Subgroups — #10948178

Order 768.

706 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	C₂×#4907	C₂	1	1	yes	maximal, characteristic
2	C₂×#4901	C₂	1	1	yes	maximal, characteristic
3	#78549	C₂	1	1	yes	maximal, characteristic
4	C₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
5	#71420	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#69882	C₂	1	1	yes	maximal, characteristic, complemented
8	#69285	C₂	1	2	yes	maximal, complemented
9	#69285	C₂	1	2	yes	maximal, complemented
10	#69284	C₂	1	2	yes	maximal, complemented
11	D₃×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
12	#33398	—	3	1	no	maximal, Sylow
13	C₂×#907	C₄	1	1	yes	characteristic
14	C₂×#901	C₂²	1	1	yes	characteristic
15	#5522	C₄	1	1	yes	characteristic
16	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#405	C₂²	1	1	yes	characteristic
19	Dic₃×M₁₆	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×#812	C₂²	1	1	yes	characteristic
22	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	#4997	C₄	1	1	yes	characteristic
24	#4907	—	2	4	no	—
25	#4901	—	2	4	no	—
26	C₂×#798	C₂²	1	2	yes	—
27	C₂×#798	C₂²	1	2	yes	—
28	C₂×D₃×C₁₆	C₂²	1	2	yes	—
29	C₂×D₃×C₁₆	C₂²	1	2	yes	—
30	#4708	C₄	1	2	yes	—
31	#4707	C₄	1	2	yes	—
32	#4689	C₂²	1	2	yes	—
33	C₄×D₃×C₈	C₂²	1	2	yes	—
34	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
36	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
37	Dic₃⋊C₁₆	C₂²	1	2	yes	—
38	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
39	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
40	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
41	Dic₃⋊C₁₆	C₂²	1	2	yes	—
42	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
43	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
44	C₂×#504	—	3	1	no	—
45	C₂×#504	—	3	1	no	—
46	C₄×#131	—	3	1	no	—
47	#2151	—	3	1	no	—
48	#2151	—	3	2	no	—
49	#2151	—	3	2	no	—
50	#2151	D₃	1	1	yes	characteristic, complemented
51	#2150	—	3	1	no	—
52	#2150	—	3	2	no	—
53	C₂×C₄⋊C₁₆	—	3	1	no	—
54	C₂×C₄⋊C₁₆	—	3	2	no	—
55	C₂×#235	C₂×C₄	1	1	yes	characteristic
56	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
58	C₃×#117	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
60	C₂×#208	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
62	#907	—	2	4	no	—
63	#901	D₄	1	4	yes	—
64	#901	Q₈	1	4	yes	—
65	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
66	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
67	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
68	C₂×D₃×C₈	C₂³	1	2	yes	—
69	#893	C₈	1	1	yes	characteristic
70	#893	C₈	1	1	yes	characteristic
71	#887	C₈	1	1	yes	characteristic
72	#887	C₈	1	1	yes	characteristic
73	#67⋊C₄	C₈	1	2	yes	—
74	#67⋊C₄	C₈	1	2	yes	—
75	#872	C₂×C₄	1	2	yes	—
76	C₄²×D₃	C₂×C₄	1	2	yes	—
77	C₃×M₃₂	—	2	2	no	—
78	C₂×C₃×C₁₆	—	2	2	no	—
79	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₁₆	C₂³	1	2	yes	—
82	C₃×#97	C₂×C₄	1	2	yes	—
83	C₃×C₄×C₈	C₂³	1	2	yes	—
84	#815	C₂×C₄	1	2	yes	—
85	C₈×Dic₃	C₂×C₄	1	2	yes	—
86	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₃	C₂³	1	1	yes	characteristic
89	C₈×Dic₃	C₂³	1	1	yes	characteristic
90	#812	—	2	2	no	—
91	C₂×C₃⋊C₁₆	—	2	2	no	—
