Subgroups — #10948810

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#79422	C₂	1	1	yes	maximal, characteristic
2	C₃×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#71409	C₂	1	2	yes	maximal, complemented
4	C₁₆×#67	C₂	1	2	yes	maximal, complemented
5	#70858	C₂	1	1	yes	maximal, characteristic, complemented
6	#69277	C₂	1	1	yes	maximal, characteristic, complemented
7	#69274	C₂	1	1	yes	maximal, characteristic, complemented
8	#69107	C₂	1	2	yes	maximal, complemented
9	#69104	C₂	1	2	yes	maximal, complemented
10	#69055	C₂	1	1	yes	maximal, characteristic, complemented
11	#69052	C₂	1	1	yes	maximal, characteristic, complemented
12	#33493	—	3	1	no	maximal, Sylow
13	C₂×#901	C₂²	1	2	yes	—
14	#5548	C₄	1	1	yes	characteristic
15	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
16	C₃×C₈×D₄	C₂²	1	1	yes	characteristic
17	C₈×#67	C₂²	1	2	yes	—
18	C₂×#812	C₂²	1	2	yes	—
19	#5012	C₄	1	1	yes	characteristic
20	C₂×#798	C₂²	1	2	yes	—
21	C₂×D₃×C₁₆	C₂²	1	2	yes	—
22	#4838	C₄	1	1	yes	characteristic
23	#4837	C₄	1	1	yes	characteristic
24	#4729	C₄	1	2	yes	—
25	#4727	C₄	1	2	yes	—
26	C₈×D₁₂	C₂²	1	1	yes	characteristic
27	C₄×D₃×C₈	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
30	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
31	#4549	C₂²	1	2	yes	—
32	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
34	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
35	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
36	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
37	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
38	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
39	Dic₃×C₁₆	C₂²	1	2	yes	—
40	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
42	#2931	—	3	1	no	—
43	#2169	—	3	1	no	—
44	#2169	—	3	2	no	—
45	#2169	—	3	1	no	—
46	D₄×C₁₆	—	3	2	no	—
47	D₄×C₁₆	—	3	1	no	—
48	D₄×C₁₆	D₃	1	1	yes	characteristic, complemented
49	#2162	—	3	1	no	—
50	#2162	—	3	1	no	—
51	#2113	—	3	2	no	—
52	#2113	—	3	2	no	—
53	C₂×#235	C₂×C₄	1	2	yes	—
54	C₂²×C₃×C₈	C₂³	1	2	yes	—
55	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
56	#940	C₈	1	1	yes	characteristic
57	#938	C₈	1	1	yes	characteristic
58	C₄×#67	C₂×C₄	1	2	yes	—
59	C₂×#208	C₂×C₄	1	2	yes	—
60	#901	—	4	2	no	—
61	#901	—	4	2	no	—
62	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
63	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
64	C₂×D₃×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×D₃×C₈	C₂³	1	2	yes	—
66	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
67	#895	C₈	1	1	yes	characteristic
68	#888	C₈	1	1	yes	characteristic
69	#885	C₈	1	2	yes	—
70	#884	C₈	1	2	yes	—
71	C₄×D₁₂	C₂×C₄	1	1	yes	characteristic
72	C₄²×D₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₃×C₁₆	—	4	2	no	—
74	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₁₆	C₂³	1	2	yes	—
76	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
77	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
78	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
