Subgroups — #392509

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1848	C₂	1	1	yes	maximal, characteristic
2	#20797	C₂	1	1	yes	maximal, characteristic
3	C₂×#1833	C₂	1	4	yes	maximal
4	C₄×#405	C₂	1	1	yes	maximal, characteristic
5	#10868	C₂	1	4	yes	maximal
6	#10848	C₂	1	2	yes	maximal
7	#10836	C₂	1	2	yes	maximal
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂²×#108	C₂²	1	2	yes	—
10	C₂×#427	C₂²	1	2	yes	—
11	C₂×#427	C₂²	1	2	yes	—
12	C₂×#423	C₂²	1	2	yes	—
13	C₂×#405	C₂²	1	2	yes	—
14	C₂×#405	C₂²	1	2	yes	—
15	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
17	#2285	C₂²	1	2	yes	—
18	C₄×#117	C₂²	1	1	yes	characteristic
19	#1848	—	2	1	no	—
20	#1848	—	2	1	no	—
21	#1848	—	2	1	no	—
22	#1848	—	2	1	no	—
23	#1846	—	2	1	no	—
24	#1846	—	2	1	no	—
25	#1845	—	2	1	no	—
26	#1844	—	2	1	no	—
27	#1833	—	2	16	no	—
28	C₂×#344	C₂²	1	8	yes	—
29	C₂×H(3,ℤ/4)	C₂²	1	8	yes	—
30	#1726	C₄	1	2	yes	complemented
31	#1726	C₄	1	2	yes	complemented
32	C₄×#97	C₄	1	2	yes	complemented
33	C₄²×C₈	C₄	1	2	yes	complemented
34	C₂²×M₁₆	C₂³	1	4	yes	—
35	C₂²×M₁₆	C₂³	1	2	yes	—
36	C₂³×C₈	C₂³	1	1	yes	characteristic
37	C₂³×C₈	C₂³	1	1	yes	characteristic
38	C₂×#117	C₂³	1	4	yes	—
39	C₂×#117	C₂³	1	2	yes	—
40	C₂²×C₄²	C₂³	1	1	yes	characteristic
41	C₂×#108	—	2	8	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	—	2	8	no	—
47	#427	—	2	8	no	—
48	#427	—	2	8	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#107	—	2	2	no	—
51	C₂×#106	—	2	2	no	—
52	C₂×#106	—	2	2	no	—
53	#423	—	2	2	no	—
54	#423	—	2	2	no	—
55	#423	—	2	4	no	—
56	C₂×C₄⋊C₈	—	2	4	no	—
57	#405	—	2	4	no	—
58	#405	—	2	4	no	—
59	#405	D₄	1	8	yes	—
60	#405	Q₈	1	8	yes	—
61	C₄×M₁₆	—	2	2	no	—
62	C₄×M₁₆	—	2	2	no	—
63	C₂×#97	C₂×C₄	1	2	yes	—
64	C₂×#97	C₂×C₄	1	4	yes	—
65	C₂×#97	C₂×C₄	1	4	yes	—
66	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
67	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	C₂×C₄×C₈	D₄	1	1	yes	characteristic
75	C₂×C₄×C₈	D₄	1	1	yes	characteristic
76	C₂×C₄×C₈	D₄	1	1	yes	characteristic
77	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
79	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
81	#384	—	2	2	no	—
82	#382	—	2	2	no	—
83	#380	—	2	4	no	—
84	C₄×C₄⋊C₄	—	2	2	no	—
85	C₄×C₄⋊C₄	—	2	4	no	—
86	C₄×C₂²⋊C₄	—	2	2	no	—
87	#376	C₂×C₄	1	2	yes	—
88	C₄³	C₂×C₄	1	2	yes	—
89	#344	—	2	16	no	—
90	H(3,ℤ/4)	—	2	16	no	—
91	C₂³×C₄	#41	1	2	yes	—
92	C₂³×C₄	C₂⁴	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	—	2	2	no	—
97	C₂×M₁₆	—	2	8	no	—
98	C₂×M₁₆	—	2	4	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂×M₁₆	—	2	8	no	—
101	C₂×M₁₆	C₂×D₄	1	4	yes	—
102	C₂×M₁₆	C₂×Q₈	1	4	yes	—
103	C₂²×C₈	—	2	4	no	—
104	C₂²×C₈	C₂²×C₄	1	4	yes	—
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	1	4	yes	—
114	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
116	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
117	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
118	C₂²×C₈	C₂×Q₈	1	4	yes	—
119	C₂²×C₈	#41	1	4	yes	—
120	C₂²×C₈	#41	1	4	yes	—
121	#117	—	2	2	no	—
122	#117	—	2	2	no	—
123	#117	—	2	8	no	—
124	#117	—	2	8	no	—
125	#117	C₂×D₄	1	4	yes	—
126	#117	C₂×Q₈	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	4	4	no	—
130	C₂×C₂²⋊C₄	—	2	2	no	—
131	C₂×C₂²⋊C₄	—	2	4	no	—
132	C₂×C₄²	—	2	4	no	—
133	C₂×C₄²	—	2	4	no	—
134	C₂×C₄²	C₂²×C₄	1	4	yes	—
135	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
140	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
141	C₂×C₄²	#41	1	2	yes	—
142	C₂×C₄²	#41	1	2	yes	—
143	C₂×C₄²	#41	1	2	yes	—
144	#108	—	4	4	no	—
145	#108	—	4	4	no	—
146	#108	—	4	4	no	—
147	#108	—	4	4	no	—
148	#107	—	4	4	no	—
149	#107	—	4	4	no	—
150	#106	—	4	4	no	—
151	#106	—	4	4	no	—
152	C₄⋊C₈	—	4	4	no	—
153	C₄⋊C₈	—	4	4	no	—
154	#97	—	2	8	no	—
155	#97	—	2	4	no	—
156	#97	C₄⋊C₄	1	16	yes	complemented
157	C₄×C₈	—	2	8	no	—
158	C₄×C₈	—	2	4	no	—
159	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₈	C₄⋊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	C₄×C₈	C₄⋊C₄	1	16	yes	complemented
