Subgroups — #149125

Order 512.

452 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1382	C₂	1	1	yes	maximal, characteristic
2	C₂×#1382	C₂	1	1	yes	maximal, characteristic
3	C₂×#1381	C₂	1	1	yes	maximal, characteristic
4	#12556	C₂	1	1	yes	maximal, characteristic
5	C₂×#1374	C₂	1	1	yes	maximal, characteristic
6	#10859	C₂	1	1	yes	maximal, characteristic
7	#10859	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂²×#108	C₂²	1	1	yes	characteristic
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×#339	—	2	1	no	—
16	C₂×#339	—	2	1	no	—
17	#1729	—	2	1	no	—
18	#1729	—	2	1	no	—
19	#1382	—	2	2	no	—
20	#1382	—	2	2	no	—
21	#1381	—	2	2	no	—
22	#1380	C₄	1	2	yes	complemented
23	#1380	C₄	1	2	yes	complemented
24	#1374	—	2	2	no	—
25	C₂²×M₁₆	—	2	1	no	—
26	C₂²×M₁₆	—	2	1	no	—
27	C₂³×C₈	D₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	—	2	1	no	—
30	C₂×#117	—	2	1	no	—
31	C₂²×C₄²	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	—	2	1	no	—
33	C₂²×C₁₆	—	2	1	no	—
34	C₂²×C₁₆	—	2	1	no	—
35	C₂²×C₁₆	—	2	1	no	—
36	C₂²×C₁₆	—	2	1	no	—
37	C₂²×C₁₆	—	2	1	no	—
38	C₂²×C₁₆	—	2	1	no	—
39	C₂²×C₁₆	—	2	1	no	—
40	C₂²×C₁₆	—	2	1	no	—
41	C₂²×C₁₆	—	2	1	no	—
42	C₂²×C₁₆	—	2	1	no	—
43	C₂²×C₁₆	—	2	1	no	—
44	C₂²×C₁₆	C₂×C₄	1	2	yes	—
45	C₂²×C₁₆	D₄	1	2	yes	—
46	C₂×#108	—	2	1	no	—
47	C₂×#108	—	2	1	no	—
48	C₂×#108	—	2	1	no	—
49	C₂×#108	—	2	1	no	—
50	C₂×#108	—	2	1	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	D₄	1	2	yes	—
57	C₂×#108	Q₈	1	2	yes	—
58	#427	—	2	1	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	C₂×#107	C₂×C₄	1	1	yes	characteristic
67	C₂×#107	C₂×C₄	1	1	yes	characteristic
68	C₂×#107	C₂×C₄	1	2	yes	—
69	C₂×#106	—	2	1	no	—
70	C₂×#106	—	2	1	no	—
71	C₂×C₄×C₈	—	2	1	no	—
72	C₂×C₄×C₈	—	2	1	no	—
73	C₂×C₄×C₈	—	2	1	no	—
74	C₂×C₄×C₈	—	2	1	no	—
75	C₂×C₄×C₈	—	2	1	no	—
76	C₂×C₄×C₈	—	2	1	no	—
77	#339	—	4	2	no	—
78	#339	—	4	2	no	—
79	#336	—	4	2	no	—
80	#336	—	4	2	no	—
81	C₂³×C₄	—	2	1	no	—
82	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	4	1	no	—
86	C₂×M₁₆	—	4	1	no	—
87	C₂×M₁₆	—	4	1	no	—
88	C₂×M₁₆	—	4	1	no	—
89	C₂×M₁₆	—	4	1	no	—
90	C₂×M₁₆	—	4	1	no	—
91	C₂×M₁₆	—	4	1	no	—
92	C₂×M₁₆	—	4	1	no	—
93	C₂×M₁₆	—	4	1	no	—
94	C₂×M₁₆	—	4	1	no	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	4	1	no	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	D₈	1	2	yes	—
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₈	#41	1	1	yes	characteristic
109	C₂²×C₈	#41	1	1	yes	characteristic
110	C₂²×C₈	#41	1	1	yes	characteristic
111	C₂²×C₈	#41	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	C₂²×C₈	#41	1	1	yes	characteristic
117	C₂²×C₈	#41	1	1	yes	characteristic
118	#117	—	4	1	no	—
119	#117	—	4	1	no	—
120	#117	—	4	1	no	—
121	#117	—	4	1	no	—
122	#117	—	4	1	no	—
123	#117	—	4	1	no	—
124	#117	—	4	1	no	—
125	#117	—	4	1	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₂²⋊C₄	—	4	1	no	—
130	C₂×C₂²⋊C₄	—	4	1	no	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	—	2	1	no	—
135	C₂×C₄²	—	2	1	no	—
