Subgroups — #120454

Order 512.

507 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21177	C₂	1	1	yes	maximal, characteristic
2	#17813	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#338	C₂	1	1	yes	maximal, characteristic
4	#8079	C₂	1	1	yes	maximal, characteristic
5	C₂×#1292	C₂	1	1	yes	maximal, characteristic
6	C₂×#1292	C₂	1	1	yes	maximal, characteristic
7	C₂×#1283	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#430	—	2	1	no	—
10	C₂×#423	C₂²	1	1	yes	characteristic
11	C₂×#423	C₂²	1	1	yes	characteristic
12	C₂×#423	C₂²	1	1	yes	characteristic
13	C₂×#423	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
16	#1950	—	2	2	no	—
17	#1950	—	2	2	no	—
18	#1738	—	2	1	no	—
19	C₂×#338	—	2	1	no	—
20	C₂×#338	—	2	1	no	—
21	#1570	—	2	2	no	—
22	#1570	C₄	1	4	yes	—
23	#1566	—	2	2	no	—
24	#1301	—	2	2	no	—
25	#1292	—	2	2	no	—
26	#1292	—	2	2	no	—
27	#1283	—	2	2	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	—	2	1	no	—
32	C₂²×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×M₁₆	D₄	1	1	yes	characteristic
34	C₂³×C₈	Q₈	1	2	yes	—
35	C₂²×C₄²	C₂³	1	1	yes	characteristic
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	C₂×C₄	1	2	yes	—
39	#431	D₄	1	2	yes	—
40	#430	—	4	2	no	—
41	#430	—	4	2	no	—
42	C₂×#108	—	2	4	no	—
43	#423	—	2	1	no	—
44	#423	—	2	1	no	—
45	#423	—	2	1	no	—
46	#423	—	2	1	no	—
47	#423	—	2	1	no	—
48	#423	—	2	1	no	—
49	#423	—	2	1	no	—
50	#423	—	2	1	no	—
51	#423	—	2	1	no	—
52	#423	—	2	1	no	—
53	#423	—	2	1	no	—
54	#423	—	2	1	no	—
55	#423	—	2	1	no	—
56	#423	C₂×C₄	1	2	yes	—
57	#423	D₄	1	2	yes	—
58	#423	D₄	1	2	yes	—
59	C₂×C₄⋊C₈	—	2	1	no	—
60	C₂×C₄⋊C₈	—	2	1	no	—
61	C₂×C₄⋊C₈	—	2	1	no	—
62	C₂×C₄⋊C₈	—	2	1	no	—
63	C₂×C₄⋊C₈	—	2	1	no	—
64	C₂×C₄⋊C₈	—	2	1	no	—
65	C₂×C₄⋊C₈	—	2	1	no	—
66	C₂×C₄⋊C₈	—	2	1	no	—
67	C₂×C₄⋊C₈	—	2	1	no	—
68	C₂×C₄⋊C₈	—	2	1	no	—
69	C₂×C₄⋊C₈	—	2	1	no	—
70	C₂×C₄⋊C₈	—	2	1	no	—
71	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₈	D₄	1	2	yes	—
73	C₂×C₄⋊C₈	Q₈	1	2	yes	—
74	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
75	C₄×M₁₆	—	2	1	no	—
76	C₄×M₁₆	—	2	1	no	—
77	C₄×M₁₆	—	2	1	no	—
78	C₄×M₁₆	—	2	1	no	—
79	C₂×#97	—	2	1	no	—
80	C₂×C₄×C₈	—	2	1	no	—
81	C₂×C₄×C₈	—	2	1	no	—
82	#341	—	2	4	no	—
83	#338	—	4	2	no	—
84	#338	—	4	2	no	—
85	#334	—	2	8	no	—
86	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
88	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	—	4	1	no	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂×M₁₆	—	4	1	no	—
113	C₂×M₁₆	—	4	1	no	—
114	C₂×M₁₆	—	4	1	no	—
115	C₂×M₁₆	—	4	1	no	—
116	C₂×M₁₆	—	4	1	no	—
117	C₂×M₁₆	—	4	1	no	—
118	C₂×M₁₆	—	4	1	no	—
119	C₂×M₁₆	—	4	1	no	—
120	C₂×M₁₆	—	4	1	no	—
121	C₂×M₁₆	—	4	1	no	—
122	C₂×M₁₆	—	4	1	no	—
123	C₂×M₁₆	SD₁₆	1	2	yes	—
