Subgroups — #109376

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1848	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×#1301	C₂	1	2	yes	maximal
5	#7901	C₂	1	2	yes	maximal
6	C₂×#431	C₂²	1	2	yes	—
7	C₂²×#108	—	2	1	no	—
8	C₂²×#108	—	2	1	no	—
9	C₂×#423	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₈	—	2	1	no	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#1848	—	2	2	no	—
15	#1848	C₄	1	4	yes	complemented
16	C₈×C₄⋊C₄	—	2	1	no	—
17	C₈×C₄⋊C₄	—	2	1	no	—
18	C₈×C₄⋊C₄	C₄	1	2	yes	—
19	C₈×C₄⋊C₄	C₄	1	2	yes	—
20	C₂×#336	—	2	1	no	—
21	#1564	—	2	2	no	—
22	#1564	C₄	1	4	yes	complemented
23	C₂×#335	—	2	1	no	—
24	C₂×#334	—	2	1	no	—
25	#1301	—	2	4	no	—
26	#1279	—	2	4	no	—
27	C₂²×M₁₆	—	2	2	no	—
28	C₂³×C₈	—	2	1	no	—
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂³×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₄⋊C₄	Q₈	1	2	yes	—
32	C₂²×C₄²	—	2	1	no	—
33	C₂²×C₄²	C₂³	1	1	yes	characteristic
34	#431	—	2	2	no	—
35	#431	—	2	2	no	—
36	#431	—	2	2	no	—
37	#431	C₂×C₄	1	4	yes	—
38	C₂×#108	—	2	4	no	—
39	C₂×#108	—	2	4	no	—
40	C₂×#108	—	4	2	no	—
41	C₂×#108	—	4	2	no	—
42	C₂×#108	—	4	2	no	—
43	C₂×#108	—	4	2	no	—
44	#423	—	2	2	no	—
45	#423	—	2	2	no	—
46	#423	—	2	2	no	—
47	#423	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₈	—	2	1	no	—
49	C₂×C₄⋊C₈	—	2	1	no	—
50	C₂×C₄⋊C₈	—	2	1	no	—
51	C₂×C₄⋊C₈	—	2	2	no	—
52	C₂×C₄⋊C₈	—	2	2	no	—
53	C₂×C₄⋊C₈	—	2	2	no	—
54	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₈	D₄	1	2	yes	—
56	C₂×C₄⋊C₈	Q₈	1	2	yes	—
57	C₂×C₂²⋊C₈	—	2	2	no	—
58	C₂×C₄×C₈	—	2	2	no	—
59	C₂×C₄×C₈	—	2	2	no	—
60	C₂×C₄×C₈	—	2	2	no	—
61	C₂×C₄×C₈	—	4	2	no	—
62	C₂×C₄×C₈	—	4	2	no	—
63	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
64	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	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	2	yes	—
70	C₂×C₄×C₈	Q₈	1	2	yes	—
71	C₄×C₄⋊C₄	—	2	1	no	—
72	C₄×C₄⋊C₄	—	2	1	no	—
73	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
74	C₄×C₄⋊C₄	D₄	1	2	yes	—
75	C₂×#95	—	2	1	no	—
76	#336	—	2	2	no	—
77	#336	—	2	2	no	—
78	#335	—	2	4	no	—
79	#334	—	2	4	no	—
80	C₂³×C₄	—	2	2	no	—
81	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂³×C₄	#41	1	1	yes	characteristic
84	C₂×M₁₆	—	2	4	no	—
85	C₂×M₁₆	—	4	2	no	—
86	C₂×M₁₆	—	4	2	no	—
87	C₂×M₁₆	—	4	2	no	—
88	C₂×M₁₆	—	4	2	no	—
89	C₂×M₁₆	—	4	2	no	—
90	C₂²×C₈	—	2	1	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	—	2	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	—	4	1	no	—
99	C₂²×C₈	—	4	1	no	—
100	C₂²×C₈	—	4	1	no	—
101	C₂²×C₈	—	4	2	no	—
102	C₂²×C₈	—	4	2	no	—
103	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
106	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
109	C₂²×C₈	C₄⋊C₄	1	2	yes	—
110	C₂²×C₈	C₄⋊C₄	1	2	yes	—
111	C₂²×C₈	D₈	1	2	yes	—
112	C₂²×C₈	SD₁₆	1	2	yes	—
113	C₂²×C₈	SD₁₆	1	2	yes	—
114	C₂²×C₈	Q₁₆	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	8	no	—
117	C₂×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
118	C₂×C₄²	—	2	2	no	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	—	4	1	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	—	4	1	no	—
