Subgroups — #149129

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1387	C₂	1	1	yes	maximal, characteristic
2	C₂×#1387	C₂	1	1	yes	maximal, characteristic
3	C₂×#1381	C₂	1	1	yes	maximal, characteristic
4	C₂×#1380	C₂	1	1	yes	maximal, characteristic
5	#12545	C₂	1	1	yes	maximal, characteristic
6	#10870	C₂	1	1	yes	maximal, characteristic
7	#10870	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	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	C₂²×#107	C₂²	1	1	yes	characteristic
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	C₂×#343	—	2	1	no	—
16	C₂×#343	—	2	1	no	—
17	#1736	—	2	1	no	—
18	#1736	—	2	1	no	—
19	#1387	—	2	2	no	—
20	#1387	C₄	1	4	yes	complemented
21	#1381	—	2	2	no	—
22	#1380	—	2	2	no	—
23	#1374	—	2	2	no	—
24	C₂²×M₁₆	—	2	1	no	—
25	C₂²×M₁₆	—	2	1	no	—
26	C₂²×M₁₆	D₄	1	1	yes	characteristic
27	C₂³×C₈	C₂³	1	1	yes	characteristic
28	C₂×#117	D₄	1	1	yes	characteristic
29	C₂²×C₄⋊C₄	—	2	1	no	—
30	C₂²×C₄⋊C₄	—	2	1	no	—
31	C₂×M₃₂	—	2	1	no	—
32	C₂×M₃₂	—	2	1	no	—
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	—	2	1	no	—
35	C₂×M₃₂	—	2	1	no	—
36	C₂×M₃₂	C₂×C₄	1	2	yes	—
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	D₄	1	2	yes	—
45	C₂×#108	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
56	C₂×#108	D₄	1	2	yes	—
57	C₂×#107	—	2	1	no	—
58	C₂×#107	—	2	1	no	—
59	C₂×#107	—	2	1	no	—
60	C₂×#107	—	2	1	no	—
61	C₂×#107	—	2	1	no	—
62	C₂×#107	—	2	1	no	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	—	2	1	no	—
65	C₂×#107	—	2	1	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	C₂×C₄	1	2	yes	—
68	C₂×#107	Q₈	1	2	yes	—
69	#405	—	2	1	no	—
70	#405	—	2	1	no	—
71	#405	—	2	1	no	—
72	#405	—	2	1	no	—
73	C₂×#97	—	2	1	no	—
74	C₂×C₄×C₈	—	2	1	no	—
75	#343	—	4	2	no	—
76	#343	—	4	2	no	—
77	#340	—	4	1	no	—
78	#340	—	4	1	no	—
79	#340	—	4	1	no	—
80	#340	—	4	1	no	—
81	C₂³×C₄	—	4	1	no	—
82	C₂³×C₄	—	4	1	no	—
83	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×M₁₆	—	2	1	no	—
85	C₂×M₁₆	—	2	1	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	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	C₂×M₁₆	—	4	1	no	—
96	C₂×M₁₆	—	4	1	no	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	SD₁₆	1	2	yes	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	4	1	no	—
104	C₂²×C₈	—	4	1	no	—
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×Q₈	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	C₂²×C₈	#41	1	1	yes	characteristic
119	#117	—	2	1	no	—
120	#117	—	2	1	no	—
121	#117	—	2	1	no	—
122	#117	—	2	1	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	4	1	no	—
126	C₂×C₄⋊C₄	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	1	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
136	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₄²	—	2	1	no	—
139	M₃₂	—	2	2	no	—
140	M₃₂	—	2	2	no	—
141	M₃₂	—	4	1	no	—
142	M₃₂	—	4	1	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	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	2	no	—
163	#107	—	2	2	no	—
164	#107	—	2	2	no	—
165	#107	—	4	1	no	—
166	#107	—	4	1	no	—
167	#107	—	4	1	no	—
168	#107	—	4	1	no	—
169	#97	—	4	1	no	—
170	#97	—	4	1	no	—
171	C₄×C₈	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	C₂⁴	#135	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	8	1	no	—
179	C₂²×C₄	—	8	1	no	—
180	C₂²×C₄	—	8	1	no	—
181	C₂²×C₄	—	8	1	no	—
182	C₂²×C₄	—	16	1	no	—
183	C₂²×C₄	—	16	1	no	—
184	C₂²×C₄	—	16	1	no	—
185	C₂²×C₄	—	16	1	no	—
186	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
187	C₂²×C₄	#121	1	1	yes	characteristic
188	C₂²×C₄	#121	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	#123	1	1	yes	characteristic
191	C₂²×C₄	#123	1	1	yes	characteristic
192	C₂²×C₄	#124	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
194	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
195	C₂²×C₄	#135	1	1	yes	characteristic
196	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂²×C₄	#137	1	1	yes	characteristic
198	C₂²×C₄	#137	1	1	yes	characteristic
199	C₂²×C₄	#137	1	1	yes	characteristic
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	1	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	8	1	no	—
209	M₁₆	—	8	1	no	—
210	M₁₆	—	8	1	no	—
211	M₁₆	—	8	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	8	1	no	—
218	C₂×C₈	—	8	1	no	—
219	C₂×C₈	—	8	1	no	—
220	C₂×C₈	—	8	1	no	—