124	C₂×M₁₆	SD₁₆	1	2	yes	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	2	1	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	4	1	no	—
135	C₂²×C₈	—	8	2	no	—
136	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
142	C₂×C₄²	#41	1	1	yes	characteristic
143	C₂×C₄²	#41	1	1	yes	characteristic
144	C₂×C₄²	#41	1	1	yes	characteristic
145	C₂×C₄²	#41	1	1	yes	characteristic
146	C₂×C₄²	#41	1	1	yes	characteristic
147	C₂×C₄²	#41	1	1	yes	characteristic
148	#108	—	4	8	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	4	1	no	—
161	C₄⋊C₈	—	4	1	no	—
162	C₄⋊C₈	—	4	1	no	—
163	C₄⋊C₈	—	4	1	no	—
164	C₄⋊C₈	—	4	1	no	—
165	C₄⋊C₈	—	4	1	no	—
166	C₄⋊C₈	—	4	1	no	—
167	C₄⋊C₈	—	4	1	no	—
168	C₄⋊C₈	—	4	1	no	—
169	C₄⋊C₈	—	4	1	no	—
170	C₄⋊C₈	C₂²⋊C₄	1	4	yes	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
173	#97	—	4	1	no	—
174	#97	—	4	1	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	4	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	—	8	1	no	—
183	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₄	#120	1	1	yes	characteristic
187	C₂²×C₄	#121	1	1	yes	characteristic
188	C₂²×C₄	#121	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂²×C₄	#123	1	1	yes	characteristic
196	C₂²×C₄	#125	1	1	yes	characteristic
197	C₂²×C₄	#125	1	1	yes	characteristic
198	C₂²×C₄	#125	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
203	C₂²×C₄	#135	1	1	yes	characteristic
204	C₂²×C₄	#135	1	1	yes	characteristic
205	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂²×C₄	#137	1	1	yes	characteristic
208	C₂²×C₄	#137	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	1	no	—
216	M₁₆	—	4	1	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	8	1	no	—
224	M₁₆	—	8	1	no	—
225	M₁₆	—	8	1	no	—
226	M₁₆	—	8	1	no	—
227	M₁₆	—	8	1	no	—
228	M₁₆	—	8	1	no	—
229	M₁₆	—	8	1	no	—
230	M₁₆	—	8	1	no	—
231	M₁₆	—	8	1	no	—
232	M₁₆	—	8	1	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	8	2	no	—
252	C₂×C₈	—	8	1	no	—
253	C₂×C₈	—	8	1	no	—
254	C₂×C₈	—	8	2	no	—
255	C₂×C₈	—	8	2	no	—
256	C₂×C₈	—	8	2	no	—
257	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₄²	#117	1	1	yes	characteristic
259	C₄²	C₄×D₄	1	1	yes	characteristic
260	C₄²	C₄×Q₈	1	1	yes	characteristic
261	C₄²	#121	1	1	yes	characteristic
262	C₄²	#121	1	1	yes	characteristic
263	C₄²	#121	1	1	yes	characteristic
264	C₄²	#121	1	1	yes	characteristic
265	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
266	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
267	C₄²	#123	1	1	yes	characteristic
268	C₄²	#123	1	1	yes	characteristic
269	C₄²	#124	1	1	yes	characteristic
270	C₄²	#125	1	1	yes	characteristic
271	C₄²	#126	1	1	yes	characteristic
272	C₄²	#126	1	1	yes	characteristic
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	#389	1	1	yes	characteristic
276	C₂³	#395	1	1	yes	characteristic
