Subgroups — #10948432

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	#83792	C₂	1	1	yes	maximal, characteristic
3	#82007	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4561	C₂	1	1	yes	maximal, characteristic, complemented
5	#71453	C₂	1	4	yes	maximal, complemented
6	#71452	C₂	1	4	yes	maximal, complemented
7	C₂×#4515	C₂	1	1	yes	maximal, characteristic, complemented
8	#71423	C₂	1	1	yes	maximal, characteristic, complemented
9	#71419	C₂	1	1	yes	maximal, characteristic, complemented
10	#33442	—	3	1	no	maximal, Sylow
11	C₂×C₃×#131	C₂²	1	1	yes	characteristic
12	#5822	C₄	1	1	yes	characteristic
13	C₂×#901	C₂²	1	1	yes	characteristic
14	C₃×#504	—	2	2	no	—
15	C₃×#504	C₂²	1	4	yes	complemented
16	C₂×C₃×M₃₂	—	2	2	no	—
17	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
19	#5236	C₄	1	1	yes	characteristic
20	#5145	C₄	1	4	yes	—
21	#5144	C₂²	1	4	yes	—
22	#5139	C₄	1	1	yes	characteristic
23	#5124	C₄	1	1	yes	characteristic
24	#5101	C₂²	1	1	yes	characteristic
25	C₂×#812	C₂²	1	1	yes	characteristic
26	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
27	#4907	—	2	2	no	—
28	D₃×M₃₂	—	2	2	no	—
29	#4901	C₂²	1	4	yes	complemented
30	#4561	C₂²	1	2	yes	complemented
31	#4561	C₂²	1	2	yes	complemented
32	#4515	C₂²	1	2	yes	complemented
33	#4515	C₂²	1	2	yes	complemented
34	#4511	C₂²	1	4	yes	complemented
35	#4502	C₂²	1	2	yes	complemented
36	Dic₃×C₁₆	C₂²	1	2	yes	complemented
37	#3407	—	3	1	no	—
38	#3407	D₃	1	1	yes	characteristic, complemented
39	#2955	—	3	1	no	—
40	#2171	—	3	4	no	—
41	#2170	—	3	4	no	—
42	#2153	—	3	1	no	—
43	C₂×#350	—	3	1	no	—
44	C₂×#350	—	3	1	no	—
45	#2108	—	3	1	no	—
46	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
47	C₂×#235	C₂×C₄	1	1	yes	characteristic
48	C₃×#131	—	2	1	no	—
49	C₃×#131	—	2	1	no	—
50	C₃×#131	C₂×C₄	1	2	yes	—
51	C₃×#131	C₂³	1	2	yes	—
52	C₂×C₃×M₁₆	—	2	1	no	—
53	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₈	—	2	1	no	—
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	#951	C₈	1	2	yes	—
57	#950	C₂×C₄	1	4	yes	—
58	#949	C₈	1	2	yes	—
59	#942	C₈	1	2	yes	—
60	#932	C₈	1	2	yes	—
61	#926	C₂×C₄	1	1	yes	characteristic
62	C₂×#208	C₂×C₄	1	1	yes	characteristic
63	C₂×#208	C₂×C₄	1	1	yes	characteristic
64	#901	—	2	1	no	—
65	#901	—	2	1	no	—
66	#901	C₂×C₄	1	2	yes	—
67	#901	C₂³	1	2	yes	—
68	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
69	C₂×D₃×C₈	—	2	1	no	—
70	C₃×M₃₂	—	2	1	no	—
71	C₃×M₃₂	—	2	4	no	—
72	C₃×M₃₂	D₄	1	2	yes	—
73	C₃×M₃₂	D₄	1	2	yes	—
74	C₃×M₃₂	C₂³	1	2	yes	—
75	C₂×C₃×C₁₆	—	2	1	no	—
76	C₂×C₃×C₁₆	—	2	4	no	—
77	C₂×C₃×C₁₆	C₂³	1	2	yes	—
78	#837	C₂×C₄	1	4	yes	—
79	#825	C₂×C₄	1	4	yes	—
80	#822	C₂×C₄	1	4	yes	—
81	#815	C₂×C₄	1	2	yes	—
82	C₈×Dic₃	C₂³	1	2	yes	—
83	#812	—	2	1	no	—
84	#812	C₂³	1	2	yes	—
85	C₂×C₃⋊C₁₆	—	2	1	no	—
86	C₂×C₃⋊C₁₆	—	2	1	no	—
87	C₂×C₃⋊C₁₆	—	2	1	no	—
