1	1	yes	characteristic
391	C₄	#1934	1	1	yes	characteristic
392	C₄	#2026	1	1	yes	characteristic
393	C₄	#2035	1	1	yes	characteristic
394	C₄	#2086	1	1	yes	characteristic
395	C₄	#2099	1	1	yes	characteristic
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	—	2	1	no	—
399	C₂	—	2	1	no	—
400	C₂	#11292	1	1	yes	characteristic
401	C₂	#12945	1	1	yes	characteristic
402	C₂	#12946	1	1	yes	characteristic
403	C₂	#13369	1	1	yes	characteristic
404	C₂	#13370	1	1	yes	characteristic
405	C₂	#13387	1	1	yes	characteristic
406	C₂	#13388	1	1	yes	characteristic

	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
23. #1382	C₄	semidirect

Maximal subgroups

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

subgroup	index
#21156	2
C₂×#1387	2
C₂×#1382	2
#12557	2
C₂×#1380	2
C₂×#1379	2
#10870	2