123	C₂×C₄²	—	4	1	no	—
124	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄²	C₂×D₄	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×Q₈	1	1	yes	characteristic
132	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
133	C₂×C₄²	#41	1	1	yes	characteristic
134	C₂×C₄²	#41	1	1	yes	characteristic
135	C₂×C₄²	#41	1	1	yes	characteristic
136	#108	—	4	8	no	—
137	#108	—	4	8	no	—
138	C₄⋊C₈	—	2	4	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	4	2	no	—
145	C₄⋊C₈	—	4	2	no	—
146	C₄⋊C₈	—	4	2	no	—
147	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₂²⋊C₈	—	4	2	no	—
150	C₄×C₈	—	2	4	no	—
151	C₄×C₈	—	4	4	no	—
152	C₄×C₈	—	4	4	no	—
153	C₄×C₈	—	4	2	no	—
154	C₄×C₈	C₄²	1	2	yes	—
155	C₄×C₈	C₄²	1	2	yes	—
156	C₄×C₈	C₂²⋊C₄	1	2	yes	—
157	C₄×C₈	C₂²⋊C₄	1	2	yes	—
158	C₄×C₈	C₄⋊C₄	1	2	yes	—
159	C₄×C₈	C₄⋊C₄	1	2	yes	—
160	C₄×C₈	C₄⋊C₄	1	2	yes	—
161	C₄×C₈	C₄⋊C₄	1	2	yes	—
162	#95	—	2	2	no	—
163	#95	—	2	2	no	—
164	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	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	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
191	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
192	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
194	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₂²×C₄	#137	1	1	yes	characteristic
197	C₂²×C₄	#137	1	1	yes	characteristic
198	M₁₆	—	4	4	no	—
199	M₁₆	—	4	4	no	—
200	M₁₆	—	8	2	no	—
201	M₁₆	—	8	2	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	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	2	no	—
217	C₂×C₈	—	4	4	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	8	4	no	—
220	C₂×C₈	—	8	2	no	—
221	C₂×C₈	—	8	2	no	—
222	C₂×C₈	#95	1	2	yes	—
223	C₂×C₈	#95	1	2	yes	—
224	C₂×C₈	#95	1	2	yes	—
225	C₂×C₈	#95	1	2	yes	—
226	C₂×C₈	#102	1	2	yes	—
227	C₂×C₈	#102	1	2	yes	—
228	C₂×C₈	#102	1	4	yes	—
229	C₂×C₈	#103	1	2	yes	—
230	C₂×C₈	#103	1	2	yes	—
231	C₂×C₈	#103	1	4	yes	—
232	C₂×C₈	#106	1	4	yes	—
233	C₂×C₈	#107	1	4	yes	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	#108	1	8	yes	complemented
237	C₄²	—	4	2	no	—
238	C₄²	—	4	2	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₂×C₄²	1	1	yes	characteristic
242	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
244	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
245	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
246	C₄²	#117	1	1	yes	characteristic
247	C₄²	C₄×D₄	1	1	yes	characteristic
248	C₄²	C₄×D₄	1	1	yes	characteristic
249	C₄²	C₄×D₄	1	1	yes	characteristic
250	C₄²	C₄×Q₈	1	1	yes	characteristic
251	C₄²	#121	1	1	yes	characteristic
252	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
253	C₄²	#124	1	1	yes	characteristic
254	C₄²	#125	1	1	yes	characteristic
255	C₄²	#127	1	1	yes	characteristic
256	C₄²	C₄⋊Q₈	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	#380	1	1	yes	characteristic
260	C₂³	#389	1	1	yes	characteristic
261	C₂³	#391	1	1	yes	characteristic
262	C₂³	#474	1	1	yes	characteristic
263	C₂³	#475	1	1	yes	characteristic
