Subgroups — #204021

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17814	C₂	1	1	yes	maximal, characteristic
2	#17809	C₂	1	1	yes	maximal, characteristic
3	#17806	C₂	1	1	yes	maximal, characteristic
4	C₂×#1574	C₂	1	1	yes	maximal, characteristic
5	#17326	C₂	1	1	yes	maximal, characteristic
6	#9636	C₂	1	4	yes	maximal, complemented
7	#9634	C₂	1	4	yes	maximal, complemented
8	#8065	C₂	1	1	yes	maximal, characteristic
9	#8061	C₂	1	1	yes	maximal, characteristic
10	#2920	C₂²	1	1	yes	characteristic
11	#2920	C₂²	1	1	yes	characteristic
12	C₂×#431	C₂²	1	1	yes	characteristic
13	C₂×#431	C₂²	1	1	yes	characteristic
14	C₂×#431	C₂²	1	1	yes	characteristic
15	C₂×#423	C₂²	1	1	yes	characteristic
16	C₂×#423	C₂²	1	1	yes	characteristic
17	#1574	—	2	1	no	—
18	#1574	—	2	1	no	—
19	#1574	C₄	1	4	yes	—
20	#1574	C₂²	1	2	yes	—
21	#1574	C₂²	1	2	yes	—
22	#1574	C₂²	1	2	yes	—
23	#1574	C₂²	1	2	yes	—
24	#1573	C₄	1	4	yes	—
25	#1571	—	2	1	no	—
26	#1571	—	2	1	no	—
27	#1571	—	2	2	no	—
28	#1571	—	2	1	no	—
29	#1570	—	2	1	no	—
30	#1567	C₂²	1	4	yes	—
31	#1565	—	2	1	no	—
32	#1565	—	2	1	no	—
33	#1564	C₂²	1	4	yes	—
34	C₂×#335	—	2	1	no	—
35	C₂×#334	—	2	1	no	—
36	#1454	—	2	2	no	—
37	#1452	—	2	1	no	—
38	#1451	C₂²	1	2	yes	—
39	#1450	—	2	1	no	—
40	C₈×M₁₆	C₂²	1	2	yes	—
41	#1299	C₂²	1	4	yes	—
42	#1299	C₂²	1	4	yes	—
43	C₂²×M₁₆	—	2	1	no	—
44	C₂²×M₁₆	—	2	1	no	—
45	C₂²×C₄²	C₂³	1	1	yes	characteristic
46	#432	—	2	1	no	—
47	#432	C₂×C₄	1	2	yes	—
48	#432	Q₈	1	2	yes	—
49	#432	Q₈	1	2	yes	—
50	#431	—	2	1	no	—
51	#431	—	2	1	no	—
52	#431	—	2	1	no	—
53	#431	—	2	1	no	—
54	#431	—	2	1	no	—
55	#431	—	2	1	no	—
56	#431	—	2	1	no	—
57	#431	C₂×C₄	1	2	yes	—
58	#431	C₂×C₄	1	2	yes	—
59	#431	C₂×C₄	1	2	yes	—
60	#431	D₄	1	1	yes	characteristic
61	#431	D₄	1	1	yes	characteristic
62	#431	D₄	1	1	yes	characteristic
63	#431	D₄	1	1	yes	characteristic
64	#431	D₄	1	2	yes	—
65	#431	C₂³	1	2	yes	—
66	#431	C₂³	1	2	yes	—
67	#431	C₂³	1	1	yes	characteristic
68	#431	C₂³	1	1	yes	characteristic
69	#423	—	2	1	no	—
70	#423	—	2	1	no	—
71	#423	—	2	1	no	—
72	#423	—	2	1	no	—
73	#423	—	2	1	no	—
74	#423	—	2	1	no	—
75	#423	—	2	1	no	—
76	#423	C₂×C₄	1	2	yes	—
77	#423	C₂³	1	2	yes	—
78	#423	C₂³	1	2	yes	—
79	C₂×C₄⋊C₈	—	2	1	no	—
80	C₂×C₄⋊C₈	—	2	1	no	—
81	C₂×C₄⋊C₈	—	2	1	no	—
82	C₂×C₄⋊C₈	—	2	1	no	—
83	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
84	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₈	Q₈	1	2	yes	—
87	#407	—	2	1	no	—
88	#407	—	2	1	no	—
89	C₂×C₂²⋊C₈	—	2	1	no	—
90	C₂×C₂²⋊C₈	—	2	1	no	—
91	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
92	C₄×M₁₆	—	2	1	no	—
93	C₄×M₁₆	C₂³	1	2	yes	—
94	C₂×C₄×C₈	—	2	1	no	—
95	C₂×C₄×C₈	—	2	1	no	—
96	C₂×C₄×C₈	C₂³	1	2	yes	—
97	#335	—	2	2	no	—
98	#335	—	2	2	no	—
99	#335	—	2	2	no	—
100	#335	—	2	2	no	—
101	#335	—	2	4	no	—
102	#335	—	2	4	no	—
103	#335	C₂×C₄	1	8	yes	—
104	#334	—	2	2	no	—
105	#334	—	2	2	no	—
106	#334	—	2	2	no	—
107	#334	—	2	2	no	—
108	#334	C₂×C₄	1	8	yes	—
109	#322	—	2	2	no	—
110	#322	—	2	2	no	—
111	#322	—	2	2	no	—
112	C₈²	—	2	2	no	—
113	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
114	C₂³×C₄	#41	1	1	yes	characteristic
115	C₂³×C₄	#41	1	1	yes	characteristic
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	4	1	no	—
121	C₂×M₁₆	—	4	1	no	—
122	C₂×M₁₆	—	8	2	no	—
123	C₂×M₁₆	—	8	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	4	1	no	—
128	C₂²×C₈	—	4	1	no	—
129	C₂²×C₈	C₂×Q₈	1	4	yes	—
130	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
137	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
138	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
139	C₂×C₄²	#41	1	1	yes	characteristic
140	C₂×C₄²	#41	1	1	yes	characteristic
141	C₂×C₄²	C₂⁴	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	4	1	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄⋊C₈	—	4	1	no	—
