Subgroups — C₄×#63481 #11859975

Order 1152.

750 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63481	C₂	1	1	yes	maximal, characteristic
2	C₄×#3701	C₂	1	2	yes	maximal, complemented
3	C₄×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#3675	C₂	1	2	yes	maximal, complemented
6	#10588560	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588552	C₂	1	1	yes	maximal, characteristic, complemented
8	#10588484	C₂	1	2	yes	maximal, complemented
9	#10588468	C₂	1	2	yes	maximal, complemented
10	C₄×D₇₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₈×D₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#919	—	3	1	no	maximal
13	C₂×#3701	C₂²	1	2	yes	—
14	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₉	C₂²	1	2	yes	—
16	Q₈×Dic₉	C₂²	1	2	yes	—
17	C₂×#3675	C₂²	1	2	yes	—
18	#63481	C₄	1	8	yes	direct product factor
19	C₂×D₇₂	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₉	C₂²	1	1	yes	characteristic
22	#63455	—	2	2	no	—
23	#63455	C₂²	1	2	yes	—
24	#63454	—	2	2	no	—
25	#63454	C₂²	1	2	yes	—
26	C₄×D₃₆	—	2	2	no	—
27	C₄×D₃₆	C₂²	1	2	yes	—
28	C₄²×D₉	—	2	2	no	—
29	C₄²×D₉	C₂²	1	1	yes	characteristic
30	C₉×#107	C₂²	1	1	yes	characteristic
31	C₉×#103	C₂²	1	2	yes	—
32	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
33	#63395	C₂²	1	2	yes	—
34	#63380	C₂²	1	2	yes	—
35	#63378	C₂²	1	1	yes	characteristic
36	C₈×Dic₉	C₂²	1	1	yes	characteristic
37	C₈×Dic₉	C₂²	1	1	yes	characteristic
38	#63369	C₂²	1	2	yes	—
39	#63367	C₂²	1	2	yes	—
40	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×#919	—	3	1	no	—
42	C₄×#239	—	3	2	no	—
43	C₃×C₄×Q₁₆	D₃	1	1	yes	characteristic
44	Dic₃×Q₁₆	—	3	1	no	—
45	C₄×#215	—	3	2	no	—
46	#4880	—	3	1	no	—
47	#4872	—	3	1	no	—
48	#4804	—	3	2	no	—
49	#4788	—	3	2	no	—
50	C₄×D₂₄	—	3	1	no	—
51	C₄×D₃×C₈	—	3	1	no	—
52	C₂×Q₈×C₉	C₂³	1	2	yes	—
53	#3701	—	2	4	no	—
54	#3701	C₂×C₄	1	8	yes	complemented
55	C₂×D₃₆	—	2	2	no	—
56	C₂×D₃₆	C₂³	1	2	yes	—
57	C₂×C₄×D₉	—	2	2	no	—
58	C₂×C₄×D₉	—	2	2	no	—
59	C₂×C₄×D₉	—	2	2	no	—
60	C₂×C₄×D₉	D₄	1	1	yes	characteristic
61	C₂×C₄×D₉	D₄	1	1	yes	characteristic
62	C₂×C₄×D₉	C₂³	1	1	yes	characteristic
63	C₉×Q₁₆	C₂×C₄	1	4	yes	complemented
64	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
66	C₉×C₄⋊C₄	—	2	2	no	—
67	C₉×C₄⋊C₄	C₂³	1	2	yes	—
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	C₂³	1	1	yes	characteristic
70	#3675	C₂×C₄	1	8	yes	complemented
71	(C₂×D₉)⋊C₄	—	2	2	no	—
72	(C₂×D₉)⋊C₄	—	2	2	no	—
73	(C₂×D₉)⋊C₄	—	2	2	no	—
74	C₄⋊Dic₉	—	2	2	no	—
75	C₄⋊Dic₉	C₂³	1	2	yes	—
76	C₄×Dic₉	—	2	2	no	—
77	C₄×Dic₉	—	2	2	no	—
78	C₄×Dic₉	D₄	1	1	yes	characteristic
79	C₄×Dic₉	D₄	1	1	yes	characteristic
80	C₄×Dic₉	C₂³	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
83	D₇₂	C₂×C₄	1	4	yes	complemented
84	C₈×D₉	—	2	2	no	—
85	C₈×D₉	C₂×C₄	1	4	yes	complemented
86	C₄×#135	—	9	1	no	Sylow
87	C₂×#239	—	3	2	no	—