264	C₂³	#476	1	1	yes	characteristic
265	C₂³	#477	1	1	yes	characteristic
266	C₂×C₄	—	2	2	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	8	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	C₂×#95	1	1	yes	characteristic
283	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
285	C₂×C₄	#380	1	1	yes	characteristic
286	C₂×C₄	#382	1	1	yes	characteristic
287	C₂×C₄	#384	1	1	yes	characteristic
288	C₂×C₄	#389	1	1	yes	characteristic
289	C₂×C₄	#389	1	1	yes	characteristic
290	C₂×C₄	#389	1	1	yes	characteristic
291	C₂×C₄	#390	1	1	yes	characteristic
292	C₂×C₄	#390	1	1	yes	characteristic
293	C₂×C₄	#391	1	1	yes	characteristic
294	C₂×C₄	C₂×#102	1	1	yes	characteristic
295	C₂×C₄	C₂×#102	1	1	yes	characteristic
296	C₂×C₄	C₂×#103	1	1	yes	characteristic
297	C₂×C₄	C₂×#103	1	1	yes	characteristic
298	C₂×C₄	#417	1	1	yes	characteristic
299	C₂×C₄	#417	1	1	yes	characteristic
300	C₂×C₄	#418	1	1	yes	characteristic
301	C₂×C₄	#419	1	1	yes	characteristic
302	C₂×C₄	C₂×#106	1	1	yes	characteristic
303	C₂×C₄	C₂×#107	1	1	yes	characteristic
304	C₂×C₄	#428	1	1	yes	characteristic
305	C₂×C₄	#428	1	1	yes	characteristic
306	C₂×C₄	C₂×#108	1	4	yes	—
307	C₂×C₄	#459	1	1	yes	characteristic
308	C₂×C₄	#459	1	1	yes	characteristic
309	C₂×C₄	#460	1	1	yes	characteristic
310	C₂×C₄	#460	1	1	yes	characteristic
311	C₂×C₄	#461	1	1	yes	characteristic
312	C₂×C₄	#461	1	1	yes	characteristic
313	C₂×C₄	#462	1	1	yes	characteristic
314	C₂×C₄	#462	1	1	yes	characteristic
315	C₂×C₄	#474	1	1	yes	characteristic
316	C₂×C₄	#475	1	1	yes	characteristic
317	C₂×C₄	#476	1	1	yes	characteristic
318	C₂×C₄	#477	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₈	—	4	4	no	—
322	C₈	—	8	4	no	—
323	C₈	—	8	4	no	—
324	C₈	#340	1	4	yes	—
325	C₈	#340	1	4	yes	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	C₄⋊#95	1	1	yes	characteristic
335	C₂²	#1802	1	1	yes	characteristic
336	C₂²	#1803	1	1	yes	characteristic
337	C₂²	#1810	1	2	yes	—
338	C₂²	#1814	1	2	yes	—
339	C₂²	#1947	1	1	yes	characteristic
340	C₂²	#1948	1	1	yes	characteristic
341	C₂²	#1987	1	1	yes	characteristic
342	C₂²	#1988	1	1	yes	characteristic
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	4	8	no	—
350	C₄	C₂×#340	1	1	yes	characteristic
351	C₄	#1736	1	1	yes	characteristic
352	C₄	C₄×#108	1	2	yes	—
353	C₄	#1802	1	1	yes	characteristic
354	C₄	#1803	1	1	yes	characteristic
355	C₄	#1848	1	2	yes	—
356	C₄	#1947	1	1	yes	characteristic
357	C₄	#1948	1	1	yes	characteristic
358	C₄	#1981	1	1	yes	characteristic
359	C₄	#1982	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	#8658	1	2	yes	—
365	C₂	#8659	1	2	yes	—
366	C₂	#10849	1	1	yes	characteristic
367	C₂	#10864	1	1	yes	characteristic
368	C₂	#10872	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#380
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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
15. #1848	C₄	semidirect
22. #1564	C₄	semidirect
236. C₄⋊C₄	#108	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1848	2
C₂×C₈×C₄⋊C₄	2
C₂×#1564	2
C₂×#1301	2
#7901	2