Subgroups — #149470

Order 512.

677 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	#60660	C₂	1	1	yes	maximal, characteristic
2	#12567	C₂	1	1	yes	maximal, characteristic
3	C₂×#1384	C₂	1	1	yes	maximal, characteristic
4	C₂×#1384	C₂	1	1	yes	maximal, characteristic
5	C₂×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×#1378	C₂	1	1	yes	maximal, characteristic
7	#10871	C₂	1	1	yes	maximal, characteristic
8	#3458	—	2	1	no	—
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×M₃₂	C₂²	1	1	yes	characteristic
11	#3164	—	2	2	no	—
12	#3164	—	2	2	no	—
13	#3163	—	2	2	no	—
14	#3163	C₄	1	4	yes	complemented
15	#3162	—	2	2	no	—
16	#3160	—	2	2	no	—
17	#2911	—	2	1	no	—
18	C₂×#427	C₂²	1	1	yes	characteristic
19	C₂×#427	C₂²	1	1	yes	characteristic
20	C₂×#427	C₂²	1	1	yes	characteristic
21	C₂×#427	C₂²	1	1	yes	characteristic
22	C₂×#405	C₂²	1	1	yes	characteristic
23	#1740	—	2	1	no	—
24	#1733	—	2	1	no	—
25	#1386	—	2	2	no	—
26	#1384	—	2	2	no	—
27	#1384	—	2	2	no	—
28	#1384	—	2	2	no	—
29	#1378	—	2	2	no	—
30	C₂²×M₁₆	D₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	#544	—	2	4	no	—
33	#544	—	2	4	no	—
34	#544	—	4	2	no	—
35	#543	—	4	2	no	—
36	#533	—	2	4	no	—
37	#533	—	4	2	no	—
38	#533	—	4	2	no	—
39	C₂×#117	—	2	1	no	—
40	C₂×#117	—	2	1	no	—
41	C₂×#117	—	2	1	no	—
42	C₂×#117	—	2	1	no	—
43	C₂×#117	D₄	1	1	yes	characteristic
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	—	2	1	no	—
54	C₂×M₃₂	—	2	1	no	—
55	C₂×M₃₂	Q₈	1	2	yes	complemented
56	C₂²×C₁₆	—	2	1	no	—
57	C₂²×C₁₆	Q₈	1	2	yes	complemented
58	#501	—	2	8	no	—
59	#501	—	2	8	no	—
60	#500	—	2	4	no	—
61	#499	—	2	4	no	—
62	#499	—	2	4	no	—
63	#498	—	2	4	no	—
64	#428	—	2	4	no	—
65	#428	—	2	4	no	—
66	#427	—	2	4	no	—
67	#427	—	2	1	no	—
68	#427	—	2	1	no	—
69	#427	—	2	1	no	—
70	#427	—	2	1	no	—
71	#427	—	2	1	no	—
72	#427	—	2	1	no	—
73	#427	—	2	1	no	—
74	#427	—	2	1	no	—
75	#427	—	2	1	no	—
76	#427	—	2	1	no	—
77	#427	—	2	1	no	—
78	#427	C₂×C₄	1	2	yes	—
79	#427	D₄	1	2	yes	—
80	#427	D₄	1	2	yes	—
81	#427	D₄	1	2	yes	—
82	#427	Q₈	1	2	yes	complemented
83	C₂×#107	—	2	1	no	—
84	C₂×#107	—	2	1	no	—
85	C₂×#107	—	2	1	no	—
86	C₂×#107	—	2	1	no	—
87	C₂×#106	—	2	1	no	—
88	C₂×#106	C₂×C₄	1	2	yes	—
89	C₂×#106	D₄	1	2	yes	—
90	C₂×#106	Q₈	1	2	yes	complemented
91	#405	—	2	1	no	—
92	#405	—	2	1	no	—
93	#405	C₂×C₄	1	2	yes	—
94	#405	D₄	1	2	yes	—
95	C₂×#97	—	2	1	no	—
96	C₂×C₄×C₈	—	2	1	no	—
97	#342	—	4	2	no	—
98	#339	—	4	2	no	—
99	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
105	C₂×M₁₆	Q₁₆	1	2	yes	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
114	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
116	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
117	C₂²×C₈	#41	1	1	yes	characteristic
118	C₂²×C₈	#41	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	C₂²×C₈	#41	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	8	no	—
122	C₄⋊Q₈	—	4	4	no	—
123	C₄⋊Q₈	—	4	4	no	—
124	#125	—	2	8	no	—
125	#125	—	4	4	no	—
126	#125	—	4	4	no	—
127	C₂²⋊Q₈	—	4	8	no	—
128	C₂²⋊Q₈	—	4	8	no	—
129	C₂²⋊Q₈	—	4	8	no	—
130	C₄×Q₈	—	4	8	no	—
131	#117	—	2	1	no	—
132	#117	—	2	1	no	—
133	#117	—	2	2	no	—
134	#117	—	2	2	no	—
135	#117	—	2	2	no	—
136	#117	—	2	2	no	—
137	#117	—	2	2	no	—
