Subgroups — #204009

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17817	C₂	1	2	yes	maximal
2	C₂×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#17323	C₂	1	1	yes	maximal, characteristic
5	#9636	C₂	1	4	yes	maximal, complemented
6	#9634	C₂	1	4	yes	maximal, complemented
7	#8065	C₂	1	2	yes	maximal
8	#2920	C₂²	1	1	yes	characteristic
9	#2920	C₂²	1	1	yes	characteristic
10	C₂×#431	C₂²	1	2	yes	—
11	C₂×#423	C₂²	1	2	yes	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#1574	C₂²	1	8	yes	—
14	#1573	—	2	4	no	—
15	#1567	—	2	2	no	—
16	#1567	—	2	1	no	—
17	#1567	—	2	1	no	—
18	#1567	C₂²	1	2	yes	—
19	#1567	C₂²	1	2	yes	—
20	#1566	—	2	2	no	—
21	#1564	C₄	1	2	yes	—
22	#1564	C₄	1	2	yes	—
23	#1564	C₂²	1	2	yes	—
24	#1564	C₂²	1	2	yes	—
25	C₂×#335	—	2	1	no	—
26	C₂×#335	—	2	1	no	—
27	C₂×#335	C₄	1	2	yes	—
28	C₂×#334	C₄	1	2	yes	—
29	#1452	—	2	1	no	—
30	#1451	—	2	1	no	—
31	#1451	C₂²	1	2	yes	—
32	#1450	—	2	1	no	—
33	C₈×M₁₆	—	2	1	no	—
34	C₈×M₁₆	C₂²	1	2	yes	—
35	#1299	C₂²	1	8	yes	—
36	C₂²×M₁₆	—	2	2	no	—
37	C₂³×C₈	Q₈	1	1	yes	characteristic
38	C₂³×C₈	Q₈	1	1	yes	characteristic
39	C₂²×C₄²	C₂³	1	1	yes	characteristic
40	#432	—	2	1	no	—
41	#432	D₄	1	2	yes	—
42	#432	Q₈	1	2	yes	—
43	#432	Q₈	1	2	yes	—
44	#431	—	2	2	no	—
45	#431	—	2	1	no	—
46	#431	—	2	1	no	—
47	#431	—	2	1	no	—
48	#431	—	2	2	no	—
49	#431	C₂×C₄	1	4	yes	—
50	#431	C₂³	1	4	yes	—
51	#431	C₂³	1	2	yes	—
52	#423	—	2	1	no	—
53	#423	—	2	2	no	—
54	#423	—	2	2	no	—
55	#423	—	2	2	no	—
56	#423	D₄	1	1	yes	characteristic
57	#423	D₄	1	1	yes	characteristic
58	#423	C₂³	1	4	yes	—
59	C₂×C₄⋊C₈	—	2	2	no	—
60	C₂×C₄⋊C₈	—	2	2	no	—
61	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
62	#407	—	2	1	no	—
63	#407	—	2	1	no	—
64	C₂×C₂²⋊C₈	—	2	2	no	—
65	C₄×M₁₆	—	2	1	no	—
66	C₄×M₁₆	C₂³	1	2	yes	—
67	C₂×C₄×C₈	—	2	2	no	—
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	D₄	1	1	yes	characteristic
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	Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
80	#335	—	2	8	no	—
81	#335	—	2	2	no	—
82	#335	—	2	2	no	—
83	#335	—	2	2	no	—
84	#335	—	2	2	no	—
85	#335	C₂×C₄	1	4	yes	—
86	#335	C₂×C₄	1	4	yes	—
87	#334	—	2	8	no	—
88	#334	C₂×C₄	1	4	yes	—
89	#334	C₂×C₄	1	4	yes	—
90	#322	—	2	2	no	—
91	#322	—	2	2	no	—
92	#322	—	2	2	no	—
93	C₈²	—	2	2	no	—
94	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
95	C₂³×C₄	#41	1	1	yes	characteristic
96	C₂³×C₄	#41	1	1	yes	characteristic
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	2	4	no	—
101	C₂×M₁₆	—	4	2	no	—
102	C₂×M₁₆	—	8	4	no	—
103	C₂²×C₈	—	2	4	no	—
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	—	4	2	no	—
106	C₂²×C₈	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₈	C₄⋊C₄	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	1	yes	characteristic
119	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄²	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	4	2	no	—
140	C₄⋊C₈	C₂²×C₄	1	8	yes	—
141	C₄⋊C₈	C₂×D₄	1	2	yes	—
142	C₄⋊C₈	C₂×D₄	1	2	yes	—
143	C₂²⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	2	2	no	—
145	C₂²⋊C₈	—	2	2	no	—
146	C₂²⋊C₈	—	2	2	no	—
147	C₂²⋊C₈	—	2	4	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	#97	—	2	2	no	—
150	#97	C₂×Q₈	1	4	yes	—
151	C₄×C₈	—	2	4	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	4	no	—
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	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
