Subgroups — C₈×#3715 #11777197

Order 1152.

397 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#3715	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589098	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589083	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589080	C₂	1	1	yes	maximal, characteristic
8	C₈×#215	—	3	1	no	maximal
9	C₈×#214	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	—	2	1	no	—
11	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
12	C₃×C₄×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×#3715	C₄	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃×(C₂×D₃)⋊C₈	—	2	1	no	—
18	C₃×Dic₃⋊C₈	—	2	1	no	—
19	C₃×C₈×Dic₃	—	2	1	no	—
20	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	#63574	C₄	1	1	yes	characteristic
23	#63565	C₄	1	1	yes	characteristic
24	#63562	C₄	1	1	yes	characteristic
25	C₃×C₈×Q₈	—	3	1	no	—
26	C₃×C₈×D₄	—	3	1	no	—
27	C₄×#215	—	3	1	no	—
28	C₄×#214	—	3	1	no	—
29	C₈×D₁₂	D₃	1	1	yes	characteristic
30	C₈×Dic₆	D₃	1	1	yes	characteristic, complemented
31	#4535	—	3	1	no	—
32	D₄⋊(C₃⋊C₈)	—	3	1	no	—
33	#4484	—	3	1	no	—
34	#4483	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4480	—	3	1	no	—
37	C₈×C₃⋊C₈	—	3	1	no	—
38	C₈×C₃⋊C₈	—	3	1	no	—
39	C₈×C₃⋊C₈	—	6	1	no	—
40	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	—	2	1	no	—
42	C₂×C₃×Dic₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
45	C₃²×C₄²	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
47	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
49	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
50	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₃×Dic₃⋊C₄	—	2	1	no	—
53	C₃×C₄×Dic₃	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	—	2	1	no	—
56	C₃×D₃×C₈	—	4	2	no	—
57	#3715	C₈	1	4	yes	direct product factor
58	C₈×SD₁₆	—	9	1	no	Sylow
59	C₂²×C₃×C₈	—	6	1	no	—
60	C₃×C₄×Q₈	—	3	1	no	—
61	C₃×C₄×D₄	—	3	1	no	—
62	C₂×#215	—	3	1	no	—
63	C₂×#214	—	3	1	no	—
64	C₂×D₃×C₈	—	2	1	no	—
65	C₄×D₁₂	C₂×D₃	1	1	yes	characteristic
66	C₄×Dic₆	C₂×D₃	1	1	yes	characteristic
67	C₃×C₄⋊C₈	—	3	1	no	—
68	C₃×C₄⋊C₈	—	3	1	no	—
69	C₃×C₄⋊C₈	—	6	1	no	—
70	C₃×C₂²⋊C₈	—	6	1	no	—
71	C₃×C₄×C₈	—	2	1	no	—
72	C₃×C₄×C₈	—	6	1	no	—
73	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
75	#835	—	3	1	no	—
76	#832	—	3	1	no	—
77	(C₂×D₃)⋊C₈	—	2	1	no	—
78	Dic₃⋊C₈	—	2	1	no	—
79	C₈×Dic₃	—	2	1	no	—
80	#810	—	3	1	no	—
81	#809	—	3	1	no	—
82	#808	—	3	1	no	—
83	#808	—	3	1	no	—
84	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
85	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
86	C₄×C₃⋊C₈	—	3	1	no	—
87	C₄×C₃⋊C₈	—	3	1	no	—
