Subgroups — #148705

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1942	C₂	1	1	yes	maximal, characteristic
2	C₂×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×#1380	C₂	1	2	yes	maximal
4	C₂×#1380	C₂	1	2	yes	maximal
5	C₂×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₆	C₂²	1	2	yes	—
7	C₂²×#107	—	2	1	no	—
8	C₂²×#107	C₂²	1	1	yes	characteristic
9	C₂²×#107	C₂²	1	2	yes	—
10	C₂²×#107	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#389	—	2	1	no	—
13	C₂×#384	—	2	1	no	—
14	#1942	—	2	2	no	—
15	#1942	C₄	1	4	yes	complemented
16	#1845	—	2	4	no	—
17	C₂×#340	—	2	1	no	—
18	#1380	—	2	4	no	—
19	#1380	—	2	4	no	—
20	#1374	—	2	2	no	—
21	C₂³×C₈	D₄	1	1	yes	characteristic
22	C₂³×C₈	C₂³	1	1	yes	characteristic
23	C₂²×C₄⋊C₄	—	2	1	no	—
24	C₂²×C₄⋊C₄	—	2	2	no	—
25	C₂²×C₄⋊C₄	—	2	1	no	—
26	C₂²×C₄⋊C₄	—	4	1	no	—
27	C₂²×C₄⋊C₄	—	4	1	no	—
28	C₂²×C₄⋊C₄	—	4	1	no	—
29	C₂²×C₄²	D₄	1	1	yes	characteristic
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	—	2	2	no	—
32	C₂²×C₁₆	—	2	2	no	—
33	C₂²×C₁₆	—	2	2	no	—
34	C₂²×C₁₆	—	2	2	no	—
35	C₂²×C₁₆	—	2	2	no	—
36	C₂²×C₁₆	Q₈	1	4	yes	—
37	C₂×#107	—	2	1	no	—
38	C₂×#107	—	2	1	no	—
39	C₂×#107	—	2	1	no	—
40	C₂×#107	—	2	2	no	—
41	C₂×#107	—	2	2	no	—
42	C₂×#107	—	2	2	no	—
43	C₂×#107	—	2	2	no	—
44	C₂×#107	—	2	2	no	—
45	C₂×#107	—	2	2	no	—
46	C₂×#107	—	2	4	no	—
47	C₂×#107	—	2	2	no	—
48	C₂×#107	—	2	1	no	—
49	C₂×#107	—	4	2	no	—
50	C₂×#107	—	4	2	no	—
51	C₂×#107	C₂×C₄	1	2	yes	—
52	C₂×#107	C₂×C₄	1	2	yes	—
53	C₂×#107	D₄	1	4	yes	—
54	C₂×#107	D₄	1	2	yes	—
55	C₂×#107	Q₈	1	2	yes	—
56	C₂×C₄×C₈	—	2	2	no	—
57	C₂×C₄×C₈	—	2	2	no	—
58	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₈	D₄	1	2	yes	—
60	#389	—	2	4	no	—
61	#389	—	4	2	no	—
62	#384	—	2	4	no	—
63	#384	—	4	2	no	—
64	C₂×#95	—	4	1	no	—
65	#340	—	2	4	no	—
66	C₂³×C₄	—	4	1	no	—
67	C₂³×C₄	—	4	2	no	—
68	C₂³×C₄	—	4	1	no	—
69	C₂³×C₄	D₈	1	1	yes	characteristic
70	C₂³×C₄	D₈	1	1	yes	characteristic
71	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₈	SD₁₆	1	2	yes	—
78	C₂²×C₈	Q₁₆	1	2	yes	—
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₄	1	2	yes	—
82	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₈	C₂×Q₈	1	2	yes	—
85	C₂²×C₈	#41	1	1	yes	characteristic
86	C₂²×C₈	#41	1	1	yes	characteristic
87	C₂²×C₈	#41	1	2	yes	—
88	C₂²×C₈	#41	1	2	yes	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	2	2	no	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	4	1	no	—
95	C₂×C₄⋊C₄	—	4	1	no	—
96	C₂×C₄⋊C₄	—	4	1	no	—
97	C₂×C₄⋊C₄	—	4	1	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₄⋊C₄	—	4	1	no	—