164	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
165	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
166	C₄×C₈	C₂²×C₄	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	C₂⁴	#125	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
182	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
183	C₂²×C₄	#121	1	1	yes	characteristic
184	C₂²×C₄	#121	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	#125	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×#41	1	1	yes	characteristic
194	C₂²×C₄	C₂×#41	1	1	yes	characteristic
195	M₁₆	—	4	4	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	8	4	no	—
199	M₁₆	—	8	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	8	4	no	—
217	C₂×C₈	—	8	4	no	—
218	C₂×C₈	#108	1	4	yes	—
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	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
227	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
228	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₄²	C₄×D₄	1	1	yes	characteristic
230	C₄²	C₄×D₄	1	1	yes	characteristic
231	C₄²	C₄×Q₈	1	1	yes	characteristic
232	C₄²	C₄×Q₈	1	1	yes	characteristic
233	C₄²	#121	1	1	yes	characteristic
234	C₄²	#121	1	1	yes	characteristic
235	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
236	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
237	C₄²	C₄⋊Q₈	1	1	yes	characteristic
238	C₄²	C₄⋊Q₈	1	1	yes	characteristic
239	C₄²	C₂³×C₄	1	1	yes	characteristic
240	C₄²	C₂²×D₄	1	1	yes	characteristic
241	C₄²	C₂²×D₄	1	1	yes	characteristic
242	C₄²	C₂²×Q₈	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	4	1	no	—
246	C₂³	#389	1	1	yes	characteristic
247	C₂³	#389	1	1	yes	characteristic
248	C₂³	C₂×#125	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	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	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	#389	1	1	yes	characteristic
270	C₂×C₄	#389	1	1	yes	characteristic
271	C₂×C₄	#389	1	1	yes	characteristic
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	C₂×#108	1	2	yes	—
276	C₂×C₄	#430	1	2	yes	—
277	C₂×C₄	#472	1	2	yes	—
278	C₂×C₄	#473	1	2	yes	—
279	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
282	C₂×C₄	C₂×#121	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
285	C₈	—	2	1	no	—
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	—	4	4	no	—
292	C₈	—	4	8	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	8	4	no	—
296	C₈	C₂×#108	1	2	yes	—
297	C₈	C₂×#108	1	2	yes	—
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	—	2	1	no	—
305	C₂²	—	4	4	no	—
306	C₂²	#1950	1	2	yes	—
307	C₂²	#1952	1	2	yes	—
308	C₂²	C₂×#389	1	1	yes	characteristic
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	#1950	1	2	yes	—
317	C₄	#1952	1	2	yes	—
318	C₄	C₂²×#108	1	1	yes	characteristic
319	C₄	C₂×#430	1	1	yes	characteristic
320	C₄	#2957	1	2	yes	—
321	C₄	C₂×#472	1	1	yes	characteristic
322	C₄	C₂×#473	1	1	yes	characteristic
323	C₂	—	2	2	no	—
324	C₂	—	2	2	no	—
325	C₂	—	4	1	no	—
326	C₂	#21163	1	1	yes	characteristic
327	C₂	C₂×#1950	1	1	yes	characteristic
328	C₂	C₂×#1952	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	#125
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. #9636	C₂	semidirect
6. #9634	C₂	semidirect

Maximal subgroups

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

subgroup	index
#17817	2
C₂×#1567	2
C₂×#1564	2
#17323	2
#9636	2
#9634	2
#8065	2