Subgroups — #203729

Order 512.

655 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#108	C₂	1	1	yes	maximal, characteristic
2	#17796	C₂	1	2	yes	maximal
3	C₂×#1566	C₂	1	1	yes	maximal, characteristic
4	#17323	C₂	1	1	yes	maximal, characteristic
5	#9609	C₂	1	4	yes	maximal, complemented
6	#9607	C₂	1	4	yes	maximal, complemented
7	#7938	C₂	1	2	yes	maximal
8	#2920	C₂²	1	1	yes	characteristic
9	#2920	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₄	1	2	yes	—
11	C₂×#423	C₂²	1	2	yes	—
12	C₂×C₄×M₁₆	C₂²	1	2	yes	—
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×#108	C₄	1	1	yes	characteristic
15	C₄×#108	C₄	1	2	yes	—
16	C₄×#108	C₄	1	1	yes	characteristic
17	C₄×#108	C₂²	1	2	yes	—
18	C₄×#108	C₂²	1	2	yes	—
19	C₂×#341	C₄	1	2	yes	—
20	#1569	—	2	4	no	—
21	#1567	—	2	2	no	—
22	#1566	—	2	2	no	—
23	#1566	—	2	1	no	—
24	#1566	—	2	1	no	—
25	#1566	—	2	1	no	—
26	#1566	—	2	1	no	—
27	#1565	C₂²	1	8	yes	—
28	C₂×#334	C₂²	1	2	yes	—
29	C₂×#334	C₂²	1	2	yes	—
30	#1452	C₂²	1	2	yes	—
31	#1451	—	2	1	no	—
32	#1451	—	2	1	no	—
33	#1450	C₂²	1	2	yes	—
34	C₈×M₁₆	—	2	1	no	—
35	C₈×M₁₆	—	2	1	no	—
36	#1283	C₂²	1	8	yes	—
37	C₂²×M₁₆	—	2	2	no	—
38	C₂²×M₁₆	C₂×C₄	1	4	yes	—
39	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄²	C₂³	1	1	yes	characteristic
42	#432	—	2	1	no	—
43	#432	—	2	1	no	—
44	#432	D₄	1	2	yes	—
45	#432	Q₈	1	2	yes	—
46	#431	—	2	1	no	—
47	#431	—	2	1	no	—
48	#431	C₂³	1	2	yes	—
49	#431	C₂³	1	2	yes	—
50	C₂×#108	—	2	8	no	—
51	C₂×#108	C₂×C₄	1	4	yes	—
52	C₂×#108	C₂×C₄	1	2	yes	—
53	C₂×#108	C₂×C₄	1	2	yes	—
54	#423	—	2	1	no	—
55	#423	—	2	2	no	—
56	#423	—	2	2	no	—
57	#423	—	2	2	no	—
58	#423	—	2	2	no	—
59	#423	D₄	1	2	yes	—
60	C₂×C₄⋊C₈	—	2	2	no	—
61	C₂×C₄⋊C₈	C₂³	1	4	yes	—
62	#407	—	2	1	no	—
63	#407	Q₈	1	2	yes	—
64	C₄×M₁₆	—	2	2	no	—
65	C₄×M₁₆	—	2	1	no	—
66	C₄×M₁₆	—	2	1	no	—
67	C₄×M₁₆	—	2	2	no	—
68	C₄×M₁₆	C₂×C₄	1	2	yes	—
69	C₄×M₁₆	C₂×C₄	1	2	yes	—
70	C₄×M₁₆	C₂³	1	4	yes	—
71	C₂×#97	—	2	2	no	—
72	C₂×C₄×C₈	—	2	2	no	—
73	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
78	C₂×C₄×C₈	D₄	1	1	yes	characteristic
79	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
81	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
82	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
85	#341	C₂×C₄	1	2	yes	—
86	#341	C₂×C₄	1	4	yes	—
87	#341	C₂×C₄	1	2	yes	—
88	#335	—	2	8	no	—
89	#334	—	2	8	no	—
90	#334	—	2	2	no	—
91	#334	—	2	2	no	—
92	#334	—	2	2	no	—
93	#334	—	2	2	no	—
94	#322	—	2	2	no	—
95	#322	—	2	2	no	—
96	#322	—	2	2	no	—
97	C₈²	—	2	2	no	—
98	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
99	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
100	C₂³×C₄	#41	1	1	yes	characteristic
101	C₂×M₁₆	—	2	4	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	2	8	no	—
105	C₂×M₁₆	—	4	4	no	—
106	C₂×M₁₆	—	4	2	no	—
107	C₂×M₁₆	—	4	2	no	—
108	C₂×M₁₆	—	8	4	no	—
109	C₂×M₁₆	C₂²×C₄	1	4	yes	—
110	C₂×M₁₆	C₂²×C₄	1	4	yes	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	—	2	4	no	—
113	C₂²×C₈	—	2	4	no	—
114	C₂²×C₈	—	4	2	no	—
115	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
133	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
134	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
135	C₂×C₄²	#41	1	1	yes	characteristic
136	C₂×C₄²	#41	1	1	yes	characteristic
137	C₂×C₄²	#41	1	1	yes	characteristic
138	C₂×C₄²	C₂⁴	1	1	yes	characteristic
139	#108	—	2	8	no	—
140	#108	—	2	4	no	—
141	#108	M₁₆	1	8	yes	complemented
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	4	no	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	4	2	no	—
