Subgroups — #109168

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#7924	C₂	1	2	yes	maximal
5	#7882	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	2	yes	—
7	C₂×#431	C₂²	1	2	yes	—
8	C₂×#405	C₄	1	2	yes	—
9	C₂×#405	C₄	1	2	yes	—
10	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×#117	C₂²	1	1	yes	characteristic
13	#1726	—	2	1	no	—
14	#1726	—	2	1	no	—
15	C₄×#97	—	2	1	no	—
16	C₄²×C₈	—	2	1	no	—
17	#1564	—	2	1	no	—
18	#1564	—	2	1	no	—
19	#1564	—	2	1	no	—
20	#1564	—	2	1	no	—
21	#1564	—	2	1	no	—
22	#1564	—	2	1	no	—
23	#1564	C₄	1	2	yes	complemented
24	#1564	C₄	1	2	yes	complemented
25	C₂×#335	—	2	1	no	—
26	C₂×#335	C₄	1	2	yes	complemented
27	C₂×#334	—	2	1	no	—
28	C₂×#334	C₄	1	2	yes	complemented
29	#1282	—	2	4	no	—
30	#1278	—	2	4	no	—
31	C₂²×M₁₆	C₂×C₄	1	2	yes	—
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×#117	C₂×C₄	1	2	yes	—
35	C₂²×C₄²	C₂³	1	1	yes	characteristic
36	#431	—	2	2	no	—
37	#431	—	2	2	no	—
38	#431	—	2	2	no	—
39	#431	—	2	2	no	—
40	#431	—	2	2	no	—
41	#431	—	2	2	no	—
42	#431	—	2	2	no	—
43	#431	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₈	—	2	2	no	—
45	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₂×C₂²⋊C₈	—	2	2	no	—
47	C₂×C₂²⋊C₈	—	2	2	no	—
48	#405	—	2	4	no	—
49	#405	—	2	4	no	—
50	#405	—	2	4	no	—
51	#405	C₈	1	8	yes	complemented
52	C₄×M₁₆	—	2	1	no	—
53	C₄×M₁₆	—	2	1	no	—
54	C₄×M₁₆	D₄	1	2	yes	—
55	C₄×M₁₆	Q₈	1	2	yes	—
56	C₂×#97	—	2	2	no	—
57	C₂×#97	—	2	2	no	—
58	C₂×#97	—	2	1	no	—
59	C₂×C₄×C₈	—	2	2	no	—
60	C₂×C₄×C₈	—	2	2	no	—
61	C₂×C₄×C₈	—	2	1	no	—
62	C₂×C₄×C₈	—	2	2	no	—
63	C₂×C₄×C₈	—	2	2	no	—
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
76	C₄×C₄⋊C₄	—	2	1	no	—
77	C₄×C₄⋊C₄	—	2	1	no	—
78	C₄×C₂²⋊C₄	D₄	1	2	yes	—
79	C₄×C₂²⋊C₄	Q₈	1	2	yes	—
80	#376	—	2	1	no	—
81	C₄³	—	2	1	no	—
82	#335	—	2	2	no	—
83	#335	—	2	2	no	—
84	#335	—	2	2	no	—
85	#335	—	2	2	no	—
86	#334	—	2	2	no	—
87	#334	—	2	2	no	—
88	#334	—	2	2	no	—
89	#334	—	2	2	no	—
90	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₄	C₂×Q₈	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	C₄⋊C₄	1	4	yes	—
97	C₂×M₁₆	C₂×C₈	1	4	yes	—
98	C₂×M₁₆	M₁₆	1	4	yes	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	2	4	no	—
103	C₂²×C₈	—	4	2	no	—
104	C₂²×C₈	C₄²	1	1	yes	characteristic
105	C₂²×C₈	C₄²	1	1	yes	characteristic
106	C₂²×C₈	C₄²	1	1	yes	characteristic
107	C₂²×C₈	C₄²	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₈	1	4	yes	—
117	C₂²×C₈	M₁₆	1	4	yes	—
118	#117	—	2	2	no	—
119	#117	—	2	2	no	—
120	#117	C₂×C₈	1	4	yes	—
121	#117	M₁₆	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄²	C₂×D₄	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×Q₈	1	1	yes	characteristic
134	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
135	C₂×C₄²	#41	1	1	yes	characteristic
136	C₂×C₄²	#41	1	1	yes	characteristic
137	C₂×C₄²	#41	1	1	yes	characteristic
138	C₂×C₄²	#41	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	4	2	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	4	2	no	—
145	C₂²⋊C₈	—	4	2	no	—
146	C₂²⋊C₈	—	4	2	no	—
147	#97	—	2	8	no	—
148	#97	—	2	8	no	—
149	#97	—	2	2	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	8	no	—
152	C₄×C₈	—	2	8	no	—
153	C₄×C₈	—	2	4	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	4	2	no	—
157	C₄×C₈	—	4	2	no	—
158	C₄×C₈	—	4	2	no	—
159	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
168	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
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	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₂⁴	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	#117	1	1	yes	characteristic
186	C₂²×C₄	#117	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
188	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
189	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