88	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
90	#798	—	2	4	no	—
91	D₃×C₁₆	—	2	4	no	—
92	#575	—	3	1	no	—
93	C₂×#131	—	3	1	no	—
94	C₂×#131	C₂×D₃	1	1	yes	characteristic
95	#504	—	2	2	no	—
96	#504	—	3	4	no	—
97	#504	—	6	2	no	—
98	#504	C₂×D₃	1	4	yes	complemented
99	C₂×M₃₂	—	2	2	no	—
100	C₂×M₃₂	—	3	1	no	—
101	C₂×M₃₂	—	6	2	no	—
102	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
103	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
104	C₂²×C₁₆	—	3	1	no	—
105	#444	—	3	4	no	—
106	#443	—	3	4	no	—
107	#430	—	3	1	no	—
108	#421	—	3	1	no	—
109	#421	—	3	1	no	—
110	#405	—	3	1	no	—
111	#364	—	3	4	no	—
112	#350	—	3	2	no	—
113	#350	—	3	2	no	—
114	#350	—	3	2	no	—
115	#350	—	3	2	no	—
116	#346	—	3	2	no	—
117	C₄×C₁₆	—	3	2	no	—
118	C₃×#41	—	2	1	no	—
119	C₃×#41	—	2	1	no	—
120	C₃×#41	C₂×C₈	1	2	yes	—
121	C₃×#41	C₂²×C₄	1	2	yes	—
122	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
127	C₂×#67	—	2	1	no	—
128	#235	—	2	1	no	—
129	#235	—	2	1	no	—
130	#235	C₂×C₈	1	2	yes	—
131	#235	C₂²×C₄	1	2	yes	—
132	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
133	C₂×C₄×D₃	—	2	1	no	—
134	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
135	C₃×M₁₆	—	2	1	no	—
136	C₃×M₁₆	—	2	2	no	—
137	C₃×M₁₆	—	2	2	no	—
138	C₃×M₁₆	C₂²×C₄	1	2	yes	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₈	#41	1	1	yes	characteristic
146	C₂×C₃×C₈	#41	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
148	#217	—	2	1	no	—
149	#216	C₂×C₈	1	4	yes	—
150	#215	C₂×C₈	1	4	yes	—
151	#214	C₂×C₈	1	4	yes	—
152	#213	C₂×C₈	1	4	yes	—
153	C₄⋊Dic₃	—	2	1	no	—
154	C₄×Dic₃	C₂²×C₄	1	2	yes	—
155	#208	—	2	1	no	—
156	#208	C₂×C₈	1	2	yes	—
157	#208	C₂×C₈	1	2	yes	—
158	#208	C₂²×C₄	1	2	yes	—
159	C₂×C₃⋊C₈	—	2	1	no	—
160	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
161	ΓL(1,𝔽₂₅)	—	2	2	no	—
162	ΓL(1,𝔽₂₅)	—	2	2	no	—
163	D₃×C₈	—	2	2	no	—
164	D₃×C₈	—	2	2	no	—
165	C₃×C₁₆	—	2	2	no	—
166	C₃×C₁₆	C₂×D₄	1	4	yes	—
167	C₃⋊C₁₆	—	2	2	no	—
168	C₃⋊C₁₆	—	2	1	no	—
169	C₃⋊C₁₆	—	2	1	no	—
170	C₂×#41	—	3	1	no	—
171	C₂×#41	C₄×D₃	1	1	yes	characteristic
172	#137	—	3	2	no	—
173	#137	—	3	2	no	—
174	AGL(1,ℤ/8)	—	3	2	no	—
175	AGL(1,ℤ/8)	—	3	2	no	—
176	#135	—	3	4	no	—
177	#131	—	2	1	no	—
178	#131	—	2	1	no	—
179	#131	—	3	2	no	—
180	#131	—	3	2	no	—
181	#131	—	6	1	no	—
182	#131	—	6	1	no	—
183	#131	C₄×D₃	1	2	yes	—
184	#131	C₂²×D₃	1	2	yes	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	3	1	no	—
187	C₂×M₁₆	—	3	1	no	—
188	C₂×M₁₆	—	6	1	no	—
189	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	6	1	no	—
192	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
193	#117	—	3	1	no	—
194	M₃₂	—	2	1	no	—
195	M₃₂	—	2	4	no	—
196	M₃₂	—	3	2	no	—
197	M₃₂	—	6	1	no	—
198	M₃₂	—	6	4	no	—
199	M₃₂	#67	1	2	yes	—
200	M₃₂	#67	1	2	yes	—
201	M₃₂	C₂²×D₃	1	2	yes	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	2	4	no	—
204	C₂×C₁₆	—	3	1	no	—
