Subgroups — #10948663

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#509	C₂	1	1	yes	maximal, characteristic
2	#83910	C₂	1	1	yes	maximal, characteristic, complemented
3	#83823	C₂	1	1	yes	maximal, characteristic, complemented
4	#72076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#4563	C₂	1	1	yes	maximal, characteristic, complemented
6	#71515	C₂	1	8	yes	maximal, complemented
7	#71427	C₂	1	1	yes	maximal, characteristic, complemented
8	#71425	C₂	1	1	yes	maximal, characteristic, complemented
9	#33471	—	3	1	no	maximal, Sylow
10	C₂×C₃×#135	C₂²	1	1	yes	characteristic
11	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
12	#5819	—	2	1	no	—
13	#5798	—	2	1	no	—
14	C₃×#509	C₄	1	4	yes	complemented
15	C₃×#509	C₂²	1	4	yes	—
16	C₂×C₃×SD₃₂	C₄	1	2	yes	complemented
17	C₂×C₃×D₁₆	C₄	1	2	yes	complemented
18	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
19	#5236	—	2	1	no	—
20	#5215	—	2	1	no	—
21	#5197	C₂²	1	4	yes	complemented
22	#5196	—	2	2	no	—
23	#5152	C₂²	1	4	yes	complemented
24	Dic₃×D₈	—	2	2	no	—
25	C₂×#819	C₂²	1	1	yes	characteristic
26	#5106	C₂²	1	1	yes	characteristic
27	#5101	C₂²	1	1	yes	characteristic
28	C₂×#812	C₂²	1	1	yes	characteristic
29	#4566	C₂²	1	4	yes	complemented
30	#4563	C₂²	1	4	yes	complemented
31	#4513	C₂²	1	4	yes	complemented
32	#4512	C₂²	1	4	yes	complemented
33	C₂×#509	D₃	1	1	yes	characteristic
34	#2957	—	3	1	no	—
35	#2948	—	3	1	no	—
36	AGL(1,ℤ/16)	—	3	8	no	—
37	#2155	—	3	1	no	—
38	C₂×#361	—	3	1	no	—
39	#2145	—	3	1	no	—
40	#2110	—	3	1	no	—
41	C₂×C₃×#41	—	2	1	no	—
42	C₂²×C₃×D₄	—	2	1	no	—
43	C₃×#135	—	2	1	no	—
44	C₃×#135	—	2	1	no	—
45	C₃×#135	C₂×C₄	1	2	yes	complemented
46	C₃×#135	C₂³	1	2	yes	—
47	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
48	C₂×C₃×SD₁₆	—	2	1	no	—
49	C₂×C₃×D₈	—	2	1	no	—
50	C₂×C₃×D₈	—	2	1	no	—
51	C₂×C₃×D₈	—	2	1	no	—
52	C₂×C₃×D₈	C₂×C₄	1	2	yes	—
53	C₂×C₃×D₈	C₂×C₄	1	2	yes	—
54	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂×C₃×D₈	C₂³	1	2	yes	—
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₂×#217	—	4	2	no	—
58	#948	—	2	4	no	—
59	#948	—	4	2	no	—
60	D₄×Dic₃	—	2	4	no	—
61	D₄×Dic₃	—	4	2	no	—
62	#926	—	2	1	no	—
63	#926	D₄	1	1	yes	characteristic
64	C₂×#208	—	2	1	no	—
65	C₂×#208	—	4	2	no	—
66	C₂×#208	D₄	1	1	yes	characteristic
67	C₃×SD₃₂	C₂×C₄	1	8	yes	complemented
68	C₃×D₁₆	C₂×C₄	1	8	yes	complemented
69	C₃×M₃₂	C₂×C₄	1	2	yes	complemented
70	C₃×M₃₂	C₂³	1	2	yes	—
71	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	complemented
72	#837	—	2	4	no	—
73	#832	—	2	4	no	—
74	#819	—	2	1	no	—
75	#819	C₂³	1	2	yes	—
76	#818	—	2	1	no	—
77	#817	C₂³	1	2	yes	—
78	#815	C₂³	1	2	yes	—
79	C₈×Dic₃	D₄	1	2	yes	—
80	#812	—	2	1	no	—
81	#812	C₂³	1	2	yes	—
82	C₂×C₃⋊C₁₆	—	2	1	no	—
