Subgroups — #147123

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33324	C₂	1	1	yes	maximal, characteristic
2	C₂×#1773	C₂	1	1	yes	maximal, characteristic
3	#12554	C₂	1	1	yes	maximal, characteristic
4	#12554	C₂	1	1	yes	maximal, characteristic
5	#12551	C₂	1	1	yes	maximal, characteristic
6	C₂×#1377	C₂	1	1	yes	maximal, characteristic
7	C₂×#1375	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×#108	C₄	1	1	yes	characteristic
13	C₂×#427	C₄	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	C₂×#405	C₂²	1	1	yes	characteristic
16	C₂²×#97	C₂²	1	1	yes	characteristic
17	#2108	—	2	2	no	—
18	#2108	—	2	2	no	—
19	C₄×M₃₂	—	2	2	no	—
20	#1773	—	2	2	no	—
21	#1773	—	2	2	no	—
22	C₂×#339	C₄	1	1	yes	characteristic
23	C₂×#339	C₄	1	1	yes	characteristic
24	#1379	—	2	2	no	—
25	#1379	—	2	2	no	—
26	#1377	—	2	2	no	—
27	#1377	C₄	1	4	yes	complemented
28	#1375	C₄	1	4	yes	complemented
29	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	C₂×C₄	1	1	yes	characteristic
34	C₂×#117	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
46	C₂×M₃₂	—	2	1	no	—
47	C₂×M₃₂	—	2	1	no	—
48	C₂×M₃₂	—	2	1	no	—
49	C₂×M₃₂	—	2	1	no	—
50	C₂×M₃₂	—	2	1	no	—
51	C₂×M₃₂	—	2	1	no	—
52	C₂×M₃₂	—	2	1	no	—
53	C₂×M₃₂	C₂×C₄	1	2	yes	—
54	C₂×M₃₂	C₂×C₄	1	2	yes	—
55	C₂×M₃₂	C₂×C₄	1	2	yes	—
56	C₂×M₃₂	C₂×C₄	1	2	yes	—
57	C₂×M₃₂	D₄	1	2	yes	—
58	C₂×M₃₂	Q₈	1	2	yes	—
59	C₂×M₃₂	Q₈	1	2	yes	—
60	C₂²×C₁₆	—	2	1	no	—
61	C₂²×C₁₆	—	2	1	no	—
62	C₂²×C₁₆	—	2	1	no	—
63	C₂²×C₁₆	D₄	1	2	yes	—
64	C₂×#108	—	2	2	no	—
65	C₂×#108	—	2	2	no	—
66	C₂×#108	C₈	1	4	yes	—
67	#427	—	2	2	no	—
68	#427	—	2	2	no	—
69	C₂×#107	C₈	1	2	yes	—
70	C₂×#106	C₈	1	2	yes	—
71	#405	—	2	1	no	—
72	#405	—	2	1	no	—
73	#405	—	2	1	no	—
74	#405	—	2	1	no	—
75	#405	—	2	1	no	—
76	#405	—	2	1	no	—
77	#405	D₄	1	2	yes	—
78	#405	D₄	1	2	yes	—
79	C₂×#97	—	2	1	no	—
80	C₂×#97	—	2	1	no	—
81	C₂×#97	—	2	1	no	—
82	C₂×#97	—	2	1	no	—
83	C₂×#97	—	2	1	no	—
84	C₂×#97	—	2	1	no	—
85	C₂×#97	C₂×C₄	1	2	yes	—
86	C₂×#97	C₂×C₄	1	2	yes	—
87	C₂×C₄×C₈	—	2	1	no	—
88	C₂×C₄×C₈	—	2	1	no	—
89	#346	—	2	4	no	—
90	C₄×C₁₆	—	2	4	no	—
91	#339	—	2	2	no	—
92	#339	—	2	2	no	—
93	C₂³×C₄	—	2	1	no	—
94	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
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	1	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	C₂²⋊C₄	1	2	yes	—
104	C₂×M₁₆	C₂×C₈	1	2	yes	—
105	C₂×M₁₆	C₂×C₈	1	2	yes	—
106	C₂×M₁₆	M₁₆	1	2	yes	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₄²	1	2	yes	—
114	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₈	1	2	yes	—
117	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₄	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×Q₈	1	1	yes	characteristic
124	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂²×C₈	#41	1	1	yes	characteristic
127	C₂²×C₈	#41	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	#117	—	2	1	no	—
132	#117	—	2	1	no	—
133	#117	—	2	1	no	—
134	#117	—	2	1	no	—
135	#117	—	2	1	no	—
136	#117	—	2	1	no	—
137	#117	C₂²⋊C₄	1	2	yes	—
138	#117	M₁₆	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
140	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
141	C₂×C₂²⋊C₄	—	2	1	no	—
142	C₂×C₂²⋊C₄	—	2	1	no	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	C₄²	1	2	yes	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	—	2	2	no	—
160	M₃₂	—	2	2	no	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	4	1	no	—
163	M₃₂	—	4	1	no	—
164	M₃₂	—	4	1	no	—
165	M₃₂	—	4	1	no	—
166	M₃₂	C₄⋊C₄	1	4	yes	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	4	1	no	—
175	C₂×C₁₆	—	4	1	no	—
176	C₂×C₁₆	—	4	1	no	—
177	C₂×C₁₆	—	4	1	no	—
178	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
179	#108	—	2	8	no	—
180	#107	—	2	4	no	—
181	#106	—	2	4	no	—
182	#97	—	2	2	no	—
183	#97	—	2	2	no	—
184	#97	—	2	2	no	—
185	#97	—	2	2	no	—
186	#97	—	2	2	no	—
187	#97	—	2	2	no	—
188	#97	—	4	1	no	—
189	#97	—	4	1	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	—	4	1	no	—
193	C₄×C₈	—	4	1	no	—
194	C₂⁴	#131	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	8	2	no	—
201	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	#117	1	1	yes	characteristic
204	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
207	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