221	C₂×C₈	#117	1	1	yes	characteristic
222	C₂×C₈	#117	1	1	yes	characteristic
223	C₂×C₈	#117	1	1	yes	characteristic
224	C₂×C₈	C₄×D₄	1	1	yes	characteristic
225	C₂×C₈	C₄×D₄	1	1	yes	characteristic
226	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
227	C₂×C₈	#121	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₈	#123	1	1	yes	characteristic
231	C₂×C₈	#123	1	1	yes	characteristic
232	C₂×C₈	#123	1	1	yes	characteristic
233	C₂×C₈	#123	1	1	yes	characteristic
234	C₂×C₈	#124	1	1	yes	characteristic
235	C₂×C₈	#124	1	1	yes	characteristic
236	C₂×C₈	#125	1	1	yes	characteristic
237	C₂×C₈	#125	1	1	yes	characteristic
238	C₂×C₈	#125	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	C₂×C₈	#126	1	1	yes	characteristic
248	C₂×C₈	#126	1	1	yes	characteristic
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	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	—	8	1	no	—
258	C₄⋊C₄	—	8	1	no	—
259	C₄⋊C₄	—	8	1	no	—
260	C₄⋊C₄	—	8	1	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₄²	—	4	1	no	—
265	C₁₆	—	4	2	no	—
266	C₁₆	—	4	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	C₄×SD₁₆	1	1	yes	characteristic
272	C₂³	#464	1	1	yes	characteristic
273	C₂³	#466	1	1	yes	characteristic
274	C₂³	#478	1	1	yes	characteristic
275	C₂³	#485	1	1	yes	characteristic
276	C₂³	#485	1	1	yes	characteristic
277	C₂³	#488	1	1	yes	characteristic
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	—	16	2	no	—
288	C₂×C₄	—	16	1	no	—
289	C₂×C₄	—	16	1	no	—
290	C₂×C₄	—	16	1	no	—
291	C₂×C₄	—	16	1	no	—
292	C₂×C₄	—	16	2	no	—
293	C₂×C₄	—	16	2	no	—
294	C₂×C₄	#387	1	1	yes	characteristic
295	C₂×C₄	#388	1	1	yes	characteristic
296	C₂×C₄	#388	1	1	yes	characteristic
297	C₂×C₄	#396	1	1	yes	characteristic
298	C₂×C₄	#397	1	1	yes	characteristic
299	C₂×C₄	#397	1	1	yes	characteristic
300	C₂×C₄	#400	1	1	yes	characteristic
301	C₂×C₄	C₄×D₈	1	1	yes	characteristic
302	C₂×C₄	#440	1	1	yes	characteristic
303	C₂×C₄	#441	1	1	yes	characteristic
304	C₂×C₄	#459	1	1	yes	characteristic
305	C₂×C₄	#461	1	1	yes	characteristic
306	C₂×C₄	#464	1	1	yes	characteristic
307	C₂×C₄	#465	1	1	yes	characteristic
308	C₂×C₄	#468	1	1	yes	characteristic
309	C₂×C₄	#470	1	1	yes	characteristic
310	C₂×C₄	#474	1	1	yes	characteristic
311	C₂×C₄	#475	1	1	yes	characteristic
312	C₂×C₄	#479	1	1	yes	characteristic
313	C₂×C₄	#480	1	1	yes	characteristic
314	C₂×C₄	#480	1	1	yes	characteristic
315	C₂×C₄	#481	1	1	yes	characteristic
316	C₂×C₄	#482	1	1	yes	characteristic
317	C₂×C₄	#483	1	1	yes	characteristic
318	C₂×C₄	#485	1	1	yes	characteristic
319	C₂×C₄	#486	1	1	yes	characteristic
320	C₂×C₄	#489	1	1	yes	characteristic
321	C₂×C₄	#491	1	1	yes	characteristic
322	C₈	—	4	2	no	—
323	C₈	—	8	2	no	—
324	C₈	—	8	2	no	—
325	C₈	#383	1	1	yes	characteristic
326	C₈	#387	1	1	yes	characteristic
327	C₈	#387	1	1	yes	characteristic
328	C₈	#388	1	1	yes	characteristic
329	C₈	#397	1	1	yes	characteristic
330	C₈	#400	1	1	yes	characteristic
331	C₈	#400	1	1	yes	characteristic
332	C₈	#401	1	1	yes	characteristic
333	C₂²	—	2	1	no	—
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	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	#1919	1	1	yes	characteristic
346	C₂²	#1927	1	1	yes	characteristic
347	C₂²	#1934	1	1	yes	characteristic
348	C₂²	#2067	1	1	yes	characteristic
349	C₂²	#2074	1	1	yes	characteristic
350	C₂²	#2075	1	1	yes	characteristic
351	C₂²	#2101	1	1	yes	characteristic
352	C₂²	#2221	1	1	yes	characteristic
353	C₂²	#2222	1	1	yes	characteristic
354	C₂²	#2223	1	1	yes	characteristic
355	C₂²	#2223	1	1	yes	characteristic
356	C₄	—	8	1	no	—
357	C₄	—	8	1	no	—
358	C₄	—	16	2	no	—
359	C₄	—	16	2	no	—
360	C₄	#1443	1	1	yes	characteristic
361	C₄	#1918	1	1	yes	characteristic
362	C₄	#1925	1	1	yes	characteristic
363	C₄	#1935	1	1	yes	characteristic
364	C₄	#2066	1	1	yes	characteristic
365	C₄	#2071	1	1	yes	characteristic
366	C₄	#2077	1	1	yes	characteristic
367	C₄	#2099	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	#11315	1	1	yes	characteristic
373	C₂	#13144	1	1	yes	characteristic
374	C₂	#13145	1	1	yes	characteristic
375	C₂	#13470	1	1	yes	characteristic
376	C₂	#13471	1	1	yes	characteristic
377	C₂	#13495	1	1	yes	characteristic
378	C₂	#13496	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
20. #1387	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1387	2
C₂×#1381	2
C₂×#1380	2
#12545	2
#10870	2