Subgroups — #147213

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1945	C₂	1	1	yes	maximal, characteristic
2	C₂×#1772	C₂	1	1	yes	maximal, characteristic
3	C₂×#1382	C₂	1	1	yes	maximal, characteristic
4	C₂×#1382	C₂	1	1	yes	maximal, characteristic
5	C₂×#1381	C₂	1	1	yes	maximal, characteristic
6	#12556	C₂	1	1	yes	maximal, characteristic
7	C₂×#1374	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂²×#108	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	C₂×#405	—	2	1	no	—
16	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
17	#1945	—	2	2	no	—
18	#1945	C₄	1	4	yes	complemented
19	#1772	—	2	2	no	—
20	#1772	—	2	2	no	—
21	C₂×#339	—	2	1	no	—
22	C₂×#339	—	2	1	no	—
23	#1382	—	2	2	no	—
24	#1382	C₄	1	4	yes	complemented
25	#1381	—	2	2	no	—
26	#1380	C₄	1	2	yes	complemented
27	#1380	C₄	1	2	yes	complemented
28	#1374	—	2	2	no	—
29	C₂²×M₁₆	—	2	1	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	C₂²×C₄²	D₄	1	1	yes	characteristic
36	C₂²×C₁₆	—	2	1	no	—
37	C₂²×C₁₆	—	2	1	no	—
38	C₂²×C₁₆	—	2	1	no	—
39	C₂²×C₁₆	—	2	1	no	—
40	C₂²×C₁₆	—	2	1	no	—
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	C₂²×C₁₆	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	—	2	1	no	—
57	C₂×#108	—	2	1	no	—
58	C₂×#108	C₂×C₄	1	2	yes	—
59	C₂×#108	C₂×C₄	1	2	yes	—
60	C₂×#108	D₄	1	2	yes	—
61	C₂×#108	Q₈	1	2	yes	—
62	#427	—	2	1	no	—
63	#427	—	2	1	no	—
64	#427	—	2	1	no	—
65	#427	—	2	1	no	—
66	#427	C₂×C₄	1	2	yes	—
67	#427	C₂×C₄	1	2	yes	—
68	#427	D₄	1	2	yes	—
69	#427	Q₈	1	2	yes	—
70	C₂×#107	C₂×C₄	1	1	yes	characteristic
71	C₂×#107	C₂×C₄	1	1	yes	characteristic
72	C₂×#107	C₂×C₄	1	2	yes	—
73	C₂×#106	—	2	1	no	—
74	C₂×#106	—	2	1	no	—
75	#424	—	2	4	no	—
76	#424	—	4	2	no	—
77	C₂×C₄⋊C₈	—	4	1	no	—
78	C₂×C₄⋊C₈	—	4	1	no	—
79	#405	—	2	4	no	—
80	#405	—	4	2	no	—
81	C₂×#97	—	4	1	no	—
82	C₂×C₄×C₈	—	2	1	no	—
83	C₂×C₄×C₈	—	2	1	no	—
84	C₂×C₄×C₈	—	2	1	no	—
85	C₂×C₄×C₈	—	2	1	no	—
86	C₂×C₄×C₈	—	4	1	no	—
87	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
88	C₂×C₄×C₈	D₄	1	2	yes	—
89	#339	—	2	4	no	—
90	#339	—	2	4	no	—
91	C₂³×C₄	—	2	1	no	—
92	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	4	1	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	—	4	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	4	1	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
112	C₂²×C₈	D₈	1	1	yes	characteristic
113	C₂²×C₈	D₈	1	1	yes	characteristic
114	C₂²×C₈	SD₁₆	1	2	yes	—
115	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
122	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
123	C₂²×C₈	#41	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂²×C₈	#41	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	C₂²×C₈	#41	1	1	yes	characteristic
129	#117	—	2	2	no	—
130	#117	—	2	2	no	—
131	#117	—	2	2	no	—
132	#117	—	2	2	no	—
133	#117	—	4	1	no	—
134	#117	—	4	1	no	—
135	#117	—	4	1	no	—
136	#117	—	4	1	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₂²⋊C₄	—	4	1	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	4	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	4	1	no	—
157	C₂×C₁₆	—	4	1	no	—
158	C₂×C₁₆	—	4	1	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	#108	—	2	2	no	—
165	#108	—	2	2	no	—
166	#108	—	2	2	no	—
167	#108	—	2	2	no	—
168	#108	—	2	2	no	—
169	#108	—	4	1	no	—
170	#108	—	4	1	no	—
171	#108	C₄⋊C₄	1	4	yes	—
172	#107	—	2	1	no	—
173	#107	—	2	1	no	—
174	#107	—	2	2	no	—
175	#107	C₄²	1	2	yes	—
176	#107	C₄²	1	2	yes	—
177	#107	C₄⋊C₄	1	4	yes	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	#106	—	2	2	no	—
181	#106	—	2	2	no	—
182	C₄⋊C₈	—	4	4	no	—
183	C₄⋊C₈	—	4	4	no	—
184	#97	—	4	4	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	4	4	no	—
190	C₂⁴	#135	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
202	C₂²×C₄	#120	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
205	C₂²×C₄	#123	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
208	C₂²×C₄	#135	1	1	yes	characteristic
209	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂²×C₄	#137	1	1	yes	characteristic
211	C₂²×C₄	#137	1	1	yes	characteristic
212	C₂²×C₄	#137	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	8	1	no	—
