Subgroups — #109399

Order 512.

573 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20476	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#108	C₂	1	1	yes	maximal, characteristic
3	#17793	C₂	1	1	yes	maximal, characteristic
4	#8087	C₂	1	2	yes	maximal
5	#7973	C₂	1	2	yes	maximal
6	#2920	C₂²	1	2	yes	—
7	C₂×#432	C₂²	1	1	yes	characteristic
8	C₂²×#108	—	2	1	no	—
9	C₂×C₄×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2283	C₂²	1	1	yes	characteristic
12	C₄×#108	—	2	1	no	—
13	C₄×#108	—	2	1	no	—
14	C₄×#108	C₄	1	2	yes	complemented
15	C₄×#108	C₄	1	2	yes	complemented
16	#1770	C₄	1	2	yes	—
17	C₈×C₄⋊C₄	C₄	1	2	yes	—
18	#1752	—	2	1	no	—
19	C₈×C₂²⋊C₄	—	2	1	no	—
20	#1746	—	2	1	no	—
21	C₂×#341	—	2	1	no	—
22	#1726	—	2	1	no	—
23	#1568	C₄	1	2	yes	complemented
24	#1568	C₄	1	2	yes	complemented
25	#1567	—	2	1	no	—
26	#1566	—	2	1	no	—
27	#1564	—	2	1	no	—
28	#1564	—	2	1	no	—
29	#1302	—	2	4	no	—
30	#1288	—	2	4	no	—
31	C₂²×M₁₆	—	2	2	no	—
32	C₂³×C₈	D₄	1	1	yes	characteristic
33	C₂³×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#432	—	2	1	no	—
36	#432	—	2	1	no	—
37	#432	—	2	2	no	—
38	#432	C₂×C₄	1	4	yes	—
39	#432	D₄	1	2	yes	—
40	#432	Q₈	1	2	yes	—
41	#431	—	2	2	no	—
42	#431	—	2	2	no	—
43	C₂×#108	—	2	8	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	2	no	—
46	#423	—	2	2	no	—
47	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	#407	—	2	2	no	—
49	C₂×C₂²⋊C₈	—	2	1	no	—
50	C₄×M₁₆	—	2	2	no	—
51	C₄×M₁₆	—	2	2	no	—
52	C₄×M₁₆	—	2	2	no	—
53	C₄×M₁₆	—	2	2	no	—
54	C₄×M₁₆	C₂×C₄	1	4	yes	—
55	C₂×#97	—	2	1	no	—
56	C₂×#97	—	2	2	no	—
57	C₂×#97	—	2	2	no	—
58	C₂×C₄×C₈	—	2	2	no	—
59	C₂×C₄×C₈	—	2	2	no	—
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
72	#383	D₄	1	2	yes	—
73	#382	—	2	1	no	—
74	#381	Q₈	1	2	yes	—
75	#380	—	2	1	no	—
76	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
77	C₄×C₂²⋊C₄	—	2	1	no	—
78	#376	—	2	1	no	—
79	#341	—	2	2	no	—
80	#341	—	2	2	no	—
81	#336	—	2	2	no	—
82	#336	—	2	2	no	—
83	#335	—	2	2	no	—
84	#335	—	2	2	no	—
85	#334	—	2	2	no	—
86	#334	—	2	2	no	—
87	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
89	C₂³×C₄	#41	1	1	yes	characteristic
90	C₂×M₁₆	—	2	8	no	—
91	C₂×M₁₆	—	2	4	no	—
92	C₂×M₁₆	—	4	4	no	—
93	C₂×M₁₆	—	4	2	no	—
94	C₂×M₁₆	—	4	2	no	—
95	C₂²×C₈	—	2	4	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	4	2	no	—
98	C₂²×C₈	—	4	2	no	—
99	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊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	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₄²	—	2	2	no	—
114	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄²	C₂×D₄	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×Q₈	1	1	yes	characteristic
122	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
123	C₂×C₄²	#41	1	1	yes	characteristic
124	C₂×C₄²	#41	1	1	yes	characteristic
125	C₂×C₄²	#41	1	1	yes	characteristic
126	#108	—	2	8	no	—
127	#108	—	2	8	no	—
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	4	2	no	—
131	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	2	4	no	—
135	C₂²⋊C₈	—	4	2	no	—
136	#97	—	2	4	no	—
137	#97	—	2	4	no	—
138	#97	—	2	2	no	—
139	#97	—	2	2	no	—
140	#97	—	4	4	no	—
141	#97	—	4	2	no	—
142	C₄×C₈	—	2	4	no	—
143	C₄×C₈	—	4	4	no	—
144	C₄×C₈	—	4	2	no	—
145	C₄×C₈	—	4	2	no	—
146	C₄×C₈	C₄²	1	1	yes	characteristic
147	C₄×C₈	C₄²	1	1	yes	characteristic
148	C₄×C₈	C₄²	1	1	yes	characteristic
