Subgroups — #148718

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1952	C₂	1	1	yes	maximal, characteristic
2	#20807	C₂	1	1	yes	maximal, characteristic
3	C₂×#1388	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	1	yes	maximal, characteristic
5	C₂×#1387	C₂	1	1	yes	maximal, characteristic
6	C₂×#1385	C₂	1	1	yes	maximal, characteristic
7	#12554	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×#430	—	2	1	no	—
11	C₂²×#108	C₂²	1	1	yes	characteristic
12	C₂²×#108	C₂²	1	1	yes	characteristic
13	C₂²×#108	C₂²	1	1	yes	characteristic
14	C₂×#428	—	2	1	no	—
15	C₂²×#107	C₂²	1	1	yes	characteristic
16	#2905	—	2	1	no	—
17	C₂×#405	C₂²	1	1	yes	characteristic
18	#1952	—	2	2	no	—
19	#1952	C₄	1	4	yes	—
20	#1849	—	2	2	no	—
21	#1848	—	2	2	no	—
22	C₂×#343	—	2	1	no	—
23	#1388	—	2	2	no	—
24	#1387	—	2	2	no	—
25	#1387	—	2	2	no	—
26	#1385	—	2	2	no	—
27	#1379	—	2	2	no	—
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	D₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	D₄	1	1	yes	characteristic
34	C₂²×C₄⋊C₄	—	2	1	no	—
35	C₂×M₃₂	—	2	1	no	—
36	C₂×M₃₂	—	2	1	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂×M₃₂	—	2	1	no	—
45	C₂×M₃₂	Q₈	1	2	yes	—
46	C₂×M₃₂	Q₈	1	2	yes	—
47	C₂²×C₁₆	—	2	1	no	—
48	C₂²×C₁₆	—	2	1	no	—
49	#430	—	4	2	no	—
50	#430	—	4	2	no	—
51	C₂×#108	—	2	1	no	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	—	2	1	no	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	—	2	1	no	—
57	C₂×#108	—	2	1	no	—
58	C₂×#108	—	2	1	no	—
59	C₂×#108	—	2	1	no	—
60	C₂×#108	—	2	1	no	—
61	C₂×#108	—	2	1	no	—
62	C₂×#108	—	2	4	no	—
63	C₂×#108	—	2	1	no	—
64	C₂×#108	—	2	1	no	—
65	C₂×#108	C₂×C₄	1	2	yes	—
66	C₂×#108	D₄	1	2	yes	—
67	C₂×#108	D₄	1	2	yes	—
68	C₂×#108	D₄	1	2	yes	—
69	C₂×#108	Q₈	1	2	yes	—
70	#428	—	2	4	no	—
71	#428	—	4	2	no	—
72	C₂×#107	—	2	1	no	—
73	C₂×#107	—	2	1	no	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	—	2	1	no	—
76	C₂×#107	—	2	1	no	—
77	C₂×#107	—	4	1	no	—
78	C₂×#107	C₂×C₄	1	2	yes	—
79	C₂×#106	—	4	1	no	—
80	#424	—	4	2	no	—
81	#424	—	4	2	no	—
82	#423	—	2	4	no	—
83	#405	—	2	1	no	—
84	#405	—	2	1	no	—
85	#405	C₂×C₄	1	2	yes	—
86	#405	D₄	1	2	yes	—
87	C₂×#97	—	2	1	no	—
88	C₂×C₄×C₈	—	2	1	no	—
89	#343	—	2	4	no	—
90	C₂³×C₄	—	4	1	no	—
91	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	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	C₂²⋊C₄	1	2	yes	—
114	C₂×M₁₆	SD₁₆	1	2	yes	—
115	C₂×M₁₆	Q₁₆	1	2	yes	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	4	1	no	—
118	C₂²×C₈	—	4	1	no	—
119	C₂²×C₈	—	4	1	no	—
120	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
126	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
127	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
128	C₂²×C₈	#41	1	1	yes	characteristic
129	C₂²×C₈	#41	1	1	yes	characteristic
130	C₂²×C₈	#41	1	1	yes	characteristic
131	C₂²×C₈	#41	1	1	yes	characteristic
132	C₂²×C₈	#41	1	1	yes	characteristic
