Subgroups — #203994

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17806	C₂	1	2	yes	maximal
2	#17793	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#9634	C₂	1	4	yes	maximal, complemented
6	#9633	C₂	1	4	yes	maximal, complemented
7	#8060	C₂	1	2	yes	maximal
8	#2920	C₂²	1	2	yes	—
9	C₂×#431	C₂²	1	2	yes	—
10	C₂×#431	C₂²	1	1	yes	characteristic
11	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#1574	C₂²	1	8	yes	—
14	#1573	—	2	4	no	—
15	#1568	—	2	1	no	—
16	#1568	—	2	1	no	—
17	#1567	C₂²	1	2	yes	—
18	#1566	C₂²	1	2	yes	—
19	#1564	—	2	4	no	—
20	#1564	—	2	1	no	—
21	#1564	—	2	1	no	—
22	#1564	C₄	1	2	yes	—
23	#1564	C₄	1	2	yes	—
24	#1564	C₂²	1	2	yes	—
25	#1564	C₂²	1	2	yes	—
26	C₂×#335	C₄	1	2	yes	—
27	C₂×#334	C₄	1	2	yes	—
28	C₈×M₁₆	—	2	1	no	—
29	C₈×M₁₆	—	2	1	no	—
30	C₈×M₁₆	C₂²	1	2	yes	—
31	C₈×M₁₆	C₂²	1	2	yes	—
32	C₂×#322	—	2	1	no	—
33	C₂×C₈²	—	2	1	no	—
34	#1299	C₂²	1	8	yes	—
35	C₂²×M₁₆	—	2	1	no	—
36	C₂³×C₈	Q₈	1	1	yes	characteristic
37	C₂³×C₈	Q₈	1	1	yes	characteristic
38	C₂²×C₄²	C₂³	1	1	yes	characteristic
39	#432	—	2	2	no	—
40	#432	—	2	2	no	—
41	#431	—	2	2	no	—
42	#431	—	2	1	no	—
43	#431	—	2	2	no	—
44	#431	—	2	1	no	—
45	#431	—	2	2	no	—
46	#431	—	2	2	no	—
47	#431	C₂×C₄	1	4	yes	—
48	#431	D₄	1	2	yes	—
49	#431	C₂³	1	4	yes	—
50	#431	C₂³	1	2	yes	—
51	#423	C₂³	1	4	yes	—
52	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₂×C₄⋊C₈	D₄	1	2	yes	—
54	#407	—	2	2	no	—
55	C₂×C₂²⋊C₈	—	2	2	no	—
56	C₂×C₂²⋊C₈	—	2	1	no	—
57	C₄×M₁₆	—	2	1	no	—
58	C₄×M₁₆	—	2	1	no	—
59	C₄×M₁₆	—	2	2	no	—
60	C₄×M₁₆	Q₈	1	2	yes	—
61	C₄×M₁₆	C₂³	1	2	yes	—
62	C₂×#97	Q₈	1	2	yes	—
63	C₂×C₄×C₈	—	2	2	no	—
64	C₂×C₄×C₈	—	2	1	no	—
65	C₂×C₄×C₈	—	2	1	no	—
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	D₄	1	1	yes	characteristic
73	C₂×C₄×C₈	D₄	1	1	yes	characteristic
74	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
78	#335	—	2	8	no	—
79	#335	—	2	2	no	—
80	#335	—	2	2	no	—
81	#335	C₂×C₄	1	4	yes	—
82	#335	C₂×C₄	1	4	yes	—
83	#334	—	2	8	no	—
84	#334	—	2	2	no	—
85	#334	—	2	2	no	—
86	#334	C₂×C₄	1	4	yes	—
87	#334	C₂×C₄	1	4	yes	—
88	#322	—	2	2	no	—
89	#322	—	2	2	no	—
90	C₈²	—	2	2	no	—
91	C₈²	—	2	2	no	—
92	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
93	C₂³×C₄	#41	1	1	yes	characteristic
94	C₂³×C₄	#41	1	1	yes	characteristic
95	C₂×M₁₆	—	2	8	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	4	no	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	—	2	4	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	—	4	2	no	—
104	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
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	C₂×Q₈	1	1	yes	characteristic
113	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
114	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
123	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
124	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
125	C₂×C₄²	#41	1	1	yes	characteristic
126	C₂×C₄²	#41	1	1	yes	characteristic
127	C₂×C₄²	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	4	no	—
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	C₂²×C₄	1	8	yes	—
133	C₄⋊C₈	C₂×D₄	1	4	yes	—
134	C₂²⋊C₈	—	2	4	no	—
135	C₂²⋊C₈	—	2	2	no	—
136	C₂²⋊C₈	—	2	4	no	—
137	C₂²⋊C₈	—	2	2	no	—
138	C₂²⋊C₈	—	4	2	no	—
139	C₂²⋊C₈	—	4	2	no	—
140	#97	—	2	2	no	—
141	#97	—	4	2	no	—
142	#97	C₂×Q₈	1	4	yes	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	4	no	—