191	C₂²×C₄	#121	1	1	yes	characteristic
192	C₂²×C₄	#121	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂²×C₄	#127	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₈	1	2	yes	—
200	C₂²×C₄	C₂×M₁₆	1	2	yes	—
201	C₂²×C₄	C₂×M₁₆	1	2	yes	—
202	C₂²×C₄	C₂×M₁₆	1	2	yes	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	C₄⋊C₈	1	8	yes	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	8	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	C₂×C₈	C₄×C₈	1	2	yes	—
231	C₂×C₈	C₄×C₈	1	2	yes	—
232	C₂×C₈	#97	1	2	yes	—
233	C₂×C₈	#97	1	2	yes	—
234	C₂×C₈	C₂²⋊C₈	1	4	yes	—
235	C₂×C₈	C₂²⋊C₈	1	4	yes	—
236	C₂×C₈	C₄⋊C₈	1	4	yes	—
237	C₂×C₈	C₄⋊C₈	1	4	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	C₄⋊C₈	1	8	yes	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	4	no	—
245	C₄²	—	4	4	no	—
246	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
248	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
249	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
250	C₄²	C₄×D₄	1	1	yes	characteristic
251	C₄²	C₄×D₄	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	#121	1	1	yes	characteristic
256	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
257	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
258	C₄²	#124	1	1	yes	characteristic
259	C₄²	#124	1	1	yes	characteristic
260	C₄²	#125	1	1	yes	characteristic
261	C₄²	#125	1	1	yes	characteristic
262	C₂³	—	4	1	no	—
263	C₂³	—	4	1	no	—
264	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
265	C₂³	#380	1	1	yes	characteristic
266	C₂³	#384	1	1	yes	characteristic
267	C₂³	C₂×C₄⋊C₈	1	2	yes	—
268	C₂³	#423	1	2	yes	—
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	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	C₂×#95	1	1	yes	characteristic
285	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
286	C₂×C₄	#380	1	1	yes	characteristic
287	C₂×C₄	#382	1	1	yes	characteristic
288	C₂×C₄	#389	1	1	yes	characteristic
289	C₂×C₄	#389	1	1	yes	characteristic
290	C₂×C₄	#389	1	1	yes	characteristic
291	C₂×C₄	#389	1	1	yes	characteristic
292	C₂×C₄	#390	1	1	yes	characteristic
293	C₂×C₄	#390	1	1	yes	characteristic
294	C₂×C₄	#391	1	1	yes	characteristic
295	C₂×C₄	#391	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
297	C₂×C₄	C₂×#97	1	1	yes	characteristic
298	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
299	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊C₈	1	2	yes	—
301	C₂×C₄	#407	1	2	yes	—
302	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
304	C₂×C₄	#423	1	2	yes	—
305	C₂×C₄	#423	1	2	yes	—
306	C₂×C₄	#431	1	2	yes	—
307	C₂×C₄	#431	1	2	yes	—
308	C₂×C₄	#432	1	2	yes	—
309	C₂×C₄	#432	1	2	yes	—
310	C₈	—	2	8	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	8	4	no	—
316	C₈	—	8	4	no	—
317	C₈	#336	1	2	yes	—
318	C₈	#336	1	2	yes	—
319	C₈	#336	1	2	yes	—
320	C₈	#336	1	2	yes	—
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	C₄×C₄⋊C₈	1	1	yes	characteristic
331	C₂²	#1767	1	1	yes	characteristic
332	C₂²	#1797	1	2	yes	—
333	C₂²	#1842	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	—	4	8	no	—
341	C₄	C₂×#336	1	1	yes	characteristic
342	C₄	#1728	1	1	yes	characteristic
343	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
344	C₄	#1767	1	1	yes	characteristic
345	C₄	#1797	1	2	yes	—
346	C₄	#1831	1	2	yes	—
347	C₄	#1953	1	2	yes	—
348	C₄	#1976	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	2	2	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	#10832	1	1	yes	characteristic
354	C₂	#10871	1	1	yes	characteristic
355	C₂	#10873	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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
23. #1564	C₄	semidirect
24. #1564	C₄	semidirect
26. C₂×#335	C₄	semidirect
28. C₂×#334	C₄	semidirect
51. #405	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#405	2
C₂×#1564	2
#7924	2
#7882	2