Subgroups — #203966

Order 512.

595 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1837	C₂	1	1	yes	maximal, characteristic
2	#20476	C₂	1	1	yes	maximal, characteristic
3	#17822	C₂	1	2	yes	maximal
4	#17793	C₂	1	1	yes	maximal, characteristic
5	#9632	C₂	1	4	yes	maximal, complemented
6	#9629	C₂	1	4	yes	maximal, complemented
7	#8093	C₂	1	2	yes	maximal
8	#2920	C₂²	1	2	yes	—
9	C₂×#432	C₂²	1	1	yes	characteristic
10	C₂×#423	C₂²	1	2	yes	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#2283	C₂²	1	1	yes	characteristic
13	#1837	—	2	1	no	—
14	#1837	—	2	1	no	—
15	#1837	C₂²	1	2	yes	—
16	#1837	C₂²	1	2	yes	—
17	#1770	C₂²	1	2	yes	—
18	C₈×C₄⋊C₄	C₂²	1	2	yes	—
19	#1752	—	2	1	no	—
20	C₈×C₂²⋊C₄	—	2	1	no	—
21	#1746	—	2	1	no	—
22	C₂×#341	—	2	1	no	—
23	C₂×#341	—	2	1	no	—
24	#1726	—	2	1	no	—
25	#1573	—	2	4	no	—
26	#1571	C₂²	1	4	yes	—
27	#1570	C₂²	1	4	yes	—
28	#1569	—	2	4	no	—
29	#1568	C₄	1	2	yes	complemented
30	#1568	C₄	1	2	yes	complemented
31	#1567	C₄	1	2	yes	complemented
32	#1566	C₄	1	2	yes	complemented
33	#1564	C₂²	1	2	yes	—
34	#1564	C₂²	1	2	yes	—
35	#1303	C₂²	1	8	yes	—
36	C₂²×M₁₆	—	2	2	no	—
37	C₂³×C₈	D₄	1	1	yes	characteristic
38	C₂³×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₄²	C₂³	1	1	yes	characteristic
40	#432	—	2	1	no	—
41	#432	—	2	1	no	—
42	#432	—	2	2	no	—
43	#432	C₂×C₄	1	4	yes	—
44	#432	D₄	1	2	yes	—
45	#432	D₄	1	2	yes	—
46	#431	—	2	2	no	—
47	#431	C₂³	1	4	yes	—
48	#423	—	2	2	no	—
49	#423	—	2	2	no	—
50	#423	—	2	2	no	—
51	#423	C₂×C₄	1	4	yes	—
52	#423	C₂³	1	4	yes	—
53	C₂×C₄⋊C₈	—	2	2	no	—
54	C₂×C₄⋊C₈	—	2	2	no	—
55	C₂×C₄⋊C₈	C₂³	1	2	yes	—
56	#407	—	2	2	no	—
57	C₂×C₂²⋊C₈	—	2	1	no	—
58	C₄×M₁₆	—	2	2	no	—
59	C₂×#97	—	2	1	no	—
60	C₂×#97	—	2	2	no	—
61	C₂×C₄×C₈	—	2	2	no	—
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	#383	Q₈	1	2	yes	—
75	#382	—	2	1	no	—
76	#381	Q₈	1	2	yes	—
77	#380	—	2	1	no	—
78	C₄×C₄⋊C₄	C₂³	1	2	yes	—
79	C₄×C₂²⋊C₄	—	2	1	no	—
80	#376	—	2	1	no	—
81	#341	—	2	2	no	—
82	#341	—	2	2	no	—
83	#341	—	2	2	no	—
84	#341	—	2	2	no	—
85	#336	—	2	2	no	—
86	#336	—	2	2	no	—
87	#335	—	2	8	no	—
88	#335	C₂×C₄	1	4	yes	—
89	#335	C₂×C₄	1	4	yes	—
90	#334	—	2	8	no	—
91	#334	C₂×C₄	1	4	yes	—
92	#334	C₂×C₄	1	4	yes	—
93	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
95	C₂³×C₄	#41	1	1	yes	characteristic
96	C₂×M₁₆	—	2	4	no	—
97	C₂×M₁₆	—	2	4	no	—
98	C₂×M₁₆	—	4	2	no	—
99	C₂×M₁₆	—	4	4	no	—
100	C₂×M₁₆	—	4	4	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	4	2	no	—
103	C₂²×C₈	—	4	2	no	—
104	C₂²×C₈	—	4	2	no	—
105	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₂²⋊C₄	—	2	2	no	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
126	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
127	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
128	C₂×C₄²	#41	1	1	yes	characteristic
129	C₂×C₄²	#41	1	1	yes	characteristic
130	C₂×C₄²	#41	1	1	yes	characteristic
131	C₂×C₄²	C₂⁴	1	1	yes	characteristic
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	4	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	C₄⋊C₈	C₂²×C₄	1	8	yes	—
139	C₄⋊C₈	C₂×D₄	1	4	yes	—
140	C₂²⋊C₈	—	2	4	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	2	2	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	#97	—	2	4	no	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	#97	—	4	4	no	—
