Subgroups — #183709

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#108	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	1	yes	maximal, characteristic
3	#17327	C₂	1	2	yes	maximal
4	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#8296	C₂	1	4	yes	maximal, complemented
6	#8293	C₂	1	2	yes	maximal, complemented
7	C₈×#108	C₂	1	2	yes	maximal, complemented
8	#7887	C₂	1	2	yes	maximal
9	#2920	C₂²	1	2	yes	—
10	C₂×#431	C₂²	1	1	yes	characteristic
11	C₂²×#108	C₄	1	2	yes	—
12	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×M₁₆	C₂²	1	2	yes	—
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×#108	C₄	1	2	yes	—
16	C₄×#108	C₄	1	2	yes	—
17	C₄×#108	C₂²	1	1	yes	characteristic
18	C₄×#108	C₂²	1	1	yes	characteristic
19	C₄×#108	C₂²	1	2	yes	—
20	C₂×#341	C₄	1	2	yes	—
21	#1568	C₄	1	2	yes	—
22	#1568	C₄	1	2	yes	—
23	#1567	C₄	1	2	yes	—
24	#1566	C₄	1	2	yes	—
25	#1564	C₂²	1	2	yes	—
26	#1564	C₂²	1	2	yes	—
27	#1453	—	2	4	no	—
28	#1451	—	2	2	no	—
29	#1451	C₂²	1	4	yes	—
30	C₈×M₁₆	—	2	2	no	—
31	C₈×M₁₆	C₄	1	2	yes	—
32	C₈×M₁₆	C₄	1	2	yes	—
33	C₈×M₁₆	C₄	1	2	yes	—
34	C₈×M₁₆	C₄	1	2	yes	—
35	C₈×M₁₆	C₂²	1	4	yes	—
36	C₂×#322	C₂²	1	2	yes	—
37	C₂×C₈²	C₂²	1	2	yes	—
38	#1278	C₂²	1	4	yes	—
39	#1278	C₂²	1	4	yes	—
40	C₂²×M₁₆	—	2	1	no	—
41	C₂²×M₁₆	C₂×C₄	1	4	yes	—
42	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄²	C₂³	1	1	yes	characteristic
45	#432	—	2	2	no	—
46	#432	C₂×C₄	1	4	yes	—
47	#431	—	2	2	no	—
48	#431	C₂×C₄	1	2	yes	—
49	#431	C₂×C₄	1	2	yes	—
50	#431	D₄	1	1	yes	characteristic
51	#431	D₄	1	1	yes	characteristic
52	#431	D₄	1	2	yes	—
53	#431	C₂³	1	4	yes	—
54	C₂×#108	—	2	8	no	—
55	C₂×#108	C₂×C₄	1	4	yes	—
56	C₂×#108	C₂×C₄	1	2	yes	—
57	C₂×#108	C₂×C₄	1	2	yes	—
58	#423	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₈	Q₈	1	2	yes	—
60	#407	—	2	2	no	—
61	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
62	C₄×M₁₆	—	2	2	no	—
63	C₄×M₁₆	—	2	2	no	—
64	C₄×M₁₆	—	2	2	no	—
65	C₄×M₁₆	—	2	1	no	—
66	C₄×M₁₆	—	2	1	no	—
67	C₄×M₁₆	C₂×C₄	1	4	yes	—
68	C₄×M₁₆	C₂×C₄	1	2	yes	—
69	C₄×M₁₆	C₂×C₄	1	2	yes	—
70	C₄×M₁₆	C₂³	1	2	yes	—
71	C₄×M₁₆	C₂³	1	2	yes	—
72	C₂×#97	—	2	1	no	—
73	C₂×#97	—	2	2	no	—
74	C₂×C₄×C₈	—	2	2	no	—
75	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂³	1	2	yes	—
86	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂³	1	2	yes	—
89	#341	C₂×C₄	1	2	yes	—
90	#341	C₂×C₄	1	4	yes	—
91	#341	C₂×C₄	1	2	yes	—
92	#335	C₂×C₄	1	4	yes	—
93	#335	C₂×C₄	1	4	yes	—
94	#334	C₂×C₄	1	4	yes	—
95	#334	C₂×C₄	1	4	yes	—
96	#322	—	2	4	no	—
97	#322	—	2	4	no	—
98	#322	—	2	4	no	—
99	#322	C₂×C₄	1	4	yes	—
100	#322	C₂×C₄	1	4	yes	—
101	C₈²	—	2	4	no	—
102	C₈²	C₂×C₄	1	4	yes	—
103	C₈²	C₂×C₄	1	4	yes	—
104	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
105	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
106	C₂³×C₄	#41	1	1	yes	characteristic
107	C₂×M₁₆	—	2	8	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	2	4	no	—
111	C₂×M₁₆	—	2	4	no	—
112	C₂×M₁₆	—	2	8	no	—
113	C₂×M₁₆	—	4	4	no	—
114	C₂×M₁₆	C₂²×C₄	1	4	yes	—
115	C₂×M₁₆	C₂²×C₄	1	4	yes	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	4	2	no	—
120	C₂²×C₈	C₄²	1	1	yes	characteristic
121	C₂²×C₈	C₄²	1	1	yes	characteristic
122	C₂²×C₈	C₄²	1	1	yes	characteristic
123	C₂²×C₈	C₄²	1	1	yes	characteristic
124	C₂²×C₈	C₄²	1	1	yes	characteristic
125	C₂²×C₈	C₄²	1	1	yes	characteristic
126	C₂²×C₈	C₄²	1	1	yes	characteristic
127	C₂²×C₈	C₄²	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₈	C₂×Q₈	1	4	yes	—
133	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
141	C₂×C₄²	#41	1	1	yes	characteristic
142	C₂×C₄²	#41	1	1	yes	characteristic
143	C₂×C₄²	#41	1	1	yes	characteristic
144	C₂×C₄²	C₂⁴	1	1	yes	characteristic
145	#108	—	2	8	no	—
146	#108	—	2	4	no	—
147	#108	M₁₆	1	8	yes	complemented
148	C₄⋊C₈	—	2	4	no	—
149	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
150	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
151	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
152	C₄⋊C₈	C₂²×C₄	1	8	yes	—