88	C₂×C₃×Q₁₆	C₂×D₃	1	1	yes	characteristic
89	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
90	Q₈×Dic₃	—	3	2	no	—
91	C₂×#215	—	3	2	no	—
92	#919	—	3	8	no	—
93	C₂×D₂₄	—	3	1	no	—
94	C₂×D₃×C₈	—	3	1	no	—
95	C₂×D₃×C₈	—	3	1	no	—
96	#893	—	3	2	no	—
97	#893	—	6	2	no	—
98	#892	—	3	2	no	—
99	#892	—	6	2	no	—
100	C₄×D₁₂	—	3	2	no	—
101	C₄×D₁₂	—	6	2	no	—
102	C₄²×D₃	—	3	1	no	—
103	C₄²×D₃	—	6	2	no	—
104	C₃×#107	C₂×D₃	1	1	yes	characteristic
105	C₃×#103	C₂×D₃	1	2	yes	—
106	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
107	#835	—	3	2	no	—
108	#821	—	3	2	no	—
109	#818	—	3	1	no	—
110	C₈×Dic₃	—	3	1	no	—
111	C₈×Dic₃	—	3	1	no	—
112	#809	—	3	2	no	—
113	#808	—	3	2	no	—
114	C₄×C₃⋊C₈	—	3	1	no	—
115	C₂²×D₉	—	2	2	no	—
116	C₂²×D₉	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₉	—	2	2	no	—
118	Q₈×C₉	C₂²×C₄	1	4	yes	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₉	—	2	2	no	—
125	C₂×Dic₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₉	C₂×D₄	1	1	yes	characteristic
128	D₃₆	—	2	8	no	—
129	D₃₆	—	2	2	no	—
130	D₃₆	C₂²×C₄	1	4	yes	—
131	C₄×D₉	—	2	2	no	—
132	C₄×D₉	—	2	8	no	—
133	C₄×D₉	—	2	2	no	—
134	C₄×D₉	—	4	4	no	—
135	C₄×D₉	—	4	4	no	—
136	C₄×D₉	C₂²×C₄	1	2	yes	—
137	C₄×D₉	#41	1	2	yes	—
138	C₈×C₉	—	2	1	no	—
139	C₈×C₉	C₂²×C₄	1	2	yes	—
140	C₉⋊C₈	—	2	1	no	—
141	C₉⋊C₈	C₂²×C₄	1	2	yes	—
142	C₂×#135	—	9	1	no	—
143	C₄×#41	—	9	2	no	—
144	C₄×Q₁₆	—	9	1	no	—
145	C₄×Q₁₆	D₉	1	1	yes	characteristic, complemented
146	C₄×SD₁₆	—	9	2	no	—
147	C₄×SD₁₆	—	9	2	no	—
148	C₄×D₈	—	9	1	no	—
149	C₄×D₈	—	9	1	no	—
150	#427	—	9	1	no	—
151	#416	—	9	2	no	—
152	C₂×C₄×C₈	—	9	1	no	—
153	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
154	#239	—	3	8	no	—
155	#239	—	6	4	no	—
156	C₂×D₁₂	—	3	2	no	—
157	C₂×D₁₂	—	6	2	no	—
158	C₂×C₄×D₃	—	3	1	no	—
159	C₂×C₄×D₃	—	3	1	no	—
160	C₂×C₄×D₃	—	3	1	no	—
161	C₂×C₄×D₃	—	6	2	no	—
162	C₂×C₄×D₃	—	6	2	no	—
163	C₂×C₄×D₃	—	6	2	no	—
164	C₃×Q₁₆	C₄×D₃	1	4	yes	—
165	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
166	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
167	C₃×C₄⋊C₄	—	2	2	no	—
168	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
169	C₃×C₄²	—	2	2	no	—
170	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
171	#215	—	3	8	no	—
172	(C₂×D₃)⋊C₄	—	6	2	no	—
173	(C₂×D₃)⋊C₄	—	6	2	no	—
174	(C₂×D₃)⋊C₄	—	6	2	no	—
175	C₄⋊Dic₃	—	3	2	no	—
176	C₄⋊Dic₃	—	6	2	no	—
177	C₄×Dic₃	—	3	1	no	—
178	C₄×Dic₃	—	3	1	no	—
179	C₄×Dic₃	—	3	1	no	—
180	C₄×Dic₃	—	6	2	no	—
181	C₄×Dic₃	—	6	2	no	—
182	C₂×C₃⋊C₈	—	3	1	no	—
183	C₂×C₃⋊C₈	—	3	1	no	—
184	D₂₄	—	3	4	no	—
185	D₃×C₈	—	3	4	no	—
186	D₃×C₈	—	6	2	no	—
187	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
188	C₂×D₉	—	2	1	no	—
189	C₂×D₉	—	2	1	no	—
190	C₂×D₉	—	2	4	no	—
191	C₂×D₉	—	4	2	no	—
192	C₂×D₉	—	4	2	no	—
193	C₂×D₉	C₄×D₄	1	2	yes	—
194	C₄×C₉	—	2	4	no	—
195	C₄×C₉	—	2	1	no	—
196	C₄×C₉	—	4	2	no	—