138	#117	—	4	1	no	—
139	#117	—	4	1	no	—
140	#117	—	4	1	no	—
141	#117	—	4	1	no	—
142	#117	—	4	1	no	—
143	#117	—	4	1	no	—
144	#117	—	4	1	no	—
145	#117	—	4	1	no	—
146	#117	—	4	1	no	—
147	#117	—	4	1	no	—
148	#117	—	4	1	no	—
149	#117	C₂²⋊C₄	1	2	yes	—
150	#117	SD₁₆	1	2	yes	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	4	1	no	—
156	C₂×C₂²⋊C₄	—	4	1	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	C₂×C₂²⋊C₄	—	4	1	no	—
159	C₂×C₂²⋊C₄	—	4	1	no	—
160	C₂×C₂²⋊C₄	D₈	1	2	yes	—
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	—	4	1	no	—
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₄²	—	4	1	no	—
165	C₂×C₄²	—	4	1	no	—
166	M₃₂	—	2	2	no	—
167	M₃₂	—	2	2	no	—
168	M₃₂	—	4	1	no	—
169	M₃₂	—	4	1	no	—
170	M₃₂	—	4	1	no	—
171	M₃₂	—	4	1	no	—
172	M₃₂	—	4	1	no	—
173	M₃₂	—	4	1	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	4	1	no	—
179	C₂×C₁₆	—	4	1	no	—
180	#107	—	2	8	no	—
181	#107	—	2	2	no	—
182	#107	—	2	2	no	—
183	#107	—	2	2	no	—
184	#107	—	2	2	no	—
185	#107	—	2	2	no	—
186	#107	—	4	1	no	—
187	#107	—	4	1	no	—
188	#107	—	4	1	no	—
189	#106	—	2	8	no	—
190	#106	—	2	2	no	—
191	#106	—	2	2	no	—
192	#106	—	2	2	no	—
193	#106	—	2	2	no	—
194	#106	—	2	2	no	—
195	#106	—	4	1	no	—
196	#106	C₂²⋊C₄	1	4	yes	—
197	#106	C₂²⋊C₄	1	4	yes	—
198	#97	—	2	2	no	—
199	#97	—	2	2	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	2	2	no	—
202	C₂⁴	C₂×D₈	1	1	yes	characteristic
203	C₂×Q₈	—	4	8	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	8	1	no	—
209	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	#120	1	1	yes	characteristic
211	C₂²×C₄	#120	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
214	C₂²×C₄	#124	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
217	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
218	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
219	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₄	#137	1	1	yes	characteristic
222	C₂²×C₄	#137	1	1	yes	characteristic
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	#103	1	4	yes	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
233	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
234	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
235	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
236	C₂×C₈	#120	1	1	yes	characteristic
237	C₂×C₈	#120	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₈	C₂²⋊Q₈	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	#124	1	1	yes	characteristic
249	C₂×C₈	#124	1	1	yes	characteristic
250	C₂×C₈	#124	1	1	yes	characteristic
251	C₂×C₈	#124	1	1	yes	characteristic
252	C₂×C₈	#124	1	1	yes	characteristic
253	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
254	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
255	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
256	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
257	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	8	no	—
263	C₄⋊C₄	—	4	8	no	—
264	C₄⋊C₄	—	4	8	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	8	1	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	8	1	no	—
278	C₂²⋊C₄	—	8	1	no	—
279	C₂²⋊C₄	—	8	1	no	—
