Subgroups — #109419

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20760	C₂	1	1	yes	maximal, characteristic
2	#20437	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#8095	C₂	1	2	yes	maximal
5	#7981	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	2	yes	—
7	C₂×#423	C₂²	1	1	yes	characteristic
8	C₂×#405	—	2	1	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2285	C₂²	1	2	yes	—
11	#2285	C₂²	1	1	yes	characteristic
12	#1835	C₄	1	2	yes	complemented
13	#1835	C₄	1	2	yes	complemented
14	#1834	—	2	1	no	—
15	#1834	—	2	1	no	—
16	#1770	—	2	1	no	—
17	#1770	C₄	1	2	yes	complemented
18	C₈×C₄⋊C₄	—	2	1	no	—
19	C₈×C₄⋊C₄	C₄	1	2	yes	complemented
20	#1735	—	2	1	no	—
21	#1735	—	2	1	no	—
22	#1729	—	2	1	no	—
23	#1564	—	2	1	no	—
24	#1564	—	2	1	no	—
25	#1564	—	2	1	no	—
26	#1564	—	2	1	no	—
27	C₂×#335	C₄	1	2	yes	complemented
28	C₂×#334	C₄	1	2	yes	complemented
29	#1303	—	2	4	no	—
30	#1289	—	2	4	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂³×C₈	D₄	1	1	yes	characteristic
33	C₂³×C₈	D₄	1	1	yes	characteristic
34	C₂×#117	—	2	1	no	—
35	C₂×#117	—	2	2	no	—
36	C₂²×C₄²	C₂³	1	1	yes	characteristic
37	#431	—	2	2	no	—
38	#431	—	2	2	no	—
39	#431	—	2	2	no	—
40	#431	—	2	2	no	—
41	#423	—	2	1	no	—
42	#423	—	2	1	no	—
43	#423	D₄	1	2	yes	—
44	#423	Q₈	1	2	yes	—
45	C₂×C₄⋊C₈	—	2	1	no	—
46	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₈	—	2	2	no	—
49	#405	—	4	4	no	—
50	C₂×#97	—	2	2	no	—
51	C₂×C₄×C₈	—	2	2	no	—
52	C₂×C₄×C₈	—	2	2	no	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	D₄	1	1	yes	characteristic
60	C₂×C₄×C₈	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	D₄	1	1	yes	characteristic
63	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
65	#384	—	2	2	no	—
66	#384	—	2	1	no	—
67	#382	C₂×C₄	1	4	yes	—
68	#382	Q₈	1	2	yes	—
69	#380	—	2	1	no	—
70	#380	—	2	2	no	—
71	#380	—	2	2	no	—
72	#380	D₄	1	2	yes	—
73	C₄×C₄⋊C₄	—	2	1	no	—
74	C₄×C₄⋊C₄	—	2	2	no	—
75	C₄×C₄⋊C₄	—	2	2	no	—
76	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
77	#340	—	2	2	no	—
78	#340	—	2	2	no	—
79	#340	—	2	2	no	—
80	#340	—	2	2	no	—
81	#336	—	2	2	no	—
82	#336	—	2	2	no	—
83	#335	—	2	2	no	—
84	#335	—	2	2	no	—
85	#334	—	2	2	no	—
86	#334	—	2	2	no	—
87	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂³×C₄	#41	1	1	yes	characteristic
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	4	2	no	—
93	C₂×M₁₆	—	4	2	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	—	2	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	2	4	no	—
100	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
112	#117	—	4	2	no	—
113	#117	—	4	2	no	—
114	#117	—	4	2	no	—
115	#117	—	4	2	no	—
116	#117	—	4	2	no	—
117	#117	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	4	2	no	—
123	C₂×C₄⋊C₄	—	4	2	no	—
124	C₂×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₂²⋊C₄	—	4	2	no	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	—	4	2	no	—
128	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
135	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
136	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
137	C₂×C₄²	#41	1	1	yes	characteristic
138	C₂×C₄²	#41	1	1	yes	characteristic
139	C₂×C₄²	#41	1	1	yes	characteristic
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
146	C₂²⋊C₈	—	4	2	no	—
147	C₂²⋊C₈	—	4	2	no	—
148	#97	—	4	4	no	—
149	C₄×C₈	—	4	4	no	—
150	C₄×C₈	—	4	2	no	—
151	C₄×C₈	—	4	2	no	—
152	C₄×C₈	C₄²	1	1	yes	characteristic
153	C₄×C₈	C₄²	1	1	yes	characteristic
154	C₄×C₈	C₄²	1	1	yes	characteristic
155	C₄×C₈	C₄²	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
167	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
168	#95	—	2	2	no	—
169	#95	—	2	4	no	—
170	#95	—	2	4	no	—
171	#95	C₄⋊C₄	1	4	yes	—
172	C₂⁴	#121	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	#117	1	1	yes	characteristic
182	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
183	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