149	C₄×C₈	—	4	2	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×D₄	1	1	yes	characteristic
163	C₄×C₈	C₂×D₄	1	1	yes	characteristic
164	C₄×C₈	C₂×D₄	1	1	yes	characteristic
165	C₄×C₈	C₂×D₄	1	1	yes	characteristic
166	C₂⁴	#125	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
175	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
176	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₄	#121	1	1	yes	characteristic
178	C₂²×C₄	#121	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
181	C₂²×C₄	#125	1	1	yes	characteristic
182	C₂²×C₄	#125	1	1	yes	characteristic
183	C₂²×C₄	#125	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
187	C₂²×C₄	C₂×#41	1	1	yes	characteristic
188	C₂²×C₄	C₂×#41	1	1	yes	characteristic
189	M₁₆	—	2	8	no	—
190	M₁₆	—	2	8	no	—
191	M₁₆	—	8	4	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	8	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	#108	1	4	yes	—
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	1	yes	characteristic
219	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
220	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₄²	C₄×D₄	1	1	yes	characteristic
222	C₄²	C₄×D₄	1	1	yes	characteristic
223	C₄²	C₄×Q₈	1	1	yes	characteristic
224	C₄²	C₄×Q₈	1	1	yes	characteristic
225	C₄²	#121	1	1	yes	characteristic
226	C₄²	#121	1	1	yes	characteristic
227	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
228	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
229	C₄²	C₄⋊Q₈	1	1	yes	characteristic
230	C₄²	C₄⋊Q₈	1	1	yes	characteristic
231	C₄²	C₂³×C₄	1	1	yes	characteristic
232	C₄²	C₂²×D₄	1	1	yes	characteristic
233	C₄²	C₂²×D₄	1	1	yes	characteristic
234	C₄²	C₂²×Q₈	1	1	yes	characteristic
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	4	1	no	—
238	C₂³	#389	1	1	yes	characteristic
239	C₂³	#389	1	1	yes	characteristic
240	C₂³	C₂×#125	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	#389	1	1	yes	characteristic
260	C₂×C₄	#389	1	1	yes	characteristic
261	C₂×C₄	#389	1	1	yes	characteristic
262	C₂×C₄	#389	1	1	yes	characteristic
263	C₂×C₄	#389	1	1	yes	characteristic
264	C₂×C₄	#389	1	1	yes	characteristic
265	C₂×C₄	C₂×#108	1	2	yes	—
266	C₂×C₄	#430	1	2	yes	—
267	C₂×C₄	#443	1	4	yes	—
268	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
271	C₂×C₄	C₂×#121	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
274	C₈	—	2	8	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	4	8	no	—
282	C₈	—	4	4	no	—
283	C₈	—	8	4	no	—
284	C₈	C₂×#108	1	2	yes	—
285	C₈	C₂×#108	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	4	4	no	—
294	C₂²	#1945	1	2	yes	—
295	C₂²	#1950	1	2	yes	—
296	C₂²	C₂×#389	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	#1945	1	2	yes	—
305	C₄	#1950	1	2	yes	—
306	C₄	C₂²×#108	1	1	yes	characteristic
307	C₄	C₂×#430	1	1	yes	characteristic
308	C₄	C₂×#443	1	2	yes	—
309	C₄	#3067	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	—	4	1	no	—
313	C₂	C₂×#1945	1	1	yes	characteristic
314	C₂	C₂×#1950	1	1	yes	characteristic
315	C₂	#21181	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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. #9634	C₂	semidirect
6. #9633	C₂	semidirect

Maximal subgroups

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

subgroup	index
#17806	2
#17793	2
C₂×#1564	2
C₂×C₈×M₁₆	2
#9634	2
#9633	2
#8060	2