Subgroups — #183719

Order 512.

699 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	#20826	C₂	1	1	yes	maximal, characteristic
2	C₂×#1564	C₂	1	1	yes	maximal, characteristic
3	#17327	C₂	1	2	yes	maximal
4	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#8297	C₂	1	4	yes	maximal, complemented
6	#8293	C₂	1	2	yes	maximal, complemented
7	C₈×#108	C₂	1	2	yes	maximal, complemented
8	#7884	C₂	1	2	yes	maximal
9	#2920	C₂²	1	2	yes	—
10	C₂×#431	C₂²	1	2	yes	—
11	C₂×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	#1848	C₄	1	2	yes	—
15	#1837	C₄	1	2	yes	—
16	#1810	C₄	1	2	yes	—
17	#1810	C₄	1	2	yes	—
18	C₄×#108	C₂²	1	2	yes	—
19	C₄×#108	C₂²	1	2	yes	—
20	#1564	C₄	1	2	yes	—
21	#1564	C₄	1	2	yes	—
22	#1564	C₂²	1	1	yes	characteristic
23	#1564	C₂²	1	1	yes	characteristic
24	#1564	C₂²	1	2	yes	—
25	C₂×#335	C₄	1	2	yes	—
26	C₂×#334	C₄	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	Q₈	1	2	yes	—
41	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄²	C₂³	1	1	yes	characteristic
44	#432	—	2	2	no	—
45	#432	—	2	2	no	—
46	#431	—	2	2	no	—
47	#431	—	2	2	no	—
48	#431	—	2	2	no	—
49	#431	—	2	2	no	—
50	#431	—	2	1	no	—
51	#431	—	2	1	no	—
52	#431	C₂×C₄	1	4	yes	—
53	#431	C₂×C₄	1	2	yes	—
54	#431	C₂×C₄	1	2	yes	—
55	#431	C₂³	1	2	yes	—
56	#431	C₂³	1	2	yes	—
57	C₂×#108	C₂×C₄	1	4	yes	—
58	C₂×#108	C₂×C₄	1	4	yes	—
59	#423	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₈	—	2	1	no	—
61	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
62	#407	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₈	—	2	1	no	—
64	C₂×C₂²⋊C₈	—	2	2	no	—
65	C₄×M₁₆	C₂×C₄	1	2	yes	—
66	C₄×M₁₆	C₂×C₄	1	2	yes	—
67	C₄×M₁₆	D₄	1	1	yes	characteristic
68	C₄×M₁₆	D₄	1	1	yes	characteristic
69	C₄×M₁₆	D₄	1	2	yes	—
70	C₄×M₁₆	C₂³	1	4	yes	—
71	C₂×#97	Q₈	1	2	yes	—
72	C₂×C₄×C₈	—	2	2	no	—
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	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	C₂³	1	2	yes	—
84	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂³	1	2	yes	—
87	#341	C₂×C₄	1	4	yes	—
88	#341	C₂×C₄	1	4	yes	—
89	#335	C₂×C₄	1	2	yes	—
90	#335	C₂×C₄	1	2	yes	—
91	#335	C₂×C₄	1	4	yes	—
92	#334	C₂×C₄	1	2	yes	—
93	#334	C₂×C₄	1	2	yes	—
94	#334	C₂×C₄	1	4	yes	—
95	#322	—	2	4	no	—
96	#322	—	2	4	no	—
97	#322	—	2	4	no	—
98	#322	C₂×C₄	1	4	yes	—
99	#322	C₂×C₄	1	4	yes	—
100	C₈²	—	2	4	no	—
101	C₈²	C₂×C₄	1	4	yes	—
102	C₈²	C₂×C₄	1	4	yes	—
103	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
104	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
105	C₂³×C₄	#41	1	1	yes	characteristic
106	C₂×M₁₆	—	2	8	no	—
107	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
108	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
109	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
110	C₂×M₁₆	C₂²×C₄	1	8	yes	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	4	no	—
114	C₂²×C₈	—	4	2	no	—
115	C₂²×C₈	C₄²	1	1	yes	characteristic
116	C₂²×C₈	C₄²	1	1	yes	characteristic
117	C₂²×C₈	C₄²	1	1	yes	characteristic
118	C₂²×C₈	C₄²	1	1	yes	characteristic
119	C₂²×C₈	C₄²	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₈	C₂×Q₈	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄²	C₂×Q₈	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	C₂×C₄²	C₂⁴	1	1	yes	characteristic
140	#108	—	2	8	no	—
141	#108	—	4	4	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	4	no	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	C₂²×C₄	1	4	yes	—
148	C₄⋊C₈	C₂²×C₄	1	4	yes	—
149	C₂²⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	2	4	no	—
152	C₂²⋊C₈	—	2	4	no	—
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	2	no	—
155	C₂²⋊C₈	—	2	2	no	—
156	#97	—	2	4	no	—
157	#97	C₄⋊C₄	1	2	yes	—
158	#97	C₄⋊C₄	1	4	yes	—
159	#97	C₄⋊C₄	1	2	yes	—