155	C₄⋊C₈	—	4	1	no	—
156	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
157	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
158	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
159	C₄⋊C₈	C₂²×C₄	1	4	yes	—
160	C₄⋊C₈	C₂²×C₄	1	4	yes	—
161	C₄⋊C₈	C₂×D₄	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	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₂²⋊C₈	—	2	1	no	—
168	C₂²⋊C₈	—	2	1	no	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₂²⋊C₈	—	2	1	no	—
173	C₂²⋊C₈	—	2	1	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
176	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
177	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
178	#97	—	2	2	no	—
179	#97	C₂×Q₈	1	4	yes	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	C₂²×C₄	1	4	yes	—
189	C₄×C₈	C₂×D₄	1	2	yes	—
190	C₄×C₈	C₂×D₄	1	2	yes	—
191	C₂⁴	#125	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
202	C₂²×C₄	#121	1	1	yes	characteristic
203	C₂²×C₄	#121	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₄	#125	1	1	yes	characteristic
207	C₂²×C₄	#125	1	1	yes	characteristic
208	C₂²×C₄	#125	1	1	yes	characteristic
209	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×#41	1	1	yes	characteristic
213	C₂²×C₄	C₂×#41	1	1	yes	characteristic
214	M₁₆	—	4	4	no	—
215	M₁₆	—	4	4	no	—
216	M₁₆	—	8	2	no	—
217	M₁₆	—	8	2	no	—
218	M₁₆	—	8	2	no	—
219	M₁₆	—	8	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	8	2	no	—
234	C₂×C₈	—	8	2	no	—
235	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
236	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
237	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
238	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
239	C₄²	C₄×D₄	1	1	yes	characteristic
240	C₄²	C₄×D₄	1	1	yes	characteristic
241	C₄²	C₄×Q₈	1	1	yes	characteristic
242	C₄²	C₄×Q₈	1	1	yes	characteristic
243	C₄²	#121	1	1	yes	characteristic
244	C₄²	#121	1	1	yes	characteristic
245	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
246	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
247	C₄²	C₄⋊Q₈	1	1	yes	characteristic
248	C₄²	C₄⋊Q₈	1	1	yes	characteristic
249	C₄²	C₂³×C₄	1	1	yes	characteristic
250	C₄²	C₂²×D₄	1	1	yes	characteristic
251	C₄²	C₂²×D₄	1	1	yes	characteristic
252	C₄²	C₂²×Q₈	1	1	yes	characteristic
253	C₂³	—	2	2	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	4	1	no	—
256	C₂³	#389	1	1	yes	characteristic
257	C₂³	#389	1	1	yes	characteristic
258	C₂³	C₂×#125	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	#389	1	1	yes	characteristic
279	C₂×C₄	#389	1	1	yes	characteristic
280	C₂×C₄	#389	1	1	yes	characteristic
281	C₂×C₄	#389	1	1	yes	characteristic
282	C₂×C₄	#389	1	1	yes	characteristic
283	C₂×C₄	#389	1	1	yes	characteristic
284	C₂×C₄	#472	1	2	yes	—
285	C₂×C₄	#473	1	2	yes	—
286	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
289	C₂×C₄	C₂×#121	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	4	no	—
294	C₈	—	4	4	no	—
295	C₈	—	4	4	no	—
296	C₈	—	4	4	no	—
297	C₈	—	4	1	no	—
298	C₈	—	4	1	no	—
299	C₈	—	4	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	8	2	no	—
302	C₈	—	8	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	#1952	1	2	yes	—
313	C₂²	C₂×#389	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	#1952	1	2	yes	—
323	C₄	#2916	1	2	yes	—
324	C₄	#2955	1	2	yes	—
325	C₄	C₂×#472	1	1	yes	characteristic
326	C₄	C₂×#473	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	4	1	no	—
329	C₂	—	4	1	no	—
330	C₂	#21164	1	1	yes	characteristic
331	C₂	#21178	1	1	yes	characteristic
332	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
6. #9636	C₂	semidirect
7. #9634	C₂	semidirect

Maximal subgroups

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

subgroup	index
#17814	2
#17809	2
#17806	2
C₂×#1574	2
#17326	2
#9636	2
#9634	2
#8065	2
#8061	2