Subgroups — C₈×#15668034 #36523808

Order 22784.

750 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083574	C₂	1	2	yes	maximal, complemented
2	C₄×#15668034	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₃₅₆	C₂	1	2	yes	maximal, complemented
4	C₈×#13083563	C₂	1	4	yes	maximal, complemented
5	C₈×D₇₁₂	C₂	1	2	yes	maximal, complemented
6	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018635	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₄	1	1	yes	characteristic
10	C₄×#13083574	C₂²	1	2	yes	—
11	C₈×Dic₁₇₈	C₂²	1	4	yes	—
12	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
13	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
14	C₄×#13083563	C₄	1	2	yes	—
15	C₄×#13083563	C₂²	1	2	yes	—
16	C₄×D₇₁₂	C₄	1	1	yes	characteristic
17	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
18	C₈×D₃₅₆	C₂²	1	4	yes	—
19	C₈×C₂²⋊D₈₉	—	2	4	no	—
20	#19151651	C₂²	1	2	yes	—
21	#19151649	—	2	2	no	—
22	#19151636	C₄	1	1	yes	characteristic
23	#19151595	C₄	1	2	yes	—
24	C₄×C₈×D₈₉	—	2	2	no	—
25	#19151351	C₂²	1	2	yes	—
26	#19151349	C₂²	1	2	yes	—
27	#19151330	C₂²	1	4	yes	—
28	#19151327	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₈²×C₈₉	C₂²	1	2	yes	—
33	C₈²×C₈₉	C₂²	1	2	yes	—
34	C₂×#13083574	C₂×C₄	1	2	yes	—
35	#15668034	C₈	1	8	yes	direct product factor
36	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083563	C₂×C₄	1	2	yes	—
38	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₈₉	—	2	4	no	—
40	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₇₈	C₂³	1	2	yes	—
42	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
43	C₄×D₃₅₆	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₈₉	—	2	2	no	—
45	C₄×C₂²⋊D₈₉	—	2	2	no	—
46	#15667971	C₂×C₄	1	2	yes	—
47	#15667969	—	2	2	no	—
48	C₄²×D₈₉	—	2	2	no	—
49	#15667946	C₂×C₄	1	1	yes	characteristic
50	#15667946	C₂×C₄	1	1	yes	characteristic
51	#15667944	C₂×C₄	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	4	yes	—
53	#15667941	—	2	4	no	—
54	#15667937	C₂×C₄	1	2	yes	—
55	#15667937	C₂×C₄	1	2	yes	—
56	#15667934	C₂×C₄	1	2	yes	—
57	#15667934	C₂×C₄	1	2	yes	—
58	C₂²⋊#10623494	—	2	2	no	—
59	#15667924	—	2	4	no	—
60	C₄×#10623494	—	2	2	no	—
61	C₈×Dic₈₉	—	2	4	no	—
62	C₂²×C₈×C₈₉	—	2	1	no	—
63	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	D₄	1	2	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	—	2	1	no	—
68	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	2	yes	—
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	2	yes	—
76	#13083574	—	2	4	no	—
77	#13083574	C₂×C₈	1	8	yes	complemented
78	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
79	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
80	C₂×C₂²⋊D₈₉	—	2	2	no	—
81	C₂×C₄×D₈₉	—	2	2	no	—
82	C₂×C₄×D₈₉	—	2	2	no	—
83	C₂×C₄×D₈₉	—	2	2	no	—
84	Dic₃₅₆	C₂×C₈	1	4	yes	complemented
85	#13083563	C₂×C₈	1	8	yes	complemented
86	D₇₁₂	C₂×C₈	1	4	yes	complemented
87	C₂×#10623494	—	2	4	no	—
88	C₈×D₈₉	—	4	8	no	—
89	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
91	#13083553	—	2	2	no	—
92	#13083553	—	2	2	no	—
93	C₂²⋊Dic₈₉	—	2	2	no	—
94	#13083551	—	2	2	no	—
95	#13083551	—	2	2	no	—
96	C₄×Dic₈₉	—	2	2	no	—
97	C₄×Dic₈₉	—	2	2	no	—
98	C₄×Dic₈₉	—	2	2	no	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₈₉	—	2	4	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	2	4	no	—
106	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	complemented
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₈₉	#131	1	2	yes	—
