Subgroups — #149267

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33409	C₂	1	1	yes	maximal, characteristic
2	#21181	C₂	1	1	yes	maximal, characteristic
3	#21157	C₂	1	1	yes	maximal, characteristic
4	C₂×#1386	C₂	1	1	yes	maximal, characteristic
5	C₂×#1383	C₂	1	1	yes	maximal, characteristic
6	#12561	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	C₂²×#108	C₂²	1	1	yes	characteristic
11	#2911	—	2	1	no	—
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂²×#106	C₂²	1	1	yes	characteristic
15	C₂×#405	C₂²	1	1	yes	characteristic
16	#2316	—	2	1	no	—
17	#2159	—	2	2	no	—
18	#2159	—	2	2	no	—
19	#2158	—	2	2	no	—
20	#1952	—	2	2	no	—
21	#1951	—	2	2	no	—
22	#1944	—	2	2	no	—
23	#1941	—	2	2	no	—
24	#1742	—	2	1	no	—
25	#1736	—	2	1	no	—
26	#1386	—	2	2	no	—
27	#1383	C₄	1	4	yes	complemented
28	#1382	—	2	2	no	—
29	#1379	—	2	2	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	—	2	1	no	—
34	C₂×#117	—	2	1	no	—
35	C₂×#117	D₄	1	1	yes	characteristic
36	C₂²×C₄⋊C₄	—	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	C₂×C₄	1	2	yes	—
47	C₂×M₃₂	Q₈	1	2	yes	—
48	C₂×M₃₂	Q₈	1	2	yes	—
49	C₂²×C₁₆	—	2	1	no	—
50	C₂²×C₁₆	D₄	1	2	yes	—
51	C₂×#108	—	2	1	no	—
52	C₂×#108	—	2	1	no	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	C₂×C₄	1	2	yes	—
55	C₂×#108	D₄	1	2	yes	—
56	C₂×#108	Q₈	1	2	yes	—
57	#428	—	2	4	no	—
58	#428	—	4	2	no	—
59	#427	—	2	1	no	—
60	#427	—	2	1	no	—
61	#427	—	2	1	no	—
62	#427	—	2	1	no	—
63	#427	—	2	1	no	—
64	#427	—	2	1	no	—
65	#427	—	2	1	no	—
66	#427	—	2	1	no	—
67	#427	—	4	2	no	—
68	C₂×#107	D₄	1	2	yes	—
69	C₂×#107	D₄	1	2	yes	—
70	C₂×#106	—	2	1	no	—
71	C₂×#106	—	2	1	no	—
72	C₂×#106	—	2	1	no	—
73	C₂×#106	—	2	1	no	—
74	C₂×#106	—	2	1	no	—
75	C₂×#106	C₂×C₄	1	2	yes	—
76	C₂×#106	D₄	1	2	yes	—
77	C₂×#106	Q₈	1	2	yes	—
78	#405	—	2	1	no	—
79	#405	—	2	1	no	—
80	#405	—	2	1	no	—
81	#405	—	2	1	no	—
82	C₂×#97	—	2	1	no	—
83	C₂×C₄×C₈	—	2	1	no	—
84	#389	—	2	4	no	—
85	#389	—	4	2	no	—
86	#380	—	4	2	no	—
87	#367	—	2	4	no	—
88	#366	—	2	4	no	—
89	#343	—	2	4	no	—
90	#340	—	2	4	no	—
91	C₂³×C₄	—	4	1	no	—
92	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	4	1	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	Q₁₆	1	2	yes	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	4	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	#117	—	2	1	no	—
122	#117	—	2	1	no	—
123	#117	—	2	1	no	—
124	#117	—	2	1	no	—
125	#117	—	4	1	no	—
126	#117	—	4	1	no	—
127	#117	—	4	1	no	—
128	#117	—	4	1	no	—
129	#117	—	4	1	no	—
130	#117	—	4	1	no	—
131	#117	—	4	1	no	—
132	#117	—	4	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	4	no	—
142	C₂×C₄⋊C₄	D₈	1	2	yes	—
143	C₂×C₂²⋊C₄	—	2	1	no	—
144	C₂×C₂²⋊C₄	—	4	1	no	—
145	C₂×C₂²⋊C₄	—	4	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	—	4	1	no	—
148	C₂×C₄²	—	4	1	no	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	4	1	no	—
156	M₃₂	—	4	1	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	4	1	no	—
161	C₂×C₁₆	—	4	1	no	—
162	C₂×C₁₆	C₄⋊C₄	1	4	yes	complemented
163	#108	—	2	2	no	—
164	#108	—	2	2	no	—
165	#108	—	2	2	no	—
166	#108	C₄⋊C₄	1	4	yes	complemented
167	#107	—	2	2	no	—
168	#107	—	2	2	no	—
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#107	—	4	4	no	—
172	#106	—	2	2	no	—
173	#106	—	2	2	no	—
174	#106	—	2	2	no	—
175	#106	—	4	4	no	—
176	#106	—	4	1	no	—
177	#106	—	4	1	no	—
178	#106	—	4	1	no	—
179	#106	—	4	1	no	—
180	#106	C₄⋊C₄	1	4	yes	—
181	#97	—	4	1	no	—
182	#97	—	4	1	no	—
183	C₄×C₈	—	4	1	no	—
184	C₄×C₈	—	4	1	no	—
185	#95	—	4	4	no	—
186	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	16	2	no	—
194	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
195	C₂²×C₄	#120	1	1	yes	characteristic
196	C₂²×C₄	#120	1	1	yes	characteristic
197	C₂²×C₄	#121	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₁₆	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	#137	1	1	yes	characteristic
207	C₂²×C₄	#137	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₈	C₄×D₄	1	1	yes	characteristic
