Subgroups — #109296

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20797	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	1	yes	maximal, characteristic
3	#17323	C₂	1	1	yes	maximal, characteristic
4	#8109	C₂	1	2	yes	maximal
5	#7953	C₂	1	2	yes	maximal
6	#2920	C₂²	1	2	yes	—
7	#2920	C₂²	1	1	yes	characteristic
8	#2920	C₂²	1	1	yes	characteristic
9	C₂×#427	C₄	1	2	yes	—
10	C₂×#427	C₄	1	2	yes	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#2285	C₂²	1	2	yes	—
13	#1846	—	2	1	no	—
14	#1846	—	2	1	no	—
15	#1845	—	2	1	no	—
16	#1844	—	2	1	no	—
17	#1568	—	2	1	no	—
18	#1568	—	2	1	no	—
19	#1567	C₄	1	2	yes	complemented
20	#1566	C₄	1	2	yes	complemented
21	#1564	—	2	1	no	—
22	#1564	—	2	1	no	—
23	#1452	—	2	1	no	—
24	#1451	—	2	1	no	—
25	#1451	C₄	1	2	yes	complemented
26	#1450	—	2	1	no	—
27	C₈×M₁₆	—	2	1	no	—
28	C₈×M₁₆	C₄	1	2	yes	complemented
29	#1305	—	2	4	no	—
30	#1285	—	2	4	no	—
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×#117	C₂×C₄	1	4	yes	—
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#432	—	2	1	no	—
36	#432	—	2	2	no	—
37	#432	—	2	2	no	—
38	#432	D₄	1	2	yes	—
39	#432	Q₈	1	2	yes	—
40	#432	Q₈	1	2	yes	—
41	#431	—	2	1	no	—
42	#431	—	2	1	no	—
43	#431	—	2	1	no	—
44	#431	—	2	2	no	—
45	#431	—	2	2	no	—
46	#431	C₂×C₄	1	2	yes	—
47	#427	—	4	4	no	—
48	#427	—	4	4	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#107	—	2	2	no	—
51	C₂×#106	—	2	2	no	—
52	C₂×#106	—	2	2	no	—
53	#423	—	2	1	no	—
54	#423	—	2	1	no	—
55	#423	C₂×C₄	1	4	yes	—
56	#407	—	2	1	no	—
57	#407	—	2	2	no	—
58	#407	D₄	1	2	yes	—
59	C₄×M₁₆	—	2	1	no	—
60	C₄×M₁₆	—	2	2	no	—
61	C₄×M₁₆	C₂×C₄	1	2	yes	—
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
74	#384	—	2	2	no	—
75	#382	—	2	2	no	—
76	#380	—	2	2	no	—
77	#380	—	2	2	no	—
78	C₄×C₄⋊C₄	—	2	2	no	—
79	C₄×C₄⋊C₄	—	2	2	no	—
80	#335	—	2	2	no	—
81	#335	—	2	2	no	—
82	#334	—	2	2	no	—
83	#334	—	2	2	no	—
84	#322	—	2	2	no	—
85	#322	—	2	2	no	—
86	#322	—	2	2	no	—
87	C₈²	—	2	2	no	—
88	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₄	#41	1	1	yes	characteristic
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	2	4	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	C₄²	1	1	yes	characteristic
99	C₂²×C₈	C₄²	1	1	yes	characteristic
100	C₂²×C₈	C₄²	1	1	yes	characteristic
101	C₂²×C₈	C₄²	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	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
110	#117	—	4	4	no	—
111	#117	—	4	4	no	—
112	C₂×C₄⋊C₄	—	2	4	no	—
113	C₂×C₄⋊C₄	—	4	2	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₂²⋊C₄	—	2	4	no	—
116	C₂×C₄²	—	2	4	no	—
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
124	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
125	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
126	C₂×C₄²	#41	1	1	yes	characteristic
127	C₂×C₄²	#41	1	1	yes	characteristic
128	C₂×C₄²	#41	1	1	yes	characteristic
129	#107	—	4	4	no	—
130	#107	—	4	4	no	—
131	#106	—	4	4	no	—
132	#106	—	4	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	—	2	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	4	2	no	—
143	C₂²⋊C₈	—	4	2	no	—
144	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
145	#97	—	2	2	no	—
146	#97	—	4	2	no	—
147	#97	C₄⋊C₄	1	4	yes	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	4	2	no	—
151	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊C₄	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	C₄⋊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	#95	—	4	2	no	—