83	C₂×#135	C₂×D₃	1	1	yes	characteristic
84	C₂²×D₈	C₂×D₃	1	1	yes	characteristic
85	#568	—	6	1	no	—
86	#518	—	6	1	no	—
87	#509	Dic₃	1	4	yes	complemented
88	#509	C₂×D₃	1	4	yes	—
89	C₂×SD₃₂	Dic₃	1	2	yes	complemented
90	C₂×D₁₆	Dic₃	1	2	yes	complemented
91	C₂×M₃₂	—	3	1	no	—
92	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
93	#444	—	3	4	no	—
94	#443	—	6	2	no	—
95	#442	—	3	4	no	—
96	C₄×D₈	—	6	2	no	—
97	C₂×#108	—	3	1	no	—
98	#427	—	3	1	no	—
99	#421	—	6	1	no	—
100	#418	—	6	1	no	—
101	#405	—	3	1	no	—
102	#365	—	3	4	no	—
103	#361	—	3	4	no	—
104	#360	—	3	4	no	—
105	#347	—	3	4	no	—
106	C₂⁴×C₃	—	4	1	no	—
107	C₃×#41	—	2	2	no	—
108	C₃×#41	—	2	2	no	—
109	C₃×#41	—	4	1	no	—
110	C₃×#41	—	4	1	no	—
111	C₂×C₃×Q₈	—	2	1	no	—
112	C₂×C₃×D₄	—	2	2	no	—
113	C₂×C₃×D₄	—	2	2	no	—
114	C₂×C₃×D₄	—	2	1	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	—	4	1	no	—
117	C₂×C₃×D₄	—	4	1	no	—
118	C₂×C₃×D₄	—	4	1	no	—
119	C₂×C₃×D₄	—	4	1	no	—
120	C₂²×C₃×C₄	—	4	1	no	—
121	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
122	C₂²×Dic₃	—	4	4	no	—
123	C₃×Q₁₆	—	2	1	no	—
124	C₃×Q₁₆	C₂²×C₄	1	2	yes	—
125	C₃×SD₁₆	—	2	2	no	—
126	C₃×SD₁₆	—	2	2	no	—
127	C₃×D₈	—	2	2	no	—
128	C₃×D₈	—	2	2	no	—
129	C₃×D₈	—	2	1	no	—
130	C₃×D₈	—	2	2	no	—
131	C₃×D₈	C₂²×C₄	1	2	yes	—
132	C₃×D₈	C₂²×C₄	1	4	yes	—
133	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	#41	1	1	yes	characteristic
137	C₂×C₃×C₈	#41	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
139	#217	—	2	1	no	—
140	#217	—	4	4	no	—
141	#217	—	4	1	no	—
142	C₄⋊Dic₃	—	2	1	no	—
143	C₄⋊Dic₃	—	2	2	no	—
144	C₄×Dic₃	—	4	1	no	—
145	C₄×Dic₃	C₂×D₄	1	2	yes	—
146	#208	—	2	1	no	—
147	#208	—	2	1	no	—
148	#208	—	2	2	no	—
149	#208	—	4	4	no	—
150	#208	—	4	1	no	—
151	C₂×C₃⋊C₈	—	4	4	no	—
152	C₂×C₃⋊C₈	—	4	1	no	—
153	C₂×C₃⋊C₈	C₂×D₄	1	2	yes	—
154	C₃×C₁₆	C₂²×C₄	1	4	yes	—
155	C₃⋊C₁₆	—	2	2	no	—
156	C₂×#41	—	2	1	no	—
157	C₂²×D₄	—	2	1	no	—
158	#135	—	2	1	no	—
159	#135	—	2	1	no	—
160	#135	C₂×Dic₃	1	2	yes	complemented
161	#135	C₂²×D₃	1	2	yes	—
162	C₂×Q₁₆	C₂×Dic₃	1	1	yes	characteristic, complemented
163	C₂×SD₁₆	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	C₂×Dic₃	1	2	yes	—
168	C₂×D₈	C₂×Dic₃	1	2	yes	—
169	C₂×D₈	C₂×Dic₃	1	1	yes	characteristic, complemented
170	C₂×D₈	C₂²×D₃	1	2	yes	—
171	#131	—	6	4	no	—
172	#131	—	12	2	no	—
173	C₂×M₁₆	—	3	1	no	—
174	C₂×M₁₆	—	6	1	no	—
175	C₂×M₁₆	—	12	2	no	—
176	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
177	C₄×D₄	—	6	4	no	—
178	C₄×D₄	—	12	2	no	—
179	#117	—	3	1	no	—
180	#117	—	6	1	no	—
181	C₂×C₂²⋊C₄	—	12	2	no	—
182	SD₃₂	C₂×Dic₃	1	8	yes	complemented
183	D₁₆	C₂×Dic₃	1	8	yes	complemented
184	M₃₂	—	3	2	no	—
185	M₃₂	—	6	1	no	—
186	M₃₂	C₂×Dic₃	1	2	yes	complemented
187	M₃₂	C₂²×D₃	1	2	yes	—