100	C₂×C₄⋊C₄	—	4	2	no	—
101	C₂×C₄⋊C₄	—	4	2	no	—
102	C₂×C₄⋊C₄	—	4	1	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄⋊C₄	—	4	1	no	—
105	C₂×C₄⋊C₄	—	4	4	no	—
106	C₂×C₄⋊C₄	—	4	4	no	—
107	C₂×C₄⋊C₄	—	4	4	no	—
108	C₂×C₄⋊C₄	—	4	1	no	—
109	C₂×C₄⋊C₄	—	4	2	no	—
110	C₂×C₄⋊C₄	—	8	2	no	—
111	C₂×C₄⋊C₄	—	8	2	no	—
112	C₂×C₄⋊C₄	—	8	2	no	—
113	C₂×C₄⋊C₄	—	8	2	no	—
114	C₂×C₄⋊C₄	—	8	2	no	—
115	C₂×C₄⋊C₄	—	8	2	no	—
116	C₂×C₄²	—	2	1	no	—
117	C₂×C₄²	—	2	1	no	—
118	C₂×C₄²	—	2	1	no	—
119	C₂×C₄²	—	2	1	no	—
120	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
121	C₂×C₄²	SD₁₆	1	2	yes	—
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	—	2	4	no	—
124	C₂×C₁₆	—	2	4	no	—
125	C₂×C₁₆	—	4	2	no	—
126	C₂×C₁₆	—	4	2	no	—
127	C₂×C₁₆	—	4	2	no	—
128	C₂×C₁₆	—	4	2	no	—
129	#107	—	2	2	no	—
130	#107	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	—	2	4	no	—
133	#107	—	2	4	no	—
134	#107	—	2	4	no	—
135	#107	—	4	8	no	—
136	#107	—	4	2	no	—
137	#107	—	4	2	no	—
138	#107	—	4	2	no	—
139	#107	C₄⋊C₄	1	4	yes	complemented
140	C₄×C₈	—	2	4	no	—
141	C₄×C₈	—	2	4	no	—
142	#95	—	4	4	no	—
143	C₂⁴	C₂×D₈	1	1	yes	characteristic
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	—	8	1	no	—
158	C₂²×C₄	—	8	1	no	—
159	C₂²×C₄	—	8	1	no	—
160	C₂²×C₄	—	8	1	no	—
161	C₂²×C₄	—	8	1	no	—
162	C₂²×C₄	—	8	2	no	—
163	C₂²×C₄	—	8	1	no	—
164	C₂²×C₄	—	8	2	no	—
165	C₂²×C₄	—	8	1	no	—
166	C₂²×C₄	—	8	2	no	—
167	C₂²×C₄	—	8	1	no	—
168	C₂²×C₄	—	8	2	no	—
169	C₂²×C₄	—	8	2	no	—
170	C₂²×C₄	—	8	2	no	—
171	C₂²×C₄	—	8	1	no	—
172	C₂²×C₄	—	8	1	no	—
173	C₂²×C₄	#102	1	2	yes	—
174	C₂²×C₄	#102	1	2	yes	—
175	C₂²×C₄	D₁₆	1	2	yes	—
176	C₂²×C₄	SD₃₂	1	2	yes	—
177	C₂²×C₄	SD₃₂	1	2	yes	—
178	C₂²×C₄	Q₃₂	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	#120	1	1	yes	characteristic
181	C₂²×C₄	#121	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
183	C₂²×C₄	#123	1	1	yes	characteristic
184	C₂²×C₄	#124	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
186	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
187	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
188	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
189	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂²×C₄	#137	1	1	yes	characteristic
192	C₂²×C₄	#137	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	#103	1	4	yes	—
200	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×C₈	#117	1	1	yes	characteristic
202	C₂×C₈	C₄×D₄	1	2	yes	—
203	C₂×C₈	C₄×Q₈	1	2	yes	—
204	C₂×C₈	#121	1	2	yes	—
205	C₂×C₈	#121	1	1	yes	characteristic
206	C₂×C₈	#121	1	2	yes	—
