Subgroups — #10948300

Order 768.

590 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	#78549	C₂	1	1	yes	maximal, characteristic
2	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
3	#71432	C₂	1	1	yes	maximal, characteristic
4	#71420	C₂	1	1	yes	maximal, characteristic, complemented
5	#71409	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#69881	C₂	1	1	yes	maximal, characteristic, complemented
8	#69074	C₂	1	2	yes	maximal, complemented
9	#69073	C₂	1	2	yes	maximal, complemented
10	#69053	C₂	1	2	yes	maximal, complemented
11	#69052	C₂	1	2	yes	maximal, complemented
12	#33423	—	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	#4708	C₄	1	2	yes	—
25	#4707	C₄	1	2	yes	—
26	#4689	C₂²	1	2	yes	—
27	C₄×D₃×C₈	C₂²	1	2	yes	—
28	C₃×#346	C₂²	1	2	yes	—
29	C₃×C₄×C₁₆	C₂²	1	2	yes	—
30	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
31	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
32	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
34	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
35	Dic₃⋊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	#4461	C₂²	1	2	yes	—
41	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
42	C₄×#131	—	3	1	no	—
43	#2164	—	3	2	no	—
44	#2164	—	3	2	no	—
45	#2163	—	3	2	no	—
46	#2162	—	3	2	no	—
47	#2150	—	3	1	no	—
48	C₂×C₄⋊C₁₆	—	3	1	no	—
49	#2113	—	3	1	no	—
50	#2113	—	3	1	no	—
51	#2108	—	3	1	no	—
52	#2108	D₃	1	1	yes	characteristic, complemented
53	C₂×#235	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₃×#117	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
58	C₂×#208	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
60	#907	—	2	4	no	—
61	#901	—	2	2	no	—
62	#901	—	2	2	no	—
63	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
64	C₂×ΓL(1,𝔽₂₅)	C₂×C₄	1	2	yes	—
65	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
66	C₂×D₃×C₈	C₂³	1	2	yes	—
67	#893	C₈	1	1	yes	characteristic
68	#893	C₈	1	1	yes	characteristic
69	#887	C₈	1	1	yes	characteristic
70	#887	C₈	1	1	yes	characteristic
71	#67⋊C₄	C₈	1	2	yes	—
72	#67⋊C₄	C₈	1	2	yes	—
73	#872	C₂×C₄	1	2	yes	—
74	C₄²×D₃	C₂×C₄	1	2	yes	—
75	C₃×M₃₂	—	2	2	no	—
76	C₂×C₃×C₁₆	—	2	2	no	—
77	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₁₆	C₂³	1	2	yes	—
80	C₃×#97	C₂×C₄	1	2	yes	—
81	C₃×C₄×C₈	C₂³	1	2	yes	—
82	#815	C₂×C₄	1	2	yes	—
83	C₈×Dic₃	C₂×C₄	1	2	yes	—
84	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₃	C₂³	1	1	yes	characteristic
87	C₈×Dic₃	C₂³	1	1	yes	characteristic
88	#812	—	2	2	no	—
89	C₂×C₃⋊C₁₆	—	2	2	no	—
90	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
91	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
92	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
93	#803	C₂×C₄	1	2	yes	—
94	C₄×C₃⋊C₈	C₂×C₄	1	2	yes	—
95	C₂×#131	—	3	1	no	—
96	C₂×#131	—	3	1	no	—
97	C₄×#41	—	3	1	no	—
98	C₂×M₃₂	—	3	1	no	—
99	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
100	C₂²×C₁₆	—	3	1	no	—
101	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
102	#405	—	3	1	no	—
103	#405	—	3	2	no	—
104	#405	—	3	2	no	—
105	#405	C₂×D₃	1	1	yes	characteristic
106	C₄×M₁₆	—	3	1	no	—
107	C₄×M₁₆	—	3	2	no	—
108	C₂×C₄×C₈	—	3	1	no	—
109	C₂×C₄×C₈	—	3	2	no	—
110	C₄⋊C₁₆	—	3	1	no	—
111	C₄⋊C₁₆	—	3	1	no	—
112	C₄⋊C₁₆	—	3	1	no	—
113	C₄⋊C₁₆	—	3	1	no	—
114	C₄⋊C₁₆	—	3	2	no	—
115	C₄⋊C₁₆	—	3	2	no	—
116	C₂²⋊C₁₆	—	3	2	no	—
117	C₂²⋊C₁₆	—	3	2	no	—
118	C₂²⋊C₁₆	—	3	2	no	—
119	C₂²⋊C₁₆	—	3	2	no	—
120	#346	—	3	2	no	—
121	#346	C₂×D₃	1	2	yes	—
122	C₄×C₁₆	—	3	2	no	—
123	C₄×C₁₆	C₂×D₃	1	2	yes	—
124	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
125	C₂×#67	C₂×C₈	1	2	yes	—
126	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
127	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
128	#235	—	2	2	no	—
129	#235	—	2	2	no	—
130	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