92	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
93	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
94	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
95	#803	C₂×C₄	1	2	yes	—
96	C₄×C₃⋊C₈	C₂×C₄	1	2	yes	—
97	#798	—	2	4	no	—
98	#798	—	2	4	no	—
99	D₃×C₁₆	—	2	4	no	—
100	D₃×C₁₆	—	2	4	no	—
101	C₂×#131	—	3	1	no	—
102	C₂×#131	—	3	1	no	—
103	C₄×#41	—	3	1	no	—
104	#504	—	6	4	no	—
105	#504	—	6	4	no	—
106	C₂×M₃₂	—	3	1	no	—
107	C₂×M₃₂	—	3	2	no	—
108	C₂×M₃₂	—	3	2	no	—
109	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
110	C₂²×C₁₆	—	3	1	no	—
111	C₂²×C₁₆	—	3	2	no	—
112	C₂²×C₁₆	—	3	2	no	—
113	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
114	#405	—	3	1	no	—
115	#405	—	3	2	no	—
116	#405	—	3	2	no	—
117	#405	C₂×D₃	1	1	yes	characteristic
118	C₄×M₁₆	—	3	1	no	—
119	C₄×M₁₆	—	3	2	no	—
120	C₂×C₄×C₈	—	3	1	no	—
121	C₂×C₄×C₈	—	3	2	no	—
122	C₄⋊C₁₆	—	3	1	no	—
123	C₄⋊C₁₆	—	3	2	no	—
124	C₄⋊C₁₆	—	3	1	no	—
125	C₄⋊C₁₆	—	3	1	no	—
126	C₄⋊C₁₆	—	3	1	no	—
127	C₄⋊C₁₆	—	3	2	no	—
128	C₄⋊C₁₆	—	3	2	no	—
129	C₄⋊C₁₆	—	3	2	no	—
130	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
131	C₄⋊C₁₆	C₂×D₃	1	2	yes	—
132	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
133	C₂×#67	C₂×C₈	1	2	yes	—
134	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
135	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
136	#235	C₄⋊C₄	1	4	yes	—
137	#235	M₁₆	1	4	yes	—
138	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
139	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
140	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
141	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
142	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
143	C₃×M₁₆	—	2	2	no	—
144	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
145	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
147	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
148	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
149	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
150	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
151	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
152	C₃×C₄²	C₂²×C₄	1	2	yes	—
153	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
154	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
155	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
156	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
157	C₄×Dic₃	C₂×C₈	1	2	yes	—
158	C₄×Dic₃	C₂×C₈	1	2	yes	—
159	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
160	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
161	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
162	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
163	#208	C₄⋊C₄	1	4	yes	—
164	C₂×C₃⋊C₈	—	2	2	no	—
165	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
168	ΓL(1,𝔽₂₅)	—	2	4	no	—
169	ΓL(1,𝔽₂₅)	C₄⋊C₄	1	8	yes	—
170	D₃×C₈	—	2	4	no	—
171	D₃×C₈	C₂×D₄	1	4	yes	—
172	D₃×C₈	C₂×Q₈	1	4	yes	—
173	C₃×C₁₆	—	2	1	no	—
174	C₃×C₁₆	—	2	2	no	—
175	C₃×C₁₆	—	2	1	no	—
176	C₃⋊C₁₆	—	2	1	no	—
177	C₃⋊C₁₆	—	2	2	no	—
178	C₃⋊C₁₆	—	2	1	no	—
179	C₂×#41	—	3	1	no	—
180	#131	—	3	4	no	—
181	#131	—	3	4	no	—
182	#131	—	6	4	no	—
