Subgroups — #225773

Order 512.

528 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#18061	C₂	1	2	yes	maximal
2	C₈×#135	C₂	1	1	yes	maximal, characteristic
3	#17835	C₂	1	1	yes	maximal, characteristic, complemented
4	#17834	C₂	1	1	yes	maximal, characteristic, complemented
5	#9636	C₂	1	2	yes	maximal, complemented
6	#8901	C₂	1	2	yes	maximal, complemented
7	#8900	C₂	1	2	yes	maximal, complemented
8	#8824	C₂	1	1	yes	maximal, characteristic, complemented
9	#8670	C₂	1	1	yes	maximal, characteristic, complemented
10	#8670	C₂	1	1	yes	maximal, characteristic, complemented
11	#8293	C₂	1	1	yes	maximal, characteristic, complemented
12	#2991	C₂²	1	2	yes	—
13	#2990	C₂²	1	2	yes	—
14	C₈×#41	C₂²	1	1	yes	characteristic
15	C₈×#41	C₂²	1	1	yes	characteristic
16	C₄×#135	C₂²	1	1	yes	characteristic
17	#1837	C₂²	1	2	yes	—
18	C₄×#108	C₂²	1	1	yes	characteristic
19	#1592	—	2	2	no	—
20	#1584	—	2	2	no	—
21	#1581	—	2	2	no	—
22	SD₁₆⋊C₈	—	2	2	no	—
23	C₈×Q₁₆	C₄	1	1	yes	characteristic
24	C₈×Q₁₆	C₄	1	1	yes	characteristic
25	C₈×SD₁₆	C₄	1	1	yes	characteristic
26	C₈×SD₁₆	C₄	1	1	yes	characteristic
27	C₈×SD₁₆	C₄	1	1	yes	characteristic
28	C₈×SD₁₆	C₄	1	1	yes	characteristic
29	C₈×D₈	C₄	1	1	yes	characteristic
30	C₈×D₈	C₄	1	1	yes	characteristic
31	#1574	—	2	2	no	—
32	#1574	C₂²	1	2	yes	—
33	#1574	C₂²	1	2	yes	—
34	#1573	—	2	2	no	—
35	#1568	—	2	1	no	—
36	#1567	C₂²	1	2	yes	—
37	#1566	—	2	1	no	—
38	#1566	C₂²	1	1	yes	characteristic
39	#1566	C₂²	1	1	yes	characteristic
40	#1564	—	2	1	no	—
41	#1564	C₂²	1	1	yes	characteristic
42	#1564	C₂²	1	1	yes	characteristic
43	C₂×#334	—	2	1	no	—
44	#1489	C₂²	1	2	yes	—
45	#1489	C₂²	1	2	yes	—
46	#1476	C₂²	1	1	yes	characteristic
47	#1476	C₂²	1	1	yes	characteristic
48	#1451	C₂²	1	2	yes	—
49	C₂×C₈²	C₂²	1	1	yes	characteristic
50	#1299	C₂²	1	2	yes	—
51	#1299	C₂²	1	2	yes	—
52	#1278	C₂²	1	2	yes	—
53	C₂×#135	Q₈	1	1	yes	characteristic
54	C₄×#41	C₂³	1	1	yes	characteristic
55	C₄×#41	C₂³	1	1	yes	characteristic
56	#446	—	2	2	no	—
57	C₈×Q₈	—	2	2	no	—
58	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
59	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
60	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
61	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
63	C₄×Q₁₆	D₄	1	1	yes	characteristic
64	C₄×SD₁₆	C₂×C₄	1	1	yes	characteristic
65	C₄×SD₁₆	C₂×C₄	1	1	yes	characteristic
66	C₄×SD₁₆	D₄	1	1	yes	characteristic
67	C₄×SD₁₆	D₄	1	1	yes	characteristic
68	C₄×D₈	C₂×C₄	1	1	yes	characteristic
69	C₄×D₈	D₄	1	1	yes	characteristic
70	#436	—	2	2	no	—
71	#436	—	4	2	no	—
72	#435	—	4	2	no	—
73	C₈×D₄	—	2	1	no	—
74	C₈×D₄	—	2	1	no	—
75	C₈×D₄	—	2	1	no	—
76	C₈×D₄	—	2	1	no	—
77	C₈×D₄	—	2	2	no	—
78	C₈×D₄	C₂×C₄	1	1	yes	characteristic
79	C₈×D₄	C₂×C₄	1	1	yes	characteristic
80	C₈×D₄	C₂×C₄	1	1	yes	characteristic
81	C₈×D₄	C₂×C₄	1	1	yes	characteristic
82	#432	—	2	2	no	—
83	#431	—	2	1	no	—
84	#431	—	2	1	no	—
85	#431	—	2	2	no	—
86	#431	C₂³	1	1	yes	characteristic
87	#431	C₂³	1	1	yes	characteristic
88	#431	C₂³	1	2	yes	—
89	C₂×#108	—	2	1	no	—
90	#427	Q₈	1	1	yes	characteristic
91	#423	—	2	2	no	—
92	#423	C₂³	1	2	yes	—
93	#423	C₂³	1	2	yes	—
94	C₂×C₄⋊C₈	—	2	2	no	—
95	#416	Q₈	1	1	yes	characteristic
96	#416	Q₈	1	1	yes	characteristic
97	C₄×M₁₆	C₂³	1	2	yes	—
98	C₂×C₄×C₈	—	2	1	no	—
99	C₂×C₄×C₈	—	2	1	no	—
100	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
103	#341	—	2	1	no	—
104	#341	—	4	2	no	—
105	#335	—	2	2	no	—
106	#335	—	2	2	no	—
107	#335	—	2	2	no	—
108	#335	—	2	2	no	—
109	#335	C₂×C₄	1	1	yes	characteristic