220	M₁₆	—	8	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	8	1	no	—
234	C₂×C₈	—	8	1	no	—
235	C₂×C₈	#102	1	2	yes	—
236	C₂×C₈	#102	1	2	yes	—
237	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₈	#117	1	1	yes	characteristic
241	C₂×C₈	#117	1	1	yes	characteristic
242	C₂×C₈	#117	1	1	yes	characteristic
243	C₂×C₈	C₄×D₄	1	1	yes	characteristic
244	C₂×C₈	C₄×D₄	1	1	yes	characteristic
245	C₂×C₈	C₄×D₄	1	1	yes	characteristic
246	C₂×C₈	C₄×D₄	1	1	yes	characteristic
247	C₂×C₈	C₄×D₄	1	1	yes	characteristic
248	C₂×C₈	C₄×D₄	1	1	yes	characteristic
249	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
250	C₂×C₈	C₄×Q₈	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	#121	1	1	yes	characteristic
254	C₂×C₈	#121	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₈	#123	1	1	yes	characteristic
258	C₂×C₈	#123	1	1	yes	characteristic
259	C₂×C₈	#124	1	1	yes	characteristic
260	C₂×C₈	#125	1	1	yes	characteristic
261	C₂×C₈	#125	1	1	yes	characteristic
262	C₂×C₈	#126	1	1	yes	characteristic
263	C₂×C₈	#126	1	1	yes	characteristic
264	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	4	1	no	—
283	C₁₆	—	4	2	no	—
284	C₁₆	—	4	2	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	#416	1	1	yes	characteristic
288	C₂³	C₄×D₈	1	1	yes	characteristic
289	C₂³	C₄×SD₁₆	1	1	yes	characteristic
290	C₂³	#452	1	1	yes	characteristic
291	C₂³	#478	1	1	yes	characteristic
292	C₂³	#479	1	1	yes	characteristic
293	C₂³	#485	1	1	yes	characteristic
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄	#385	1	1	yes	characteristic
308	C₂×C₄	#385	1	1	yes	characteristic
309	C₂×C₄	#386	1	1	yes	characteristic
310	C₂×C₄	#387	1	1	yes	characteristic
311	C₂×C₄	#387	1	1	yes	characteristic
312	C₂×C₄	#391	1	1	yes	characteristic
313	C₂×C₄	C₂×#102	1	1	yes	characteristic
314	C₂×C₄	#417	1	1	yes	characteristic
315	C₂×C₄	#419	1	1	yes	characteristic
316	C₂×C₄	C₄×D₈	1	1	yes	characteristic
317	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
318	C₂×C₄	#440	1	1	yes	characteristic
319	C₂×C₄	#440	1	1	yes	characteristic
320	C₂×C₄	#441	1	1	yes	characteristic
321	C₂×C₄	#441	1	1	yes	characteristic
322	C₂×C₄	#447	1	1	yes	characteristic
323	C₂×C₄	#448	1	1	yes	characteristic
324	C₂×C₄	#452	1	1	yes	characteristic
325	C₂×C₄	#474	1	1	yes	characteristic
326	C₂×C₄	#474	1	1	yes	characteristic
327	C₂×C₄	#475	1	1	yes	characteristic
328	C₂×C₄	#475	1	1	yes	characteristic
329	C₂×C₄	#478	1	1	yes	characteristic
330	C₂×C₄	#479	1	1	yes	characteristic
331	C₂×C₄	#480	1	1	yes	characteristic
332	C₂×C₄	#481	1	1	yes	characteristic
333	C₂×C₄	#485	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	4	no	—
337	C₈	—	8	2	no	—
338	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
339	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
340	C₈	#387	1	1	yes	characteristic
341	C₈	#387	1	1	yes	characteristic
342	C₈	#388	1	1	yes	characteristic
343	C₈	#388	1	1	yes	characteristic
344	C₈	#389	1	1	yes	characteristic
345	C₈	#389	1	1	yes	characteristic
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	C₄×#102	1	1	yes	characteristic
355	C₂²	#1862	1	1	yes	characteristic
356	C₂²	#1865	1	1	yes	characteristic
357	C₂²	#1873	1	1	yes	characteristic
358	C₂²	#1918	1	1	yes	characteristic
359	C₂²	#1919	1	1	yes	characteristic
360	C₂²	#1989	1	1	yes	characteristic
361	C₂²	#2178	1	2	yes	—
362	C₂²	#2179	1	2	yes	—
363	C₄	—	4	4	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	4	no	—
367	C₄	#1436	1	1	yes	characteristic
368	C₄	#1763	1	1	yes	characteristic
369	C₄	#1862	1	1	yes	characteristic
370	C₄	#1865	1	1	yes	characteristic
371	C₄	#1874	1	1	yes	characteristic
372	C₄	#1918	1	1	yes	characteristic
373	C₄	#1919	1	1	yes	characteristic
374	C₄	#1988	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	#10884	1	1	yes	characteristic
380	C₂	#12641	1	1	yes	characteristic
381	C₂	#12642	1	1	yes	characteristic
382	C₂	#12939	1	2	yes	—
383	C₂	#13113	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ 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
18. #1945	C₄	semidirect
24. #1382	C₄	semidirect
26. #1380	C₄	semidirect
27. #1380	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1945	2
C₂×#1772	2
C₂×#1382	2
C₂×#1381	2
#12556	2
C₂×#1374	2