Subgroups — #149468

Order 512.

694 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3164	C₂	1	1	yes	maximal, characteristic
2	#12569	C₂	1	1	yes	maximal, characteristic
3	C₂×#1387	C₂	1	1	yes	maximal, characteristic
4	#12567	C₂	1	1	yes	maximal, characteristic
5	#12557	C₂	1	1	yes	maximal, characteristic
6	#12554	C₂	1	1	yes	maximal, characteristic
7	#10872	C₂	1	1	yes	maximal, characteristic
8	C₂×#544	—	2	1	no	—
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×M₃₂	C₂²	1	1	yes	characteristic
11	#3164	—	2	2	no	—
12	#3164	—	2	2	no	—
13	#3164	—	2	2	no	—
14	#3164	C₄	1	4	yes	complemented
15	C₂×#501	—	2	2	no	—
16	C₂×#500	—	2	1	no	—
17	C₂×#499	—	2	1	no	—
18	C₂²×#108	C₂²	1	1	yes	characteristic
19	C₂×#428	—	2	1	no	—
20	C₂×#427	C₂²	1	1	yes	characteristic
21	C₂×#427	C₂²	1	1	yes	characteristic
22	C₂²×#107	C₂²	1	1	yes	characteristic
23	C₂×#405	C₂²	1	1	yes	characteristic
24	C₂×#344	—	2	1	no	—
25	#1733	—	2	1	no	—
26	#1387	—	2	2	no	—
27	#1387	—	2	1	no	—
28	#1387	—	2	1	no	—
29	#1386	—	2	2	no	—
30	#1380	—	2	2	no	—
31	#1379	—	2	2	no	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂²×M₁₆	D₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	#544	—	2	4	no	—
36	#544	—	2	4	no	—
37	#544	—	4	2	no	—
38	#544	—	4	2	no	—
39	C₂×C₄⋊Q₈	—	2	2	no	—
40	C₂×C₄⋊Q₈	—	4	1	no	—
41	C₂×#125	—	2	2	no	—
42	C₂×#125	—	4	1	no	—
43	C₂×C₂²⋊Q₈	—	4	2	no	—
44	C₂×#117	—	2	1	no	—
45	C₂×#117	—	2	1	no	—
46	C₂×#117	D₄	1	1	yes	characteristic
47	C₂²×C₄⋊C₄	—	2	1	no	—
48	C₂×M₃₂	—	2	1	no	—
49	C₂×M₃₂	—	2	1	no	—
50	C₂×M₃₂	—	2	1	no	—
51	C₂×M₃₂	—	2	1	no	—
52	C₂×M₃₂	—	2	1	no	—
53	C₂×M₃₂	—	2	1	no	—
54	C₂×M₃₂	—	2	1	no	—
55	C₂×M₃₂	—	2	1	no	—
56	C₂×M₃₂	—	2	1	no	—
57	C₂×M₃₂	—	2	1	no	—
58	C₂×M₃₂	Q₈	1	2	yes	complemented
59	C₂×M₃₂	Q₈	1	2	yes	complemented
60	C₂²×C₁₆	—	2	1	no	—
61	C₂²×C₁₆	—	2	1	no	—
62	#501	—	2	8	no	—
63	#501	—	2	8	no	—
64	#500	—	2	4	no	—
65	#500	—	2	4	no	—
66	#499	—	2	4	no	—
67	#499	—	2	4	no	—
68	C₂×#108	—	2	1	no	—
69	C₂×#108	—	2	1	no	—
70	C₂×#108	—	2	1	no	—
71	C₂×#108	—	2	1	no	—
72	C₂×#108	—	2	1	no	—
73	C₂×#108	Q₈	1	2	yes	complemented
74	#428	—	2	4	no	—
75	#428	—	2	4	no	—
76	#427	—	2	1	no	—
77	#427	—	2	1	no	—
78	#427	—	2	1	no	—
79	#427	—	2	1	no	—
80	#427	—	2	1	no	—
81	#427	C₂×C₄	1	2	yes	—
82	#427	D₄	1	2	yes	—
83	#427	Q₈	1	2	yes	complemented
84	C₂×#107	—	2	2	no	—
85	C₂×#107	—	2	1	no	—
86	C₂×#107	—	2	1	no	—
87	C₂×#107	—	2	1	no	—
88	C₂×#107	—	2	1	no	—
89	C₂×#107	C₂×C₄	1	2	yes	—
90	C₂×#107	D₄	1	1	yes	characteristic
91	C₂×#107	D₄	1	1	yes	characteristic
92	C₂×#107	D₄	1	2	yes	—
93	C₂×#107	D₄	1	2	yes	—
94	C₂×#106	—	2	2	no	—
95	C₂×#106	—	2	1	no	—
96	C₂×#106	—	2	1	no	—
97	#405	—	2	1	no	—
98	#405	—	2	1	no	—
99	#405	C₂×C₄	1	2	yes	—
100	#405	D₄	1	2	yes	—
101	C₂×#97	—	2	1	no	—
102	C₂×C₄×C₈	—	2	1	no	—
103	#344	—	4	2	no	—
104	#339	—	4	2	no	—
105	C₂²×Q₈	—	4	2	no	—
106	C₂³×C₄	—	4	1	no	—
107	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×M₁₆	—	2	1	no	—
109	C₂×M₁₆	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	4	1	no	—
113	C₂×M₁₆	—	4	1	no	—
114	C₂×M₁₆	—	4	1	no	—
115	C₂×M₁₆	—	4	1	no	—
116	C₂×M₁₆	—	4	1	no	—
117	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
118	C₂×M₁₆	SD₁₆	1	2	yes	—
119	C₂×M₁₆	Q₁₆	1	2	yes	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	—	4	1	no	—
122	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
129	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
130	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
131	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
132	C₂²×C₈	#41	1	1	yes	characteristic
133	C₂²×C₈	#41	1	1	yes	characteristic
134	C₂²×C₈	#41	1	1	yes	characteristic
