Subgroups — #225736

Order 512.

585 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22802	C₂	1	1	yes	maximal, characteristic
2	#22688	C₂	1	1	yes	maximal, characteristic
3	#17886	C₂	1	1	yes	maximal, characteristic
4	#17832	C₂	1	2	yes	maximal, complemented
5	#9628	C₂	1	1	yes	maximal, characteristic, complemented
6	#9627	C₂	1	1	yes	maximal, characteristic, complemented
7	#8905	C₂	1	2	yes	maximal, complemented
8	#8904	C₂	1	2	yes	maximal, complemented
9	#8822	C₂	1	1	yes	maximal, characteristic, complemented
10	#8668	C₂	1	2	yes	maximal, complemented
11	C₈×#107	C₂	1	1	yes	maximal, characteristic, complemented
12	#2965	C₂²	1	2	yes	—
13	C₂×#443	—	2	1	no	—
14	C₄×#135	C₂²	1	1	yes	characteristic
15	#2925	C₂²	1	2	yes	—
16	#2906	C₂²	1	2	yes	—
17	#1974	—	2	1	no	—
18	#1945	—	2	1	no	—
19	#1929	—	2	2	no	—
20	#1866	—	2	2	no	—
21	#1848	C₂²	1	1	yes	characteristic
22	#1848	C₂²	1	1	yes	characteristic
23	#1806	C₂²	1	2	yes	—
24	#1806	C₂²	1	2	yes	—
25	C₄×#107	C₂²	1	1	yes	characteristic
26	#1772	—	2	1	no	—
27	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	#1591	C₄	1	1	yes	characteristic, complemented
30	#1591	C₄	1	1	yes	characteristic, complemented
31	#1590	C₄	1	2	yes	complemented
32	#1590	C₄	1	2	yes	complemented
33	#1589	C₄	1	1	yes	characteristic, complemented
34	#1589	C₄	1	1	yes	characteristic, complemented
35	#1574	—	2	2	no	—
36	#1573	—	2	2	no	—
37	#1571	C₂²	1	2	yes	—
38	#1570	C₂²	1	2	yes	—
39	#1569	—	2	2	no	—
40	#1566	C₂²	1	2	yes	—
41	#1566	C₂²	1	2	yes	—
42	#1565	—	2	2	no	—
43	#1564	C₂²	1	1	yes	characteristic
44	C₂×#335	C₂²	1	1	yes	characteristic
45	#1476	C₂²	1	2	yes	—
46	C₂×C₈²	C₂²	1	1	yes	characteristic
47	#1303	C₂²	1	2	yes	—
48	#1303	C₂²	1	2	yes	—
49	#1279	C₂²	1	1	yes	characteristic
50	#1279	C₂²	1	1	yes	characteristic
51	C₂×#135	D₄	1	1	yes	characteristic
52	C₂×#131	—	2	2	no	—
53	C₄×#41	C₂³	1	2	yes	—
54	#446	C₂×C₄	1	2	yes	—
55	#446	C₂×C₄	1	2	yes	—
56	#443	—	4	4	no	—
57	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₁₆	Q₈	1	1	yes	characteristic
59	C₄×SD₁₆	C₂×C₄	1	2	yes	—
60	C₄×SD₁₆	Q₈	1	2	yes	—
61	C₄×D₈	C₂×C₄	1	1	yes	characteristic
62	C₄×D₈	Q₈	1	1	yes	characteristic
63	#436	C₂×C₄	1	2	yes	—
64	#436	C₂×C₄	1	2	yes	—
65	#435	—	2	2	no	—
66	#435	—	2	2	no	—
67	#433	—	4	2	no	—
68	#432	—	2	2	no	—
69	#431	—	2	2	no	—
70	#431	—	2	2	no	—
71	#431	C₂³	1	2	yes	—
72	C₂×#108	—	2	1	no	—
73	C₂×#108	—	2	1	no	—
74	C₂×#108	—	2	1	no	—
75	C₂×#108	—	2	1	no	—
76	#427	D₄	1	1	yes	characteristic
77	C₂×#107	—	2	1	no	—
78	#424	—	2	2	no	—
79	#424	—	4	2	no	—
80	#423	—	2	2	no	—
81	#423	C₂³	1	2	yes	—
82	#423	C₂³	1	2	yes	—
83	C₂×C₄⋊C₈	—	2	2	no	—
84	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×#104	—	2	2	no	—
87	C₂×#104	—	2	2	no	—
88	#416	D₄	1	2	yes	—
89	#408	—	2	2	no	—
90	#405	—	2	2	no	—
91	C₄×M₁₆	—	2	2	no	—
92	C₂×C₄×C₈	—	2	1	no	—
93	C₂×C₄×C₈	—	2	1	no	—
94	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
97	C₄×C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₄×C₄⋊C₄	C₂³	1	1	yes	characteristic
99	#340	—	2	1	no	—
