Subgroups — C₂×Q₈×#10621884 #25977015

Order 11008.

2058 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	Q₈×#10621884	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583295	C₂	1	3	yes	maximal, complemented
4	C₂×#14583184	C₂	1	3	yes	maximal, complemented
5	C₂×#14583186	C₂	1	3	yes	maximal
6	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#10621884	C₂	1	3	yes	maximal
8	C₂×Q₈²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₄₃	—	2	1	no	—
10	C₂²×#10621884	C₂²	1	1	yes	characteristic
11	Q₈×D₁₇₂	—	2	4	no	—
12	Q₈×D₁₇₂	C₂²	1	8	yes	—
13	C₂×#13063325	C₂²	1	3	yes	—
14	C₂×#13063305	—	2	3	no	—
15	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
16	#14583295	C₂²	1	24	yes	complemented
17	#14583186	C₂²	1	24	yes	—
18	#14583184	C₂²	1	24	yes	complemented
19	Q₈×C₄₃⋊C₈	C₂²	1	8	yes	complemented
20	C₄×#10621884	C₂²	1	24	yes	—
21	C₂×#13063256	C₂²	1	3	yes	—
22	C₂×#13063256	C₂²	1	6	yes	—
23	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
24	C₂×#13063249	C₂²	1	3	yes	—
25	C₂×#13063245	C₂²	1	3	yes	—
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
27	Q₈²×C₄₃	—	2	4	no	—
28	Q₈²×C₄₃	C₂²	1	8	yes	complemented
29	C₂×C₄⋊Q₈×C₄₃	—	2	3	no	—
30	C₂×C₄⋊Q₈×C₄₃	C₂²	1	3	yes	—
31	C₂×C₄×Q₈×C₄₃	—	2	1	no	—
32	C₂×C₄×Q₈×C₄₃	C₂²	1	3	yes	—
33	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂²×Dic₈₆	—	2	3	no	—
35	C₂×Q₈×D₄₃	—	2	8	no	—
36	C₂×Q₈×D₄₃	—	4	8	no	—
37	C₂²×D₁₇₂	C₂³	1	1	yes	characteristic
38	C₂²×C₄×D₄₃	—	2	3	no	—
39	C₂×#10621884	Q₈	1	8	yes	direct product factor
40	C₂×#10621884	C₂³	1	4	yes	—
41	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
42	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	#13063325	—	2	6	no	—
44	#13063325	C₂³	1	12	yes	—
45	#13063305	—	2	24	no	—
46	C₄×D₁₇₂	—	2	6	no	—
47	C₄×D₁₇₂	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₄₃	—	2	3	no	—
49	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
50	C₂×(C₂×D₄₃)⋊C₄	—	2	3	no	—
51	#13063256	C₂³	1	24	yes	—
52	#13063256	C₂³	1	12	yes	—
53	C₄⋊(C₄₃⋊C₈)	C₂³	1	12	yes	—
54	#13063249	C₂³	1	12	yes	—
55	#13063245	C₂³	1	12	yes	—
56	C₄×C₄₃⋊C₈	C₂³	1	12	yes	—
57	C₂²×Q₈×C₄₃	—	2	3	no	—
58	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₄₃	—	2	24	no	—
61	C₄⋊Q₈×C₄₃	—	2	6	no	—
62	C₄⋊Q₈×C₄₃	C₂³	1	12	yes	—
63	C₄×Q₈×C₄₃	—	2	8	no	—
64	C₄×Q₈×C₄₃	—	2	6	no	—
65	C₄×Q₈×C₄₃	D₄	1	4	yes	—
66	C₄×Q₈×C₄₃	C₂³	1	12	yes	—
67	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
69	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
70	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
71	C₂×C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
72	C₂×C₄²×C₄₃	—	2	3	no	—
73	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
74	C₂³×D₄₃	—	2	1	no	—
75	C₂×Q₈×C₄₃	—	2	1	no	—
76	C₂×Q₈×C₄₃	—	2	12	no	—
77	C₂×Q₈×C₄₃	—	2	2	no	—
78	C₂×Q₈×C₄₃	—	8	6	no	—
79	C₂×Q₈×C₄₃	SD₁₆	1	8	yes	complemented
80	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