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
156	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
157	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
158	#97	—	2	2	no	—
159	#97	—	2	2	no	—
160	#97	—	2	4	no	—
161	#97	—	2	4	no	—
162	#97	—	2	2	no	—
163	#97	—	2	2	no	—
164	C₄×C₈	—	2	4	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	4	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₄²	1	1	yes	characteristic
170	C₄×C₈	C₄²	1	1	yes	characteristic
171	C₄×C₈	C₄²	1	1	yes	characteristic
172	C₄×C₈	C₄²	1	1	yes	characteristic
173	C₄×C₈	C₄²	1	1	yes	characteristic
174	C₄×C₈	C₄²	1	1	yes	characteristic
175	C₄×C₈	C₄²	1	1	yes	characteristic
176	C₄×C₈	C₄²	1	1	yes	characteristic
177	C₄×C₈	C₂²×C₄	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
183	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
184	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
185	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
186	C₄×C₈	C₂²×C₄	1	4	yes	—
187	C₄×C₈	C₂×D₄	1	2	yes	—
188	C₄×C₈	C₂×D₄	1	2	yes	—
189	C₂⁴	C₄×Q₈	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	#117	1	1	yes	characteristic
199	C₂²×C₄	#117	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	C₄×D₄	1	1	yes	characteristic
203	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
206	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
208	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
209	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
211	C₂²×C₄	C₂×#41	1	1	yes	characteristic
212	M₁₆	—	2	8	no	—
213	M₁₆	—	2	8	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	8	no	—
218	M₁₆	—	4	8	no	—
219	M₁₆	C₂×M₁₆	1	8	yes	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	—	4	8	no	—
238	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
247	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
248	C₂×C₈	C₂×M₁₆	1	4	yes	—
249	C₄²	C₂×C₄²	1	1	yes	characteristic
250	C₄²	C₂×C₄²	1	1	yes	characteristic
251	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
252	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
253	C₄²	#117	1	1	yes	characteristic
254	C₄²	#117	1	1	yes	characteristic
255	C₄²	C₄×D₄	1	1	yes	characteristic
256	C₄²	C₄×D₄	1	1	yes	characteristic
257	C₄²	C₄×D₄	1	1	yes	characteristic
258	C₄²	C₄×D₄	1	1	yes	characteristic
259	C₄²	C₄×Q₈	1	1	yes	characteristic
260	C₄²	C₄×Q₈	1	1	yes	characteristic
261	C₄²	C₂³×C₄	1	1	yes	characteristic
262	C₄²	C₂³×C₄	1	1	yes	characteristic
263	C₄²	C₂²×D₄	1	1	yes	characteristic
264	C₄²	C₂×#41	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	4	1	no	—
268	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
269	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
270	C₂³	C₂×C₄×Q₈	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
295	C₂×C₄	#443	1	4	yes	—
296	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₂×#117	1	1	yes	characteristic
299	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
302	C₂×C₄	C₂²×M₁₆	1	2	yes	—
303	C₂×C₄	#568	1	2	yes	—
304	C₈	—	2	8	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	4	no	—
312	C₈	—	2	4	no	—
313	C₈	—	2	8	no	—
314	C₈	—	4	4	no	—
315	C₈	—	4	8	no	—
316	C₈	C₄×M₁₆	1	2	yes	—
317	C₈	C₄×M₁₆	1	2	yes	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	4	no	—
326	C₂²	#1772	1	2	yes	—
327	C₂²	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
328	C₂²	Q₈×M₁₆	1	2	yes	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	#1772	1	2	yes	—
337	C₄	C₂×C₄×M₁₆	1	1	yes	characteristic
338	C₄	#2873	1	1	yes	characteristic
339	C₄	#2916	1	2	yes	—
340	C₄	D₄×M₁₆	1	2	yes	—
341	C₄	C₂×#443	1	2	yes	—
342	C₂	—	2	2	no	—
343	C₂	—	2	2	no	—
344	C₂	—	4	1	no	—
345	C₂	C₄⋊C₄×M₁₆	1	1	yes	characteristic
346	C₂	C₂×#1772	1	1	yes	characteristic
347	C₂	#20460	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #8296	C₂	semidirect
6. #8293	C₂	semidirect
7. C₈×#108	C₂	semidirect
147. #108	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#108	2
#17793	2
#17327	2
C₂×C₈×M₁₆	2
#8296	2
#8293	2
C₈×#108	2
#7887	2