Subgroups — #147102

Order 512.

528 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	C₂²×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×#2108	C₂	1	1	yes	maximal, characteristic
4	C₂×#1389	C₂	1	2	yes	maximal
5	#10872	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#405	C₂²	1	1	yes	characteristic
11	C₂×#368	—	2	1	no	—
12	C₂×#368	—	2	1	no	—
13	C₂×#368	—	2	2	no	—
14	C₂×#368	C₄	1	4	yes	complemented
15	#2158	—	2	2	no	—
16	#2158	C₄	1	4	yes	complemented
17	C₂×#367	—	2	1	no	—
18	C₂×#366	—	2	1	no	—
19	#2108	—	2	1	no	—
20	#2108	—	2	1	no	—
21	#2108	C₄	1	2	yes	complemented
22	#2108	C₄	1	2	yes	complemented
23	C₂×#346	—	2	1	no	—
24	C₂×C₄×C₁₆	—	2	1	no	—
25	C₂×#344	—	2	2	no	—
26	#1733	—	2	2	no	—
27	#1389	—	2	4	no	—
28	C₂²×M₁₆	—	2	2	no	—
29	C₂²×M₁₆	D₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	1	yes	characteristic
31	C₂×#117	—	2	2	no	—
32	C₂×#117	D₄	1	1	yes	characteristic
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	—	2	1	no	—
35	C₂×M₃₂	—	2	1	no	—
36	C₂×M₃₂	C₂×C₄	1	2	yes	—
37	C₂×M₃₂	D₄	1	2	yes	—
38	C₂×M₃₂	Q₈	1	2	yes	—
39	C₂²×C₁₆	—	2	1	no	—
40	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₁₆	C₂×C₄	1	2	yes	—
43	C₂²×C₁₆	C₂×C₄	1	2	yes	—
44	C₂²×C₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₁₆	D₄	1	1	yes	characteristic
47	C₂²×C₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₁₆	D₄	1	2	yes	—
49	C₂²×C₁₆	Q₈	1	2	yes	—
50	C₂×#108	—	2	2	no	—
51	C₂×#108	—	2	2	no	—
52	C₂×#108	—	2	2	no	—
53	C₂×#108	—	2	2	no	—
54	C₂×#108	—	2	2	no	—
55	C₂×#108	C₂×C₄	1	4	yes	—
56	#427	—	2	2	no	—
57	#427	—	2	2	no	—
58	#427	—	2	2	no	—
59	#427	C₂×C₄	1	4	yes	—
60	C₂×#107	—	2	2	no	—
61	C₂×#106	—	2	2	no	—
62	#405	—	2	1	no	—
63	#405	—	2	1	no	—
64	#405	C₂×C₄	1	2	yes	—
65	#405	Q₈	1	2	yes	—
66	C₂×#97	—	2	1	no	—
67	C₂×C₄×C₈	—	2	1	no	—
68	#368	—	2	4	no	—
69	#368	—	2	4	no	—
70	#367	—	2	4	no	—
71	#366	—	2	4	no	—
72	#346	—	2	2	no	—
73	#346	—	2	2	no	—
74	C₄×C₁₆	—	2	2	no	—
75	C₄×C₁₆	—	2	2	no	—
76	#344	—	4	4	no	—
77	#339	—	4	4	no	—
78	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×M₁₆	—	2	1	no	—
80	C₂×M₁₆	—	2	1	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂×M₁₆	—	2	4	no	—
83	C₂×M₁₆	—	4	2	no	—
84	C₂×M₁₆	—	4	2	no	—
85	C₂×M₁₆	—	4	2	no	—
86	C₂×M₁₆	—	4	2	no	—
87	C₂×M₁₆	—	4	2	no	—
88	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
89	C₂×M₁₆	D₈	1	2	yes	—
90	C₂×M₁₆	Q₁₆	1	2	yes	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	—	4	2	no	—
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
103	C₂²×C₈	#41	1	1	yes	characteristic
104	C₂²×C₈	#41	1	1	yes	characteristic
105	C₂²×C₈	#41	1	1	yes	characteristic
106	C₂²×C₈	#41	1	1	yes	characteristic
107	#117	—	2	1	no	—
108	#117	—	2	1	no	—
109	#117	—	2	4	no	—
110	#117	—	4	2	no	—
111	#117	—	4	2	no	—
112	#117	—	4	2	no	—
113	#117	C₄⋊C₄	1	2	yes	—
114	#117	D₈	1	2	yes	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄⋊C₄	Q₁₆	1	2	yes	—
117	C₂×C₂²⋊C₄	—	2	1	no	—
118	C₂×C₂²⋊C₄	—	4	2	no	—
119	C₂×C₄²	—	2	1	no	—
120	C₂×C₄²	—	4	2	no	—
121	M₃₂	—	2	2	no	—
122	M₃₂	—	2	2	no	—
123	M₃₂	—	2	2	no	—
124	M₃₂	C₄⋊C₄	1	4	yes	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	C₄²	1	2	yes	—
129	C₂×C₁₆	C₄²	1	2	yes	—
130	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
135	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
136	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
137	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
138	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
139	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
140	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
141	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
142	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
143	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
144	#108	—	2	4	no	—
145	#108	—	2	4	no	—
146	#108	—	4	2	no	—
147	#108	—	4	2	no	—
148	#107	—	2	4	no	—
149	#107	—	4	2	no	—
150	#106	—	2	4	no	—
151	#106	—	4	2	no	—
152	#97	—	2	2	no	—
153	#97	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂⁴	#135	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	8	2	no	—
159	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
161	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
162	C₂²×C₄	#121	1	1	yes	characteristic
163	C₂²×C₄	#121	1	1	yes	characteristic