160	C₄×C₈	—	2	4	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	4	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	C₄²	1	1	yes	characteristic
166	C₄×C₈	C₄²	1	1	yes	characteristic
167	C₄×C₈	C₄²	1	1	yes	characteristic
168	C₄×C₈	C₄²	1	1	yes	characteristic
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	C₂²×C₄	1	4	yes	—
174	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
175	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
176	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
177	C₄×C₈	C₂²×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	C₂²×C₄	1	4	yes	—
183	C₄×C₈	C₂×D₄	1	2	yes	—
184	C₄×C₈	C₂×D₄	1	2	yes	—
185	C₂⁴	C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	#117	1	1	yes	characteristic
195	C₂²×C₄	#117	1	1	yes	characteristic
196	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₄×D₄	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	C₄×Q₈	1	1	yes	characteristic
203	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
204	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×#41	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	8	no	—
210	M₁₆	—	2	8	no	—
211	M₁₆	—	2	8	no	—
212	M₁₆	#108	1	8	yes	complemented
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	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	8	4	no	—
231	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
240	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
241	C₂×C₈	#131	1	4	yes	—
242	C₄²	C₂×C₄²	1	1	yes	characteristic
243	C₄²	C₂×C₄²	1	1	yes	characteristic
244	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₄²	#117	1	1	yes	characteristic
247	C₄²	#117	1	1	yes	characteristic
248	C₄²	C₄×D₄	1	1	yes	characteristic
249	C₄²	C₄×D₄	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	C₂³×C₄	1	1	yes	characteristic
255	C₄²	C₂³×C₄	1	1	yes	characteristic
256	C₄²	C₂²×D₄	1	1	yes	characteristic
257	C₄²	C₂×#41	1	1	yes	characteristic
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	4	1	no	—
261	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
262	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
263	C₂³	C₂×C₄×Q₈	1	1	yes	characteristic
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	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	—	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	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₄×C₄⋊C₄	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	C₂×#108	1	4	yes	—
289	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂×#117	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
295	C₂×C₄	C₂×#131	1	2	yes	—
296	C₂×C₄	#568	1	2	yes	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	4	no	—
304	C₈	—	4	4	no	—
305	C₈	—	4	8	no	—
306	C₈	—	4	4	no	—
307	C₈	—	4	4	no	—
308	C₈	#405	1	2	yes	—
309	C₈	#405	1	2	yes	—
310	C₈	C₂×#108	1	8	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	C₄×#108	1	2	yes	—
320	C₂²	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
321	C₂²	#2962	1	2	yes	—
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	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	C₄×#108	1	2	yes	—
330	C₄	C₂×#405	1	1	yes	characteristic
331	C₄	#2873	1	1	yes	characteristic
332	C₄	C₂²×#108	1	2	yes	—
333	C₄	#2935	1	2	yes	—
334	C₄	#2957	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	2	2	no	—
337	C₂	—	4	1	no	—
338	C₂	#20441	1	1	yes	characteristic
339	C₂	#20446	1	1	yes	characteristic
340	C₂	C₂×C₄×#108	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×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #8297	C₂	semidirect
6. #8293	C₂	semidirect
7. C₈×#108	C₂	semidirect
212. M₁₆	#108	semidirect

Maximal subgroups

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

subgroup	index
#20826	2
C₂×#1564	2
#17327	2
C₂×C₈×M₁₆	2
#8297	2
#8293	2
C₈×#108	2
#7884	2