205	C₂×C₁₆	—	3	1	no	—
206	C₂×C₁₆	—	6	1	no	—
207	C₂×C₁₆	—	6	1	no	—
208	C₂×C₁₆	—	6	1	no	—
209	C₂×C₁₆	—	6	4	no	—
210	C₂×C₁₆	C₂²×D₃	1	2	yes	—
211	#108	—	3	4	no	—
212	#104	—	3	4	no	—
213	#104	—	3	4	no	—
214	#97	—	3	2	no	—
215	C₄×C₈	—	3	2	no	—
216	C₂³×C₃	—	2	1	no	—
217	C₂³×C₃	#131	1	1	yes	characteristic
218	C₂²×D₃	—	2	1	no	—
219	C₃×Q₈	—	2	1	no	—
220	C₃×Q₈	C₂²×C₈	1	2	yes	—
221	C₃×D₄	—	2	2	no	—
222	C₃×D₄	—	2	2	no	—
223	C₃×D₄	—	2	1	no	—
224	C₃×D₄	C₂²×C₈	1	2	yes	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
232	C₂×C₃×C₄	#131	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
234	#67	—	2	2	no	—
235	#67	—	2	2	no	—
236	C₂×Dic₃	—	2	1	no	—
237	C₂×Dic₃	—	2	2	no	—
238	D₁₂	—	2	1	no	—
239	D₁₂	C₂²×C₈	1	2	yes	—
240	C₄×D₃	—	2	2	no	—
241	C₄×D₃	—	2	2	no	—
242	Dic₆	—	2	1	no	—
243	Dic₆	C₂²×C₈	1	2	yes	—
244	C₃×C₈	—	2	2	no	—
245	C₃×C₈	C₄×D₄	1	1	yes	characteristic
246	C₃×C₈	C₄×D₄	1	1	yes	characteristic
247	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
248	C₃×C₈	C₂×#41	1	1	yes	characteristic
249	C₃⋊C₈	—	2	2	no	—
250	C₃⋊C₈	C₂²×C₈	1	4	yes	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	3	2	no	—
254	#41	—	3	2	no	—
255	#41	—	6	1	no	—
256	#41	—	6	1	no	—
257	#41	D₃×C₈	1	2	yes	—
258	#41	C₂×C₄×D₃	1	2	yes	—
259	C₂×Q₈	—	3	1	no	—
260	C₂×Q₈	D₃×C₈	1	1	yes	characteristic
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	3	1	no	—
263	C₂×D₄	—	6	1	no	—
264	C₂×D₄	D₃×C₈	1	1	yes	characteristic
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	6	1	no	—
267	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
268	Q₁₆	—	3	4	no	—
269	SD₁₆	—	3	4	no	—
270	SD₁₆	—	3	4	no	—
271	D₈	—	3	4	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	3	2	no	—
276	M₁₆	—	3	2	no	—
277	M₁₆	—	3	2	no	—
278	M₁₆	—	6	1	no	—
279	M₁₆	—	6	2	no	—
280	M₁₆	—	6	2	no	—
281	M₁₆	C₂×C₄×D₃	1	2	yes	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	3	2	no	—
286	C₂×C₈	—	6	1	no	—
287	C₂×C₈	—	6	2	no	—
288	C₂×C₈	—	6	2	no	—
289	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
290	C₂×C₈	#235	1	1	yes	characteristic
291	C₂×C₈	#235	1	1	yes	characteristic
292	C₂×C₈	C₂×#67	1	1	yes	characteristic
293	C₂×C₈	C₂×#67	1	1	yes	characteristic
294	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
295	C₄⋊C₄	—	6	1	no	—
296	C₂²⋊C₄	—	6	1	no	—
297	C₄²	—	3	2	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	—	6	1	no	—
300	C₁₆	—	6	1	no	—
301	C₁₆	—	6	2	no	—
302	C₁₆	C₂×#67	1	4	yes	—
303	C₂²×C₃	—	2	2	no	—
304	C₂²×C₃	—	4	1	no	—
305	C₂²×C₃	—	8	1	no	—
306	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
307	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
308	C₂²×C₃	C₂×#131	1	1	yes	characteristic
309	C₂×D₃	—	2	2	no	—
310	C₂×D₃	—	8	1	no	—
311	C₃×C₄	—	2	2	no	—
312	C₃×C₄	C₈×D₄	1	1	yes	characteristic
313	C₃×C₄	C₈×D₄	1	1	yes	characteristic
314	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