88	C₄×C₃⋊C₈	—	3	1	no	—
89	C₄×C₃⋊C₈	—	3	1	no	—
90	C₄×C₃⋊C₈	—	6	1	no	—
91	C₄×C₃⋊C₈	—	6	1	no	—
92	C₂²×C₃×D₃	—	2	1	no	—
93	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	#41	1	1	yes	characteristic
96	C₂×C₃×Dic₃	—	2	1	no	—
97	C₂×C₃×Dic₃	—	2	1	no	—
98	C₂×C₃×Dic₃	—	2	1	no	—
99	C₃×D₁₂	—	2	1	no	—
100	C₃×D₁₂	C₂×C₈	1	2	yes	—
101	C₃×C₄×D₃	—	4	1	no	—
102	C₃×C₄×D₃	—	4	1	no	—
103	C₃×Dic₆	—	2	1	no	—
104	C₃×Dic₆	C₂×C₈	1	2	yes	complemented
105	C₃²×C₈	—	2	1	no	—
106	C₃²×C₈	SD₁₆	1	2	yes	complemented
107	C₃⋊(C₃⋊C₈)	—	2	2	no	—
108	C₃⋊(C₃⋊C₈)	—	2	1	no	—
109	C₃⋊(C₃⋊C₈)	C₂×C₈	1	2	yes	complemented
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	—	4	1	no	—
112	C₈×Q₈	—	3	1	no	—
113	C₄×SD₁₆	—	9	1	no	—
114	C₈×D₄	—	3	1	no	—
115	#334	—	9	1	no	—
116	Q₈⋊C₈	—	9	1	no	—
117	D₄⋊C₈	—	9	1	no	—
118	C₈²	—	9	1	no	—
119	C₂×C₃×Q₈	—	3	1	no	—
120	C₂×C₃×D₄	—	3	1	no	—
121	C₂²×C₃×C₄	—	6	1	no	—
122	C₂×D₁₂	C₄×D₃	1	1	yes	characteristic
123	C₂×C₄×D₃	—	2	1	no	—
124	C₂×Dic₆	C₄×D₃	1	1	yes	characteristic
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	6	1	no	—
128	C₂×C₃×C₈	—	6	1	no	—
129	C₂×C₃×C₈	—	12	2	no	—
130	C₂×C₃×C₈	#67	1	1	yes	characteristic
131	C₂×C₃×C₈	#67	1	1	yes	characteristic
132	C₂×C₃×C₈	#67	1	1	yes	characteristic
133	C₂×C₃×C₈	#67	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	3	1	no	—
135	C₃×C₄⋊C₄	—	3	1	no	—
136	C₃×C₄⋊C₄	—	6	1	no	—
137	C₃×C₂²⋊C₄	—	6	1	no	—
138	C₃×C₄²	—	2	1	no	—
139	C₃×C₄²	—	6	1	no	—
140	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
141	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
142	#215	—	3	4	no	—
143	#214	—	3	4	no	—
144	(C₂×D₃)⋊C₄	—	2	1	no	—
145	C₄⋊Dic₃	C₄×D₃	1	1	yes	characteristic
146	C₄⋊Dic₃	C₄×D₃	1	1	yes	characteristic
147	Dic₃⋊C₄	—	2	1	no	—
148	C₄×Dic₃	—	2	1	no	—
149	C₂×C₃⋊C₈	—	2	1	no	—
150	C₂×C₃⋊C₈	—	2	1	no	—
151	C₂×C₃⋊C₈	—	3	1	no	—
152	C₂×C₃⋊C₈	—	3	1	no	—
153	C₂×C₃⋊C₈	—	3	1	no	—
154	C₂×C₃⋊C₈	—	3	1	no	—
155	C₂×C₃⋊C₈	—	6	1	no	—
156	C₂×C₃⋊C₈	—	6	1	no	—
157	C₂×C₃⋊C₈	—	6	2	no	—
158	C₂×C₃⋊C₈	—	6	1	no	—
159	C₂×C₃⋊C₈	—	6	1	no	—
160	D₃×C₈	—	4	2	no	—
161	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
162	C₂×C₃×D₃	—	2	2	no	—
163	C₂×C₃×D₃	—	4	1	no	—
164	C₂×C₃×D₃	—	4	1	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
167	C₃²×C₄	#131	1	1	yes	characteristic
168	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
169	C₃²×C₄	#135	1	1	yes	characteristic
170	C₃×Dic₃	—	2	2	no	—
171	C₃×Dic₃	—	4	1	no	—
172	C₃×Dic₃	—	4	1	no	—
173	C₂×SD₁₆	—	9	1	no	—
174	C₂²×C₈	—	6	1	no	—
175	C₄×Q₈	—	3	1	no	—
176	C₄×D₄	—	3	1	no	—
177	#106	—	9	1	no	—