197	C₄×C₉	#135	1	2	yes	complemented
198	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
199	C₄×C₉	C₂×#41	1	1	yes	characteristic
200	Dic₉	—	2	1	no	—
201	Dic₉	—	4	2	no	—
202	Dic₉	C₄×D₄	1	2	yes	—
203	Dic₉	#135	1	2	yes	complemented
204	C₂×#41	—	9	2	no	—
205	#135	—	9	8	no	—
206	C₂×Q₁₆	C₂×D₉	1	1	yes	characteristic
207	C₂×SD₁₆	—	9	2	no	—
208	C₂×D₈	—	9	1	no	—
209	C₂²×C₈	—	9	1	no	—
210	C₂²×C₈	—	9	1	no	—
211	C₄×Q₈	—	9	2	no	—
212	C₄×Q₈	C₂×D₉	1	2	yes	—
213	C₄×D₄	—	9	2	no	—
214	C₄×D₄	—	9	2	no	—
215	C₄×D₄	—	18	2	no	—
216	C₄×D₄	—	18	2	no	—
217	#117	—	9	2	no	—
218	#117	—	18	2	no	—
219	C₂×C₄²	—	9	1	no	—
220	C₂×C₄²	—	18	2	no	—
221	#107	—	9	1	no	—
222	#107	C₂×D₉	1	1	yes	characteristic
223	#106	—	9	2	no	—
224	#103	—	9	2	no	—
225	#103	C₂×D₉	1	2	yes	—
226	#102	—	9	2	no	—
227	#102	—	9	2	no	—
228	C₄×C₈	—	9	1	no	—
229	C₄×C₈	—	9	1	no	—
230	C₄×C₈	—	9	1	no	—
231	C₄×C₈	C₂×D₉	1	1	yes	characteristic
232	C₂²×D₃	—	3	1	no	—
233	C₂²×D₃	—	6	2	no	—
234	C₃×Q₈	—	2	2	no	—
235	C₃×Q₈	C₂×C₄×D₃	1	4	yes	—
236	C₂×C₃×C₄	—	2	2	no	—
237	C₂×C₃×C₄	—	2	2	no	—
238	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
239	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
241	C₂×Dic₃	—	3	1	no	—
242	C₂×Dic₃	—	3	1	no	—
243	C₂×Dic₃	—	3	1	no	—
244	C₂×Dic₃	—	6	2	no	—
245	D₁₂	—	3	4	no	—
246	D₁₂	—	6	8	no	—
247	D₁₂	—	6	2	no	—
248	C₄×D₃	—	3	2	no	—
249	C₄×D₃	—	3	2	no	—
250	C₄×D₃	—	6	2	no	—
251	C₄×D₃	—	6	8	no	—
252	C₄×D₃	—	6	2	no	—
253	C₄×D₃	—	12	4	no	—
254	C₄×D₃	—	12	4	no	—
255	C₃×C₈	—	2	1	no	—
256	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
257	C₃⋊C₈	—	3	2	no	—
258	C₃⋊C₈	—	6	1	no	—
259	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
260	C₂×C₉	C₂×#135	1	1	yes	characteristic
261	C₂×C₉	C₂×#135	1	1	yes	characteristic
262	D₉	—	2	2	no	—
263	D₉	—	4	4	no	—
264	#41	—	9	8	no	—
265	#41	—	18	4	no	—
266	C₂×Q₈	C₂²×D₉	1	2	yes	—
267	C₂×D₄	—	9	2	no	—
268	C₂×D₄	—	18	2	no	—
269	C₂²×C₄	—	9	1	no	—
270	C₂²×C₄	—	9	1	no	—
271	C₂²×C₄	—	9	1	no	—
272	C₂²×C₄	—	18	2	no	—
273	C₂²×C₄	—	18	2	no	—
274	C₂²×C₄	—	18	2	no	—
275	Q₁₆	C₄×D₉	1	4	yes	complemented
276	SD₁₆	—	9	8	no	—
277	D₈	—	9	4	no	—
278	C₂×C₈	—	9	1	no	—
279	C₂×C₈	—	9	4	no	—
280	C₂×C₈	—	9	1	no	—
281	C₂×C₈	—	18	2	no	—
282	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
283	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	9	2	no	—
286	C₄⋊C₄	—	18	2	no	—
287	C₄⋊C₄	C₂²×D₉	1	2	yes	—
288	C₂²⋊C₄	—	18	2	no	—
289	C₂²⋊C₄	—	18	2	no	—
290	C₂²⋊C₄	—	18	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	9	1	no	—
293	C₄²	—	9	1	no	—
294	C₄²	—	9	1	no	—
295	C₄²	—	18	2	no	—
296	C₄²	—	18	2	no	—
297	C₄²	C₂²×D₉	1	1	yes	characteristic
298	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
299	C₂×D₃	—	3	2	no	—
300	C₂×D₃	—	6	1	no	—
301	C₂×D₃	—	6	1	no	—
302	C₂×D₃	—	6	4	no	—
303	C₂×D₃	—	12	2	no	—
304	C₂×D₃	—	12	2	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	2	4	no	—