148	C₄×C₈	—	2	4	no	—
149	C₄×C₈	—	4	4	no	—
150	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
161	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
162	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
163	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
164	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
165	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
166	#95	—	2	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	C₂×Q₈	1	4	yes	—
170	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
181	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
182	C₂²×C₄	#121	1	1	yes	characteristic
183	C₂²×C₄	#121	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₄	#125	1	1	yes	characteristic
188	C₂²×C₄	#125	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×#41	1	1	yes	characteristic
194	M₁₆	—	4	4	no	—
195	M₁₆	—	4	4	no	—
196	M₁₆	—	8	4	no	—
197	M₁₆	—	8	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	8	4	no	—
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	1	yes	characteristic
219	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
224	C₄⋊C₄	—	4	4	no	—
225	C₂²⋊C₄	—	4	4	no	—
226	C₄²	—	4	2	no	—
227	C₄²	—	4	2	no	—
228	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
230	C₄²	C₄×D₄	1	1	yes	characteristic
231	C₄²	C₄×D₄	1	1	yes	characteristic
232	C₄²	C₄×Q₈	1	1	yes	characteristic
233	C₄²	C₄×Q₈	1	1	yes	characteristic
234	C₄²	#121	1	1	yes	characteristic
235	C₄²	#121	1	1	yes	characteristic
236	C₄²	#125	1	1	yes	characteristic
237	C₄²	#125	1	1	yes	characteristic
238	C₄²	C₄⋊Q₈	1	1	yes	characteristic
239	C₄²	C₄⋊Q₈	1	1	yes	characteristic
240	C₄²	C₂³×C₄	1	1	yes	characteristic
241	C₄²	C₂²×D₄	1	1	yes	characteristic
242	C₄²	C₂²×Q₈	1	1	yes	characteristic
243	C₄²	C₂×#41	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	4	1	no	—
247	C₂³	#389	1	1	yes	characteristic
248	C₂³	#389	1	1	yes	characteristic
249	C₂³	C₂×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	#389	1	1	yes	characteristic
273	C₂×C₄	#389	1	1	yes	characteristic
274	C₂×C₄	#389	1	1	yes	characteristic
275	C₂×C₄	#389	1	1	yes	characteristic
276	C₂×C₄	#389	1	1	yes	characteristic
277	C₂×C₄	#389	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
281	C₂×C₄	C₂×#121	1	1	yes	characteristic
282	C₂×C₄	C₂×#125	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
284	C₂×C₄	#575	1	2	yes	—
285	C₂×C₄	#575	1	2	yes	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	4	8	no	—
295	C₈	—	4	4	no	—
296	C₈	—	8	4	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	4	no	—
305	C₂²	#1953	1	2	yes	—
306	C₂²	C₂×#389	1	1	yes	characteristic
307	C₂²	#3080	1	2	yes	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	4	no	—
318	C₄	#2911	1	1	yes	characteristic
319	C₄	#2911	1	1	yes	characteristic
320	C₄	#2957	1	2	yes	—
321	C₄	#3040	1	2	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	2	2	no	—
324	C₂	—	4	1	no	—
325	C₂	#21163	1	1	yes	characteristic
326	C₂	#21172	1	1	yes	characteristic
327	C₂	C₂×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #9632	C₂	semidirect
6. #9629	C₂	semidirect
29. #1568	C₄	semidirect
30. #1568	C₄	semidirect
31. #1567	C₄	semidirect
32. #1566	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1837	2
#20476	2
#17822	2
#17793	2
#9632	2
#9629	2
#8093	2