184	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₄	#121	1	1	yes	characteristic
187	C₂²×C₄	#121	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	#124	1	1	yes	characteristic
192	C₂²×C₄	#125	1	1	yes	characteristic
193	C₂²×C₄	#127	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
195	C₂²×C₄	#135	1	2	yes	—
196	C₂²×C₄	#135	1	2	yes	—
197	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
198	C₂²×C₄	#137	1	2	yes	—
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	8	4	no	—
222	C₂×C₈	#95	1	1	yes	characteristic
223	C₂×C₈	#95	1	1	yes	characteristic
224	C₂×C₈	#95	1	1	yes	characteristic
225	C₂×C₈	#95	1	1	yes	characteristic
226	C₂×C₈	#95	1	1	yes	characteristic
227	C₂×C₈	#95	1	1	yes	characteristic
228	C₂×C₈	#95	1	1	yes	characteristic
229	C₂×C₈	#95	1	1	yes	characteristic
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	8	4	no	—
236	C₄⋊C₄	—	8	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	8	2	no	—
239	C₂²⋊C₄	—	8	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	8	2	no	—
243	C₄²	—	8	2	no	—
244	C₄²	C₂×C₄²	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
248	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
249	C₄²	#117	1	1	yes	characteristic
250	C₄²	C₄×D₄	1	1	yes	characteristic
251	C₄²	C₄×Q₈	1	1	yes	characteristic
252	C₄²	C₄×Q₈	1	1	yes	characteristic
253	C₄²	C₄×Q₈	1	1	yes	characteristic
254	C₄²	#121	1	1	yes	characteristic
255	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
256	C₄²	#124	1	1	yes	characteristic
257	C₄²	#125	1	1	yes	characteristic
258	C₄²	C₄⋊Q₈	1	1	yes	characteristic
259	C₄²	C₄⋊Q₈	1	1	yes	characteristic
260	C₂³	—	4	1	no	—
261	C₂³	—	4	1	no	—
262	C₂³	#380	1	1	yes	characteristic
263	C₂³	#389	1	1	yes	characteristic
264	C₂³	#390	1	1	yes	characteristic
265	C₂³	#463	1	2	yes	—
266	C₂³	#464	1	2	yes	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	8	4	no	—
283	C₂×C₄	—	8	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	C₂×#95	1	1	yes	characteristic
286	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
288	C₂×C₄	#380	1	1	yes	characteristic
289	C₂×C₄	#382	1	1	yes	characteristic
290	C₂×C₄	#384	1	1	yes	characteristic
291	C₂×C₄	#389	1	1	yes	characteristic
292	C₂×C₄	#389	1	1	yes	characteristic
293	C₂×C₄	#389	1	1	yes	characteristic
294	C₂×C₄	#390	1	1	yes	characteristic
295	C₂×C₄	#391	1	1	yes	characteristic
296	C₂×C₄	#391	1	1	yes	characteristic
297	C₂×C₄	#416	1	1	yes	characteristic
298	C₂×C₄	#416	1	1	yes	characteristic
299	C₂×C₄	#416	1	2	yes	—
300	C₂×C₄	#417	1	2	yes	—
301	C₂×C₄	#418	1	1	yes	characteristic
302	C₂×C₄	#419	1	1	yes	characteristic
303	C₂×C₄	#427	1	2	yes	—
304	C₂×C₄	#428	1	2	yes	—
305	C₂×C₄	#463	1	2	yes	—
306	C₂×C₄	#464	1	2	yes	—
307	C₂×C₄	#478	1	2	yes	—
308	C₂×C₄	#478	1	2	yes	—
309	C₂×C₄	#479	1	2	yes	—
310	C₂×C₄	#479	1	2	yes	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	4	4	no	—
320	C₈	—	8	4	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	C₄⋊#95	1	1	yes	characteristic
330	C₂²	#1804	1	1	yes	characteristic
331	C₂²	#1805	1	1	yes	characteristic
332	C₂²	#1949	1	2	yes	—
333	C₂²	#1983	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	4	no	—
341	C₄	—	16	4	no	—
342	C₄	#1735	1	1	yes	characteristic
343	C₄	#1736	1	1	yes	characteristic
344	C₄	#1804	1	1	yes	characteristic
345	C₄	#1805	1	1	yes	characteristic
346	C₄	#1833	1	2	yes	—
347	C₄	#1949	1	2	yes	—
348	C₄	#1976	1	2	yes	—
349	C₄	#1989	1	2	yes	—
350	C₂	—	2	1	no	—
351	C₂	—	2	2	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	#10850	1	1	yes	characteristic
355	C₂	#10868	1	1	yes	characteristic
356	C₂	#10871	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #1835	C₄	semidirect
13. #1835	C₄	semidirect
17. #1770	C₄	semidirect
19. C₈×C₄⋊C₄	C₄	semidirect
27. C₂×#335	C₄	semidirect
28. C₂×#334	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20760	2
#20437	2
C₂×#1564	2
#8095	2
#7981	2