188	C₂×C₁₆	—	6	1	no	—
189	C₂×C₁₆	C₂×Dic₃	1	2	yes	complemented
190	#108	—	3	2	no	—
191	#108	—	6	1	no	—
192	#107	—	6	1	no	—
193	#106	—	3	2	no	—
194	#104	—	6	4	no	—
195	#102	—	6	4	no	—
196	#97	—	3	2	no	—
197	C₄×C₈	—	3	2	no	—
198	C₂³×C₃	—	4	2	no	—
199	C₂³×C₃	—	4	1	no	—
200	C₂³×C₃	—	4	2	no	—
201	C₂³×C₃	—	4	2	no	—
202	C₂³×C₃	—	16	2	no	—
203	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₃×Q₈	—	2	2	no	—
205	C₃×Q₈	—	4	1	no	—
206	C₃×D₄	—	2	4	no	—
207	C₃×D₄	—	2	2	no	—
208	C₃×D₄	—	4	2	no	—
209	C₃×D₄	—	4	2	no	—
210	C₃×D₄	—	4	2	no	—
211	C₃×D₄	—	4	2	no	—
212	C₃×D₄	—	4	2	no	—
213	C₃×D₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
223	C₂×Dic₃	—	2	2	no	—
224	C₂×Dic₃	—	4	2	no	—
225	C₂×Dic₃	—	16	4	no	—
226	C₃×C₈	C₄×D₄	1	1	yes	characteristic
227	C₃×C₈	C₄×D₄	1	1	yes	characteristic
228	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
229	C₃×C₈	C₂×#41	1	1	yes	characteristic
230	C₃⋊C₈	—	2	2	no	—
231	C₃⋊C₈	—	4	2	no	—
232	C₂⁴	—	4	1	no	—
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	12	4	no	—
248	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
249	Q₁₆	—	2	1	no	—
250	Q₁₆	C₂²×Dic₃	1	2	yes	—
251	SD₁₆	—	2	2	no	—
252	SD₁₆	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	2	no	—
257	D₈	C₂²×Dic₃	1	2	yes	—
258	D₈	C₂²×Dic₃	1	4	yes	—
259	M₁₆	—	6	1	no	—
260	M₁₆	—	6	1	no	—
261	M₁₆	—	6	2	no	—
262	M₁₆	—	12	4	no	—
263	M₁₆	—	12	1	no	—
264	C₂×C₈	—	3	2	no	—
265	C₂×C₈	—	12	4	no	—
266	C₂×C₈	—	12	1	no	—
267	C₂×C₈	D₃×D₄	1	1	yes	characteristic
268	C₂×C₈	#237	1	1	yes	characteristic
269	C₂×C₈	#237	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
272	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
273	C₄⋊C₄	—	6	1	no	—
274	C₄⋊C₄	—	6	2	no	—
275	C₂²⋊C₄	—	6	1	no	—
276	C₂²⋊C₄	—	12	4	no	—
277	C₂²⋊C₄	—	12	1	no	—
278	C₄²	—	3	2	no	—
279	C₄²	—	12	1	no	—
280	C₁₆	—	6	2	no	—
281	C₁₆	C₂²×Dic₃	1	4	yes	—
282	C₂²×C₃	—	4	1	no	—
283	C₂²×C₃	—	4	2	no	—
284	C₂²×C₃	—	4	4	no	—
285	C₂²×C₃	—	8	4	no	—
286	C₂²×C₃	—	16	1	no	—
287	C₂²×C₃	—	16	2	no	—
288	C₂²×C₃	—	16	2	no	—
289	C₂²×C₃	#442	1	1	yes	characteristic
290	C₂²×C₃	#442	1	1	yes	characteristic
291	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
292	C₃×C₄	—	4	2	no	—
293	C₃×C₄	#442	1	1	yes	characteristic
294	C₃×C₄	#442	1	1	yes	characteristic
295	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
296	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
297	Dic₃	—	4	2	no	—
298	Dic₃	—	8	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	1	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	16	2	no	—
304	C₂³	#911	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	4	1	no	—
307	D₄	—	2	4	no	—
308	D₄	—	2	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	6	2	no	—
319	C₂×C₄	—	12	2	no	—