178	C₄⋊C₈	—	3	1	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₄⋊C₈	—	6	1	no	—
181	#103	—	9	1	no	—
182	#102	—	9	1	no	—
183	C₂²⋊C₈	—	6	1	no	—
184	C₄×C₈	—	6	1	no	—
185	C₄×C₈	—	9	1	no	—
186	C₄×C₈	—	9	1	no	—
187	C₄×C₈	D₃²	1	1	yes	characteristic
188	C₂³×C₃	—	6	1	no	—
189	C₂²×D₃	—	2	1	no	—
190	C₃×Q₈	—	3	2	no	—
191	C₃×Q₈	—	6	1	no	—
192	C₃×D₄	—	3	2	no	—
193	C₃×D₄	—	6	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	6	1	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	6	1	no	—
200	C₂×C₃×C₄	—	12	1	no	—
201	C₂×C₃×C₄	—	12	1	no	—
202	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	#235	1	1	yes	characteristic
205	C₂×C₃×C₄	#235	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
208	C₂×Dic₃	—	2	1	no	—
209	C₂×Dic₃	—	2	1	no	—
210	C₂×Dic₃	—	2	1	no	—
211	D₁₂	—	2	1	no	—
212	D₁₂	D₃×C₈	1	2	yes	—
213	C₄×D₃	—	4	1	no	—
214	C₄×D₃	—	4	1	no	—
215	Dic₆	—	2	1	no	—
216	Dic₆	D₃×C₈	1	2	yes	complemented
217	C₃×C₈	—	2	2	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	2	1	no	—
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	—	12	1	no	—
222	C₃×C₈	—	12	1	no	—
223	C₃×C₈	#214	1	2	yes	complemented
224	C₃×C₈	#215	1	2	yes	complemented
225	C₃⋊C₈	—	3	2	no	—
226	C₃⋊C₈	—	3	2	no	—
227	C₃⋊C₈	—	4	1	no	—
228	C₃⋊C₈	—	4	1	no	—
229	C₃⋊C₈	—	6	2	no	—
230	C₃⋊C₈	—	6	4	no	—
231	C₃⋊C₈	—	6	2	no	—
232	C₃⋊C₈	—	6	2	no	—
233	C₃⋊C₈	—	6	2	no	—
234	C₃⋊C₈	—	6	1	no	—
235	C₃⋊C₈	—	6	1	no	—
236	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
237	C₂×C₃²	C₄×SD₁₆	1	1	yes	characteristic
238	C₂×C₃²	#443	1	1	yes	characteristic
239	C₃×D₃	—	4	2	no	—
240	C₂×Q₈	—	3	1	no	—
241	C₂×D₄	—	3	1	no	—
242	C₂²×C₄	—	6	1	no	—
243	SD₁₆	—	9	4	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	6	1	no	—
246	C₂×C₈	—	9	1	no	—
247	C₂×C₈	—	9	1	no	—
248	C₂×C₈	—	12	2	no	—
249	C₂×C₈	—	18	1	no	—
250	C₂×C₈	#625	1	1	yes	characteristic
251	C₂×C₈	#625	1	1	yes	characteristic
252	C₄⋊C₄	—	3	1	no	—
253	C₄⋊C₄	—	3	1	no	—
254	C₄⋊C₄	—	6	1	no	—
255	C₂²⋊C₄	—	6	1	no	—
256	C₄²	—	6	1	no	—
257	C₄²	C₂×D₃²	1	1	yes	characteristic
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	6	2	no	—
260	C₂²×C₃	—	12	1	no	—
261	C₂²×C₃	—	12	1	no	—
262	C₂²×C₃	C₄×#67	1	1	yes	characteristic
263	C₂²×C₃	C₄×#67	1	1	yes	characteristic
264	C₂×D₃	—	2	2	no	—
265	C₂×D₃	—	4	1	no	—
266	C₂×D₃	—	4	1	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	6	2	no	—
275	C₃×C₄	—	12	1	no	—
276	C₃×C₄	—	12	1	no	—
277	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
278	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
279	C₃×C₄	#901	1	1	yes	characteristic
280	C₃×C₄	#901	1	1	yes	characteristic
281	C₃×C₄	C₂×#214	1	1	yes	characteristic