135	C₂²×C₈	#41	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	8	no	—
137	C₄⋊Q₈	—	4	4	no	—
138	C₄⋊Q₈	—	4	4	no	—
139	C₄⋊Q₈	—	4	4	no	—
140	#125	—	2	8	no	—
141	#125	—	4	4	no	—
142	#125	—	4	4	no	—
143	#125	—	4	4	no	—
144	C₂²⋊Q₈	—	4	8	no	—
145	C₂²⋊Q₈	—	4	8	no	—
146	#117	—	2	1	no	—
147	#117	—	2	1	no	—
148	#117	—	2	2	no	—
149	#117	—	2	2	no	—
150	#117	—	2	2	no	—
151	#117	—	4	1	no	—
152	#117	—	4	1	no	—
153	#117	—	4	1	no	—
154	#117	—	4	1	no	—
155	#117	—	4	1	no	—
156	#117	C₂²⋊C₄	1	2	yes	—
157	#117	D₈	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	4	2	no	—
164	C₂×C₄⋊C₄	—	4	2	no	—
165	C₂×C₄⋊C₄	—	4	2	no	—
166	C₂×C₄⋊C₄	—	4	1	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	4	1	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
171	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	—	4	1	no	—
177	C₂×C₄²	—	4	1	no	—
178	M₃₂	—	2	2	no	—
179	M₃₂	—	2	2	no	—
180	M₃₂	—	2	2	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	—	4	1	no	—
183	M₃₂	—	4	1	no	—
184	M₃₂	—	4	1	no	—
185	M₃₂	—	4	1	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	4	1	no	—
189	C₂×C₁₆	—	4	1	no	—
190	C₂×C₁₆	—	4	1	no	—
191	C₂×C₁₆	—	4	1	no	—
192	#108	—	2	2	no	—
193	#108	—	2	2	no	—
194	#108	—	4	1	no	—
195	#108	—	4	1	no	—
196	#107	—	2	8	no	—
197	#107	—	2	2	no	—
198	#107	—	2	2	no	—
199	#107	—	2	2	no	—
200	#107	—	2	1	no	—
201	#107	—	2	1	no	—
202	#107	—	2	2	no	—
203	#107	—	4	1	no	—
204	#107	C₂²⋊C₄	1	4	yes	—
205	#107	C₂²⋊C₄	1	2	yes	—
206	#107	C₂²⋊C₄	1	2	yes	—
207	#106	—	2	8	no	—
208	#106	—	2	2	no	—
209	#106	—	2	2	no	—
210	#106	—	2	2	no	—
211	#106	—	4	1	no	—
212	#97	—	2	2	no	—
213	#97	—	2	2	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	—	2	2	no	—
216	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂×Q₈	—	4	8	no	—
218	C₂×Q₈	—	16	4	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	8	1	no	—
227	C₂²×C₄	—	16	2	no	—
228	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	#120	1	1	yes	characteristic
230	C₂²×C₄	#120	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
233	C₂²×C₄	#124	1	1	yes	characteristic
234	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
236	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
237	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
239	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂²×C₄	#137	1	1	yes	characteristic
241	C₂²×C₄	#137	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	8	1	no	—
248	M₁₆	—	8	1	no	—
249	M₁₆	#103	1	4	yes	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	8	1	no	—
255	C₂×C₈	—	8	1	no	—
256	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
259	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
260	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
261	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
262	C₂×C₈	#120	1	1	yes	characteristic
263	C₂×C₈	#120	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
265	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
266	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
272	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
273	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
274	C₂×C₈	#124	1	1	yes	characteristic
275	C₂×C₈	#124	1	1	yes	characteristic
276	C₂×C₈	#124	1	1	yes	characteristic
277	C₂×C₈	#124	1	1	yes	characteristic
278	C₂×C₈	#124	1	1	yes	characteristic
279	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
280	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
281	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
282	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