126	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	2	8	no	—
130	C₂×Dic₈₉	—	2	2	no	—
131	C₂×Dic₈₉	—	2	2	no	—
132	C₂×Dic₈₉	—	2	2	no	—
133	D₃₅₆	—	2	2	no	—
134	D₃₅₆	C₂²×C₈	1	4	yes	—
135	C₄×D₈₉	—	2	8	no	—
136	C₄×D₈₉	—	4	4	no	—
137	C₄×D₈₉	—	4	4	no	—
138	Dic₁₇₈	—	2	2	no	—
139	Dic₁₇₈	C₂²×C₈	1	4	yes	—
140	C₈×C₈₉	—	2	4	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	2	no	—
144	C₈×C₈₉	C₂²×C₈	1	2	yes	—
145	C₈×C₈₉	C₂²×C₈	1	2	yes	—
146	C₈×C₈₉	#135	1	4	yes	complemented
147	#10623494	—	4	4	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	C₈×D₄	1	2	yes	—
151	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×D₈₉	—	2	4	no	—
153	C₂×D₈₉	—	4	2	no	—
154	C₂×D₈₉	—	4	2	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	C₈×D₄	1	2	yes	—
158	C₄×C₈₉	#443	1	2	yes	—
159	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
160	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#135	1	2	yes	—
162	Dic₈₉	—	2	4	no	—
163	Dic₈₉	—	4	2	no	—
164	Dic₈₉	—	4	2	no	—
165	C₈×#135	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	2	2	no	—
167	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
168	C₂×C₈₉	C₄×#135	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
170	D₈₉	—	4	4	no	—
171	C₈×#41	—	89	2	no	—
172	C₄×#135	—	89	1	no	—
173	C₈×Q₁₆	—	89	2	no	—
174	C₈×SD₁₆	—	89	4	no	—
175	C₈×D₈	—	89	2	no	—
176	#1564	—	89	1	no	—
177	#1476	—	89	2	no	—
178	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
179	C₈₉	C₈×#135	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	—	89	1	no	—
181	C₄×#41	—	89	2	no	—
182	C₈×Q₈	—	89	4	no	—
183	C₄×Q₁₆	—	89	1	no	—
184	C₄×Q₁₆	—	89	1	no	—
185	C₄×SD₁₆	—	89	2	no	—
186	C₄×SD₁₆	—	89	2	no	—
187	C₄×D₈	—	89	1	no	—
188	C₄×D₈	—	89	1	no	—
189	C₈×D₄	—	89	4	no	—
190	C₈×D₄	—	178	4	no	—
191	#431	—	89	2	no	—
192	#431	—	178	2	no	—
193	#427	—	89	1	no	—
194	#416	—	89	2	no	—
195	C₂×C₄×C₈	—	178	2	no	—
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
199	#335	—	89	2	no	—
200	#334	—	89	2	no	—
201	Q₈⋊C₈	—	89	4	no	—
202	D₄⋊C₈	—	89	4	no	—
203	C₈²	C₂×D₈₉	1	2	yes	—
204	C₈²	C₂×D₈₉	1	2	yes	—
205	C₂×#41	—	89	2	no	—
206	#135	—	89	8	no	—
207	C₂×Q₁₆	—	89	1	no	—
208	C₂×SD₁₆	—	89	2	no	—
209	C₂×D₈	—	89	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	178	4	no	—
212	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
213	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
214	C₂²×C₈	D₃₅₆	1	2	yes	—
215	C₄×Q₈	—	89	2	no	—
216	C₄×Q₈	—	89	2	no	—
217	C₄×D₄	—	89	2	no	—
218	C₄×D₄	—	89	2	no	—
219	C₄×D₄	—	178	2	no	—
220	C₄×D₄	—	178	2	no	—
221	#117	—	89	2	no	—
222	#117	—	178	2	no	—
223	C₂×C₄²	—	178	2	no	—
224	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
225	#107	—	89	1	no	—
226	#107	—	89	1	no	—
227	#106	—	89	2	no	—
228	C₄⋊C₈	—	89	4	no	—
229	C₄⋊C₈	—	178	4	no	—
230	#103	—	89	2	no	—
231	#103	—	89	2	no	—
232	#102	—	89	2	no	—
233	#102	—	89	2	no	—
234	C₂²⋊C₈	—	178	2	no	—
235	C₂²⋊C₈	—	178	4	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	178	4	no	—
238	C₄×C₈	—	178	2	no	—
239	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
240	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
241	C₄×C₈	D₃₅₆	1	1	yes	characteristic
242	C₄×C₈	D₃₅₆	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	2	yes	—
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₈₉	1	2	yes	—
247	#41	—	89	8	no	—
248	#41	—	178	4	no	—
249	C₂×Q₈	—	89	2	no	—
