Subgroups — C₈×#3716 #11777196

Order 1152.

411 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#634	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₄×#3716	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589097	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589085	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589081	C₂	1	1	yes	maximal, characteristic
8	C₈×#214	—	3	1	no	maximal
9	C₈×#204	—	3	1	no	maximal
10	C₄×#634	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₃×C₈	—	2	1	no	—
12	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×#3716	C₄	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#63641	—	2	1	no	—
16	C₈×C₃⋊Dic₃	—	2	1	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
18	C₃×(C₂×D₃)⋊C₈	—	2	1	no	—
19	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#63573	C₄	1	1	yes	characteristic
23	#63567	C₄	1	1	yes	characteristic
24	#63563	C₄	1	1	yes	characteristic
25	C₃×C₈×D₄	—	3	1	no	—
26	C₄×#214	—	3	1	no	—
27	C₄×#204	—	3	1	no	—
28	C₈×D₁₂	D₃	1	1	yes	characteristic
29	C₈×Dic₆	—	3	1	no	—
30	C₈×Dic₆	—	3	1	no	—
31	C₈×Dic₆	—	6	1	no	—
32	C₃×C₈²	—	3	1	no	—
33	D₄⋊(C₃⋊C₈)	—	3	1	no	—
34	#4484	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4459	—	3	1	no	—
37	#4457	—	3	1	no	—
38	#4456	—	3	1	no	—
39	C₈×C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
40	C₂×#634	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	—	2	1	no	—
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	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
47	#3750	—	2	1	no	—
48	C₄×C₃⋊Dic₃	—	2	1	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
50	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
51	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₃×D₃×C₈	—	4	2	no	—
57	#3716	C₈	1	4	yes	direct product factor
58	C₈×SD₁₆	—	9	1	no	Sylow
59	C₂²×C₃×C₈	—	6	1	no	—
60	C₃×C₄×D₄	—	3	1	no	—
61	C₂×#214	—	3	1	no	—
62	C₂×#204	—	3	1	no	—
63	C₂×D₃×C₈	—	2	1	no	—
64	C₄×D₁₂	C₂×D₃	1	1	yes	characteristic
65	C₄×Dic₆	—	3	1	no	—
66	C₄×Dic₆	—	3	1	no	—
67	C₄×Dic₆	—	6	1	no	—
68	C₃×C₄⋊C₈	—	3	1	no	—
69	C₃×C₂²⋊C₈	—	6	1	no	—
70	C₃×C₄×C₈	—	2	1	no	—
71	C₃×C₄×C₈	—	3	1	no	—
72	C₃×C₄×C₈	—	3	1	no	—
73	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
75	#832	—	3	1	no	—
76	#821	—	3	1	no	—
77	(C₂×D₃)⋊C₈	—	2	1	no	—
78	#817	—	3	1	no	—
79	#816	—	3	1	no	—
80	Dic₃⋊C₈	—	6	2	no	—
81	Dic₃⋊C₈	—	6	1	no	—
82	Dic₃⋊C₈	—	6	1	no	—
83	C₈×Dic₃	—	6	2	no	—
84	C₈×Dic₃	—	6	1	no	—
85	C₈×Dic₃	—	6	1	no	—
86	#810	—	3	1	no	—
87	#808	—	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)	C₂×D₃	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