81	C₂×Q₈×C₄₃	C₂×Q₈	1	4	yes	complemented
82	C₂×Q₈×C₄₃	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₄₃	—	2	1	no	—
84	C₂²×C₄×C₄₃	—	2	3	no	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
87	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	2	3	no	—
89	Q₈×D₄₃	—	4	32	no	—
90	C₂×D₁₇₂	—	2	1	no	—
91	C₂×D₁₇₂	—	2	2	no	—
92	C₂×D₁₇₂	C₂×Q₈	1	4	yes	—
93	C₂×D₁₇₂	C₂⁴	1	2	yes	—
94	C₂×C₄×D₄₃	—	2	12	no	—
95	C₂×C₄×D₄₃	—	4	12	no	—
96	C₂×Dic₈₆	—	2	12	no	—
97	C₂×Dic₈₆	—	4	12	no	—
98	C₄⋊C₄×C₄₃	—	2	12	no	—
99	C₄⋊C₄×C₄₃	—	2	12	no	—
100	C₄⋊C₄×C₄₃	—	2	3	no	—
101	C₄⋊C₄×C₄₃	C₂×D₄	1	12	yes	—
102	C₄⋊C₄×C₄₃	C₂⁴	1	6	yes	—
103	C₄²×C₄₃	—	2	12	no	—
104	C₄²×C₄₃	C₂×D₄	1	6	yes	—
105	C₄²×C₄₃	C₂⁴	1	6	yes	—
106	#10621884	C₂×Q₈	1	16	yes	direct product factor
107	(C₂×D₄₃)⋊C₄	—	2	12	no	—
108	C₄⋊Dic₄₃	—	2	12	no	—
109	C₄⋊Dic₄₃	—	2	3	no	—
110	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
111	C₂×C₄₃⋊C₈	—	2	6	no	—
112	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	complemented
113	C₂×C₄₃⋊C₈	C₂⁴	1	6	yes	—
114	C₂×C₄₃⋊C₈	C₂⁴	1	2	yes	—
115	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×D₄₃	—	2	4	no	—
117	C₂²×D₄₃	—	2	2	no	—
118	C₂²×D₄₃	—	4	2	no	—
119	C₂²×D₄₃	—	4	2	no	—
120	Q₈×C₄₃	—	2	4	no	—
121	Q₈×C₄₃	—	2	2	no	—
122	Q₈×C₄₃	—	8	12	no	—
123	Q₈×C₄₃	C₂×SD₁₆	1	16	yes	complemented
124	Q₈×C₄₃	C₂²×Q₈	1	4	yes	complemented
125	C₂×C₄×C₄₃	—	2	4	no	—
126	C₂×C₄×C₄₃	—	2	6	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	6	no	—
129	C₂×C₄×C₄₃	—	4	6	no	—
130	C₂×C₄×C₄₃	C₂×SD₁₆	1	12	yes	—
131	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
134	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
135	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
136	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
137	C₂×Dic₄₃	—	2	6	no	—
138	C₂×Dic₄₃	—	4	6	no	—
139	D₁₇₂	—	2	4	no	—
140	D₁₇₂	—	2	2	no	—
141	D₁₇₂	C₂²×Q₈	1	4	yes	—
142	C₄×D₄₃	—	4	24	no	—
143	Dic₈₆	—	4	24	no	—
144	C₄₃⋊C₈	—	2	6	no	—
145	C₄₃⋊C₈	C₂²×Q₈	1	4	yes	complemented
146	C₂×Q₈×SD₁₆	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	D₄×Q₈	1	2	yes	—
148	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
149	C₂²×C₄₃	#577	1	2	yes	—
150	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
151	C₂×D₄₃	—	2	4	no	—
152	C₂×D₄₃	—	4	8	no	—
153	C₂×D₄₃	—	4	2	no	—
154	C₂×D₄₃	—	4	2	no	—
155	C₄×C₄₃	—	2	4	no	—
156	C₄×C₄₃	—	2	6	no	—
157	C₄×C₄₃	—	4	6	no	—
158	C₄×C₄₃	D₄×Q₈	1	2	yes	—
159	C₄×C₄₃	C₂²×SD₁₆	1	12	yes	—
160	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
161	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₄₃	—	4	6	no	—
163	C₂×Q₈²	D₄₃	1	1	yes	characteristic, complemented
164	C₂×D₄×Q₈	—	43	1	no	—
165	Q₈×SD₁₆	—	43	16	no	—
166	C₂×#498	—	43	3	no	—
167	C₂×#475	—	43	3	no	—
168	C₂×#476	—	43	3	no	—
169	C₂×C₈×Q₈	—	43	1	no	—
170	C₂×C₄×SD₁₆	—	43	3	no	—
171	C₂×C₄₃	Q₈×SD₁₆	1	4	yes	complemented
172	C₂×C₄₃	C₂×D₄×Q₈	1	1	yes	characteristic