226	C₂×C₈	C₄×D₄	1	1	yes	characteristic
227	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
228	C₂×C₈	#120	1	1	yes	characteristic
229	C₂×C₈	#121	1	1	yes	characteristic
230	C₂×C₈	#121	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₈	#123	1	1	yes	characteristic
239	C₂×C₈	#123	1	1	yes	characteristic
240	C₂×C₈	#123	1	1	yes	characteristic
241	C₂×C₈	#125	1	1	yes	characteristic
242	C₂×C₈	#125	1	1	yes	characteristic
243	C₂×C₈	#125	1	1	yes	characteristic
244	C₂×C₈	#127	1	1	yes	characteristic
245	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
246	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
247	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
248	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
249	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	16	4	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	8	1	no	—
264	C₂²⋊C₄	—	8	1	no	—
265	C₂²⋊C₄	—	8	1	no	—
266	C₂²⋊C₄	—	8	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	8	1	no	—
270	C₄²	—	8	1	no	—
271	C₄²	—	8	1	no	—
272	C₄²	—	8	1	no	—
273	C₁₆	—	2	4	no	—
274	C₁₆	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	#440	1	1	yes	characteristic
277	C₂³	#447	1	1	yes	characteristic
278	C₂³	#449	1	1	yes	characteristic
279	C₂³	#469	1	1	yes	characteristic
280	C₂³	#477	1	1	yes	characteristic
281	C₂³	#478	1	1	yes	characteristic
282	C₂³	#501	1	1	yes	characteristic
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	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	#385	1	1	yes	characteristic
295	C₂×C₄	#385	1	1	yes	characteristic
296	C₂×C₄	#389	1	1	yes	characteristic
297	C₂×C₄	#392	1	1	yes	characteristic
298	C₂×C₄	#395	1	1	yes	characteristic
299	C₂×C₄	#395	1	1	yes	characteristic
300	C₂×C₄	#398	1	1	yes	characteristic
301	C₂×C₄	C₄×D₈	1	1	yes	characteristic
302	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
303	C₂×C₄	#440	1	1	yes	characteristic
304	C₂×C₄	#447	1	1	yes	characteristic
305	C₂×C₄	#449	1	1	yes	characteristic
306	C₂×C₄	#451	1	1	yes	characteristic
307	C₂×C₄	#451	1	1	yes	characteristic
308	C₂×C₄	#452	1	1	yes	characteristic
309	C₂×C₄	#452	1	1	yes	characteristic
310	C₂×C₄	#466	1	1	yes	characteristic
311	C₂×C₄	#467	1	1	yes	characteristic
312	C₂×C₄	#470	1	1	yes	characteristic
313	C₂×C₄	#474	1	1	yes	characteristic
314	C₂×C₄	#474	1	1	yes	characteristic
315	C₂×C₄	#477	1	1	yes	characteristic
316	C₂×C₄	#478	1	1	yes	characteristic
317	C₂×C₄	#479	1	1	yes	characteristic
318	C₂×C₄	#479	1	1	yes	characteristic
319	C₂×C₄	#499	1	1	yes	characteristic
320	C₂×C₄	#500	1	1	yes	characteristic
321	C₂×C₄	#501	1	1	yes	characteristic
322	C₈	—	4	4	no	—
323	C₈	—	4	2	no	—
324	C₈	#384	1	1	yes	characteristic
325	C₈	#385	1	1	yes	characteristic
326	C₈	#389	1	1	yes	characteristic
327	C₈	#389	1	1	yes	characteristic
328	C₈	#395	1	1	yes	characteristic
329	C₈	#398	1	1	yes	characteristic
330	C₈	#398	1	1	yes	characteristic
331	C₈	#399	1	1	yes	characteristic
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	#1862	1	1	yes	characteristic
340	C₂²	#1863	1	1	yes	characteristic
341	C₂²	#1944	1	1	yes	characteristic
342	C₂²	#1999	1	1	yes	characteristic
343	C₂²	#2057	1	1	yes	characteristic
344	C₂²	#2059	1	1	yes	characteristic
345	C₂²	#2087	1	1	yes	characteristic
346	C₄	—	8	4	no	—
347	C₄	—	8	2	no	—
348	C₄	—	16	2	no	—
349	C₄	—	16	2	no	—
350	C₄	#1444	1	1	yes	characteristic
351	C₄	#1862	1	1	yes	characteristic
352	C₄	#1863	1	1	yes	characteristic
353	C₄	#1942	1	1	yes	characteristic
354	C₄	#2001	1	1	yes	characteristic
355	C₄	#2057	1	1	yes	characteristic
356	C₄	#2059	1	1	yes	characteristic
357	C₄	#2086	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	#11328	1	1	yes	characteristic
365	C₂	#13318	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
27. #1383	C₄	semidirect
162. C₂×C₁₆	C₄⋊C₄	semidirect
166. #108	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#33409	2
#21181	2
#21157	2
C₂×#1386	2
C₂×#1383	2
#12561	2
#12554	2