79	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
80	#830	C₂×C₄	1	2	yes	—
81	(C₂×D₃)⋊C₈	C₂×C₄	1	2	yes	—
82	(C₂×D₃)⋊C₈	C₂×C₄	1	2	yes	—
83	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
84	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₃	C₂³	1	1	yes	characteristic
86	#812	—	4	2	no	—
87	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
88	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
89	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
90	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
91	C₄×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
92	#798	—	4	2	no	—
93	D₃×C₁₆	—	4	2	no	—
94	C₂×#131	—	3	2	no	—
95	#532	—	3	1	no	—
96	C₂×M₃₂	—	3	2	no	—
97	C₂×M₃₂	—	3	2	no	—
98	C₂²×C₁₆	—	3	2	no	—
99	C₂²×C₁₆	C₂×D₃	1	2	yes	—
100	#436	—	3	1	no	—
101	#436	—	3	1	no	—
102	#436	—	3	2	no	—
103	C₈×D₄	—	3	2	no	—
104	C₈×D₄	—	3	1	no	—
105	C₈×D₄	C₂×D₃	1	1	yes	characteristic
106	#423	—	3	1	no	—
107	#408	—	3	2	no	—
108	C₂×C₄×C₈	—	3	1	no	—
109	C₄⋊C₁₆	—	3	1	no	—
110	C₄⋊C₁₆	—	3	1	no	—
111	C₄⋊C₁₆	—	3	1	no	—
112	C₄⋊C₁₆	C₂×D₃	1	1	yes	characteristic
113	C₂²⋊C₁₆	—	3	2	no	—
114	C₂²⋊C₁₆	—	3	1	no	—
115	C₂²⋊C₁₆	—	3	2	no	—
116	C₂²⋊C₁₆	—	3	1	no	—
117	C₂²⋊C₁₆	—	3	2	no	—
118	C₂²⋊C₁₆	—	3	2	no	—
119	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
120	C₄×C₁₆	—	3	2	no	—
121	C₄×C₁₆	—	3	1	no	—
122	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
123	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
125	C₂×#67	C₂×C₈	1	2	yes	—
126	C₂×#67	C₂×C₈	1	2	yes	—
127	C₂×#67	C₂×C₈	1	2	yes	—
128	#235	—	4	2	no	—
129	#235	—	4	2	no	—
130	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
131	C₂×D₁₂	C₂×C₈	1	2	yes	—
132	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
133	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
134	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
135	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×Dic₆	C₂×C₈	1	2	yes	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
141	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
142	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
143	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
144	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
145	#217	C₂×C₈	1	2	yes	—
146	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
147	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
148	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
149	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
150	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
151	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
152	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
153	#208	—	4	2	no	—
154	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
155	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
157	ΓL(1,𝔽₂₅)	—	4	2	no	—
158	ΓL(1,𝔽₂₅)	—	4	2	no	—
159	D₃×C₈	—	2	2	no	—
160	D₃×C₈	—	2	2	no	—
161	D₃×C₈	—	2	1	no	—
162	D₃×C₈	—	2	1	no	—
163	D₃×C₈	—	4	1	no	—
164	C₃×C₁₆	—	2	2	no	—
165	C₃×C₁₆	—	2	1	no	—
