Subgroups — #497829

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33443	C₂	1	2	yes	maximal
2	#33401	C₂	1	2	yes	maximal, complemented
3	C₄×#443	C₂	1	1	yes	maximal, characteristic
4	#13164	C₂	1	2	yes	maximal, complemented
5	#12918	C₂	1	4	yes	maximal, complemented
6	#12900	C₂	1	1	yes	maximal, characteristic, complemented
7	#12792	C₂	1	2	yes	maximal, complemented
8	C₄×#364	C₂	1	1	yes	maximal, characteristic, complemented
9	#3407	C₂²	1	4	yes	—
10	C₂×#443	C₄	1	1	yes	characteristic
11	C₂×#443	C₂²	1	1	yes	characteristic
12	C₄×#135	C₄	1	1	yes	characteristic
13	C₄×#131	C₂²	1	2	yes	—
14	#2171	—	2	4	no	—
15	#2170	—	2	4	no	—
16	#2153	—	2	4	no	—
17	#2153	C₂²	1	4	yes	—
18	C₂×#364	—	2	4	no	—
19	C₂×#364	C₂²	1	2	yes	—
20	C₂×#364	C₂²	1	2	yes	—
21	C₂×#364	C₂²	1	1	yes	characteristic
22	C₂×#364	C₂²	1	1	yes	characteristic
23	#2150	C₂²	1	2	yes	—
24	#2117	C₂²	1	4	yes	—
25	C₄×M₃₂	C₂²	1	2	yes	—
26	C₄×#108	C₄	1	1	yes	characteristic
27	#1772	C₄	1	1	yes	characteristic
28	#1772	C₂²	1	1	yes	characteristic
29	#1764	C₄	1	2	yes	—
30	#1764	C₂²	1	2	yes	—
31	C₄×#104	C₄	1	2	yes	—
32	C₄²×C₈	C₂²	1	1	yes	characteristic
33	#1378	C₂²	1	4	yes	—
34	#1376	C₂²	1	2	yes	—
35	C₂×#135	C₂×C₄	1	1	yes	characteristic
36	C₂×#131	C₂×C₄	1	2	yes	—
37	C₂×#131	C₂³	1	2	yes	—
38	C₄×#41	C₂×C₄	1	2	yes	—
39	#504	—	2	8	no	—
40	#504	—	4	4	no	—
41	C₂×M₃₂	—	2	8	no	—
42	C₂×M₃₂	C₂³	1	2	yes	—
43	C₂×M₃₂	C₂³	1	4	yes	—
44	C₂×M₃₂	C₂³	1	2	yes	—
45	#443	C₈	1	2	yes	—
46	#443	C₈	1	2	yes	—
47	#443	C₈	1	2	yes	—
48	#443	C₈	1	2	yes	—
49	#443	D₄	1	2	yes	—
50	#443	D₄	1	2	yes	—
51	#443	Q₈	1	2	yes	—
52	#443	Q₈	1	2	yes	—
53	C₄×Q₁₆	—	2	1	no	—
54	C₄×SD₁₆	—	2	2	no	—
55	C₄×D₈	—	2	1	no	—
56	C₂×#108	C₂×C₄	1	1	yes	characteristic
57	C₂×#108	C₂×C₄	1	1	yes	characteristic
58	#427	C₂×C₄	1	1	yes	characteristic
59	C₂×#104	C₂×C₄	1	2	yes	—
60	C₂×#104	C₂×C₄	1	2	yes	—
61	#416	C₂×C₄	1	2	yes	—
62	#405	—	2	2	no	—
63	#405	—	2	2	no	—
64	#405	C₂×C₄	1	2	yes	—
65	#405	C₂³	1	2	yes	—
66	C₄×M₁₆	—	2	2	no	—
67	C₄×M₁₆	C₂×C₄	1	2	yes	—
68	C₂×C₄×C₈	—	2	2	no	—
69	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	C₄³	C₂×C₄	1	1	yes	characteristic
76	#364	—	2	8	no	—
77	#364	—	2	2	no	—
78	#364	—	2	2	no	—
79	#364	—	2	2	no	—
80	#364	—	2	2	no	—
81	#364	—	2	1	no	—
82	#364	—	2	1	no	—
83	#364	D₄	1	2	yes	—
84	#364	D₄	1	2	yes	—
85	C₄⋊C₁₆	—	4	2	no	—
86	#350	—	2	8	no	—
87	#346	—	2	2	no	—
88	C₄×C₁₆	—	2	2	no	—
89	#341	C₂×C₄	1	2	yes	—
90	#339	C₂×C₄	1	2	yes	—
91	#339	C₂×C₄	1	2	yes	—
92	C₂×#41	C₂²×C₄	1	2	yes	—
93	#135	C₄⋊C₄	1	2	yes	—
94	#135	C₄⋊C₄	1	2	yes	—
95	#135	C₂×C₈	1	2	yes	—
96	#135	C₂×C₈	1	2	yes	—
97	C₂×Q₁₆	M₁₆	1	1	yes	characteristic
98	C₂×SD₁₆	M₁₆	1	2	yes	—
99	C₂×D₈	M₁₆	1	1	yes	characteristic
100	#131	—	2	2	no	—
101	#131	—	2	2	no	—
102	#131	—	2	2	no	—
103	#131	—	2	2	no	—
104	#131	C₂×C₈	1	4	yes	—
105	#131	C₂×C₈	1	4	yes	—
106	#131	C₂×D₄	1	4	yes	—
107	#131	C₂×Q₈	1	4	yes	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	C₂²×C₄	1	2	yes	—
111	C₂×M₁₆	C₂²×C₄	1	2	yes	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₈	C₂⁴	1	1	yes	characteristic
118	C₄×Q₈	—	2	2	no	—
119	C₄×D₄	—	2	2	no	—
120	C₄×D₄	—	4	2	no	—
121	#117	—	2	2	no	—
122	#117	C₂²×C₄	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	—	4	1	no	—
125	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	4	no	—
130	M₃₂	—	2	4	no	—
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	2	no	—
133	M₃₂	—	4	8	no	—
134	M₃₂	C₂×D₄	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	—	4	8	no	—
139	#108	C₄⋊C₄	1	2	yes	—
140	#108	C₄⋊C₄	1	2	yes	—
141	#108	C₂×C₈	1	2	yes	—
142	#108	C₂×C₈	1	2	yes	—
143	#107	—	2	1	no	—
144	#106	—	2	1	no	—
145	#104	C₄⋊C₄	1	4	yes	—
146	#104	C₄⋊C₄	1	4	yes	—
147	#104	C₂×C₈	1	4	yes	—
148	#104	C₂×C₈	1	4	yes	—
149	#103	—	2	2	no	—
150	#102	—	2	2	no	—
151	#97	—	2	2	no	—
152	#97	—	2	2	no	—