176	#95	—	2	4	no	—
177	#95	—	4	4	no	—
178	C₂⁴	C₂×#41	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	#117	1	2	yes	—
183	C₂²×C₄	#117	1	2	yes	—
184	C₂²×C₄	#124	1	2	yes	—
185	C₂²×C₄	#124	1	2	yes	—
186	C₂²×C₄	#125	1	2	yes	—
187	C₂²×C₄	#125	1	2	yes	—
188	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₄	1	2	yes	—
191	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Q₈	1	2	yes	—
193	C₂²×C₄	C₂×#41	1	2	yes	—
194	C₂²×C₄	C₂×#41	1	2	yes	—
195	C₂²×C₄	C₂×#41	1	1	yes	characteristic
196	C₂²×C₄	C₂×#41	1	1	yes	characteristic
197	C₂²×C₄	C₂×#41	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	M₁₆	—	4	4	no	—
204	M₁₆	—	4	8	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	8	no	—
209	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
210	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
219	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
220	C₂×C₈	#117	1	4	yes	—
221	C₂×C₈	#117	1	4	yes	—
222	C₂×C₈	#124	1	2	yes	—
223	C₂×C₈	#124	1	2	yes	—
224	C₂×C₈	#124	1	2	yes	—
225	C₂×C₈	#124	1	2	yes	—
226	C₂×C₈	#125	1	4	yes	—
227	C₂×C₈	#125	1	4	yes	—
228	C₂×C₈	#127	1	2	yes	—
229	C₂×C₈	#127	1	2	yes	—
230	C₂×C₈	C₄⋊Q₈	1	4	yes	—
231	C₂×C₈	C₄⋊Q₈	1	4	yes	—
232	C₂×C₈	C₄⋊Q₈	1	2	yes	—
233	C₂×C₈	C₄⋊Q₈	1	2	yes	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	8	8	no	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	4	8	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	4	8	no	—
244	C₄²	—	4	8	no	—
245	C₄²	C₂×C₄⋊C₄	1	8	yes	—
246	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₄²	C₂×C₄⋊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	#117	1	2	yes	—
251	C₄²	#117	1	2	yes	—
252	C₄²	#124	1	2	yes	—
253	C₄²	#124	1	2	yes	—
254	C₄²	#125	1	2	yes	—
255	C₄²	#125	1	2	yes	—
256	C₂³	—	4	2	no	—
257	C₂³	C₂×#117	1	1	yes	characteristic
258	C₂³	C₂×#124	1	1	yes	characteristic
259	C₂³	C₂×#125	1	1	yes	characteristic
260	C₂³	#532	1	2	yes	—
261	C₂³	#533	1	2	yes	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	—	8	8	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	#382	1	2	yes	—
273	C₂×C₄	#382	1	2	yes	—
274	C₂×C₄	#382	1	2	yes	—
275	C₂×C₄	#382	1	2	yes	—
276	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₄⋊C₄	1	2	yes	—
278	C₂×C₄	C₂×#117	1	2	yes	—
279	C₂×C₄	C₂×#117	1	1	yes	characteristic
280	C₂×C₄	C₂×#124	1	1	yes	characteristic
281	C₂×C₄	C₂×#124	1	1	yes	characteristic
282	C₂×C₄	C₂×#124	1	1	yes	characteristic
283	C₂×C₄	C₂×#125	1	1	yes	characteristic
284	C₂×C₄	C₂×#125	1	2	yes	—
285	C₂×C₄	C₂×#127	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
288	C₂×C₄	#532	1	2	yes	—
289	C₂×C₄	#532	1	2	yes	—
290	C₂×C₄	#532	1	2	yes	—
291	C₂×C₄	#533	1	2	yes	—
292	C₂×C₄	#533	1	2	yes	—
293	C₂×C₄	#533	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₈	—	8	4	no	—
296	C₈	#382	1	2	yes	—
297	C₈	#382	1	2	yes	—
298	C₈	#382	1	2	yes	—
299	C₈	#382	1	2	yes	—
300	C₈	#384	1	2	yes	—
301	C₈	#384	1	2	yes	—
302	C₈	#384	1	2	yes	—
303	C₈	#384	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₂×#382	1	1	yes	characteristic
310	C₂²	#2285	1	2	yes	—
311	C₂²	#2858	1	1	yes	characteristic
312	C₂²	#2859	1	2	yes	—
313	C₂²	#2862	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	8	no	—
318	C₄	—	16	4	no	—
319	C₄	#1833	1	4	yes	complemented
320	C₄	C₂×#382	1	1	yes	characteristic
321	C₄	C₂×#384	1	1	yes	characteristic
322	C₄	#2285	1	2	yes	—
323	C₄	#2858	1	1	yes	characteristic
324	C₄	#2859	1	2	yes	—
325	C₄	#2862	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	1	no	—
330	C₂	#14687	1	1	yes	characteristic
331	C₂	C₂×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
30. #1726	C₄	semidirect
31. #1726	C₄	semidirect
32. C₄×#97	C₄	semidirect
33. C₄²×C₈	C₄	semidirect
156. #97	C₄⋊C₄	semidirect
175. C₄×C₈	C₄⋊C₄	semidirect
319. C₄	#1833	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1848	2
#20797	2
C₂×#1833	2
C₄×#405	2
#10868	2
#10848	2
#10836	2