149	C₄×C₈	C₄²	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊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	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	#95	—	2	2	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	#95	C₄⋊C₄	1	4	yes	—
166	C₂⁴	C₂²⋊Q₈	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	—	4	2	no	—
172	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	#117	1	1	yes	characteristic
177	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
178	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
179	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
180	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
181	C₂²×C₄	#121	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	#124	1	1	yes	characteristic
187	C₂²×C₄	#125	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
190	M₁₆	—	2	8	no	—
191	M₁₆	—	4	8	no	—
192	M₁₆	—	4	4	no	—
193	M₁₆	—	8	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	4	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	8	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	4	no	—
211	C₂×C₈	#95	1	1	yes	characteristic
212	C₂×C₈	#95	1	1	yes	characteristic
213	C₂×C₈	#95	1	1	yes	characteristic
214	C₂×C₈	#95	1	1	yes	characteristic
215	C₂×C₈	#95	1	1	yes	characteristic
216	C₂×C₈	#95	1	1	yes	characteristic
217	C₂×C₈	#95	1	1	yes	characteristic
218	C₂×C₈	#95	1	1	yes	characteristic
219	C₂×C₈	#104	1	4	yes	—
220	C₄⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	4	2	no	—
223	C₄²	—	4	2	no	—
224	C₄²	C₂×C₄²	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₄²	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	#117	1	1	yes	characteristic
230	C₄²	C₄×D₄	1	1	yes	characteristic
231	C₄²	C₄×D₄	1	1	yes	characteristic
232	C₄²	C₄×D₄	1	1	yes	characteristic
233	C₄²	C₄×Q₈	1	1	yes	characteristic
234	C₄²	#121	1	1	yes	characteristic
235	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
236	C₄²	#124	1	1	yes	characteristic
237	C₄²	#125	1	1	yes	characteristic
238	C₄²	#127	1	1	yes	characteristic
239	C₄²	C₄⋊Q₈	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	4	1	no	—
243	C₂³	#380	1	1	yes	characteristic
244	C₂³	#389	1	1	yes	characteristic
245	C₂³	#391	1	1	yes	characteristic
246	C₂×C₄	—	2	2	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	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₂×#95	1	1	yes	characteristic
269	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	#380	1	1	yes	characteristic
272	C₂×C₄	#382	1	1	yes	characteristic
273	C₂×C₄	#384	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	#390	1	1	yes	characteristic
278	C₂×C₄	#390	1	1	yes	characteristic
279	C₂×C₄	#391	1	1	yes	characteristic
280	C₂×C₄	C₂×#104	1	2	yes	—
281	C₂×C₄	#421	1	2	yes	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	4	8	no	—
289	C₈	—	4	4	no	—
290	C₈	—	8	4	no	—
291	C₈	#339	1	2	yes	—
292	C₈	#339	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	4	4	no	—
301	C₂²	C₄⋊#95	1	1	yes	characteristic
302	C₂²	#1806	1	2	yes	—
303	C₂²	#1953	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
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	—	8	4	no	—
314	C₄	C₂×#339	1	1	yes	characteristic
315	C₄	#1733	1	1	yes	characteristic
316	C₄	#1806	1	2	yes	—
317	C₄	#1833	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	#10852	1	1	yes	characteristic
322	C₂	#10868	1	1	yes	characteristic
323	C₂	#10873	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
14. C₄×#108	C₄	semidirect
15. C₄×#108	C₄	semidirect
23. #1568	C₄	semidirect
24. #1568	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20476	2
C₂×C₄×#108	2
#17793	2
#8087	2
#7973	2