166	C₃×C₁₆	—	2	1	no	—
167	C₃⋊C₁₆	—	2	1	no	—
168	C₃⋊C₁₆	—	2	2	no	—
169	C₃⋊C₁₆	—	2	1	no	—
170	C₂×#41	—	3	2	no	—
171	#131	—	12	2	no	—
172	#131	—	12	2	no	—
173	C₂×M₁₆	—	3	2	no	—
174	C₂×M₁₆	—	3	2	no	—
175	C₂×M₁₆	—	3	2	no	—
176	C₂²×C₈	—	3	1	no	—
177	C₂²×C₈	—	3	2	no	—
178	C₂²×C₈	—	3	1	no	—
179	C₂²×C₈	C₂²×D₃	1	2	yes	—
180	C₄⋊Q₈	—	3	1	no	—
181	#127	—	3	1	no	—
182	#124	—	3	2	no	—
183	#121	—	3	1	no	—
184	#121	—	3	1	no	—
185	#121	—	3	2	no	—
186	C₄×D₄	—	3	2	no	—
187	C₄×D₄	—	3	1	no	—
188	C₄×D₄	C₄×D₃	1	1	yes	characteristic
189	C₂×C₄²	—	3	1	no	—
190	M₃₂	—	12	2	no	—
191	M₃₂	—	12	2	no	—
192	C₂×C₁₆	—	3	1	no	—
193	C₂×C₁₆	—	3	2	no	—
194	C₂×C₁₆	—	3	1	no	—
195	C₂×C₁₆	—	4	2	no	—
196	C₂×C₁₆	—	12	2	no	—
197	C₂×C₁₆	C₂²×D₃	1	2	yes	—
198	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
199	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
200	C₄⋊C₈	—	3	2	no	—
201	C₄⋊C₈	—	3	1	no	—
202	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
203	C₂²⋊C₈	—	3	2	no	—
204	C₂²⋊C₈	—	3	2	no	—
205	C₂²⋊C₈	—	3	2	no	—
206	C₂²⋊C₈	C₄×D₃	1	2	yes	—
207	C₄×C₈	—	3	1	no	—
208	C₄×C₈	—	3	1	no	—
209	C₄×C₈	—	3	1	no	—
210	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
211	C₂³×C₃	C₂²×C₈	1	2	yes	—
212	C₂²×D₃	C₂²×C₈	1	2	yes	—
213	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
214	C₃×D₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
219	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
221	#67	—	2	4	no	—
222	#67	—	4	2	no	—
223	#67	—	4	2	no	—
224	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
225	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
226	C₂×Dic₃	C₂²×C₈	1	2	yes	—
227	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
228	D₁₂	—	2	2	no	—
229	D₁₂	—	4	2	no	—
230	C₄×D₃	—	2	1	no	—
231	C₄×D₃	—	2	2	no	—
232	C₄×D₃	—	2	1	no	—
233	C₄×D₃	—	2	2	no	—
234	C₄×D₃	—	2	2	no	—
235	C₄×D₃	—	2	2	no	—
236	Dic₆	—	4	2	no	—
237	C₃×C₈	—	2	2	no	—
238	C₃×C₈	—	2	1	no	—
239	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
240	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
241	C₃⋊C₈	—	2	1	no	—
242	C₃⋊C₈	—	2	2	no	—
243	C₃⋊C₈	—	2	1	no	—
244	#41	—	12	2	no	—
245	#41	—	12	2	no	—
246	C₂×Q₈	—	3	2	no	—
247	C₂×D₄	—	3	2	no	—
248	C₂×D₄	—	3	2	no	—
249	C₂×D₄	—	3	2	no	—
250	C₂×D₄	—	3	1	no	—
251	C₂×D₄	—	3	2	no	—
252	C₂×D₄	D₃×C₈	1	1	yes	characteristic
253	C₂²×C₄	—	3	1	no	—
254	C₂²×C₄	—	3	1	no	—
255	C₂²×C₄	—	3	2	no	—
256	C₂²×C₄	—	3	1	no	—
257	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
258	M₁₆	—	12	2	no	—
259	M₁₆	—	12	2	no	—
260	M₁₆	—	12	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	3	1	no	—
264	C₂×C₈	—	3	1	no	—
265	C₂×C₈	—	3	2	no	—
266	C₂×C₈	—	6	2	no	—
267	C₂×C₈	—	6	1	no	—
268	C₂×C₈	—	6	2	no	—
269	C₂×C₈	—	6	1	no	—
270	C₂×C₈	—	12	1	no	—
271	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
273	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
274	C₄⋊C₄	—	3	2	no	—
275	C₄⋊C₄	—	3	1	no	—
276	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
277	C₂²⋊C₄	—	3	2	no	—
278	C₂²⋊C₄	—	3	2	no	—