320	C₂×C₄	—	48	4	no	—
321	C₂×C₄	#911	1	1	yes	characteristic
322	C₂×C₄	#911	1	1	yes	characteristic
323	C₂×C₄	#911	1	1	yes	characteristic
324	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
325	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
326	C₂×C₄	C₂×D₃×D₄	1	1	yes	characteristic
327	C₈	—	6	2	no	—
328	C₈	—	12	2	no	—
329	C₈	D₄×Dic₃	1	1	yes	characteristic
330	C₈	D₄×Dic₃	1	1	yes	characteristic
331	C₈	C₂×#237	1	1	yes	characteristic
332	C₈	C₂³×Dic₃	1	1	yes	characteristic
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	1	no	—
336	C₂×C₃	—	8	4	no	—
337	C₂×C₃	—	8	2	no	—
338	C₂×C₃	C₂×#442	1	1	yes	characteristic
339	C₂²	—	4	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	2	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	16	1	no	—
346	C₂²	#5152	1	1	yes	characteristic
347	C₂²	#5152	1	1	yes	characteristic
348	C₂²	C₂×#911	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	12	2	no	—
351	C₄	—	24	2	no	—
352	C₄	#5152	1	1	yes	characteristic
353	C₄	#5152	1	1	yes	characteristic
354	C₄	C₂×#911	1	1	yes	characteristic
355	C₄	C₂×D₄×Dic₃	1	1	yes	characteristic
356	C₃	#33471	1	1	yes	characteristic, complemented, Sylow
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#5152	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#5152
commutator	C₃×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₃
Fitting, prime radical	C₂×C₃×#509	C₂
socle	C₂×C₃	C₂×#442
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×#509	D₃
hypocenter	C₃	#33471
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#509

Complements

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

N	H ≅ G/N	type
2. #83910	C₂	semidirect
3. #83823	C₂	semidirect
4. #72076	C₂	semidirect
5. C₂×#4563	C₂	semidirect
6. #71515	C₂	semidirect
7. #71427	C₂	semidirect
8. #71425	C₂	semidirect
14. C₃×#509	C₄	semidirect
16. C₂×C₃×SD₃₂	C₄	semidirect
17. C₂×C₃×D₁₆	C₄	semidirect
21. #5197	C₂²	semidirect
23. #5152	C₂²	semidirect
29. #4566	C₂²	semidirect
30. #4563	C₂²	semidirect
31. #4513	C₂²	semidirect
32. #4512	C₂²	semidirect
45. C₃×#135	C₂×C₄	semidirect
47. C₂×C₃×Q₁₆	C₂×C₄	semidirect
54. C₂×C₃×D₈	C₂×C₄	semidirect
67. C₃×SD₃₂	C₂×C₄	semidirect
68. C₃×D₁₆	C₂×C₄	semidirect
69. C₃×M₃₂	C₂×C₄	semidirect
71. C₂×C₃×C₁₆	C₂×C₄	semidirect
87. #509	Dic₃	semidirect
89. C₂×SD₃₂	Dic₃	semidirect
90. C₂×D₁₆	Dic₃	semidirect
160. #135	C₂×Dic₃	semidirect
162. C₂×Q₁₆	C₂×Dic₃	semidirect
169. C₂×D₈	C₂×Dic₃	semidirect
182. SD₃₂	C₂×Dic₃	semidirect
183. D₁₆	C₂×Dic₃	semidirect
186. M₃₂	C₂×Dic₃	semidirect
189. C₂×C₁₆	C₂×Dic₃	semidirect
356. C₃	#33471	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#509	2
#83910	2
#83823	2
#72076	2
C₂×#4563	2
#71515	2
#71427	2
#71425	2
#33471	3