153	#97	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	C₂×C₈	1	1	yes	characteristic
164	C₄×C₈	C₂×C₈	1	1	yes	characteristic
165	C₄×C₈	C₂×C₈	1	1	yes	characteristic
166	C₄×C₈	C₂×C₈	1	1	yes	characteristic
167	C₄×C₈	C₂×D₄	1	1	yes	characteristic
168	C₄×C₈	C₂×D₄	1	1	yes	characteristic
169	C₄×C₈	C₂×D₄	1	1	yes	characteristic
170	C₄×C₈	C₂×D₄	1	1	yes	characteristic
171	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
172	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
173	C₄×C₈	#41	1	1	yes	characteristic
174	C₄×C₈	#41	1	1	yes	characteristic
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	C₂×C₄⋊C₄	1	4	yes	—
178	#41	C₂²×C₈	1	4	yes	—
179	C₂×Q₈	C₂×M₁₆	1	2	yes	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	C₂×M₁₆	1	2	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
188	Q₁₆	C₄⋊C₈	1	2	yes	—
189	Q₁₆	C₄⋊C₈	1	2	yes	—
190	SD₁₆	C₄⋊C₈	1	4	yes	—
191	SD₁₆	C₄⋊C₈	1	4	yes	—
192	D₈	C₄⋊C₈	1	2	yes	—
193	D₈	C₄⋊C₈	1	2	yes	—
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	C₂×C₄⋊C₄	1	4	yes	—
201	M₁₆	C₂²×C₈	1	4	yes	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₈	C₄×D₄	1	1	yes	characteristic
214	C₂×C₈	C₄×D₄	1	1	yes	characteristic
215	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
216	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
218	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
219	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
220	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
221	C₂×C₈	#131	1	1	yes	characteristic
222	C₂×C₈	#131	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
226	C₂×C₈	C₂×#41	1	1	yes	characteristic
227	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	4	no	—
239	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
241	C₄²	C₄×D₄	1	1	yes	characteristic
242	C₄²	C₄×D₄	1	1	yes	characteristic
243	C₄²	C₂²×C₈	1	1	yes	characteristic
244	C₄²	C₂²×C₈	1	1	yes	characteristic
245	C₁₆	—	2	4	no	—
246	C₁₆	—	4	4	no	—
247	C₁₆	—	4	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	#568	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	C₂×C₄⋊C₈	1	4	yes	—
252	D₄	—	2	4	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	C₂×C₄⋊C₈	1	4	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
272	C₂×C₄	#520	1	1	yes	characteristic
273	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
275	C₂×C₄	C₂×#131	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₈	—	4	1	no	—
279	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
280	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
281	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
282	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
283	C₈	C₈×D₄	1	1	yes	characteristic
284	C₈	C₈×D₄	1	1	yes	characteristic
285	C₈	C₈×D₄	1	1	yes	characteristic
286	C₈	C₈×D₄	1	1	yes	characteristic
287	C₈	D₄²	1	1	yes	characteristic
288	C₈	#547	1	1	yes	characteristic
289	C₈	D₄×Q₈	1	1	yes	characteristic
290	C₈	#554	1	1	yes	characteristic
291	C₂²	—	2	4	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	8	2	no	—
294	C₂²	#2906	1	1	yes	characteristic
295	C₂²	D₄×M₁₆	1	1	yes	characteristic
296	C₂²	#2988	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	4	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	8	2	no	—
305	C₄	D₄×C₄⋊C₄	1	1	yes	characteristic
306	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
307	C₄	C₂×C₈×D₄	1	1	yes	characteristic
308	C₄	#2990	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	4	4	no	—
313	C₂	D₄×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×C₄⋊C₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #33401	C₂	semidirect
4. #13164	C₂	semidirect
5. #12918	C₂	semidirect
6. #12900	C₂	semidirect
7. #12792	C₂	semidirect
8. C₄×#364	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33443	2
#33401	2
C₄×#443	2
#13164	2
#12918	2
#12900	2
#12792	2
C₄×#364	2