Subgroups — #109357

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1845	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×#335	C₂	1	1	yes	maximal, characteristic
4	#10849	C₂	1	2	yes	maximal
5	C₂×#1296	C₂	1	2	yes	maximal
6	C₂²×#107	—	2	1	no	—
7	C₂²×#107	—	2	1	no	—
8	C₂²×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₈	—	2	1	no	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#384	C₂²	1	2	yes	—
13	C₂×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#1845	—	2	2	no	—
15	#1845	C₄	1	4	yes	complemented
16	C₈×C₄⋊C₄	—	2	2	no	—
17	C₈×C₄⋊C₄	C₄	1	4	yes	complemented
18	C₂×#340	—	2	4	no	—
19	C₂×#336	—	2	1	no	—
20	C₂×#335	—	2	2	no	—
21	C₂×#335	—	2	2	no	—
22	C₂×#335	C₄	1	4	yes	complemented
23	#1296	—	2	4	no	—
24	C₂³×C₈	—	2	1	no	—
25	C₂³×C₈	—	2	2	no	—
26	C₂³×C₈	D₄	1	1	yes	characteristic
27	C₂³×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₄⋊C₄	—	2	2	no	—
29	C₂²×C₄⋊C₄	—	2	2	no	—
30	C₂²×C₄⋊C₄	—	2	2	no	—
31	C₂²×C₄⋊C₄	D₄	1	2	yes	—
32	C₂²×C₄²	—	2	1	no	—
33	C₂²×C₄²	C₂³	1	1	yes	characteristic
34	C₂×#107	—	2	4	no	—
35	C₂×#107	—	2	4	no	—
36	C₂×#107	—	4	2	no	—
37	C₂×#107	—	4	2	no	—
38	C₂×#107	—	4	2	no	—
39	C₂×#107	—	4	2	no	—
40	C₂×C₄⋊C₈	—	2	1	no	—
41	C₂×C₄⋊C₈	—	2	1	no	—
42	C₂×C₄⋊C₈	—	2	1	no	—
43	C₂×C₄⋊C₈	—	2	2	no	—
44	C₂×C₄⋊C₈	—	2	2	no	—
45	C₂×C₄⋊C₈	—	2	2	no	—
46	C₂×C₄⋊C₈	—	2	2	no	—
47	C₂×C₄⋊C₈	—	2	2	no	—
48	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₈	D₄	1	2	yes	—
51	C₂×C₄⋊C₈	Q₈	1	2	yes	—
52	C₂×C₄×C₈	—	2	4	no	—
53	C₂×C₄×C₈	—	4	2	no	—
54	C₂×C₄×C₈	—	4	2	no	—
55	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
57	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
58	C₂×C₄×C₈	D₄	1	2	yes	—
59	C₂×C₄×C₈	D₄	1	2	yes	—
60	C₂×C₄×C₈	Q₈	1	2	yes	—
61	#384	—	2	2	no	—
62	#384	—	2	2	no	—
63	#384	—	2	2	no	—
64	#384	C₂×C₄	1	4	yes	—
65	C₄×C₄⋊C₄	—	2	1	no	—
66	C₄×C₄⋊C₄	—	2	1	no	—
67	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×C₄⋊C₄	Q₈	1	2	yes	—
69	C₂×#95	—	2	1	no	—
70	#340	—	4	8	no	—
71	#336	—	2	4	no	—
72	#335	—	2	8	no	—
73	C₂³×C₄	—	2	2	no	—
74	C₂³×C₄	—	4	2	no	—
75	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₄	#41	1	1	yes	characteristic
78	C₂²×C₈	—	2	1	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	2	2	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	4	no	—
86	C₂²×C₈	—	4	1	no	—
87	C₂²×C₈	—	4	1	no	—
88	C₂²×C₈	—	4	1	no	—
89	C₂²×C₈	—	4	1	no	—
90	C₂²×C₈	—	4	2	no	—
91	C₂²×C₈	—	4	2	no	—
92	C₂²×C₈	—	4	4	no	—
93	C₂²×C₈	—	4	4	no	—
94	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
95	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
96	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
97	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
98	C₂²×C₈	C₄⋊C₄	1	2	yes	—
99	C₂²×C₈	C₄⋊C₄	1	2	yes	—
100	C₂²×C₈	D₈	1	2	yes	—
101	C₂²×C₈	SD₁₆	1	2	yes	—
102	C₂²×C₈	SD₁₆	1	2	yes	—
103	C₂²×C₈	Q₁₆	1	2	yes	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	—	4	4	no	—
111	C₂×C₄⋊C₄	—	4	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	4	4	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄⋊C₄	—	4	2	no	—
117	C₂×C₄⋊C₄	—	4	2	no	—
118	C₂×C₄⋊C₄	—	4	2	no	—
119	C₂×C₄⋊C₄	—	8	4	no	—
120	C₂×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
121	C₂×C₄⋊C₄	D₈	1	4	yes	—
122	C₂×C₄⋊C₄	Q₁₆	1	4	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	—	4	1	no	—
126	C₂×C₄²	—	4	1	no	—
127	C₂×C₄²	—	4	1	no	—
128	C₂×C₄²	—	4	1	no	—
129	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
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×Q₈	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	#41	1	1	yes	characteristic