183	C₂×M₁₆	—	3	1	no	—
184	C₂×M₁₆	—	3	2	no	—
185	C₂×M₁₆	—	3	2	no	—
186	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
187	C₂²×C₈	—	3	2	no	—
188	C₂²×C₈	—	3	1	no	—
189	C₂²×C₈	—	3	2	no	—
190	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
191	C₄×Q₈	—	3	1	no	—
192	C₄×Q₈	—	3	1	no	—
193	C₄×D₄	—	3	1	no	—
194	C₄×D₄	—	3	1	no	—
195	C₄×D₄	—	3	2	no	—
196	C₄×D₄	—	3	2	no	—
197	#117	—	3	2	no	—
198	#117	C₄×D₃	1	1	yes	characteristic
199	C₂×C₄²	—	3	1	no	—
200	C₂×C₄²	—	3	2	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	6	4	no	—
203	M₃₂	—	6	4	no	—
204	M₃₂	—	6	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	3	1	no	—
207	C₂×C₁₆	—	3	2	no	—
208	C₂×C₁₆	—	3	1	no	—
209	C₂×C₁₆	—	6	4	no	—
210	C₂×C₁₆	—	6	4	no	—
211	C₂×C₁₆	—	6	2	no	—
212	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
213	C₂×C₁₆	C₂²×D₃	1	2	yes	—
214	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
215	#97	—	3	2	no	—
216	#97	—	3	2	no	—
217	#97	C₄×D₃	1	2	yes	—
218	C₄×C₈	—	3	2	no	—
219	C₄×C₈	—	3	1	no	—
220	C₄×C₈	—	3	1	no	—
221	C₄×C₈	—	3	1	no	—
222	C₄×C₈	—	3	1	no	—
223	C₄×C₈	—	3	2	no	—
224	C₄×C₈	C₂²×D₃	1	2	yes	—
225	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
226	C₂²×D₃	C₂²×C₈	1	2	yes	—
227	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
228	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
229	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
230	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
232	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
233	#67	C₄⋊C₈	1	8	yes	—
234	C₂×Dic₃	—	2	2	no	—
235	C₂×Dic₃	—	4	1	no	—
236	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
237	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
238	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
239	C₂×Dic₃	C₂²×C₈	1	2	yes	—
240	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
241	D₁₂	C₄⋊C₈	1	4	yes	—
242	C₄×D₃	—	2	4	no	—
243	C₄×D₃	—	4	2	no	—
244	C₄×D₃	C₂×C₄⋊C₄	1	4	yes	—
245	C₄×D₃	C₂×M₁₆	1	4	yes	—
246	Dic₆	C₄⋊C₈	1	4	yes	—
247	C₃×C₈	—	2	2	no	—
248	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
249	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
250	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
251	C₃⋊C₈	—	2	1	no	—
252	C₃⋊C₈	—	2	1	no	—
253	C₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
254	#41	—	3	4	no	—
255	#41	—	3	4	no	—
256	C₂×Q₈	—	3	1	no	—
257	C₂×D₄	—	3	2	no	—
258	C₂×D₄	—	3	1	no	—
259	C₂²×C₄	—	3	2	no	—
260	C₂²×C₄	—	3	1	no	—
261	C₂²×C₄	—	3	1	no	—
262	C₂²×C₄	—	3	2	no	—
263	C₂²×C₄	—	3	2	no	—
264	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
265	M₁₆	—	2	2	no	—
266	M₁₆	—	3	4	no	—
267	M₁₆	—	3	8	no	—
268	M₁₆	—	6	4	no	—
269	C₂×C₈	—	3	2	no	—
270	C₂×C₈	—	3	4	no	—
271	C₂×C₈	—	3	4	no	—
272	C₂×C₈	—	3	1	no	—
273	C₂×C₈	—	3	1	no	—
274	C₂×C₈	—	6	4	no	—
275	C₂×C₈	—	6	2	no	—
276	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
277	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
278	C₂×C₈	D₃×D₄	1	2	yes	—
279	C₂×C₈	D₃×Q₈	1	2	yes	—
280	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
281	C₄⋊C₄	—	3	2	no	—
282	C₄⋊C₄	—	3	2	no	—
283	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