210	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
212	C₂²×C₄	#131	1	1	yes	characteristic
213	C₂²×C₄	#131	1	1	yes	characteristic
214	C₂²×C₄	#131	1	1	yes	characteristic
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	C₂²⋊C₈	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	C₄×C₈	1	2	yes	—
238	C₂×C₈	C₄×C₈	1	2	yes	—
239	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₈	#117	1	1	yes	characteristic
242	C₂×C₈	#117	1	1	yes	characteristic
243	C₂×C₈	C₄×D₄	1	1	yes	characteristic
244	C₂×C₈	C₄×D₄	1	1	yes	characteristic
245	C₂×C₈	C₄×D₄	1	1	yes	characteristic
246	C₂×C₈	C₄×D₄	1	1	yes	characteristic
247	C₂×C₈	C₄×D₄	1	1	yes	characteristic
248	C₂×C₈	C₄×D₄	1	1	yes	characteristic
249	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
250	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
251	C₂×C₈	#120	1	1	yes	characteristic
252	C₂×C₈	#121	1	1	yes	characteristic
253	C₂×C₈	#121	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₈	#123	1	1	yes	characteristic
259	C₂×C₈	#123	1	1	yes	characteristic
260	C₂×C₈	#123	1	1	yes	characteristic
261	C₂×C₈	#124	1	1	yes	characteristic
262	C₂×C₈	#125	1	1	yes	characteristic
263	C₂×C₈	#125	1	1	yes	characteristic
264	C₂×C₈	#126	1	1	yes	characteristic
265	C₂×C₈	#126	1	1	yes	characteristic
266	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂²⋊C₈	1	4	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₁₆	—	2	4	no	—
284	C₁₆	—	2	4	no	—
285	C₁₆	—	4	2	no	—
286	C₁₆	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	#405	1	1	yes	characteristic
289	C₂³	#408	1	1	yes	characteristic
290	C₂³	#433	1	1	yes	characteristic
291	C₂³	C₈×D₄	1	1	yes	characteristic
292	C₂³	#435	1	1	yes	characteristic
293	C₂³	C₈×Q₈	1	1	yes	characteristic
294	C₂³	#446	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
306	C₂×C₄	#386	1	1	yes	characteristic
307	C₂×C₄	#387	1	1	yes	characteristic
308	C₂×C₄	#387	1	1	yes	characteristic
309	C₂×C₄	#391	1	1	yes	characteristic
310	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
311	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
312	C₂×C₄	#405	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
314	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
315	C₂×C₄	#408	1	1	yes	characteristic
316	C₂×C₄	#431	1	1	yes	characteristic
317	C₂×C₄	#431	1	1	yes	characteristic
318	C₂×C₄	#433	1	1	yes	characteristic
319	C₂×C₄	C₈×D₄	1	1	yes	characteristic
320	C₂×C₄	C₈×D₄	1	1	yes	characteristic
321	C₂×C₄	C₈×D₄	1	1	yes	characteristic
322	C₂×C₄	#435	1	1	yes	characteristic
323	C₂×C₄	#435	1	1	yes	characteristic
324	C₂×C₄	#435	1	1	yes	characteristic
325	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
326	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
327	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
328	C₂×C₄	#446	1	1	yes	characteristic
329	C₂×C₄	#446	1	1	yes	characteristic
330	C₂×C₄	#446	1	1	yes	characteristic
331	C₈	—	2	4	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	#385	1	1	yes	characteristic
336	C₈	#385	1	1	yes	characteristic
337	C₈	#387	1	1	yes	characteristic
338	C₈	#387	1	1	yes	characteristic
339	C₈	#388	1	1	yes	characteristic
340	C₈	#388	1	1	yes	characteristic
341	C₈	#389	1	1	yes	characteristic
342	C₈	#389	1	1	yes	characteristic
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	#1750	1	1	yes	characteristic
351	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
352	C₂²	#1771	1	1	yes	characteristic
353	C₂²	#1869	1	1	yes	characteristic
354	C₂²	#1916	1	1	yes	characteristic
355	C₂²	#1916	1	1	yes	characteristic
356	C₂²	#1984	1	1	yes	characteristic
357	C₄	—	4	4	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	2	no	—
360	C₄	#1436	1	1	yes	characteristic
361	C₄	C₄×C₂²⋊C₈	1	1	yes	characteristic
362	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
363	C₄	#1771	1	1	yes	characteristic
364	C₄	#1869	1	1	yes	characteristic
365	C₄	#1916	1	1	yes	characteristic
366	C₄	#1916	1	1	yes	characteristic
367	C₄	#1984	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	#10874	1	1	yes	characteristic
375	C₂	#12986	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
27. #1377	C₄	semidirect
28. #1375	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33324	2
C₂×#1773	2
#12554	2
#12551	2
C₂×#1377	2
C₂×#1375	2