131	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
132	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
133	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
134	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
135	C₃×M₁₆	M₁₆	1	4	yes	—
136	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
138	C₂×C₃×C₈	#41	1	2	yes	—
139	C₂×C₃×C₈	#41	1	2	yes	—
140	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
141	C₃×C₄⋊C₄	C₂×C₈	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	2	yes	—
145	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
146	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
147	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
148	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
149	C₄×Dic₃	C₂×C₈	1	2	yes	—
150	C₄×Dic₃	C₂×C₈	1	2	yes	—
151	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
152	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
153	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
154	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
155	#208	—	2	2	no	—
156	C₂×C₃⋊C₈	—	2	2	no	—
157	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
159	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
160	ΓL(1,𝔽₂₅)	—	2	4	no	—
161	ΓL(1,𝔽₂₅)	—	2	4	no	—
162	D₃×C₈	—	2	4	no	—
163	D₃×C₈	—	2	2	no	—
164	D₃×C₈	—	2	2	no	—
165	C₃×C₁₆	—	2	1	no	—
166	C₃×C₁₆	—	2	1	no	—
167	C₃×C₁₆	—	2	2	no	—
168	C₃⋊C₁₆	—	2	1	no	—
169	C₃⋊C₁₆	—	2	1	no	—
170	C₃⋊C₁₆	—	2	2	no	—
171	C₂×#41	—	3	1	no	—
172	#131	—	6	4	no	—
173	#131	—	6	2	no	—
174	#131	—	6	2	no	—
175	C₂×M₁₆	—	3	1	no	—
176	C₂×M₁₆	—	3	2	no	—
177	C₂×M₁₆	—	3	2	no	—
178	C₂×M₁₆	C₄×D₃	1	1	yes	characteristic
179	C₂²×C₈	—	3	1	no	—
180	C₂²×C₈	—	3	2	no	—
181	C₂²×C₈	—	3	2	no	—
182	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
183	C₄×Q₈	—	3	1	no	—
184	C₄×Q₈	—	3	1	no	—
185	C₄×D₄	—	3	1	no	—
186	C₄×D₄	—	3	1	no	—
187	C₄×D₄	—	3	2	no	—
188	C₄×D₄	—	3	2	no	—
189	#117	—	3	2	no	—
190	#117	C₄×D₃	1	1	yes	characteristic
191	C₂×C₄²	—	3	1	no	—
192	C₂×C₄²	—	3	2	no	—
193	M₃₂	—	2	2	no	—
194	M₃₂	—	6	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	3	1	no	—
197	C₂×C₁₆	—	3	2	no	—
198	C₂×C₁₆	—	3	1	no	—
199	C₂×C₁₆	—	6	2	no	—
200	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
201	C₂×C₁₆	C₂²×D₃	1	2	yes	—
202	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
203	#97	—	3	2	no	—
204	#97	—	3	2	no	—
205	#97	C₄×D₃	1	2	yes	—
206	C₄×C₈	—	3	1	no	—
207	C₄×C₈	—	3	2	no	—
208	C₄×C₈	—	3	1	no	—
209	C₄×C₈	—	3	1	no	—
210	C₄×C₈	—	3	1	no	—
211	C₄×C₈	—	3	2	no	—
212	C₄×C₈	C₂²×D₃	1	2	yes	—
213	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
214	C₂²×D₃	C₂²×C₈	1	2	yes	—
215	C₂×C₃×C₄	#117	1	2	yes	—
216	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
217	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
218	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
220	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
221	#67	—	2	4	no	—
222	C₂×Dic₃	—	2	2	no	—
223	C₂×Dic₃	—	4	1	no	—
224	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
225	C₂×Dic₃	C₂²×C₈	1	2	yes	—
226	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
227	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
228	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
229	D₁₂	—	2	2	no	—
230	C₄×D₃	—	2	2	no	—
231	C₄×D₃	—	2	2	no	—
232	C₄×D₃	—	2	4	no	—
233	C₄×D₃	—	4	2	no	—
234	Dic₆	—	2	2	no	—
235	C₃×C₈	#117	1	2	yes	—
236	C₃×C₈	C₂×M₁₆	1	4	yes	—
237	C₃×C₈	C₂×#41	1	1	yes	characteristic
238	C₃×C₈	C₂×#41	1	1	yes	characteristic
239	C₃⋊C₈	—	2	2	no	—
240	C₃⋊C₈	—	2	1	no	—
241	C₃⋊C₈	—	2	1	no	—
242	#41	—	6	2	no	—
243	#41	—	6	2	no	—
244	C₂×Q₈	—	3	1	no	—
245	C₂×D₄	—	3	2	no	—
246	C₂×D₄	—	3	1	no	—
247	C₂²×C₄	—	3	1	no	—
248	C₂²×C₄	—	3	1	no	—
249	C₂²×C₄	—	3	2	no	—
250	C₂²×C₄	—	3	2	no	—
