Subgroups — #203963

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20826	C₂	1	1	yes	maximal, characteristic
2	#20437	C₂	1	1	yes	maximal, characteristic
3	#17820	C₂	1	2	yes	maximal
4	#17793	C₂	1	1	yes	maximal, characteristic
5	#9632	C₂	1	4	yes	maximal, complemented
6	#9628	C₂	1	2	yes	maximal, complemented
7	#9627	C₂	1	2	yes	maximal, complemented
8	#8094	C₂	1	2	yes	maximal
9	#2920	C₂²	1	2	yes	—
10	#2920	C₂²	1	2	yes	—
11	C₂×#423	C₂²	1	1	yes	characteristic
12	C₂×#405	—	2	1	no	—
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#2285	C₂²	1	1	yes	characteristic
15	#1848	C₂²	1	1	yes	characteristic
16	#1848	C₂²	1	1	yes	characteristic
17	#1837	C₂²	1	2	yes	—
18	#1810	—	2	1	no	—
19	#1810	—	2	1	no	—
20	C₄×#108	—	2	1	no	—
21	C₄×#108	—	2	1	no	—
22	#1770	—	2	1	no	—
23	#1770	C₂²	1	2	yes	—
24	C₈×C₄⋊C₄	—	2	1	no	—
25	C₈×C₄⋊C₄	C₂²	1	2	yes	—
26	#1729	—	2	1	no	—
27	#1574	—	2	4	no	—
28	#1571	C₂²	1	4	yes	—
29	#1570	C₂²	1	4	yes	—
30	#1569	—	2	4	no	—
31	#1568	C₄	1	2	yes	complemented
32	#1568	C₄	1	2	yes	complemented
33	#1567	C₄	1	2	yes	complemented
34	#1566	C₄	1	2	yes	complemented
35	#1564	C₂²	1	2	yes	—
36	#1564	C₂²	1	2	yes	—
37	#1303	C₂²	1	4	yes	—
38	#1303	C₂²	1	4	yes	—
39	C₂²×M₁₆	—	2	1	no	—
40	C₂³×C₈	D₄	1	1	yes	characteristic
41	C₂³×C₈	D₄	1	1	yes	characteristic
42	C₂×#117	—	2	1	no	—
43	C₂²×C₄²	C₂³	1	1	yes	characteristic
44	#432	—	2	2	no	—
45	#432	—	2	2	no	—
46	#432	—	2	2	no	—
47	#432	C₂×C₄	1	4	yes	—
48	#431	—	2	2	no	—
49	#431	—	2	2	no	—
50	#431	—	2	2	no	—
51	#431	C₂³	1	4	yes	—
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	#423	—	2	1	no	—
57	#423	—	2	1	no	—
58	#423	C₂×C₄	1	4	yes	—
59	#423	D₄	1	2	yes	—
60	#423	D₄	1	2	yes	—
61	#423	C₂³	1	2	yes	—
62	#423	C₂³	1	2	yes	—
63	C₂×C₄⋊C₈	—	2	1	no	—
64	C₂×C₄⋊C₈	C₂³	1	2	yes	—
65	#407	—	2	2	no	—
66	#407	—	2	2	no	—
67	#405	—	4	4	no	—
68	C₄×M₁₆	—	2	2	no	—
69	C₄×M₁₆	—	2	2	no	—
70	C₂×#97	—	2	2	no	—
71	C₂×C₄×C₈	—	2	2	no	—
72	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈	D₄	1	1	yes	characteristic
77	C₂×C₄×C₈	D₄	1	1	yes	characteristic
78	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
79	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
81	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
84	#384	—	2	1	no	—
85	#382	Q₈	1	2	yes	—
86	#380	—	2	1	no	—
87	#380	Q₈	1	2	yes	—
88	C₄×C₄⋊C₄	—	2	1	no	—
89	C₄×C₄⋊C₄	C₂³	1	2	yes	—
90	#341	—	2	2	no	—
91	#341	—	2	2	no	—
92	#336	—	2	2	no	—
93	#336	—	2	2	no	—
94	#335	—	2	8	no	—
95	#335	C₂×C₄	1	4	yes	—
96	#335	C₂×C₄	1	4	yes	—
97	#334	—	2	8	no	—
98	#334	C₂×C₄	1	4	yes	—
99	#334	C₂×C₄	1	4	yes	—
100	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
102	C₂³×C₄	#41	1	1	yes	characteristic
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	—	2	4	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	—	4	2	no	—
109	C₂×M₁₆	—	4	2	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	—	4	1	no	—
115	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₈	C₄⋊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	#117	—	4	2	no	—
128	#117	—	4	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₂²⋊C₄	—	2	2	no	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
140	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
141	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
142	C₂×C₄²	#41	1	1	yes	characteristic
143	C₂×C₄²	#41	1	1	yes	characteristic
144	C₂×C₄²	#41	1	1	yes	characteristic
145	C₂×C₄²	C₂⁴	1	1	yes	characteristic
146	#108	—	4	2	no	—
147	#108	—	4	2	no	—
148	#108	—	4	2	no	—
149	#108	—	4	2	no	—
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	C₂²×C₄	1	8	yes	—
159	C₄⋊C₈	C₂×D₄	1	4	yes	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	—	2	4	no	—
162	C₂²⋊C₈	—	4	2	no	—
163	C₂²⋊C₈	—	4	2	no	—
164	#97	—	2	4	no	—
165	#97	—	4	4	no	—
166	#97	—	4	2	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	4	4	no	—