279	C₂²⋊C₄	—	3	2	no	—
280	C₂²⋊C₄	D₃×C₈	1	2	yes	—
281	C₄²	—	3	1	no	—
282	C₄²	—	3	1	no	—
283	C₄²	—	3	1	no	—
284	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	6	1	no	—
289	C₁₆	—	6	2	no	—
290	C₁₆	—	6	1	no	—
291	C₂²×C₃	—	2	2	no	—
292	C₂²×C₃	—	2	2	no	—
293	C₂²×C₃	—	2	2	no	—
294	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
295	C₂×D₃	—	2	1	no	—
296	C₂×D₃	—	2	1	no	—
297	C₂×D₃	—	2	2	no	—
298	C₂×D₃	—	2	2	no	—
299	C₂×D₃	—	2	1	no	—
300	C₂×D₃	—	2	2	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	2	no	—
303	C₃×C₄	#504	1	2	yes	—
304	C₃×C₄	#518	1	1	yes	characteristic
305	C₃×C₄	#568	1	1	yes	characteristic
306	Dic₃	—	2	1	no	—
307	Dic₃	—	2	1	no	—
308	Dic₃	—	2	2	no	—
309	Dic₃	#504	1	2	yes	—
310	C₂³	—	3	2	no	—
311	C₂³	—	3	1	no	—
312	C₂³	C₂×D₃×C₈	1	2	yes	—
313	Q₈	—	12	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	6	4	no	—
316	D₄	—	6	2	no	—
317	D₄	—	12	2	no	—
318	D₄	—	12	2	no	—
319	D₄	—	12	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	3	1	no	—
323	C₂×C₄	—	3	1	no	—
324	C₂×C₄	—	3	1	no	—
325	C₂×C₄	—	3	2	no	—
326	C₂×C₄	—	6	2	no	—
327	C₂×C₄	—	6	2	no	—
328	C₂×C₄	—	6	1	no	—
329	C₂×C₄	—	6	2	no	—
330	C₂×C₄	—	6	1	no	—
331	C₂×C₄	—	6	2	no	—
332	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
333	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
334	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	6	1	no	—
339	C₈	—	6	1	no	—
340	C₈	—	6	2	no	—
341	C₈	#1004	1	1	yes	characteristic
342	C₈	#1009	1	1	yes	characteristic
343	C₂×C₃	—	4	2	no	—
344	C₂×C₃	#2972	1	1	yes	characteristic
345	C₂×C₃	C₂×#504	1	1	yes	characteristic
346	C₂×C₃	C₂×#504	1	1	yes	characteristic
347	D₃	—	4	1	no	—
348	D₃	—	4	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	6	1	no	—
353	C₂²	—	6	2	no	—
354	C₂²	—	6	1	no	—
355	C₂²	—	6	2	no	—
356	C₂²	—	6	1	no	—
357	C₂²	—	6	2	no	—
358	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	2	1	no	—
361	C₄	—	3	2	no	—
362	C₄	—	6	1	no	—
363	C₄	—	6	2	no	—
364	C₄	—	6	1	no	—
365	C₄	#4901	1	2	yes	—
366	C₄	#5545	1	1	yes	characteristic
367	C₄	#5750	1	1	yes	characteristic
368	C₃	#33493	1	1	yes	characteristic, complemented, Sylow
369	C₂	—	4	2	no	—
370	C₂	—	12	1	no	—
371	C₂	—	12	2	no	—
372	C₂	#79426	1	1	yes	characteristic
373	C₂	C₂×#4901	1	1	yes	characteristic
374	C₂	D₃×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×D₄×C₁₆	C₂
prime radical	#5548	C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	D₄×C₁₆	D₃
hypocenter	C₃	#33493
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×D₄×C₁₆	C₂	semidirect
3. #71409	C₂	semidirect
4. C₁₆×#67	C₂	semidirect
5. #70858	C₂	semidirect
6. #69277	C₂	semidirect
7. #69274	C₂	semidirect
8. #69107	C₂	semidirect
9. #69104	C₂	semidirect
10. #69055	C₂	semidirect
11. #69052	C₂	semidirect
48. D₄×C₁₆	D₃	semidirect
368. C₃	#33493	semidirect

Maximal subgroups

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

subgroup	index
#79422	2
C₃×D₄×C₁₆	2
#71409	2
C₁₆×#67	2
#70858	2
#69277	2
#69274	2
#69107	2
#69104	2
#69055	2
#69052	2
#33493	3