250	C₂×D₄	—	89	2	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂²×C₄	—	178	2	no	—
253	C₂²×C₄	—	178	2	no	—
254	C₂²×C₄	—	178	2	no	—
255	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
257	C₂²×C₄	#13083574	1	1	yes	characteristic
258	Q₁₆	—	89	4	no	—
259	SD₁₆	—	89	8	no	—
260	D₈	—	89	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	356	8	no	—
266	C₂×C₈	—	178	4	no	—
267	C₂×C₈	C₈×D₈₉	1	4	yes	complemented
268	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
272	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
273	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
274	C₄⋊C₄	—	89	2	no	—
275	C₄⋊C₄	—	89	2	no	—
276	C₄⋊C₄	—	178	2	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₂²⋊C₄	—	178	2	no	—
279	C₂²⋊C₄	—	178	2	no	—
280	C₂²⋊C₄	—	178	2	no	—
281	C₄²	—	178	2	no	—
282	C₄²	—	178	2	no	—
283	C₄²	—	178	2	no	—
284	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
286	C₄²	#13083574	1	1	yes	characteristic
287	C₄²	C₂³×D₈₉	1	1	yes	characteristic
288	C₂³	—	178	2	no	—
289	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
290	Q₈	—	89	4	no	—
291	Q₈	—	178	2	no	—
292	D₄	—	89	4	no	—
293	D₄	—	178	8	no	—
294	D₄	—	178	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	356	4	no	—
298	C₂×C₄	—	356	4	no	—
299	C₂×C₄	—	178	2	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	—	178	2	no	—
302	C₂×C₄	—	178	8	no	—
303	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
304	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
305	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
307	C₂×C₄	#15668018	1	2	yes	—
308	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
310	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	356	4	no	—
316	C₈	C₂×C₈×D₈₉	1	2	yes	—
317	C₈	C₂×C₈×D₈₉	1	2	yes	—
318	C₈	#15668034	1	4	yes	direct product factor
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	356	2	no	—
322	C₂²	—	356	2	no	—
323	C₂²	—	178	4	no	—
324	C₂²	C₈×D₃₅₆	1	2	yes	—
325	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	356	2	no	—
329	C₄	—	356	2	no	—
330	C₄	—	178	4	no	—
331	C₄	C₈×D₃₅₆	1	2	yes	—
332	C₄	#19151716	1	2	yes	—
333	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
334	C₄	C₂×#15668018	1	1	yes	characteristic
335	C₄	C₂×#15668034	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	356	4	no	—
338	C₂	C₂×C₈×D₃₅₆	1	1	yes	characteristic
339	C₂	C₄×#15668034	1	1	yes	characteristic
340	C₂	C₂×#19151716	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	D₃₅₆
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668034	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	C₈×#135
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. C₈×#13083574	C₂	semidirect
3. C₈×Dic₃₅₆	C₂	semidirect
4. C₈×#13083563	C₂	semidirect
5. C₈×D₇₁₂	C₂	semidirect
6. #26019000	C₂	semidirect
7. #26018635	C₂	semidirect
8. C₂×C₈²×C₈₉	C₂	semidirect
35. #15668034	C₈	direct (N × H)
77. #13083574	C₂×C₈	semidirect
84. Dic₃₅₆	C₂×C₈	semidirect
85. #13083563	C₂×C₈	semidirect
86. D₇₁₂	C₂×C₈	semidirect
106. C₂×C₈×C₈₉	C₂×C₈	semidirect
146. C₈×C₈₉	#135	semidirect
178. C₂×C₈²	D₈₉	semidirect
179. C₈₉	C₈×#135	semidirect
267. C₂×C₈	C₈×D₈₉	semidirect
318. C₈	#15668034	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#13083574	2
C₄×#15668034	2
C₈×Dic₃₅₆	2
C₈×#13083563	2
C₈×D₇₁₂	2
#26019000	2
#26018635	2
C₂×C₈²×C₈₉	2
C₈×#135	89

Subgroups — C8×#15668034 #36523808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#15668034 #36523808