168	#95	—	4	2	no	—
169	C₂⁴	C₄×D₄	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	C₂×C₄²	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×Q₈	1	1	yes	characteristic
185	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
187	C₂²×C₄	#121	1	1	yes	characteristic
188	C₂²×C₄	#121	1	1	yes	characteristic
189	C₂²×C₄	#121	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	M₁₆	—	4	4	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	8	4	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	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	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	#95	1	1	yes	characteristic
214	C₂×C₈	#95	1	1	yes	characteristic
215	C₂×C₈	#95	1	1	yes	characteristic
216	C₂×C₈	#95	1	1	yes	characteristic
217	C₂×C₈	#95	1	1	yes	characteristic
218	C₂×C₈	#95	1	1	yes	characteristic
219	C₂×C₈	#95	1	1	yes	characteristic
220	C₂×C₈	#95	1	1	yes	characteristic
221	C₂×C₈	#100	1	2	yes	—
222	C₂×C₈	#100	1	2	yes	—
223	C₂×C₈	#101	1	2	yes	—
224	C₂×C₈	#101	1	2	yes	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	8	4	no	—
228	C₄⋊C₄	—	8	4	no	—
229	C₂²⋊C₄	—	8	4	no	—
230	C₄²	—	8	4	no	—
231	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
234	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₄²	#117	1	1	yes	characteristic
237	C₄²	C₄×D₄	1	1	yes	characteristic
238	C₄²	C₄×Q₈	1	1	yes	characteristic
239	C₄²	#121	1	1	yes	characteristic
240	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
241	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
242	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
243	C₄²	#124	1	1	yes	characteristic
244	C₄²	#125	1	1	yes	characteristic
245	C₄²	C₄⋊Q₈	1	1	yes	characteristic
246	C₄²	C₄⋊Q₈	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
252	C₂³	#389	1	1	yes	characteristic
253	C₂³	#390	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	C₂×#95	1	1	yes	characteristic
280	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₄	#380	1	1	yes	characteristic
282	C₂×C₄	#380	1	1	yes	characteristic
283	C₂×C₄	#382	1	1	yes	characteristic
284	C₂×C₄	#384	1	1	yes	characteristic
285	C₂×C₄	#389	1	1	yes	characteristic
286	C₂×C₄	#389	1	1	yes	characteristic
287	C₂×C₄	#389	1	1	yes	characteristic
288	C₂×C₄	#390	1	1	yes	characteristic
289	C₂×C₄	#391	1	1	yes	characteristic
290	C₂×C₄	#391	1	1	yes	characteristic
291	C₂×C₄	C₂×#100	1	1	yes	characteristic
292	C₂×C₄	C₂×#100	1	1	yes	characteristic
293	C₂×C₄	C₂×#101	1	1	yes	characteristic
294	C₂×C₄	C₂×#101	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	4	4	no	—
300	C₈	—	4	4	no	—
301	C₈	—	8	4	no	—
302	C₈	#343	1	2	yes	—
303	C₈	#343	1	2	yes	—
304	C₈	#343	1	2	yes	—
305	C₈	#343	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₄⋊#95	1	1	yes	characteristic
317	C₂²	#1965	1	1	yes	characteristic
318	C₂²	#1966	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	16	4	no	—
328	C₄	C₂×#343	1	1	yes	characteristic
329	C₄	C₂×#343	1	1	yes	characteristic
330	C₄	#1833	1	2	yes	—
331	C₄	#1976	1	2	yes	—
332	C₄	#1997	1	1	yes	characteristic
333	C₄	#1997	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	4	1	no	—
337	C₂	—	4	1	no	—
338	C₂	#10857	1	1	yes	characteristic
339	C₂	#10868	1	1	yes	characteristic
340	C₂	#10871	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
19. #1567	C₄	semidirect
20. #1566	C₄	semidirect
25. #1451	C₄	semidirect
28. C₈×M₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20797	2
#17793	2
#17323	2
#8109	2
#7953	2