139	C₂×C₄²	#41	1	1	yes	characteristic
140	C₂×C₄²	#41	1	1	yes	characteristic
141	#107	—	4	8	no	—
142	#107	—	4	8	no	—
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	4	2	no	—
149	C₄⋊C₈	—	4	2	no	—
150	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
151	C₄×C₈	—	4	8	no	—
152	C₄×C₈	C₄²	1	4	yes	—
153	C₄×C₈	C₂²⋊C₄	1	4	yes	—
154	C₄×C₈	C₄⋊C₄	1	4	yes	—
155	C₄×C₈	C₄⋊C₄	1	4	yes	—
156	#95	—	2	2	no	—
157	#95	—	2	2	no	—
158	C₂⁴	#121	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	—	8	2	no	—
170	C₂²×C₄	—	8	2	no	—
171	C₂²×C₄	—	8	4	no	—
172	C₂²×C₄	—	8	4	no	—
173	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	#117	1	1	yes	characteristic
178	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₄×Q₈	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	C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₄	#124	1	1	yes	characteristic
188	C₂²×C₄	#125	1	1	yes	characteristic
189	C₂²×C₄	#127	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₈	1	2	yes	—
193	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
194	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
195	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
196	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
197	C₂²×C₄	#135	1	2	yes	—
198	C₂²×C₄	#135	1	2	yes	—
199	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₂²×C₄	#137	1	1	yes	characteristic
202	C₂²×C₄	#137	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	—	4	8	no	—
218	C₂×C₈	—	8	4	no	—
219	C₂×C₈	—	8	2	no	—
220	C₂×C₈	—	8	2	no	—
221	C₂×C₈	#95	1	4	yes	—
222	C₂×C₈	#95	1	4	yes	—
223	C₂×C₈	#102	1	4	yes	—
224	C₂×C₈	#102	1	4	yes	—
225	C₂×C₈	#103	1	4	yes	—
226	C₂×C₈	#103	1	4	yes	—
227	C₂×C₈	#106	1	4	yes	—
228	C₂×C₈	#107	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	8	no	—
235	C₄⋊C₄	—	8	4	no	—
236	C₄⋊C₄	—	8	4	no	—
237	C₄⋊C₄	—	8	2	no	—
238	C₄⋊C₄	—	8	2	no	—
239	C₄⋊C₄	—	8	2	no	—
240	C₄⋊C₄	—	8	2	no	—
241	C₄⋊C₄	#107	1	8	yes	complemented
242	C₄²	—	4	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	C₂×C₄²	1	1	yes	characteristic
247	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
248	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
249	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
250	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
251	C₄²	#117	1	1	yes	characteristic
252	C₄²	C₄×D₄	1	1	yes	characteristic
253	C₄²	C₄×Q₈	1	1	yes	characteristic
254	C₄²	C₄×Q₈	1	1	yes	characteristic
255	C₄²	C₄×Q₈	1	1	yes	characteristic
256	C₄²	#121	1	1	yes	characteristic
257	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
258	C₄²	#124	1	1	yes	characteristic
259	C₄²	#125	1	1	yes	characteristic
260	C₄²	C₄⋊Q₈	1	1	yes	characteristic
261	C₄²	C₄⋊Q₈	1	1	yes	characteristic
262	C₂³	#380	1	1	yes	characteristic
263	C₂³	#389	1	1	yes	characteristic
264	C₂³	#390	1	1	yes	characteristic
265	C₂³	#459	1	1	yes	characteristic
266	C₂³	#460	1	1	yes	characteristic
267	C₂³	#461	1	1	yes	characteristic
268	C₂³	#462	1	1	yes	characteristic
269	C₂³	#466	1	2	yes	—
270	C₂³	#467	1	2	yes	—
271	C₂³	#472	1	2	yes	—
272	C₂³	#473	1	2	yes	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	16	4	no	—
292	C₂×C₄	—	16	4	no	—
293	C₂×C₄	C₂×#95	1	1	yes	characteristic
294	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
296	C₂×C₄	#380	1	1	yes	characteristic
297	C₂×C₄	#382	1	1	yes	characteristic
298	C₂×C₄	#384	1	1	yes	characteristic
299	C₂×C₄	#389	1	1	yes	characteristic
300	C₂×C₄	#389	1	1	yes	characteristic
301	C₂×C₄	#389	1	1	yes	characteristic