93	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
94	C₂²×C₃×D₃	—	2	1	no	—
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	#41	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	—	2	1	no	—
99	C₂×C₃⋊Dic₃	—	2	1	no	—
100	#634	—	2	1	no	—
101	#634	C₂×C₈	1	2	yes	complemented
102	C₂×C₃×Dic₃	—	2	1	no	—
103	C₃×D₁₂	—	2	1	no	—
104	C₃×D₁₂	C₂×C₈	1	2	yes	—
105	C₃×C₄×D₃	—	4	1	no	—
106	C₃×C₄×D₃	—	4	1	no	—
107	C₃²×C₈	—	2	1	no	—
108	C₃²×C₈	SD₁₆	1	2	yes	complemented
109	C₃⋊(C₃⋊C₈)	—	4	1	no	—
110	C₃×C₃⋊C₈	—	2	2	no	—
111	C₃×C₃⋊C₈	—	2	1	no	—
112	C₃×C₃⋊C₈	—	4	1	no	—
113	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	complemented
114	C₈×Q₈	—	9	1	no	—
115	C₄×SD₁₆	—	9	1	no	—
116	C₈×D₄	—	3	1	no	—
117	#334	—	9	1	no	—
118	Q₈⋊C₈	—	9	1	no	—
119	D₄⋊C₈	—	9	1	no	—
120	C₈²	—	3	1	no	—
121	C₂×C₃×D₄	—	3	1	no	—
122	C₂²×C₃×C₄	—	6	1	no	—
123	C₂×D₁₂	C₄×D₃	1	1	yes	characteristic
124	C₂×C₄×D₃	—	2	1	no	—
125	C₂×Dic₆	—	3	1	no	—
126	C₂×Dic₆	—	3	1	no	—
127	C₂×Dic₆	—	6	1	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	3	1	no	—
131	C₂×C₃×C₈	—	3	1	no	—
132	C₂×C₃×C₈	—	6	1	no	—
133	C₂×C₃×C₈	—	6	1	no	—
134	C₂×C₃×C₈	—	12	2	no	—
135	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
136	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
137	C₂×C₃×C₈	#67	1	1	yes	characteristic
138	C₂×C₃×C₈	#67	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	3	1	no	—
140	C₃×C₂²⋊C₄	—	6	1	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
143	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
144	#214	—	3	4	no	—
145	(C₂×D₃)⋊C₄	—	2	1	no	—
146	C₄⋊Dic₃	—	3	1	no	—
147	C₄⋊Dic₃	—	3	1	no	—
148	C₄⋊Dic₃	—	6	1	no	—
149	C₄⋊Dic₃	C₄×D₃	1	1	yes	characteristic
150	Dic₃⋊C₄	—	6	2	no	—
151	Dic₃⋊C₄	—	6	1	no	—
152	Dic₃⋊C₄	—	6	1	no	—
153	C₄×Dic₃	—	6	2	no	—
154	C₄×Dic₃	—	6	1	no	—
155	C₄×Dic₃	—	6	1	no	—
156	C₂×C₃⋊C₈	—	2	1	no	—
157	C₂×C₃⋊C₈	—	2	1	no	—
158	C₂×C₃⋊C₈	—	6	2	no	—
159	C₂×C₃⋊C₈	—	6	1	no	—
160	C₂×C₃⋊C₈	—	6	1	no	—
161	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
162	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
163	#204	—	3	4	no	—
164	D₃×C₈	—	4	2	no	—
165	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
166	C₂×C₃×D₃	—	2	2	no	—
167	C₂×C₃×D₃	—	4	1	no	—
168	C₂×C₃×D₃	—	4	1	no	—
169	C₃²×C₄	—	2	1	no	—
170	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
171	C₃²×C₄	#131	1	1	yes	characteristic
172	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
173	C₃²×C₄	#135	1	1	yes	characteristic
174	C₃⋊Dic₃	—	2	2	no	—
175	C₃⋊Dic₃	—	4	1	no	—
176	C₃×Dic₃	—	4	1	no	—
177	C₂×SD₁₆	—	9	1	no	—
178	C₂²×C₈	—	6	1	no	—
179	C₄×Q₈	—	9	1	no	—
180	C₄×D₄	—	3	1	no	—