283	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	8	no	—
290	C₄⋊C₄	—	4	8	no	—
291	C₄⋊C₄	—	4	8	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	—	8	1	no	—
299	C₄⋊C₄	#102	1	2	yes	—
300	C₄⋊C₄	#102	1	2	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₂²⋊C₄	—	4	2	no	—
306	C₂²⋊C₄	—	8	1	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	4	2	no	—
310	C₄²	—	4	2	no	—
311	C₄²	—	4	2	no	—
312	C₄²	—	8	1	no	—
313	C₁₆	—	4	2	no	—
314	C₁₆	—	4	2	no	—
315	C₂³	—	4	1	no	—
316	C₂³	—	4	1	no	—
317	C₂³	#418	1	1	yes	characteristic
318	C₂³	#447	1	1	yes	characteristic
319	C₂³	#450	1	1	yes	characteristic
320	C₂³	#476	1	1	yes	characteristic
321	C₂³	#477	1	1	yes	characteristic
322	C₂³	#489	1	1	yes	characteristic
323	C₂³	#501	1	1	yes	characteristic
324	Q₈	—	16	8	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	16	2	no	—
337	C₂×C₄	—	16	2	no	—
338	C₂×C₄	—	16	1	no	—
339	C₂×C₄	—	16	1	no	—
340	C₂×C₄	#379	1	1	yes	characteristic
341	C₂×C₄	#391	1	1	yes	characteristic
342	C₂×C₄	#391	1	1	yes	characteristic
343	C₂×C₄	#393	1	1	yes	characteristic
344	C₂×C₄	#393	1	1	yes	characteristic
345	C₂×C₄	#395	1	1	yes	characteristic
346	C₂×C₄	#395	1	1	yes	characteristic
347	C₂×C₄	C₂×#102	1	1	yes	characteristic
348	C₂×C₄	C₂×#103	1	1	yes	characteristic
349	C₂×C₄	#419	1	1	yes	characteristic
350	C₂×C₄	#447	1	1	yes	characteristic
351	C₂×C₄	#448	1	1	yes	characteristic
352	C₂×C₄	#448	1	1	yes	characteristic
353	C₂×C₄	#450	1	1	yes	characteristic
354	C₂×C₄	#451	1	1	yes	characteristic
355	C₂×C₄	#451	1	1	yes	characteristic
356	C₂×C₄	#474	1	1	yes	characteristic
357	C₂×C₄	#474	1	1	yes	characteristic
358	C₂×C₄	#475	1	1	yes	characteristic
359	C₂×C₄	#475	1	1	yes	characteristic
360	C₂×C₄	#476	1	1	yes	characteristic
361	C₂×C₄	#477	1	1	yes	characteristic
362	C₂×C₄	#486	1	1	yes	characteristic
363	C₂×C₄	#487	1	1	yes	characteristic
364	C₂×C₄	#489	1	1	yes	characteristic
365	C₂×C₄	#498	1	1	yes	characteristic
366	C₂×C₄	#500	1	1	yes	characteristic
367	C₂×C₄	#501	1	1	yes	characteristic
368	C₈	—	2	4	no	—
369	C₈	—	8	2	no	—
370	C₈	#391	1	1	yes	characteristic
371	C₈	#391	1	1	yes	characteristic
372	C₈	#391	1	1	yes	characteristic
373	C₈	#391	1	1	yes	characteristic
374	C₈	#393	1	1	yes	characteristic
375	C₈	#393	1	1	yes	characteristic
376	C₈	#395	1	1	yes	characteristic
377	C₈	#395	1	1	yes	characteristic
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	#1786	1	1	yes	characteristic
389	C₂²	#1987	1	1	yes	characteristic
390	C₂²	#1993	1	1	yes	characteristic
391	C₂²	#2023	1	1	yes	characteristic
392	C₂²	#2024	1	1	yes	characteristic
393	C₂²	#2057	1	1	yes	characteristic
394	C₂²	#2058	1	1	yes	characteristic
395	C₄	—	4	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	8	4	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	2	no	—
400	C₄	—	16	2	no	—
401	C₄	#1446	1	1	yes	characteristic
402	C₄	#1787	1	1	yes	characteristic
403	C₄	#1988	1	1	yes	characteristic
404	C₄	#1990	1	1	yes	characteristic
405	C₄	#2023	1	1	yes	characteristic
406	C₄	#2024	1	1	yes	characteristic
407	C₄	#2057	1	1	yes	characteristic
408	C₄	#2058	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	#11384	1	1	yes	characteristic
416	C₂	#12750	1	1	yes	characteristic
417	C₂	#12751	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1786
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 ◁ C₂² ◁ 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
14. #3164	C₄	semidirect
58. C₂×M₃₂	Q₈	semidirect
59. C₂×M₃₂	Q₈	semidirect
73. C₂×#108	Q₈	semidirect
83. #427	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3164	2
#12569	2
C₂×#1387	2
#12567	2
#12557	2
#12554	2
#10872	2