100	#339	—	2	2	no	—
101	#339	—	2	2	no	—
102	#336	—	2	1	no	—
103	#336	—	2	1	no	—
104	#335	—	2	2	no	—
105	#335	—	2	2	no	—
106	#335	C₂×C₄	1	1	yes	characteristic
107	#335	C₂×C₄	1	1	yes	characteristic
108	#335	D₄	1	1	yes	characteristic
109	#335	D₄	1	1	yes	characteristic
110	#335	D₄	1	1	yes	characteristic
111	#335	D₄	1	1	yes	characteristic
112	#334	—	2	2	no	—
113	#334	—	2	2	no	—
114	#334	—	2	2	no	—
115	#334	—	2	2	no	—
116	#334	—	4	2	no	—
117	#334	C₂×C₄	1	2	yes	—
118	Q₈⋊C₈	C₂×C₄	1	2	yes	—
119	Q₈⋊C₈	C₂×C₄	1	2	yes	—
120	D₄⋊C₈	C₂×C₄	1	2	yes	—
121	D₄⋊C₈	C₂×C₄	1	2	yes	—
122	C₈²	C₂×C₄	1	1	yes	characteristic
123	C₈²	C₂×C₄	1	1	yes	characteristic
124	C₈²	C₂×C₄	1	1	yes	characteristic
125	C₈²	C₂×C₄	1	1	yes	characteristic
126	C₂×#41	C₂×D₄	1	2	yes	—
127	#135	—	2	2	no	—
128	#135	—	2	2	no	—
129	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
130	C₂×SD₁₆	C₄⋊C₄	1	2	yes	—
131	C₂×D₈	C₄⋊C₄	1	1	yes	characteristic
132	#131	—	4	4	no	—
133	#131	—	8	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	4	2	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	4	2	no	—
143	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₈	#41	1	1	yes	characteristic
146	C₂²×C₈	#41	1	1	yes	characteristic
147	C₄×Q₈	C₂²×C₄	1	2	yes	—
148	C₄×Q₈	C₂×Q₈	1	2	yes	—
149	C₄×D₄	—	2	2	no	—
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	C₂²×C₄	1	2	yes	—
152	C₄×D₄	C₂×Q₈	1	2	yes	—
153	#117	—	2	2	no	—
154	#117	C₂×D₄	1	2	yes	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	C₂⁴	1	1	yes	characteristic
161	#108	—	4	2	no	—
162	#108	—	4	2	no	—
163	#108	—	8	1	no	—
164	#108	—	8	1	no	—
165	#107	—	2	4	no	—
166	#107	C₄⋊C₄	1	1	yes	characteristic
167	#107	C₄⋊C₄	1	1	yes	characteristic
168	#106	C₄⋊C₄	1	2	yes	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	4	2	no	—
178	C₄⋊C₈	—	4	2	no	—
179	C₄⋊C₈	C₂²×C₄	1	2	yes	—
180	C₄⋊C₈	C₂²×C₄	1	2	yes	—
181	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
182	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
184	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
185	#104	—	4	4	no	—
186	#104	—	8	2	no	—
187	#103	C₄⋊C₄	1	2	yes	—
188	#103	C₄⋊C₄	1	2	yes	—
189	#102	C₄⋊C₄	1	2	yes	—
190	#102	C₄⋊C₄	1	2	yes	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	4	2	no	—
195	C₂²⋊C₈	—	4	2	no	—
196	#97	—	2	2	no	—
197	#97	—	2	2	no	—
198	#97	—	4	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	4	2	no	—
202	C₄×C₈	—	4	2	no	—
203	C₄×C₈	—	4	2	no	—
204	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
205	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
206	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
207	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
208	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
209	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
210	C₄×C₈	C₂×D₄	1	1	yes	characteristic
211	C₄×C₈	C₂×D₄	1	1	yes	characteristic
212	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
213	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#41	—	2	2	no	—