133	C₂²×C₈	#41	1	1	yes	characteristic
134	#117	—	2	1	no	—
135	#117	—	2	1	no	—
136	#117	C₂²⋊C₄	1	2	yes	—
137	#117	D₈	1	2	yes	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
145	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
146	C₂×C₂²⋊C₄	—	2	1	no	—
147	C₂×C₄²	—	2	1	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	4	1	no	—
153	M₃₂	—	4	1	no	—
154	M₃₂	—	4	1	no	—
155	M₃₂	—	4	1	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	4	1	no	—
159	C₂×C₁₆	—	4	1	no	—
160	C₂×C₁₆	—	4	1	no	—
161	C₂×C₁₆	—	4	1	no	—
162	#108	—	2	2	no	—
163	#108	—	2	2	no	—
164	#108	—	2	2	no	—
165	#108	—	2	2	no	—
166	#108	—	2	2	no	—
167	#108	—	2	2	no	—
168	#108	—	2	2	no	—
169	#108	—	4	8	no	—
170	#108	—	4	1	no	—
171	#108	—	4	1	no	—
172	#108	—	4	1	no	—
173	#108	—	4	1	no	—
174	#108	C₄⋊C₄	1	4	yes	—
175	#107	—	2	2	no	—
176	#107	—	2	2	no	—
177	#107	—	4	1	no	—
178	#107	—	4	1	no	—
179	#107	—	4	4	no	—
180	#106	—	4	4	no	—
181	C₄⋊C₈	—	4	8	no	—
182	#97	—	2	2	no	—
183	#97	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₂⁴	#137	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	8	1	no	—
191	C₂²×C₄	—	8	1	no	—
192	C₂²×C₄	—	16	2	no	—
193	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	#120	1	1	yes	characteristic
195	C₂²×C₄	#121	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂²×C₄	#123	1	1	yes	characteristic
198	C₂²×C₄	#124	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₈	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	C₂×Q₁₆	1	1	yes	characteristic
204	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₂²×C₄	#137	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	8	1	no	—
219	M₁₆	—	8	1	no	—
220	M₁₆	—	8	1	no	—
221	M₁₆	—	8	1	no	—
222	M₁₆	#103	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	8	1	no	—
232	C₂×C₈	—	8	1	no	—
233	C₂×C₈	—	8	1	no	—
234	C₂×C₈	—	8	1	no	—
235	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₈	#117	1	1	yes	characteristic
237	C₂×C₈	C₄×D₄	1	1	yes	characteristic
238	C₂×C₈	C₄×D₄	1	1	yes	characteristic
239	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
240	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
241	C₂×C₈	#121	1	1	yes	characteristic
242	C₂×C₈	#121	1	1	yes	characteristic
243	C₂×C₈	#121	1	1	yes	characteristic
244	C₂×C₈	#121	1	1	yes	characteristic
245	C₂×C₈	#121	1	1	yes	characteristic
246	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₈	#123	1	1	yes	characteristic
254	C₂×C₈	#123	1	1	yes	characteristic
255	C₂×C₈	#125	1	1	yes	characteristic
256	C₂×C₈	#125	1	1	yes	characteristic
257	C₂×C₈	#125	1	1	yes	characteristic
258	C₂×C₈	#125	1	1	yes	characteristic
259	C₂×C₈	#126	1	1	yes	characteristic
260	C₂×C₈	#126	1	1	yes	characteristic
261	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
262	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	8	1	no	—
268	C₄⋊C₄	—	8	1	no	—
269	C₄⋊C₄	#102	1	2	yes	—