136	C₂×C₄²	—	4	1	no	—
137	C₂×C₄²	—	4	1	no	—
138	C₂×C₄²	SD₁₆	1	2	yes	—
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	4	1	no	—
146	C₂×C₁₆	—	4	1	no	—
147	C₂×C₁₆	—	4	1	no	—
148	C₂×C₁₆	—	4	1	no	—
149	C₂×C₁₆	—	4	1	no	—
150	C₂×C₁₆	—	4	1	no	—
151	C₂×C₁₆	—	4	1	no	—
152	C₂×C₁₆	—	4	1	no	—
153	#108	—	2	2	no	—
154	#108	—	2	2	no	—
155	#108	—	2	2	no	—
156	#108	—	2	2	no	—
157	#108	—	4	1	no	—
158	#108	—	4	1	no	—
159	#108	—	4	1	no	—
160	#108	—	4	1	no	—
161	#107	—	2	2	no	—
162	#107	—	2	1	no	—
163	#107	—	2	1	no	—
164	#107	—	2	2	no	—
165	#107	—	2	1	no	—
166	#107	—	2	1	no	—
167	#106	—	4	1	no	—
168	#106	—	4	1	no	—
169	#106	—	4	1	no	—
170	#106	—	4	1	no	—
171	C₄×C₈	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	C₄×C₈	—	4	1	no	—
174	C₄×C₈	—	4	1	no	—
175	C₂⁴	#135	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
185	C₂²×C₄	#121	1	1	yes	characteristic
186	C₂²×C₄	#121	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
188	C₂²×C₄	#123	1	1	yes	characteristic
189	C₂²×C₄	#123	1	1	yes	characteristic
190	C₂²×C₄	#124	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
193	C₂²×C₄	#135	1	1	yes	characteristic
194	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂²×C₄	#137	1	1	yes	characteristic
196	C₂²×C₄	#137	1	1	yes	characteristic
197	C₂²×C₄	#137	1	1	yes	characteristic
198	M₁₆	—	4	2	no	—
199	M₁₆	—	4	2	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	M₁₆	—	8	1	no	—
203	M₁₆	—	8	1	no	—
204	M₁₆	—	8	1	no	—
205	M₁₆	—	8	1	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	8	1	no	—
216	C₂×C₈	—	8	1	no	—
217	C₂×C₈	—	8	1	no	—
218	C₂×C₈	—	8	1	no	—
219	C₂×C₈	#117	1	1	yes	characteristic
220	C₂×C₈	#117	1	1	yes	characteristic
221	C₂×C₈	#117	1	1	yes	characteristic
222	C₂×C₈	C₄×D₄	1	1	yes	characteristic
223	C₂×C₈	C₄×D₄	1	1	yes	characteristic
224	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
225	C₂×C₈	#121	1	1	yes	characteristic
226	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
227	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
228	C₂×C₈	#123	1	1	yes	characteristic
229	C₂×C₈	#123	1	1	yes	characteristic
230	C₂×C₈	#123	1	1	yes	characteristic
231	C₂×C₈	#123	1	1	yes	characteristic
232	C₂×C₈	#124	1	1	yes	characteristic
233	C₂×C₈	#124	1	1	yes	characteristic
234	C₂×C₈	#125	1	1	yes	characteristic
235	C₂×C₈	#125	1	1	yes	characteristic
236	C₂×C₈	#125	1	1	yes	characteristic
237	C₂×C₈	#126	1	1	yes	characteristic
238	C₂×C₈	#126	1	1	yes	characteristic
239	C₂×C₈	#126	1	1	yes	characteristic
240	C₂×C₈	#126	1	1	yes	characteristic
241	C₂×C₈	#126	1	1	yes	characteristic
242	C₂×C₈	#126	1	1	yes	characteristic
243	C₂×C₈	#126	1	1	yes	characteristic
244	C₂×C₈	#126	1	1	yes	characteristic
245	C₂×C₈	#126	1	1	yes	characteristic
246	C₂×C₈	#126	1	1	yes	characteristic
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	8	1	no	—
254	C₂²⋊C₄	—	8	1	no	—
255	C₂²⋊C₄	—	8	1	no	—
256	C₂²⋊C₄	—	8	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	8	1	no	—
262	C₄²	—	8	1	no	—
263	C₄²	—	8	1	no	—
264	C₄²	—	8	1	no	—