181	#106	—	9	1	no	—
182	C₄⋊C₈	—	3	1	no	—
183	C₄⋊C₈	—	9	1	no	—
184	C₄⋊C₈	—	18	1	no	—
185	#103	—	9	1	no	—
186	#102	—	9	1	no	—
187	C₂²⋊C₈	—	6	1	no	—
188	C₄×C₈	—	3	1	no	—
189	C₄×C₈	—	3	1	no	—
190	C₄×C₈	—	18	1	no	—
191	C₄×C₈	D₃²	1	1	yes	characteristic
192	C₂³×C₃	—	6	1	no	—
193	C₂²×D₃	—	2	1	no	—
194	C₃×D₄	—	3	2	no	—
195	C₃×D₄	—	6	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	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	C₂×D₁₂	1	1	yes	characteristic
205	C₂×C₃×C₄	#235	1	1	yes	characteristic
206	C₂×C₃×C₄	#235	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
208	C₂×Dic₃	—	2	1	no	—
209	C₂×Dic₃	—	6	2	no	—
210	C₂×Dic₃	—	6	1	no	—
211	C₂×Dic₃	—	6	1	no	—
212	C₂×Dic₃	—	6	2	no	—
213	C₂×Dic₃	—	6	1	no	—
214	C₂×Dic₃	—	6	1	no	—
215	D₁₂	—	2	1	no	—
216	D₁₂	D₃×C₈	1	2	yes	—
217	C₄×D₃	—	4	1	no	—
218	C₄×D₃	—	4	1	no	—
219	Dic₆	—	3	2	no	—
220	Dic₆	—	3	2	no	—
221	Dic₆	—	6	2	no	—
222	Dic₆	—	6	2	no	—
223	Dic₆	—	6	1	no	—
224	Dic₆	—	6	1	no	—
225	C₃×C₈	—	2	1	no	—
226	C₃×C₈	—	2	1	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	3	2	no	—
229	C₃×C₈	—	4	1	no	—
230	C₃×C₈	—	6	2	no	—
231	C₃×C₈	—	6	1	no	—
232	C₃×C₈	—	12	1	no	—
233	C₃×C₈	#204	1	2	yes	complemented
234	C₃×C₈	#214	1	2	yes	complemented
235	C₃⋊C₈	—	2	2	no	—
236	C₃⋊C₈	—	2	1	no	—
237	C₃⋊C₈	—	4	1	no	—
238	C₃⋊C₈	—	12	2	no	—
239	C₃⋊C₈	—	12	1	no	—
240	C₃⋊C₈	—	12	1	no	—
241	C₃⋊C₈	D₃×C₈	1	2	yes	complemented
242	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
243	C₂×C₃²	C₄×SD₁₆	1	1	yes	characteristic
244	C₂×C₃²	#443	1	1	yes	characteristic
245	C₃×D₃	—	4	2	no	—
246	C₂×Q₈	—	9	1	no	—
247	C₂×D₄	—	3	1	no	—
248	C₂²×C₄	—	6	1	no	—
249	SD₁₆	—	9	4	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	—	3	1	no	—
252	C₂×C₈	—	6	1	no	—
253	C₂×C₈	—	6	1	no	—
254	C₂×C₈	—	12	2	no	—
255	C₂×C₈	—	18	1	no	—
256	C₂×C₈	#626	1	1	yes	characteristic
257	C₂×C₈	#626	1	1	yes	characteristic
258	C₄⋊C₄	—	3	1	no	—
259	C₄⋊C₄	—	9	1	no	—
260	C₄⋊C₄	—	18	1	no	—
261	C₂²⋊C₄	—	6	1	no	—
262	C₄²	—	18	1	no	—
263	C₄²	C₂×D₃²	1	1	yes	characteristic
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	6	2	no	—
266	C₂²×C₃	—	12	1	no	—
267	C₂²×C₃	—	12	1	no	—
268	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
269	C₂²×C₃	C₄×#67	1	1	yes	characteristic
270	C₂×D₃	—	2	2	no	—
271	C₂×D₃	—	4	1	no	—
272	C₂×D₃	—	4	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	12	1	no	—
281	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
282	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
283	C₃×C₄	#901	1	1	yes	characteristic
284	C₃×C₄	#901	1	1	yes	characteristic
285	C₃×C₄	C₂×#204	1	1	yes	characteristic