315	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
316	Dic₃	—	2	2	no	—
317	Dic₃	—	4	2	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	6	1	no	—
320	C₂³	#901	1	1	yes	characteristic
321	Q₈	—	2	1	no	—
322	Q₈	—	3	2	no	—
323	Q₈	—	6	1	no	—
324	Q₈	C₂×D₃×C₈	1	2	yes	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	3	2	no	—
329	D₄	—	6	2	no	—
330	D₄	—	6	2	no	—
331	D₄	—	6	1	no	—
332	D₄	C₂×D₃×C₈	1	2	yes	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	6	1	no	—
337	C₂×C₄	—	6	2	no	—
338	C₂×C₄	—	6	2	no	—
339	C₂×C₄	—	6	2	no	—
340	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
341	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
342	C₂×C₄	#901	1	1	yes	characteristic
343	C₂×C₄	C₄×#67	1	1	yes	characteristic
344	C₂×C₄	C₄×#67	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	3	4	no	—
348	C₈	—	6	2	no	—
349	C₈	C₄×#67	1	1	yes	characteristic
350	C₈	C₄×#67	1	1	yes	characteristic
351	C₈	C₂×#235	1	1	yes	characteristic
352	C₈	C₂²×#67	1	1	yes	characteristic
353	C₂×C₃	—	2	1	no	—
354	C₂×C₃	—	2	1	no	—
355	C₂×C₃	—	2	1	no	—
356	C₂×C₃	—	4	2	no	—
357	C₂×C₃	C₂×C₈×D₄	1	1	yes	characteristic
358	D₃	—	4	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	6	2	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	24	1	no	—
364	C₂²	C₈×#67	1	1	yes	characteristic
365	C₂²	C₈×#67	1	1	yes	characteristic
366	C₂²	C₂×#901	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	6	2	no	—
369	C₄	—	12	2	no	—
370	C₄	C₈×#67	1	1	yes	characteristic
371	C₄	C₈×#67	1	1	yes	characteristic
372	C₄	C₂²×D₃×C₈	1	1	yes	characteristic
373	C₄	C₂×C₄×#67	1	1	yes	characteristic
374	C₃	#33442	1	1	yes	characteristic, complemented, Sylow
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	12	2	no	—
380	C₂	C₂×C₈×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×#67
commutator	C₃×C₄	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#3407	C₂
socle	C₂×C₃	C₂×C₈×D₄
prime radical	#5822	C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#3407	D₃
hypocenter	C₃	#33442
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #3407

Complements

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

N	H ≅ G/N	type
1. C₃×#3407	C₂	semidirect
3. #82007	C₂	semidirect
4. C₂×#4561	C₂	semidirect
5. #71453	C₂	semidirect
6. #71452	C₂	semidirect
7. C₂×#4515	C₂	semidirect
8. #71423	C₂	semidirect
9. #71419	C₂	semidirect
15. C₃×#504	C₂²	semidirect
29. #4901	C₂²	semidirect
30. #4561	C₂²	semidirect
31. #4561	C₂²	semidirect
32. #4515	C₂²	semidirect
33. #4515	C₂²	semidirect
34. #4511	C₂²	semidirect
35. #4502	C₂²	semidirect
36. Dic₃×C₁₆	C₂²	semidirect
38. #3407	D₃	semidirect
98. #504	C₂×D₃	semidirect
374. C₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3407	2
#83792	2
#82007	2
C₂×#4561	2
#71453	2
#71452	2
C₂×#4515	2
#71423	2
#71419	2
#33442	3