280	C₂²⋊C₄	#102	1	4	yes	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	—	4	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	—	4	2	no	—
288	C₄²	—	8	1	no	—
289	C₄²	—	8	1	no	—
290	C₄²	—	8	1	no	—
291	C₁₆	—	4	2	no	—
292	C₁₆	—	4	2	no	—
293	C₂³	—	8	1	no	—
294	C₂³	C₂×#102	1	1	yes	characteristic
295	C₂³	#447	1	1	yes	characteristic
296	C₂³	#447	1	1	yes	characteristic
297	C₂³	#474	1	1	yes	characteristic
298	C₂³	#474	1	1	yes	characteristic
299	C₂³	#486	1	1	yes	characteristic
300	C₂³	#500	1	1	yes	characteristic
301	Q₈	—	16	8	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	—	16	2	no	—
311	C₂×C₄	—	16	2	no	—
312	C₂×C₄	#379	1	1	yes	characteristic
313	C₂×C₄	#391	1	1	yes	characteristic
314	C₂×C₄	#391	1	1	yes	characteristic
315	C₂×C₄	#393	1	1	yes	characteristic
316	C₂×C₄	#393	1	1	yes	characteristic
317	C₂×C₄	#395	1	1	yes	characteristic
318	C₂×C₄	#395	1	1	yes	characteristic
319	C₂×C₄	C₂×#103	1	1	yes	characteristic
320	C₂×C₄	#418	1	1	yes	characteristic
321	C₂×C₄	#419	1	1	yes	characteristic
322	C₂×C₄	#448	1	1	yes	characteristic
323	C₂×C₄	#448	1	1	yes	characteristic
324	C₂×C₄	#450	1	1	yes	characteristic
325	C₂×C₄	#450	1	1	yes	characteristic
326	C₂×C₄	#451	1	1	yes	characteristic
327	C₂×C₄	#451	1	1	yes	characteristic
328	C₂×C₄	#475	1	1	yes	characteristic
329	C₂×C₄	#475	1	1	yes	characteristic
330	C₂×C₄	#476	1	1	yes	characteristic
331	C₂×C₄	#476	1	1	yes	characteristic
332	C₂×C₄	#477	1	1	yes	characteristic
333	C₂×C₄	#477	1	1	yes	characteristic
334	C₂×C₄	#487	1	1	yes	characteristic
335	C₂×C₄	#489	1	1	yes	characteristic
336	C₂×C₄	#489	1	1	yes	characteristic
337	C₂×C₄	#498	1	1	yes	characteristic
338	C₂×C₄	#501	1	1	yes	characteristic
339	C₂×C₄	#501	1	1	yes	characteristic
340	C₈	—	2	4	no	—
341	C₈	#391	1	1	yes	characteristic
342	C₈	#391	1	1	yes	characteristic
343	C₈	#391	1	1	yes	characteristic
344	C₈	#391	1	1	yes	characteristic
345	C₈	#393	1	1	yes	characteristic
346	C₈	#393	1	1	yes	characteristic
347	C₈	#395	1	1	yes	characteristic
348	C₈	#395	1	1	yes	characteristic
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	#1786	1	1	yes	characteristic
357	C₂²	#1988	1	1	yes	characteristic
358	C₂²	#1990	1	1	yes	characteristic
359	C₂²	#2023	1	1	yes	characteristic
360	C₂²	#2023	1	1	yes	characteristic
361	C₂²	#2057	1	1	yes	characteristic
362	C₂²	#2057	1	1	yes	characteristic
363	C₄	—	4	4	no	—
364	C₄	—	8	4	no	—
365	C₄	—	8	4	no	—
366	C₄	—	16	2	no	—
367	C₄	—	16	2	no	—
368	C₄	#1446	1	1	yes	characteristic
369	C₄	#1787	1	1	yes	characteristic
370	C₄	#1987	1	1	yes	characteristic
371	C₄	#1993	1	1	yes	characteristic
372	C₄	#2024	1	1	yes	characteristic
373	C₄	#2024	1	1	yes	characteristic
374	C₄	#2058	1	1	yes	characteristic
375	C₄	#2058	1	1	yes	characteristic
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	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	#11384	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2058
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11384
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. #3163	C₄	semidirect
55. C₂×M₃₂	Q₈	semidirect
57. C₂²×C₁₆	Q₈	semidirect
82. #427	Q₈	semidirect
90. C₂×#106	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#60660	2
#12567	2
C₂×#1384	2
C₂×#1378	2
#10871	2