302	C₂×C₄	#390	1	1	yes	characteristic
303	C₂×C₄	#391	1	1	yes	characteristic
304	C₂×C₄	#391	1	1	yes	characteristic
305	C₂×C₄	C₂×#102	1	1	yes	characteristic
306	C₂×C₄	C₂×#102	1	1	yes	characteristic
307	C₂×C₄	C₂×#103	1	1	yes	characteristic
308	C₂×C₄	C₂×#103	1	1	yes	characteristic
309	C₂×C₄	#417	1	1	yes	characteristic
310	C₂×C₄	#417	1	1	yes	characteristic
311	C₂×C₄	#418	1	1	yes	characteristic
312	C₂×C₄	#419	1	1	yes	characteristic
313	C₂×C₄	C₂×#106	1	1	yes	characteristic
314	C₂×C₄	C₂×#107	1	2	yes	—
315	C₂×C₄	C₂×#107	1	1	yes	characteristic
316	C₂×C₄	#427	1	2	yes	—
317	C₂×C₄	#428	1	1	yes	characteristic
318	C₂×C₄	#428	1	1	yes	characteristic
319	C₂×C₄	C₄×D₈	1	2	yes	—
320	C₂×C₄	C₄×D₈	1	2	yes	—
321	C₂×C₄	C₄×Q₁₆	1	2	yes	—
322	C₂×C₄	C₄×Q₁₆	1	2	yes	—
323	C₂×C₄	#459	1	1	yes	characteristic
324	C₂×C₄	#460	1	1	yes	characteristic
325	C₂×C₄	#461	1	1	yes	characteristic
326	C₂×C₄	#462	1	1	yes	characteristic
327	C₂×C₄	#466	1	2	yes	—
328	C₂×C₄	#467	1	2	yes	—
329	C₂×C₄	#474	1	1	yes	characteristic
330	C₂×C₄	#474	1	1	yes	characteristic
331	C₂×C₄	#475	1	1	yes	characteristic
332	C₂×C₄	#475	1	1	yes	characteristic
333	C₂×C₄	#476	1	1	yes	characteristic
334	C₂×C₄	#476	1	1	yes	characteristic
335	C₂×C₄	#477	1	1	yes	characteristic
336	C₂×C₄	#477	1	1	yes	characteristic
337	C₂×C₄	#493	1	1	yes	characteristic
338	C₂×C₄	#494	1	1	yes	characteristic
339	C₂×C₄	#495	1	1	yes	characteristic
340	C₂×C₄	#495	1	1	yes	characteristic
341	C₂×C₄	#499	1	2	yes	—
342	C₂×C₄	#500	1	2	yes	—
343	C₈	—	2	4	no	—
344	C₈	—	4	4	no	—
345	C₈	—	8	4	no	—
346	C₈	#340	1	8	yes	complemented
347	C₂²	C₄⋊#95	1	1	yes	characteristic
348	C₂²	#1667	1	2	yes	—
349	C₂²	#1668	1	2	yes	—
350	C₂²	#1669	1	2	yes	—
351	C₂²	#1670	1	2	yes	—
352	C₂²	#1673	1	2	yes	—
353	C₂²	#1674	1	2	yes	—
354	C₂²	#1675	1	2	yes	—
355	C₂²	#1676	1	2	yes	—
356	C₂²	#1802	1	1	yes	characteristic
357	C₂²	#1803	1	1	yes	characteristic
358	C₂²	#1808	1	2	yes	—
359	C₂²	#1816	1	2	yes	—
360	C₂²	#1942	1	2	yes	—
361	C₂²	#1947	1	1	yes	characteristic
362	C₂²	#1948	1	1	yes	characteristic
363	C₂²	#1952	1	2	yes	—
364	C₂²	#1969	1	1	yes	characteristic
365	C₂²	#1970	1	1	yes	characteristic
366	C₂²	#1978	1	1	yes	characteristic
367	C₂²	#1979	1	1	yes	characteristic
368	C₂²	#1981	1	1	yes	characteristic
369	C₂²	#1982	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	4	8	no	—
375	C₄	—	16	4	no	—
376	C₄	C₂×#340	1	1	yes	characteristic
377	C₄	#1736	1	1	yes	characteristic
378	C₄	C₄×#107	1	2	yes	—
379	C₄	#1802	1	1	yes	characteristic
380	C₄	#1803	1	1	yes	characteristic
381	C₄	#1845	1	1	yes	characteristic
382	C₄	#1846	1	1	yes	characteristic
383	C₄	#1942	1	2	yes	—
384	C₄	#1947	1	1	yes	characteristic
385	C₄	#1948	1	1	yes	characteristic
386	C₄	#1969	1	1	yes	characteristic
387	C₄	#1970	1	1	yes	characteristic
388	C₄	#1987	1	1	yes	characteristic
389	C₄	#1988	1	1	yes	characteristic
390	C₂	#8644	1	2	yes	—
391	C₂	#8645	1	2	yes	—
392	C₂	#9614	1	2	yes	—
393	C₂	#9615	1	2	yes	—
394	C₂	#9730	1	1	yes	characteristic
395	C₂	#9731	1	1	yes	characteristic
396	C₂	#9733	1	1	yes	characteristic
397	C₂	#9734	1	1	yes	characteristic
398	C₂	#10849	1	1	yes	characteristic
399	C₂	#10862	1	1	yes	characteristic
400	C₂	#10870	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ 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
15. #1845	C₄	semidirect
17. C₈×C₄⋊C₄	C₄	semidirect
22. C₂×#335	C₄	semidirect
241. C₄⋊C₄	#107	semidirect
346. C₈	#340	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1845	2
C₂×C₈×C₄⋊C₄	2
C₂²×#335	2
#10849	2
C₂×#1296	2