207	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
208	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
209	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
210	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
211	C₂×C₈	#123	1	2	yes	—
212	C₂×C₈	#125	1	2	yes	—
213	C₂×C₈	#125	1	2	yes	—
214	C₂×C₈	#126	1	2	yes	—
215	C₂×C₈	C₄⋊Q₈	1	2	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	8	8	no	—
225	C₄⋊C₄	—	8	8	no	—
226	C₄⋊C₄	—	8	8	no	—
227	C₄⋊C₄	—	8	1	no	—
228	C₄⋊C₄	—	8	2	no	—
229	C₄⋊C₄	—	8	1	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₁₆	—	4	4	no	—
236	C₂³	C₂×#102	1	1	yes	characteristic
237	C₂³	#447	1	1	yes	characteristic
238	C₂³	#459	1	1	yes	characteristic
239	C₂³	#474	1	1	yes	characteristic
240	C₂³	#480	1	1	yes	characteristic
241	C₂³	#486	1	1	yes	characteristic
242	C₂³	#500	1	1	yes	characteristic
243	C₂³	C₂×D₁₆	1	1	yes	characteristic
244	C₂³	C₂×SD₃₂	1	1	yes	characteristic
245	C₂³	C₂×SD₃₂	1	1	yes	characteristic
246	C₂³	C₂×Q₃₂	1	1	yes	characteristic
247	C₂³	#509	1	1	yes	characteristic
248	C₂³	#509	1	1	yes	characteristic
249	C₂³	#510	1	1	yes	characteristic
250	C₂³	#510	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	16	1	no	—
277	C₂×C₄	—	16	1	no	—
278	C₂×C₄	—	16	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	—	16	1	no	—
281	C₂×C₄	—	16	1	no	—
282	C₂×C₄	—	16	2	no	—
283	C₂×C₄	—	16	2	no	—
284	C₂×C₄	#357	1	4	yes	—
285	C₂×C₄	#358	1	4	yes	—
286	C₂×C₄	#380	1	1	yes	characteristic
287	C₂×C₄	#386	1	1	yes	characteristic
288	C₂×C₄	#391	1	1	yes	characteristic
289	C₂×C₄	#394	1	1	yes	characteristic
290	C₂×C₄	#395	1	1	yes	characteristic
291	C₂×C₄	#398	1	1	yes	characteristic
292	C₂×C₄	#400	1	1	yes	characteristic
293	C₂×C₄	C₂×#103	1	1	yes	characteristic
294	C₂×C₄	#417	1	1	yes	characteristic
295	C₂×C₄	#417	1	1	yes	characteristic
296	C₂×C₄	#448	1	1	yes	characteristic
297	C₂×C₄	#450	1	1	yes	characteristic
298	C₂×C₄	#451	1	1	yes	characteristic
299	C₂×C₄	#460	1	1	yes	characteristic
300	C₂×C₄	#461	1	1	yes	characteristic
301	C₂×C₄	#462	1	1	yes	characteristic
302	C₂×C₄	#475	1	1	yes	characteristic
303	C₂×C₄	#476	1	1	yes	characteristic
304	C₂×C₄	#477	1	1	yes	characteristic
305	C₂×C₄	#481	1	1	yes	characteristic
306	C₂×C₄	#483	1	1	yes	characteristic
307	C₂×C₄	#484	1	1	yes	characteristic
308	C₂×C₄	#487	1	1	yes	characteristic
309	C₂×C₄	#490	1	1	yes	characteristic
310	C₂×C₄	#491	1	1	yes	characteristic
311	C₂×C₄	#498	1	1	yes	characteristic
312	C₂×C₄	#501	1	1	yes	characteristic
313	C₂×C₄	#501	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	#387	1	2	yes	—
316	C₈	#389	1	2	yes	—