251	C₂²×C₄	—	3	2	no	—
252	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
253	M₁₆	—	6	2	no	—
254	M₁₆	—	6	4	no	—
255	M₁₆	—	6	4	no	—
256	M₁₆	ΓL(1,𝔽₂₅)	1	4	yes	—
257	C₂×C₈	—	3	1	no	—
258	C₂×C₈	—	3	2	no	—
259	C₂×C₈	—	3	1	no	—
260	C₂×C₈	—	6	2	no	—
261	C₂×C₈	—	6	4	no	—
262	C₂×C₈	—	6	2	no	—
263	C₂×C₈	—	6	2	no	—
264	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
266	C₂×C₈	#235	1	2	yes	—
267	C₂×C₈	#235	1	2	yes	—
268	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
269	C₄⋊C₄	—	3	2	no	—
270	C₄⋊C₄	—	3	2	no	—
271	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
272	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
273	C₂²⋊C₄	—	3	2	no	—
274	C₂²⋊C₄	—	3	2	no	—
275	C₂²⋊C₄	D₃×C₈	1	2	yes	—
276	C₄²	—	3	1	no	—
277	C₄²	—	3	2	no	—
278	C₄²	—	3	1	no	—
279	C₄²	—	3	2	no	—
280	C₄²	—	3	1	no	—
281	C₄²	—	3	1	no	—
282	C₄²	C₂×C₄×D₃	1	2	yes	—
283	C₁₆	—	2	1	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	6	1	no	—
287	C₁₆	—	6	1	no	—
288	C₁₆	—	6	2	no	—
289	C₂²×C₃	—	2	1	no	—
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	#431	1	2	yes	—
292	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
293	C₂×D₃	—	2	2	no	—
294	C₂×D₃	—	2	2	no	—
295	C₂×D₃	—	2	2	no	—
296	C₃×C₄	—	2	2	no	—
297	C₃×C₄	—	2	2	no	—
298	C₃×C₄	#431	1	2	yes	—
299	C₃×C₄	C₂×#117	1	1	yes	characteristic
300	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
301	Dic₃	—	2	1	no	—
302	Dic₃	—	2	1	no	—
303	Dic₃	—	2	1	no	—
304	Dic₃	—	2	2	no	—
305	Dic₃	#504	1	2	yes	—
306	C₂³	—	3	2	no	—
307	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
308	Q₈	—	6	2	no	—
309	D₄	—	6	4	no	—
310	D₄	—	6	2	no	—
311	C₂×C₄	—	3	1	no	—
312	C₂×C₄	—	3	1	no	—
313	C₂×C₄	—	3	1	no	—
314	C₂×C₄	—	3	1	no	—
315	C₂×C₄	—	3	2	no	—
316	C₂×C₄	—	6	4	no	—
317	C₂×C₄	—	6	2	no	—
318	C₂×C₄	—	6	2	no	—
319	C₂×C₄	—	6	2	no	—
320	C₂×C₄	—	12	2	no	—
321	C₂×C₄	—	12	1	no	—
322	C₂×C₄	#872	1	2	yes	—
323	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
324	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
325	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
326	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
327	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
328	C₈	—	6	2	no	—
329	C₈	—	6	1	no	—
330	C₈	—	6	1	no	—
331	C₈	#872	1	2	yes	—
332	C₈	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
333	C₈	C₂×#235	1	1	yes	characteristic
334	C₈	C₂×#235	1	1	yes	characteristic
335	C₂×C₃	—	2	2	no	—
336	C₂×C₃	C₂×#431	1	1	yes	characteristic
337	C₂×C₃	C₂×#504	1	1	yes	characteristic
338	C₂×C₃	#3407	1	1	yes	characteristic
339	D₃	—	4	2	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	6	2	no	—
343	C₂²	—	6	2	no	—
344	C₂²	—	6	2	no	—
345	C₂²	#4685	1	2	yes	—
346	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	3	2	no	—
350	C₄	—	6	1	no	—
351	C₄	—	6	1	no	—
352	C₄	—	6	1	no	—
353	C₄	—	6	2	no	—
354	C₄	#4685	1	2	yes	—
355	C₄	C₂×#872	1	1	yes	characteristic
356	C₄	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
357	C₃	#33423	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	2	2	no	—
359	C₂	—	12	2	no	—
360	C₂	C₂×#4685	1	1	yes	characteristic
361	C₂	D₃×#504	1	1	yes	characteristic
362	C₂	#82012	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2108	C₂
prime radical	C₂×#235	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2108	D₃
hypocenter	C₃	#33423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. C₃×#2108	C₂	semidirect
4. #71420	C₂	semidirect
6. C₂×Dic₃⋊C₁₆	C₂	semidirect
7. #69881	C₂	semidirect
8. #69074	C₂	semidirect
9. #69073	C₂	semidirect
10. #69053	C₂	semidirect
11. #69052	C₂	semidirect
52. #2108	D₃	semidirect
357. C₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#78549	2
C₃×#2108	2
#71432	2
#71420	2
#71409	2
C₂×Dic₃⋊C₁₆	2
#69881	2
#69074	2
#69073	2
#69053	2
#69052	2
#33423	3