270	C₄⋊C₄	#102	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₁₆	—	4	2	no	—
276	C₁₆	—	4	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	#417	1	1	yes	characteristic
280	C₂³	#448	1	1	yes	characteristic
281	C₂³	#462	1	1	yes	characteristic
282	C₂³	#475	1	1	yes	characteristic
283	C₂³	#480	1	1	yes	characteristic
284	C₂³	#491	1	1	yes	characteristic
285	C₂³	#501	1	1	yes	characteristic
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	16	2	no	—
292	C₂×C₄	—	16	2	no	—
293	C₂×C₄	—	16	2	no	—
294	C₂×C₄	#380	1	1	yes	characteristic
295	C₂×C₄	#386	1	1	yes	characteristic
296	C₂×C₄	#391	1	1	yes	characteristic
297	C₂×C₄	#394	1	1	yes	characteristic
298	C₂×C₄	#395	1	1	yes	characteristic
299	C₂×C₄	#398	1	1	yes	characteristic
300	C₂×C₄	#400	1	1	yes	characteristic
301	C₂×C₄	C₂×#102	1	1	yes	characteristic
302	C₂×C₄	C₂×#103	1	1	yes	characteristic
303	C₂×C₄	#417	1	1	yes	characteristic
304	C₂×C₄	#447	1	1	yes	characteristic
305	C₂×C₄	#450	1	1	yes	characteristic
306	C₂×C₄	#451	1	1	yes	characteristic
307	C₂×C₄	#459	1	1	yes	characteristic
308	C₂×C₄	#460	1	1	yes	characteristic
309	C₂×C₄	#461	1	1	yes	characteristic
310	C₂×C₄	#474	1	1	yes	characteristic
311	C₂×C₄	#476	1	1	yes	characteristic
312	C₂×C₄	#477	1	1	yes	characteristic
313	C₂×C₄	#481	1	1	yes	characteristic
314	C₂×C₄	#483	1	1	yes	characteristic
315	C₂×C₄	#484	1	1	yes	characteristic
316	C₂×C₄	#486	1	1	yes	characteristic
317	C₂×C₄	#487	1	1	yes	characteristic
318	C₂×C₄	#490	1	1	yes	characteristic
319	C₂×C₄	#498	1	1	yes	characteristic
320	C₂×C₄	#500	1	1	yes	characteristic
321	C₂×C₄	#501	1	1	yes	characteristic
322	C₈	—	2	4	no	—
323	C₈	—	4	4	no	—
324	C₈	—	8	2	no	—
325	C₈	—	8	2	no	—
326	C₈	#387	1	1	yes	characteristic
327	C₈	#387	1	1	yes	characteristic
328	C₈	#389	1	1	yes	characteristic
329	C₈	#389	1	1	yes	characteristic
330	C₈	#396	1	1	yes	characteristic
331	C₈	#396	1	1	yes	characteristic
332	C₈	#398	1	1	yes	characteristic
333	C₈	#398	1	1	yes	characteristic
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	#1803	1	1	yes	characteristic
345	C₂²	#1874	1	1	yes	characteristic
346	C₂²	#1992	1	1	yes	characteristic
347	C₂²	#2043	1	1	yes	characteristic
348	C₂²	#2058	1	1	yes	characteristic
349	C₂²	#2083	1	1	yes	characteristic
350	C₂²	#2099	1	1	yes	characteristic
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	16	2	no	—
354	C₄	#1441	1	1	yes	characteristic
355	C₄	#1802	1	1	yes	characteristic
356	C₄	#1872	1	1	yes	characteristic
357	C₄	#1990	1	1	yes	characteristic
358	C₄	#2041	1	1	yes	characteristic
359	C₄	#2057	1	1	yes	characteristic
360	C₄	#2084	1	1	yes	characteristic
361	C₄	#2100	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	#11207	1	1	yes	characteristic
369	C₂	#13006	1	1	yes	characteristic
370	C₂	#13007	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#1952	2
#20807	2
C₂×#1388	2
C₂×#1387	2
C₂×#1385	2
#12554	2