282	C₃×C₄	C₂×#215	1	1	yes	characteristic
283	C₃×C₄	#950	1	1	yes	characteristic
284	C₃×C₄	#950	1	1	yes	characteristic
285	Dic₃	—	2	2	no	—
286	Dic₃	—	4	1	no	—
287	Dic₃	—	4	1	no	—
288	C₃²	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	6	1	no	—
290	Q₈	—	3	2	no	—
291	Q₈	—	6	1	no	—
292	D₄	—	3	2	no	—
293	D₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	6	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	12	1	no	—
298	C₂×C₄	—	12	1	no	—
299	C₂×C₄	#3798	1	1	yes	characteristic
300	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
301	C₂×C₄	C₂×#625	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	9	2	no	—
304	C₈	—	12	1	no	—
305	C₈	—	12	1	no	—
306	C₈	—	18	2	no	—
307	C₈	—	18	1	no	—
308	C₈	#3715	1	2	yes	direct product factor
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	12	2	no	—
313	C₂×C₃	C₄×#214	1	1	yes	characteristic
314	C₂×C₃	C₄×#215	1	1	yes	characteristic
315	C₂×C₃	C₈×#67	1	1	yes	characteristic
316	C₂×C₃	C₈×#67	1	1	yes	characteristic
317	C₂×C₃	#5144	1	1	yes	characteristic
318	C₂×C₃	#5144	1	1	yes	characteristic
319	D₃	—	4	2	no	—
320	C₂²	—	6	2	no	—
321	C₂²	—	12	1	no	—
322	C₂²	—	12	1	no	—
323	C₂²	C₄×#625	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	6	2	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	D₃²×C₈	1	1	yes	characteristic
329	C₄	#63803	1	1	yes	characteristic
330	C₄	#63822	1	1	yes	characteristic
331	C₄	C₂×#3715	1	1	yes	characteristic
332	C₃	—	2	1	no	—
333	C₃	C₈×#214	1	1	yes	characteristic, complemented
334	C₃	C₈×#215	1	1	yes	characteristic, complemented
335	C₂	—	12	2	no	—
336	C₂	C₄×#3715	1	1	yes	characteristic
337	C₂	C₈×#625	1	1	yes	characteristic
338	C₂	#10590921	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#625
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₁₂	C₂×C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	C₈×SD₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×Dic₆	C₂	semidirect
4. C₈×C₃⋊(C₃⋊C₈)	C₂	semidirect
5. #10589098	C₂	semidirect
6. #10589083	C₂	semidirect
30. C₈×Dic₆	D₃	semidirect
57. #3715	C₈	direct (N × H)
104. C₃×Dic₆	C₂×C₈	semidirect
106. C₃²×C₈	SD₁₆	semidirect
109. C₃⋊(C₃⋊C₈)	C₂×C₈	semidirect
216. Dic₆	D₃×C₈	semidirect
223. C₃×C₈	#214	semidirect
224. C₃×C₈	#215	semidirect
288. C₃²	C₈×SD₁₆	semidirect
308. C₈	#3715	direct (N × H)
333. C₃	C₈×#214	semidirect
334. C₃	C₈×#215	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×D₁₂	2
C₃×C₈×Dic₆	2
C₄×#3715	2
C₈×C₃⋊(C₃⋊C₈)	2
#10589098	2
#10589083	2
#10589080	2
C₈×#215	3
C₈×#214	3

Subgroups — C8×#3715 #11777197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#3715 #11777197