265	C₁₆	—	4	2	no	—
266	C₁₆	—	4	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₄×D₈	1	1	yes	characteristic
270	C₂³	#464	1	1	yes	characteristic
271	C₂³	#465	1	1	yes	characteristic
272	C₂³	#479	1	1	yes	characteristic
273	C₂³	#482	1	1	yes	characteristic
274	C₂³	#485	1	1	yes	characteristic
275	C₂³	#488	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	#387	1	1	yes	characteristic
289	C₂×C₄	#388	1	1	yes	characteristic
290	C₂×C₄	#388	1	1	yes	characteristic
291	C₂×C₄	#396	1	1	yes	characteristic
292	C₂×C₄	#397	1	1	yes	characteristic
293	C₂×C₄	#397	1	1	yes	characteristic
294	C₂×C₄	#400	1	1	yes	characteristic
295	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
296	C₂×C₄	#440	1	1	yes	characteristic
297	C₂×C₄	#441	1	1	yes	characteristic
298	C₂×C₄	#459	1	1	yes	characteristic
299	C₂×C₄	#461	1	1	yes	characteristic
300	C₂×C₄	#464	1	1	yes	characteristic
301	C₂×C₄	#466	1	1	yes	characteristic
302	C₂×C₄	#468	1	1	yes	characteristic
303	C₂×C₄	#470	1	1	yes	characteristic
304	C₂×C₄	#474	1	1	yes	characteristic
305	C₂×C₄	#475	1	1	yes	characteristic
306	C₂×C₄	#478	1	1	yes	characteristic
307	C₂×C₄	#480	1	1	yes	characteristic
308	C₂×C₄	#480	1	1	yes	characteristic
309	C₂×C₄	#481	1	1	yes	characteristic
310	C₂×C₄	#483	1	1	yes	characteristic
311	C₂×C₄	#485	1	1	yes	characteristic
312	C₂×C₄	#485	1	1	yes	characteristic
313	C₂×C₄	#486	1	1	yes	characteristic
314	C₂×C₄	#489	1	1	yes	characteristic
315	C₂×C₄	#491	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	—	8	2	no	—
318	C₈	—	8	2	no	—
319	C₈	#383	1	1	yes	characteristic
320	C₈	#387	1	1	yes	characteristic
321	C₈	#387	1	1	yes	characteristic
322	C₈	#388	1	1	yes	characteristic
323	C₈	#397	1	1	yes	characteristic
324	C₈	#400	1	1	yes	characteristic
325	C₈	#400	1	1	yes	characteristic
326	C₈	#401	1	1	yes	characteristic
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	#1918	1	1	yes	characteristic
336	C₂²	#1925	1	1	yes	characteristic
337	C₂²	#1934	1	1	yes	characteristic
338	C₂²	#2066	1	1	yes	characteristic
339	C₂²	#2074	1	1	yes	characteristic
340	C₂²	#2077	1	1	yes	characteristic
341	C₂²	#2101	1	1	yes	characteristic
342	C₂²	#2178	1	2	yes	—
343	C₂²	#2179	1	2	yes	—
344	C₄	—	4	4	no	—
345	C₄	—	16	2	no	—
346	C₄	—	16	1	no	—
347	C₄	—	16	1	no	—
348	C₄	#1443	1	1	yes	characteristic
349	C₄	#1919	1	1	yes	characteristic
350	C₄	#1927	1	1	yes	characteristic
351	C₄	#1935	1	1	yes	characteristic
352	C₄	#2067	1	1	yes	characteristic
353	C₄	#2071	1	1	yes	characteristic
354	C₄	#2075	1	1	yes	characteristic
355	C₄	#2099	1	1	yes	characteristic
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	#11315	1	1	yes	characteristic
361	C₂	#13113	1	2	yes	—
362	C₂	#13134	1	2	yes	—
363	C₂	#13151	1	1	yes	characteristic
364	C₂	#13152	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×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
22. #1380	C₄	semidirect
23. #1380	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1382	2
C₂×#1381	2
#12556	2
C₂×#1374	2
#10859	2