110	#335	C₂×C₄	1	1	yes	characteristic
111	#335	D₄	1	2	yes	—
112	#334	—	2	2	no	—
113	#334	—	2	2	no	—
114	#334	—	2	1	no	—
115	#334	—	2	1	no	—
116	#334	—	2	1	no	—
117	#334	—	2	1	no	—
118	#334	C₂×C₄	1	1	yes	characteristic
119	#334	C₂×C₄	1	1	yes	characteristic
120	#334	D₄	1	1	yes	characteristic
121	#334	D₄	1	1	yes	characteristic
122	Q₈⋊C₈	—	2	2	no	—
123	Q₈⋊C₈	—	2	2	no	—
124	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
125	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
126	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
127	Q₈⋊C₈	C₂×C₄	1	1	yes	characteristic
128	D₄⋊C₈	—	2	2	no	—
129	D₄⋊C₈	—	2	2	no	—
130	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
131	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
132	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
133	D₄⋊C₈	C₂×C₄	1	1	yes	characteristic
134	#322	—	2	2	no	—
135	#322	—	2	2	no	—
136	C₈²	C₂×C₄	1	1	yes	characteristic
137	C₈²	C₂×C₄	1	1	yes	characteristic
138	C₈²	C₂×C₄	1	1	yes	characteristic
139	C₈²	C₂×C₄	1	1	yes	characteristic
140	C₂×#41	C₂×Q₈	1	1	yes	characteristic
141	C₂×#41	C₂×Q₈	1	1	yes	characteristic
142	#135	—	2	4	no	—
143	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
144	C₂×SD₁₆	C₄⋊C₄	1	1	yes	characteristic
145	C₂×SD₁₆	C₄⋊C₄	1	1	yes	characteristic
146	C₂×D₈	C₄⋊C₄	1	1	yes	characteristic
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	4	2	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	4	2	no	—
157	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
158	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
159	C₂²×C₈	#41	1	1	yes	characteristic
160	C₂²×C₈	#41	1	1	yes	characteristic
161	C₄×Q₈	C₂²×C₄	1	1	yes	characteristic
162	C₄×Q₈	C₂²×C₄	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
164	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	C₂²×C₄	1	1	yes	characteristic
170	C₄×D₄	C₂²×C₄	1	1	yes	characteristic
171	C₄×D₄	C₂×D₄	1	1	yes	characteristic
172	C₄×D₄	C₂×D₄	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	—	2	1	no	—
175	#117	C₂×Q₈	1	1	yes	characteristic
176	#117	C₂×Q₈	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂⁴	1	1	yes	characteristic
180	#108	—	4	2	no	—
181	#107	C₄⋊C₄	1	1	yes	characteristic
182	#107	C₄⋊C₄	1	1	yes	characteristic
183	#106	C₄⋊C₄	1	1	yes	characteristic
184	#106	C₄⋊C₄	1	1	yes	characteristic
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	4	2	no	—
194	C₄⋊C₈	—	4	2	no	—
195	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₈	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₄	1	2	yes	—
200	C₄⋊C₈	C₂×D₄	1	2	yes	—
201	#103	C₄⋊C₄	1	1	yes	characteristic
202	#103	C₄⋊C₄	1	1	yes	characteristic
203	#103	C₄⋊C₄	1	1	yes	characteristic
204	#103	C₄⋊C₄	1	1	yes	characteristic
205	#102	C₄⋊C₄	1	1	yes	characteristic
206	#102	C₄⋊C₄	1	1	yes	characteristic
207	#102	C₄⋊C₄	1	1	yes	characteristic
208	#102	C₄⋊C₄	1	1	yes	characteristic
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	1	no	—
213	C₂²⋊C₈	—	2	1	no	—
214	C₂²⋊C₈	—	2	1	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	4	2	no	—
217	C₂²⋊C₈	—	4	2	no	—
218	#97	—	2	2	no	—
219	#97	—	4	2	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	4	2	no	—
230	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
231	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
232	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
233	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
234	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
235	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