217	#41	—	2	2	no	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	C₂×Q₈	C₂×C₄⋊C₄	1	2	yes	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	C₂×C₄⋊C₄	1	2	yes	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×#41	1	1	yes	characteristic
230	C₂²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	SD₁₆	—	2	4	no	—
233	D₈	—	2	2	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	8	2	no	—
240	M₁₆	—	8	4	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	8	4	no	—
261	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₈	C₄×D₄	1	1	yes	characteristic
266	C₂×C₈	C₄×D₄	1	1	yes	characteristic
267	C₂×C₈	C₄×D₄	1	1	yes	characteristic
268	C₂×C₈	C₄×D₄	1	1	yes	characteristic
269	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
270	C₂×C₈	#137	1	2	yes	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
278	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₄²	—	4	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂³×C₄	1	1	yes	characteristic
290	C₄²	C₂²×D₄	1	1	yes	characteristic
291	C₄²	C₂²×Q₈	1	1	yes	characteristic
292	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
293	C₂³	—	2	2	no	—
294	C₂³	#547	1	1	yes	characteristic
295	Q₈	—	2	2	no	—
296	Q₈	—	2	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	8	1	no	—
318	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
320	C₂×C₄	#520	1	1	yes	characteristic
321	C₂×C₄	D₄²	1	1	yes	characteristic
322	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
323	C₂×C₄	#554	1	1	yes	characteristic
324	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
325	C₂×C₄	C₂×#137	1	1	yes	characteristic
326	C₂×C₄	#577	1	1	yes	characteristic
327	C₂×C₄	#577	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	1	no	—
340	C₈	—	4	1	no	—
341	C₈	—	8	2	no	—
342	C₈	—	8	2	no	—
343	C₈	#428	1	2	yes	—
344	C₈	#428	1	2	yes	—
345	C₈	#443	1	2	yes	complemented
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	4	2	no	—
351	C₂²	D₄×C₄⋊C₄	1	1	yes	characteristic
352	C₂²	#3354	1	1	yes	characteristic
353	C₂²	#3356	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	8	2	no	—
361	C₄	—	8	2	no	—
362	C₄	C₂×#428	1	1	yes	characteristic
363	C₄	#2912	1	1	yes	characteristic
364	C₄	C₂×#443	1	1	yes	characteristic
365	C₄	#2957	1	1	yes	characteristic
366	C₄	#3272	1	1	yes	characteristic
367	C₄	#3278	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	8	2	no	—
371	C₂	#22663	1	1	yes	characteristic
372	C₂	#22688	1	1	yes	characteristic
373	C₂	#22694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #17832	C₂	semidirect
5. #9628	C₂	semidirect
6. #9627	C₂	semidirect
7. #8905	C₂	semidirect
8. #8904	C₂	semidirect
9. #8822	C₂	semidirect
10. #8668	C₂	semidirect
11. C₈×#107	C₂	semidirect
29. #1591	C₄	semidirect
30. #1591	C₄	semidirect
31. #1590	C₄	semidirect
32. #1590	C₄	semidirect
33. #1589	C₄	semidirect
34. #1589	C₄	semidirect
345. C₈	#443	semidirect

Maximal subgroups

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

subgroup	index
#22802	2
#22688	2
#17886	2
#17832	2
#9628	2
#9627	2
#8905	2
#8904	2
#8822	2
#8668	2
C₈×#107	2