169	C₄×C₈	—	4	2	no	—
170	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
171	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
172	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
174	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
175	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
176	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
178	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
179	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
180	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
181	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
182	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
183	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
184	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
185	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
186	#95	—	2	2	no	—
187	#95	C₂×Q₈	1	4	yes	—
188	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
202	C₂²×C₄	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂²×C₄	#125	1	1	yes	characteristic
208	C₂²×C₄	#125	1	1	yes	characteristic
209	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×#41	1	1	yes	characteristic
214	M₁₆	—	4	4	no	—
215	M₁₆	—	4	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	8	4	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
250	C₂×C₈	#137	1	2	yes	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	2	no	—
257	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
258	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
259	C₄²	C₄×D₄	1	1	yes	characteristic
260	C₄²	C₄×D₄	1	1	yes	characteristic
261	C₄²	C₄×Q₈	1	1	yes	characteristic
262	C₄²	C₄×Q₈	1	1	yes	characteristic
263	C₄²	#121	1	1	yes	characteristic
264	C₄²	#121	1	1	yes	characteristic
265	C₄²	#125	1	1	yes	characteristic
266	C₄²	#125	1	1	yes	characteristic
267	C₄²	C₄⋊Q₈	1	1	yes	characteristic
268	C₄²	C₄⋊Q₈	1	1	yes	characteristic
269	C₄²	C₂³×C₄	1	1	yes	characteristic
270	C₄²	C₂²×D₄	1	1	yes	characteristic
271	C₄²	C₂²×Q₈	1	1	yes	characteristic
272	C₄²	C₂×#41	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	1	no	—
277	C₂³	#389	1	1	yes	characteristic
278	C₂³	#389	1	1	yes	characteristic
279	C₂³	C₂×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	#389	1	1	yes	characteristic
306	C₂×C₄	#389	1	1	yes	characteristic
307	C₂×C₄	#389	1	1	yes	characteristic
308	C₂×C₄	#389	1	1	yes	characteristic
309	C₂×C₄	#389	1	1	yes	characteristic
310	C₂×C₄	#389	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
314	C₂×C₄	C₂×#121	1	1	yes	characteristic
315	C₂×C₄	C₂×#125	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
317	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂×C₄	C₂×#137	1	1	yes	characteristic
319	C₂×C₄	#575	1	1	yes	characteristic
320	C₂×C₄	#575	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	4	8	no	—
328	C₈	—	4	4	no	—
329	C₈	—	8	2	no	—
330	C₈	—	8	2	no	—
331	C₈	#428	1	2	yes	—
332	C₈	#428	1	2	yes	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	C₂×#389	1	1	yes	characteristic
344	C₂²	#3077	1	1	yes	characteristic
345	C₂²	#3078	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	8	4	no	—
355	C₄	#1953	1	2	yes	—
356	C₄	C₂×#428	1	1	yes	characteristic
357	C₄	#2911	1	1	yes	characteristic
358	C₄	#2957	1	2	yes	—
359	C₄	#3036	1	1	yes	characteristic
360	C₄	#3037	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	4	1	no	—
364	C₂	—	4	1	no	—
365	C₂	#21163	1	1	yes	characteristic
366	C₂	#21170	1	1	yes	characteristic
367	C₂	C₂×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #9632	C₂	semidirect
6. #9628	C₂	semidirect
7. #9627	C₂	semidirect
31. #1568	C₄	semidirect
32. #1568	C₄	semidirect
33. #1567	C₄	semidirect
34. #1566	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20826	2
#20437	2
#17820	2
#17793	2
#9632	2
#9628	2
#9627	2
#8094	2