173	C₂×C₄₃	C₂³×SD₁₆	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#577	1	1	yes	characteristic
175	D₄₃	—	4	4	no	—
176	C₂³×Q₈	—	86	1	no	—
177	C₂²×SD₁₆	—	43	1	no	—
178	Q₈²	—	2	4	no	—
179	Q₈²	C₂×D₄₃	1	8	yes	complemented
180	D₄×Q₈	—	43	8	no	—
181	D₄×Q₈	—	86	4	no	—
182	C₂×C₄⋊Q₈	—	2	3	no	—
183	C₂×C₄⋊Q₈	—	43	3	no	—
184	C₂×C₄⋊Q₈	C₂×D₄₃	1	3	yes	—
185	C₂×C₂²⋊Q₈	—	86	3	no	—
186	C₂×C₄×Q₈	—	2	1	no	—
187	C₂×C₄×Q₈	C₂×D₄₃	1	3	yes	—
188	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
189	C₂×C₄×D₄	—	43	3	no	—
190	#498	—	43	24	no	—
191	#476	—	43	24	no	—
192	#475	—	43	24	no	—
193	C₈×Q₈	—	43	8	no	—
194	C₄×SD₁₆	—	43	24	no	—
195	C₂×#106	—	43	6	no	—
196	C₂×#106	—	43	3	no	—
197	C₂×C₄⋊C₈	—	43	3	no	—
198	C₂×#103	—	43	3	no	—
199	C₂×#102	—	43	3	no	—
200	C₂×C₄×C₈	—	43	3	no	—
201	C₄₃	C₂×Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	2	3	no	—
203	C₂²×Q₈	—	86	8	no	—
204	C₂²×Q₈	—	86	3	no	—
205	C₂²×Q₈	—	172	8	no	—
206	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
207	C₂²×Q₈	C₂²×D₄₃	1	1	yes	characteristic
208	C₂²×D₄	—	43	1	no	—
209	C₂³×C₄	—	86	3	no	—
210	C₂×SD₁₆	—	43	8	no	—
211	C₂×SD₁₆	—	43	4	no	—
212	C₂²×C₈	—	43	1	no	—
213	C₂²×C₈	—	43	3	no	—
214	C₄⋊Q₈	—	2	24	no	—
215	C₄⋊Q₈	—	2	6	no	—
216	C₄⋊Q₈	—	43	12	no	—
217	C₄⋊Q₈	—	86	6	no	—
218	C₄⋊Q₈	C₂²×D₄₃	1	12	yes	—
219	C₂²⋊Q₈	—	86	24	no	—
220	C₄×Q₈	—	2	8	no	—
221	C₄×Q₈	—	2	6	no	—
222	C₄×Q₈	C₂²⋊D₄₃	1	4	yes	—
223	C₄×Q₈	C₂²×D₄₃	1	12	yes	—
224	C₄×Q₈	C₂²×D₄₃	1	4	yes	—
225	C₄×D₄	—	43	12	no	—
226	C₄×D₄	—	86	6	no	—
227	C₂×C₄⋊C₄	—	2	3	no	—
228	C₂×C₄⋊C₄	—	2	3	no	—
229	C₂×C₄⋊C₄	—	43	3	no	—
230	C₂×C₄⋊C₄	—	86	3	no	—
231	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
232	C₂×C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
233	C₂×C₂²⋊C₄	—	86	3	no	—
234	C₂×C₄²	—	2	3	no	—
235	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
236	#106	—	43	12	no	—
237	#106	—	43	24	no	—
238	C₄⋊C₈	—	43	12	no	—
239	#103	—	43	12	no	—
240	#102	—	43	12	no	—
241	C₄×C₈	—	43	12	no	—
242	C₂⁴	—	86	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	12	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	8	6	no	—
247	C₂×Q₈	—	86	12	no	—
248	C₂×Q₈	—	172	32	no	—
249	C₂×Q₈	—	172	12	no	—
250	C₂×Q₈	#10621884	1	8	yes	direct product factor
251	C₂×Q₈	Q₈×D₄₃	1	4	yes	complemented
252	C₂×Q₈	C₂×C₂²⋊D₄₃	1	4	yes	—
253	C₂×Q₈	C₂³×D₄₃	1	2	yes	—
254	C₂×D₄	—	43	4	no	—
255	C₂×D₄	—	43	2	no	—
256	C₂×D₄	—	86	1	no	—
257	C₂×D₄	—	86	2	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	3	no	—
260	C₂²×C₄	—	86	3	no	—
261	C₂²×C₄	—	86	12	no	—
262	C₂²×C₄	—	172	12	no	—
263	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
265	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
266	SD₁₆	—	43	16	no	—
267	C₂×C₈	—	43	4	no	—
268	C₂×C₈	—	43	2	no	—
269	C₂×C₈	—	43	6	no	—
270	C₂×C₈	—	86	6	no	—
271	C₄⋊C₄	—	2	12	no	—
272	C₄⋊C₄	—	2	12	no	—
273	C₄⋊C₄	—	2	3	no	—
274	C₄⋊C₄	—	43	6	no	—
275	C₄⋊C₄	—	86	3	no	—
276	C₄⋊C₄	—	86	12	no	—
277	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	12	yes	—