236	C₄×C₈	C₂×D₄	1	1	yes	characteristic
237	C₄×C₈	C₂×D₄	1	1	yes	characteristic
238	C₄×C₈	C₂×D₄	1	1	yes	characteristic
239	C₄×C₈	C₂×D₄	1	1	yes	characteristic
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₂×Q₈	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂×C₄⋊C₄	1	1	yes	characteristic
249	C₂×D₄	C₂×C₄⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
257	C₂²×C₄	C₂×#41	1	1	yes	characteristic
258	C₂²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
259	Q₁₆	—	2	2	no	—
260	SD₁₆	—	2	2	no	—
261	SD₁₆	#108	1	4	yes	complemented
262	D₈	—	2	2	no	—
263	M₁₆	—	4	2	no	—
264	M₁₆	—	4	2	no	—
265	M₁₆	—	8	2	no	—
266	M₁₆	—	8	2	no	—
267	M₁₆	—	16	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	4	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	8	1	no	—
287	C₂×C₈	—	8	1	no	—
288	C₂×C₈	—	16	2	no	—
289	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
293	C₂×C₈	C₄×D₄	1	1	yes	characteristic
294	C₂×C₈	C₄×D₄	1	1	yes	characteristic
295	C₂×C₈	C₄×D₄	1	1	yes	characteristic
296	C₂×C₈	C₄×D₄	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
302	C₄⋊C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
303	C₄⋊C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
304	C₄⋊C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
305	C₂²⋊C₄	—	2	1	no	—
306	C₂²⋊C₄	—	2	1	no	—
307	C₂²⋊C₄	—	2	1	no	—
308	C₂²⋊C₄	—	2	1	no	—
309	C₂²⋊C₄	—	2	1	no	—
310	C₂²⋊C₄	—	2	1	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	1	no	—
315	C₄²	—	2	1	no	—
316	C₄²	—	2	1	no	—
317	C₄²	C₂³×C₄	1	1	yes	characteristic
318	C₄²	C₂²×D₄	1	1	yes	characteristic
319	C₄²	C₂²×D₄	1	1	yes	characteristic
320	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
321	C₂³	—	2	1	no	—
322	C₂³	—	2	1	no	—
323	C₂³	#554	1	1	yes	characteristic
324	Q₈	—	2	1	no	—
325	Q₈	—	2	1	no	—
326	Q₈	—	2	1	no	—
327	Q₈	C₂×#108	1	2	yes	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	C₂×#108	1	2	yes	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
354	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
355	C₂×C₄	#520	1	1	yes	characteristic
356	C₂×C₄	D₄²	1	1	yes	characteristic
357	C₂×C₄	#547	1	1	yes	characteristic
358	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	—	4	1	no	—
366	C₈	—	4	1	no	—
367	C₈	—	4	1	no	—
368	C₈	—	4	1	no	—
369	C₈	—	4	1	no	—
370	C₈	—	4	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	8	2	no	—
374	C₈	—	8	2	no	—
375	C₈	—	8	2	no	—
376	C₈	C₂×#108	1	2	yes	—
377	C₈	#444	1	2	yes	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	4	1	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	D₄×C₄⋊C₄	1	1	yes	characteristic
388	C₄	—	2	1	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	—	4	1	no	—
397	C₄	C₂²×#108	1	1	yes	characteristic
398	C₄	#2916	1	1	yes	characteristic
399	C₄	C₂×#444	1	1	yes	characteristic
400	C₄	#2957	1	1	yes	characteristic
401	C₄	#3290	1	2	yes	—
402	C₂	—	4	1	no	—
403	C₂	—	4	2	no	—
404	C₂	—	8	1	no	—
405	C₂	D₄×#108	1	1	yes	characteristic
406	C₂	#22686	1	1	yes	characteristic
407	C₂	#22690	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	Q₈
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
3. #17835	C₂	semidirect
4. #17834	C₂	semidirect
5. #9636	C₂	semidirect
6. #8901	C₂	semidirect
7. #8900	C₂	semidirect
8. #8824	C₂	semidirect
9. #8670	C₂	semidirect
10. #8670	C₂	semidirect
11. #8293	C₂	semidirect
261. SD₁₆	#108	semidirect

Maximal subgroups

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

subgroup	index
#18061	2
C₈×#135	2
#17835	2
#17834	2
#9636	2
#8901	2
#8900	2
#8824	2
#8670	2
#8293	2