284	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
285	C₂²⋊C₄	—	3	2	no	—
286	C₂²⋊C₄	—	3	2	no	—
287	C₂²⋊C₄	D₃×C₈	1	2	yes	—
288	C₄²	—	3	2	no	—
289	C₄²	—	3	1	no	—
290	C₄²	—	3	2	no	—
291	C₄²	—	3	1	no	—
292	C₄²	—	3	1	no	—
293	C₄²	—	3	1	no	—
294	C₄²	C₂×C₄×D₃	1	2	yes	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₁₆	—	6	1	no	—
299	C₁₆	—	6	2	no	—
300	C₁₆	—	6	1	no	—
301	C₂²×C₃	—	2	1	no	—
302	C₂²×C₃	—	2	1	no	—
303	C₂²×C₃	C₂×C₄⋊C₈	1	2	yes	—
304	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
305	C₂×D₃	—	2	2	no	—
306	C₂×D₃	—	2	2	no	—
307	C₂×D₃	C₂×C₄⋊C₈	1	4	yes	—
308	C₃×C₄	—	2	2	no	—
309	C₃×C₄	—	2	2	no	—
310	C₃×C₄	C₂×C₄⋊C₈	1	2	yes	—
311	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
312	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
313	Dic₃	—	2	1	no	—
314	Dic₃	—	2	1	no	—
315	Dic₃	—	2	1	no	—
316	Dic₃	C₂×C₄⋊C₈	1	4	yes	—
317	Dic₃	#504	1	2	yes	complemented
318	C₂³	—	3	2	no	—
319	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
320	Q₈	—	3	4	no	—
321	D₄	—	3	8	no	—
322	D₄	—	3	4	no	—
323	C₂×C₄	—	3	1	no	—
324	C₂×C₄	—	3	2	no	—
325	C₂×C₄	—	3	4	no	—
326	C₂×C₄	—	3	4	no	—
327	C₂×C₄	—	3	1	no	—
328	C₂×C₄	—	3	1	no	—
329	C₂×C₄	—	3	1	no	—
330	C₂×C₄	—	6	4	no	—
331	C₂×C₄	—	6	2	no	—
332	C₂×C₄	—	12	2	no	—
333	C₂×C₄	—	12	1	no	—
334	C₂×C₄	D₃×C₄⋊C₄	1	2	yes	—
335	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
336	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
337	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
338	C₂×C₄	D₃×M₁₆	1	2	yes	—
339	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	3	4	no	—
342	C₈	—	6	1	no	—
343	C₈	—	6	1	no	—
344	C₈	D₃×C₄⋊C₄	1	2	yes	—
345	C₈	C₂×D₃×D₄	1	1	yes	characteristic
346	C₈	C₂×D₃×Q₈	1	1	yes	characteristic
347	C₂×C₃	—	2	2	no	—
348	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
349	C₂×C₃	C₂×#504	1	1	yes	characteristic
350	C₂×C₃	#3407	1	1	yes	characteristic
351	D₃	—	2	4	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	3	4	no	—
355	C₂²	—	6	2	no	—
356	C₂²	—	6	2	no	—
357	C₂²	D₃×C₄⋊C₈	1	2	yes	—
358	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	3	2	no	—
362	C₄	—	3	4	no	—
363	C₄	—	6	1	no	—
364	C₄	—	6	1	no	—
365	C₄	—	6	1	no	—
366	C₄	D₃×C₄⋊C₈	1	2	yes	—
367	C₄	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
368	C₄	C₂×D₃×M₁₆	1	1	yes	characteristic
369	C₃	#33398	1	1	yes	characteristic, complemented, Sylow
370	C₂	—	2	2	no	—
371	C₂	—	6	4	no	—
372	C₂	C₂×D₃×C₄⋊C₈	1	1	yes	characteristic
373	C₂	D₃×#504	1	1	yes	characteristic
374	C₂	#82012	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2151	C₂
prime radical	C₂×#235	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2151	D₃
hypocenter	C₃	#33398
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
4. C₃×#2151	C₂	semidirect
5. #71420	C₂	semidirect
6. C₂×Dic₃⋊C₁₆	C₂	semidirect
7. #69882	C₂	semidirect
8. #69285	C₂	semidirect
9. #69285	C₂	semidirect
10. #69284	C₂	semidirect
11. D₃×C₄⋊C₁₆	C₂	semidirect
50. #2151	D₃	semidirect
317. Dic₃	#504	semidirect
369. C₃	#33398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4907	2
C₂×#4901	2
#78549	2
C₃×#2151	2
#71420	2
C₂×Dic₃⋊C₁₆	2
#69882	2
#69285	2
#69284	2
D₃×C₄⋊C₁₆	2
#33398	3