278	C₄⋊C₄	C₂³×D₄₃	1	6	yes	—
279	C₂²⋊C₄	—	86	12	no	—
280	C₄²	—	2	12	no	—
281	C₄²	C₂×C₂²⋊D₄₃	1	6	yes	—
282	C₄²	C₂³×D₄₃	1	6	yes	—
283	C₂³	—	86	4	no	—
284	C₂³	—	86	2	no	—
285	C₂³	—	172	2	no	—
286	C₂³	—	172	2	no	—
287	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	8	12	no	—
291	Q₈	—	172	24	no	—
292	Q₈	C₂×#10621884	1	16	yes	direct product factor
293	Q₈	C₂×Q₈×D₄₃	1	4	yes	complemented
294	D₄	—	43	4	no	—
295	D₄	—	86	4	no	—
296	D₄	—	86	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	6	no	—
301	C₂×C₄	—	4	6	no	—
302	C₂×C₄	—	86	6	no	—
303	C₂×C₄	—	172	6	no	—
304	C₂×C₄	—	172	24	no	—
305	C₂×C₄	C₂×#10621884	1	12	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
307	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
309	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
310	C₂×C₄	#13063373	1	2	yes	—
311	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
312	C₈	—	43	4	no	—
313	C₈	—	86	6	no	—
314	C₂²	—	86	4	no	—
315	C₂²	—	172	2	no	—
316	C₂²	—	172	2	no	—
317	C₂²	—	172	8	no	—
318	C₂²	Q₈×C₂²⋊D₄₃	1	2	yes	—
319	C₂²	C₂²×#10621884	1	2	yes	—
320	C₂²	#14583710	1	2	yes	—
321	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	2	6	no	—
324	C₄	—	4	6	no	—
325	C₄	—	172	6	no	—
326	C₄	Q₈×C₂²⋊D₄₃	1	2	yes	—
327	C₄	C₂²×#10621884	1	12	yes	—
328	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
329	C₄	C₂×#13063373	1	1	yes	characteristic
330	C₂	—	172	4	no	—
331	C₂	Q₈×#10621884	1	4	yes	direct product factor
332	C₂	C₂×Q₈×C₂²⋊D₄₃	1	1	yes	characteristic
333	C₂	C₂³×#10621884	1	1	yes	characteristic
334	C₂	C₂×#14583710	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×Q₈²×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₁₇₂	C₂³
hypercenter	C₂×Q₈²	D₄₃
hypocenter	C₄₃	C₂×Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
2. Q₈×#10621884	C₂	direct (N × H)
3. C₂×#14583295	C₂	semidirect
4. C₂×#14583184	C₂	semidirect
6. C₂×Q₈×C₄₃⋊C₈	C₂	semidirect
8. C₂×Q₈²×C₄₃	C₂	semidirect
16. #14583295	C₂²	semidirect
18. #14583184	C₂²	semidirect
19. Q₈×C₄₃⋊C₈	C₂²	semidirect
28. Q₈²×C₄₃	C₂²	semidirect
39. C₂×#10621884	Q₈	direct (N × H)
79. C₂×Q₈×C₄₃	SD₁₆	semidirect
81. C₂×Q₈×C₄₃	C₂×Q₈	semidirect
106. #10621884	C₂×Q₈	direct (N × H)
112. C₂×C₄₃⋊C₈	C₂×Q₈	semidirect
123. Q₈×C₄₃	C₂×SD₁₆	semidirect
124. Q₈×C₄₃	C₂²×Q₈	semidirect
145. C₄₃⋊C₈	C₂²×Q₈	semidirect
163. C₂×Q₈²	D₄₃	semidirect
171. C₂×C₄₃	Q₈×SD₁₆	semidirect
179. Q₈²	C₂×D₄₃	semidirect
201. C₄₃	C₂×Q₈×SD₁₆	semidirect
250. C₂×Q₈	#10621884	direct (N × H)
251. C₂×Q₈	Q₈×D₄₃	semidirect
292. Q₈	C₂×#10621884	direct (N × H)
293. Q₈	C₂×Q₈×D₄₃	semidirect
331. C₂	Q₈×#10621884	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×Q₈×D₁₇₂	2
Q₈×#10621884	2
C₂×#14583295	2
C₂×#14583184	2
C₂×#14583186	2
C₂×Q₈×C₄₃⋊C₈	2
C₂×C₄×#10621884	2
C₂×Q₈²×C₄₃	2
C₂×Q₈×SD₁₆	43

Subgroups — C2×Q8×#10621884 #25977015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#10621884 #25977015