317	C₈	#396	1	2	yes	—
318	C₈	#398	1	2	yes	—
319	C₂²	#1802	1	1	yes	characteristic
320	C₂²	#1874	1	1	yes	characteristic
321	C₂²	#1990	1	1	yes	characteristic
322	C₂²	#2041	1	1	yes	characteristic
323	C₂²	#2057	1	1	yes	characteristic
324	C₂²	#2083	1	1	yes	characteristic
325	C₂²	#2099	1	1	yes	characteristic
326	C₂²	C₂×#357	1	1	yes	characteristic
327	C₂²	C₂×#358	1	1	yes	characteristic
328	C₂²	#2139	1	1	yes	characteristic
329	C₂²	#2139	1	1	yes	characteristic
330	C₂²	#2184	1	1	yes	characteristic
331	C₂²	#2185	1	1	yes	characteristic
332	C₂²	#2187	1	1	yes	characteristic
333	C₂²	#2188	1	1	yes	characteristic
334	C₂²	#2206	1	1	yes	characteristic
335	C₂²	#2207	1	1	yes	characteristic
336	C₂²	#2208	1	1	yes	characteristic
337	C₂²	#2209	1	1	yes	characteristic
338	C₂²	#2224	1	1	yes	characteristic
339	C₂²	#2225	1	1	yes	characteristic
340	C₂²	#2226	1	1	yes	characteristic
341	C₂²	#2227	1	1	yes	characteristic
342	C₂²	#2232	1	1	yes	characteristic
343	C₂²	#2233	1	1	yes	characteristic
344	C₂²	#2235	1	1	yes	characteristic
345	C₂²	#2236	1	1	yes	characteristic
346	C₂²	#2240	1	1	yes	characteristic
347	C₂²	#2241	1	1	yes	characteristic
348	C₂²	#2244	1	1	yes	characteristic
349	C₂²	#2245	1	1	yes	characteristic
350	C₂²	#2252	1	1	yes	characteristic
351	C₂²	#2254	1	1	yes	characteristic
352	C₂²	#2255	1	1	yes	characteristic
353	C₂²	#2255	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	2	4	no	—
356	C₄	—	8	4	no	—
357	C₄	—	16	4	no	—
358	C₄	—	16	2	no	—
359	C₄	#1441	1	1	yes	characteristic
360	C₄	#1803	1	1	yes	characteristic
361	C₄	#1872	1	1	yes	characteristic
362	C₄	#1992	1	1	yes	characteristic
363	C₄	#2043	1	1	yes	characteristic
364	C₄	#2058	1	1	yes	characteristic
365	C₄	#2084	1	1	yes	characteristic
366	C₄	#2100	1	1	yes	characteristic
367	C₂	#11207	1	1	yes	characteristic
368	C₂	#12811	1	1	yes	characteristic
369	C₂	#12812	1	1	yes	characteristic
370	C₂	#12991	1	1	yes	characteristic
371	C₂	#12993	1	1	yes	characteristic
372	C₂	#13282	1	1	yes	characteristic
373	C₂	#13284	1	1	yes	characteristic
374	C₂	#13407	1	1	yes	characteristic
375	C₂	#13409	1	1	yes	characteristic
376	C₂	#13451	1	1	yes	characteristic
377	C₂	#13452	1	1	yes	characteristic
378	C₂	#13505	1	1	yes	characteristic
379	C₂	#13506	1	1	yes	characteristic
380	C₂	#13537	1	1	yes	characteristic
381	C₂	#13538	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ 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
15. #1942	C₄	semidirect
139. #107	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1942	2
C₂×#1845	2
C₂×#1380	2
C₂×#1374	2