286	C₃×C₄	#904	1	1	yes	characteristic
287	C₃×C₄	C₂×#214	1	1	yes	characteristic
288	C₃×C₄	#950	1	1	yes	characteristic
289	Dic₃	—	4	1	no	—
290	Dic₃	—	6	4	no	—
291	Dic₃	—	6	2	no	—
292	Dic₃	—	6	2	no	—
293	Dic₃	—	12	2	no	—
294	Dic₃	—	12	1	no	—
295	Dic₃	—	12	1	no	—
296	C₃²	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
297	C₂³	—	6	1	no	—
298	Q₈	—	9	2	no	—
299	Q₈	—	18	1	no	—
300	D₄	—	3	2	no	—
301	D₄	—	6	1	no	—
302	C₂×C₄	—	6	1	no	—
303	C₂×C₄	—	12	1	no	—
304	C₂×C₄	—	12	1	no	—
305	C₂×C₄	—	18	1	no	—
306	C₂×C₄	—	18	1	no	—
307	C₂×C₄	#3798	1	1	yes	characteristic
308	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
309	C₂×C₄	C₂×#626	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	3	2	no	—
312	C₈	—	6	2	no	—
313	C₈	—	6	1	no	—
314	C₈	—	12	1	no	—
315	C₈	—	36	1	no	—
316	C₈	#3716	1	2	yes	direct product factor
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	12	2	no	—
321	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
322	C₂×C₃	C₄×#204	1	1	yes	characteristic
323	C₂×C₃	#4700	1	1	yes	characteristic
324	C₂×C₃	C₄×#214	1	1	yes	characteristic
325	C₂×C₃	C₈×#67	1	1	yes	characteristic
326	C₂×C₃	#5144	1	1	yes	characteristic
327	D₃	—	4	2	no	—
328	C₂²	—	6	2	no	—
329	C₂²	—	12	1	no	—
330	C₂²	—	12	1	no	—
331	C₂²	C₄×#626	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	12	1	no	—
334	C₄	—	18	2	no	—
335	C₄	—	36	1	no	—
336	C₄	D₃²×C₈	1	1	yes	characteristic
337	C₄	#63803	1	1	yes	characteristic
338	C₄	C₂×#3716	1	1	yes	characteristic
339	C₄	#63829	1	1	yes	characteristic
340	C₃	—	2	1	no	—
341	C₃	C₈×#204	1	1	yes	characteristic, complemented
342	C₃	C₈×#214	1	1	yes	characteristic, complemented
343	C₂	—	12	2	no	—
344	C₂	C₄×#3716	1	1	yes	characteristic
345	C₂	C₈×#626	1	1	yes	characteristic
346	C₂	#10590923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#626
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
1. C₈×#634	C₂	semidirect
4. C₃×C₈×C₃⋊C₈	C₂	semidirect
5. #10589097	C₂	semidirect
6. #10589085	C₂	semidirect
39. C₈×C₃⋊C₈	D₃	semidirect
57. #3716	C₈	direct (N × H)
101. #634	C₂×C₈	semidirect
108. C₃²×C₈	SD₁₆	semidirect
113. C₃×C₃⋊C₈	C₂×C₈	semidirect
233. C₃×C₈	#204	semidirect
234. C₃×C₈	#214	semidirect
241. C₃⋊C₈	D₃×C₈	semidirect
296. C₃²	C₈×SD₁₆	semidirect
316. C₈	#3716	direct (N × H)
341. C₃	C₈×#204	semidirect
342. C₃	C₈×#214	semidirect

Maximal subgroups

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

subgroup	index
C₈×#634	2
C₃×C₈×D₁₂	2
C₄×#3716	2
C₃×C₈×C₃⋊C₈	2
#10589097	2
#10589085	2
#10589081	2
C₈×#214	3
C₈×#204	3

Subgroups — C8×#3716 #11777196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#3716 #11777196