148	C₄⋊C₈	—	4	2	no	—
149	C₄⋊C₈	C₂×D₄	1	4	yes	—
150	C₂²⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	C₂×Q₈	1	4	yes	—
154	#97	—	2	2	no	—
155	#97	—	2	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	2	no	—
158	#97	—	2	4	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	4	no	—
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	C₂×D₄	1	1	yes	characteristic
173	C₄×C₈	C₂×D₄	1	1	yes	characteristic
174	C₄×C₈	C₂×D₄	1	1	yes	characteristic
175	C₄×C₈	C₂×D₄	1	1	yes	characteristic
176	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
177	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
178	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
179	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
180	C₂⁴	C₄×Q₈	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
188	C₂²×C₄	C₄×D₄	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	C₄×Q₈	1	1	yes	characteristic
192	C₂²×C₄	#121	1	1	yes	characteristic
193	C₂²×C₄	#121	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×C₄	#125	1	1	yes	characteristic
197	C₂²×C₄	#125	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×#41	1	1	yes	characteristic
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	8	no	—
207	M₁₆	—	4	4	no	—
208	M₁₆	—	4	4	no	—
209	M₁₆	—	8	4	no	—
210	M₁₆	—	8	4	no	—
211	M₁₆	C₂×M₁₆	1	8	yes	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	8	no	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	8	4	no	—
230	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₈	C₂×M₁₆	1	4	yes	—
239	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
241	C₄²	C₄×D₄	1	1	yes	characteristic
242	C₄²	C₄×D₄	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	#125	1	1	yes	characteristic
248	C₄²	#125	1	1	yes	characteristic
249	C₄²	C₄⋊Q₈	1	1	yes	characteristic
250	C₄²	C₄⋊Q₈	1	1	yes	characteristic
251	C₄²	C₂²×D₄	1	1	yes	characteristic
252	C₄²	C₂²×D₄	1	1	yes	characteristic
253	C₄²	C₂²×Q₈	1	1	yes	characteristic
254	C₄²	C₂×#41	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	#389	1	1	yes	characteristic
259	C₂³	#389	1	1	yes	characteristic
260	C₂³	C₂×C₄×Q₈	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
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	#389	1	1	yes	characteristic
285	C₂×C₄	#471	1	4	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×#121	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
290	C₂×C₄	C₂×#125	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×M₁₆	1	2	yes	—
293	C₂×C₄	#568	1	2	yes	—
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	8	no	—
301	C₈	—	4	4	no	—
302	C₈	—	4	4	no	—
303	C₈	—	4	4	no	—
304	C₈	—	8	4	no	—
305	C₈	#423	1	2	yes	—
306	C₈	#423	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	#1951	1	2	yes	—
316	C₂²	C₂×#389	1	1	yes	characteristic
317	C₂²	Q₈×M₁₆	1	2	yes	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	#1951	1	2	yes	—
326	C₄	#1953	1	2	yes	—
327	C₄	C₂×#423	1	1	yes	characteristic
328	C₄	#2905	1	1	yes	characteristic
329	C₄	D₄×M₁₆	1	2	yes	—
330	C₄	C₂×#471	1	2	yes	—
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	4	1	no	—
334	C₂	#21146	1	1	yes	characteristic
335	C₂	C₂×#1951	1	1	yes	characteristic
336	C₂	C₂×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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. #9609	C₂	semidirect
6. #9607	C₂	semidirect
141. #108	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#108	2
#17796	2
C₂×#1566	2
#17323	2
#9609	2
#9607	2
#7938	2