277	C₂³	#398	1	1	yes	characteristic
278	C₂³	#498	1	1	yes	characteristic
279	C₂³	#499	1	1	yes	characteristic
280	C₂³	#501	1	1	yes	characteristic
281	C₂³	#501	1	1	yes	characteristic
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	#380	1	1	yes	characteristic
298	C₂×C₄	#386	1	1	yes	characteristic
299	C₂×C₄	#387	1	1	yes	characteristic
300	C₂×C₄	#387	1	1	yes	characteristic
301	C₂×C₄	#389	1	1	yes	characteristic
302	C₂×C₄	#391	1	1	yes	characteristic
303	C₂×C₄	#394	1	1	yes	characteristic
304	C₂×C₄	#396	1	1	yes	characteristic
305	C₂×C₄	#396	1	1	yes	characteristic
306	C₂×C₄	#398	1	1	yes	characteristic
307	C₂×C₄	#398	1	1	yes	characteristic
308	C₂×C₄	#400	1	1	yes	characteristic
309	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
310	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
311	C₂×C₄	#440	1	1	yes	characteristic
312	C₂×C₄	#440	1	1	yes	characteristic
313	C₂×C₄	#448	1	1	yes	characteristic
314	C₂×C₄	#449	1	1	yes	characteristic
315	C₂×C₄	#450	1	1	yes	characteristic
316	C₂×C₄	#452	1	1	yes	characteristic
317	C₂×C₄	#460	1	1	yes	characteristic
318	C₂×C₄	#461	1	1	yes	characteristic
319	C₂×C₄	#463	1	1	yes	characteristic
320	C₂×C₄	#464	1	1	yes	characteristic
321	C₂×C₄	#465	1	1	yes	characteristic
322	C₂×C₄	#465	1	1	yes	characteristic
323	C₂×C₄	#469	1	1	yes	characteristic
324	C₂×C₄	#470	1	1	yes	characteristic
325	C₂×C₄	#475	1	1	yes	characteristic
326	C₂×C₄	#476	1	1	yes	characteristic
327	C₂×C₄	#478	1	1	yes	characteristic
328	C₂×C₄	#479	1	1	yes	characteristic
329	C₂×C₄	#481	1	1	yes	characteristic
330	C₂×C₄	#482	1	1	yes	characteristic
331	C₂×C₄	#483	1	1	yes	characteristic
332	C₂×C₄	#485	1	1	yes	characteristic
333	C₈	—	4	4	no	—
334	C₈	—	4	4	no	—
335	C₈	—	4	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	8	2	no	—
338	C₈	—	8	2	no	—
339	C₈	—	8	2	no	—
340	C₈	—	8	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	#1441	1	1	yes	characteristic
352	C₂²	#1941	1	1	yes	characteristic
353	C₂²	#1944	1	1	yes	characteristic
354	C₂²	#2058	1	1	yes	characteristic
355	C₂²	#2059	1	1	yes	characteristic
356	C₂²	#2084	1	1	yes	characteristic
357	C₂²	#2085	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	#1877	1	1	yes	characteristic
366	C₄	#1880	1	1	yes	characteristic
367	C₄	#1919	1	1	yes	characteristic
368	C₄	#1920	1	1	yes	characteristic
369	C₄	#2034	1	1	yes	characteristic
370	C₄	#2035	1	1	yes	characteristic
371	C₄	#2065	1	1	yes	characteristic
372	C₄	#2066	1	1	yes	characteristic
373	C₄	#2069	1	1	yes	characteristic
374	C₄	#2070	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	#11220	1	1	yes	characteristic
382	C₂	#11235	1	1	yes	characteristic
383	C₂	#11252	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#21177	2
#17813	2
C₄⋊#338	2
#8079	2
C₂×#1292	2
C₂×#1283	2