307	C₃×C₄	—	4	2	no	—
308	C₃×C₄	#919	1	2	yes	—
309	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
310	C₃×C₄	D₃×#41	1	1	yes	characteristic
311	Dic₃	—	3	2	no	—
312	Dic₃	—	3	2	no	—
313	Dic₃	—	6	1	no	—
314	Dic₃	—	12	2	no	—
315	C₉	C₄×#135	1	1	yes	characteristic, complemented, Sylow
316	C₂³	—	9	1	no	—
317	C₂³	—	18	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	C₂×C₄×D₉	1	4	yes	—
320	D₄	—	9	4	no	—
321	D₄	—	18	8	no	—
322	D₄	—	18	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	9	1	no	—
326	C₂×C₄	—	9	1	no	—
327	C₂×C₄	—	9	2	no	—
328	C₂×C₄	—	9	2	no	—
329	C₂×C₄	—	9	1	no	—
330	C₂×C₄	—	18	2	no	—
331	C₂×C₄	—	18	8	no	—
332	C₂×C₄	—	18	2	no	—
333	C₂×C₄	—	18	2	no	—
334	C₂×C₄	—	36	4	no	—
335	C₂×C₄	—	36	4	no	—
336	C₂×C₄	D₄×D₉	1	1	yes	characteristic
337	C₂×C₄	D₄×D₉	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	9	2	no	—
341	C₈	—	18	1	no	—
342	C₈	C₂×C₄×D₉	1	2	yes	—
343	C₂×C₃	C₄×D₃×D₄	1	1	yes	characteristic
344	C₂×C₃	C₂×#919	1	1	yes	characteristic
345	C₂×C₃	D₃×#135	1	1	yes	characteristic
346	D₃	—	6	2	no	—
347	D₃	—	12	4	no	—
348	C₂²	—	9	2	no	—
349	C₂²	—	18	1	no	—
350	C₂²	—	18	1	no	—
351	C₂²	—	18	4	no	—
352	C₂²	—	36	2	no	—
353	C₂²	—	36	2	no	—
354	C₂²	C₂×D₄×D₉	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	2	4	no	—
357	C₄	—	4	2	no	—
358	C₄	—	9	2	no	—
359	C₄	—	9	2	no	—
360	C₄	—	18	1	no	—
361	C₄	—	36	2	no	—
362	C₄	#63481	1	2	yes	direct product factor
363	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
364	C₄	#41×D₉	1	1	yes	characteristic
365	C₃	C₄×#919	1	1	yes	characteristic
366	C₂	—	18	2	no	—
367	C₂	—	36	4	no	—
368	C₂	C₄×D₄×D₉	1	1	yes	characteristic
369	C₂	C₂×#63481	1	1	yes	characteristic
370	C₂	D₉×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₄×C₉×Q₁₆	C₂
socle	C₂²×C₃	C₂×D₃×D₄
prime radical	C₂×#63481	C₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₄×Q₁₆	D₉
hypocenter	C₉	C₄×#135
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#3701	C₂	semidirect
3. C₄×C₉×Q₁₆	C₂	semidirect
4. Q₁₆×Dic₉	C₂	semidirect
5. C₄×#3675	C₂	semidirect
6. #10588560	C₂	semidirect
7. #10588552	C₂	semidirect
8. #10588484	C₂	semidirect
9. #10588468	C₂	semidirect
10. C₄×D₇₂	C₂	semidirect
11. C₄×C₈×D₉	C₂	semidirect
18. #63481	C₄	direct (N × H)
54. #3701	C₂×C₄	semidirect
63. C₉×Q₁₆	C₂×C₄	semidirect
70. #3675	C₂×C₄	semidirect
83. D₇₂	C₂×C₄	semidirect
85. C₈×D₉	C₂×C₄	semidirect
145. C₄×Q₁₆	D₉	semidirect
197. C₄×C₉	#135	semidirect
203. Dic₉	#135	semidirect
275. Q₁₆	C₄×D₉	semidirect
315. C₉	C₄×#135	semidirect
362. C₄	#63481	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#63481	2
C₄×#3701	2
C₄×C₉×Q₁₆	2
Q₁₆×Dic₉	2
C₄×#3675	2
#10588560	2
#10588552	2
#10588484	2
#10588468	2
C₄×D₇₂	2
C₄×C₈×D₉	2
C₄×#919	3

Subgroups — C4×#63481 #11859975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#63481 #11859975