164	C₂²×C₄	#127	1	1	yes	characteristic
165	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
170	C₂²×C₄	#135	1	1	yes	characteristic
171	C₂²×C₄	#135	1	1	yes	characteristic
172	C₂²×C₄	#135	1	1	yes	characteristic
173	M₁₆	—	2	2	no	—
174	M₁₆	—	2	2	no	—
175	M₁₆	—	2	2	no	—
176	M₁₆	—	4	4	no	—
177	M₁₆	—	4	4	no	—
178	M₁₆	—	8	2	no	—
179	M₁₆	—	8	2	no	—
180	M₁₆	#107	1	4	yes	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	—	8	2	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₈	#117	1	1	yes	characteristic
196	C₂×C₈	#117	1	1	yes	characteristic
197	C₂×C₈	C₄×D₄	1	1	yes	characteristic
198	C₂×C₈	C₄×D₄	1	1	yes	characteristic
199	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
200	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
201	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
202	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
203	C₂×C₈	#121	1	1	yes	characteristic
204	C₂×C₈	#121	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
209	C₂×C₈	#124	1	1	yes	characteristic
210	C₂×C₈	#124	1	1	yes	characteristic
211	C₂×C₈	#125	1	1	yes	characteristic
212	C₂×C₈	#125	1	1	yes	characteristic
213	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	8	2	no	—
218	C₄⋊C₄	#107	1	4	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₂²⋊C₄	—	8	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	4	4	no	—
226	C₄²	—	8	2	no	—
227	C₁₆	—	2	4	no	—
228	C₁₆	#95	1	2	yes	—
229	C₁₆	#95	1	2	yes	—
230	C₁₆	#95	1	2	yes	—
231	C₁₆	#95	1	2	yes	—
232	C₂³	—	4	2	no	—
233	C₂³	#427	1	1	yes	characteristic
234	C₂³	C₄×D₈	1	1	yes	characteristic
235	C₂³	C₄×Q₁₆	1	1	yes	characteristic
236	C₂³	#466	1	1	yes	characteristic
237	C₂³	#467	1	1	yes	characteristic
238	C₂³	#495	1	1	yes	characteristic
239	C₂³	#499	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	16	4	no	—
244	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₄	#380	1	1	yes	characteristic
246	C₂×C₄	#384	1	1	yes	characteristic
247	C₂×C₄	#389	1	1	yes	characteristic
248	C₂×C₄	#389	1	1	yes	characteristic
249	C₂×C₄	#390	1	1	yes	characteristic
250	C₂×C₄	#390	1	1	yes	characteristic
251	C₂×C₄	C₂×#107	1	1	yes	characteristic
252	C₂×C₄	C₂×#107	1	1	yes	characteristic
253	C₂×C₄	#427	1	1	yes	characteristic
254	C₂×C₄	C₄×D₈	1	1	yes	characteristic
255	C₂×C₄	C₄×D₈	1	1	yes	characteristic
256	C₂×C₄	C₄×D₈	1	1	yes	characteristic
257	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
258	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
259	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
260	C₂×C₄	#466	1	1	yes	characteristic
261	C₂×C₄	#466	1	1	yes	characteristic
262	C₂×C₄	#466	1	1	yes	characteristic
263	C₂×C₄	#467	1	1	yes	characteristic
264	C₂×C₄	#467	1	1	yes	characteristic
265	C₂×C₄	#467	1	1	yes	characteristic
266	C₂×C₄	#493	1	1	yes	characteristic
267	C₂×C₄	#494	1	1	yes	characteristic
268	C₂×C₄	#495	1	1	yes	characteristic
269	C₂×C₄	#499	1	1	yes	characteristic
270	C₂×C₄	#500	1	1	yes	characteristic
271	C₂×C₄	#500	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	8	4	no	—
274	C₈	C₂×#95	1	1	yes	characteristic
275	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
276	C₈	#380	1	1	yes	characteristic
277	C₈	#382	1	1	yes	characteristic
278	C₈	#389	1	1	yes	characteristic
279	C₈	#389	1	1	yes	characteristic
280	C₈	#391	1	1	yes	characteristic
281	C₈	#391	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×#107	1	1	yes	characteristic
290	C₂²	#1808	1	1	yes	characteristic
291	C₂²	#1846	1	1	yes	characteristic
292	C₂²	#1942	1	1	yes	characteristic
293	C₂²	#1942	1	1	yes	characteristic
294	C₂²	#1969	1	1	yes	characteristic
295	C₂²	#1970	1	1	yes	characteristic
296	C₄	—	4	4	no	—
297	C₄	—	16	4	no	—
298	C₄	C₄⋊#95	1	1	yes	characteristic
299	C₄	C₄×#107	1	1	yes	characteristic
300	C₄	#1808	1	1	yes	characteristic
301	C₄	#1845	1	1	yes	characteristic
302	C₄	#1942	1	1	yes	characteristic
303	C₄	#1942	1	1	yes	characteristic
304	C₄	#1969	1	1	yes	characteristic
305	C₄	#1970	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	#10862	1	1	yes	characteristic
313	C₂	#12841	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. C₂×#368	C₄	semidirect
16. #2158	C₄	semidirect
21. #2108	C₄	semidirect
22. #2108	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#368	2
C₂×#2158	2
C₂×#2108	2
C₂×#1389	2
#10872	2