Subgroups — C₂×Q₈×#10621873 #25977020

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×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
3	Q₈×#10621873	C₂	1	16	yes	maximal, direct product factor
4	C₂×#14583297	C₂	1	3	yes	maximal, complemented
5	C₂×#14583185	C₂	1	3	yes	maximal, complemented
6	C₂×#14583190	C₂	1	3	yes	maximal
7	C₂×C₄×#10621873	C₂	1	3	yes	maximal
8	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₄₃	—	2	1	no	—
10	C₂²×#10621873	C₂²	1	1	yes	characteristic
11	Q₈×Dic₈₆	—	2	4	no	—
12	Q₈×Dic₈₆	C₂²	1	8	yes	complemented
13	Q₈×D₁₇₂	—	2	4	no	—
14	Q₈×D₁₇₂	C₂²	1	8	yes	—
15	C₂×#13063325	C₂²	1	3	yes	—
16	C₂×#13063325	C₂²	1	3	yes	—
17	C₂×#13063324	—	2	3	no	—
18	C₂×#13063305	—	2	3	no	—
19	C₂×Q₈×Dic₄₃	—	2	1	no	—
20	C₂×C₄×Dic₈₆	C₂²	1	3	yes	—
21	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
22	#14583297	C₂²	1	24	yes	complemented
23	#14583190	C₂²	1	24	yes	—
24	#14583185	C₂²	1	24	yes	complemented
25	C₄×#10621873	C₂²	1	24	yes	—
26	C₂×#13063257	C₂²	1	3	yes	—
27	C₂×#13063257	C₂²	1	6	yes	—
28	C₂×#13063250	C₂²	1	3	yes	—
29	C₂×#13063247	C₂²	1	3	yes	—
30	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₄₃	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₈×C₄₃	C₂²	1	3	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
34	C₂²×Dic₈₆	—	2	3	no	—
35	C₂²×Dic₈₆	—	2	3	no	—
36	C₂²×Dic₈₆	C₂³	1	1	yes	characteristic
37	C₂×Q₈×D₄₃	—	2	8	no	—
38	C₂×Q₈×D₄₃	—	4	8	no	—
39	C₂²×D₁₇₂	C₂³	1	1	yes	characteristic
40	C₂²×C₄×D₄₃	—	2	3	no	—
41	C₂×#10621873	Q₈	1	8	yes	direct product factor
42	C₂×#10621873	C₂³	1	4	yes	—
43	#13063325	—	2	6	no	—
44	#13063325	—	2	6	no	—
45	#13063325	C₂³	1	12	yes	—
46	#13063325	C₂³	1	12	yes	—
47	#13063324	—	2	24	no	—
48	#13063305	—	2	24	no	—
49	Q₈×Dic₄₃	—	2	8	no	—
50	C₄×Dic₈₆	—	2	6	no	—
51	C₄×Dic₈₆	C₂³	1	12	yes	—
52	C₄×D₁₇₂	—	2	6	no	—
53	C₄×D₁₇₂	C₂³	1	12	yes	—
54	C₂×C₄⋊Dic₄₃	—	2	3	no	—
55	C₂×C₄⋊Dic₄₃	—	2	3	no	—
56	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
57	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
58	C₂×Dic₄₃⋊C₄	—	2	3	no	—
59	C₂×(C₂×D₄₃)⋊C₄	—	2	3	no	—
60	C₂×C₄×Dic₄₃	—	2	3	no	—
61	#13063257	C₂³	1	12	yes	—
62	#13063257	C₂³	1	24	yes	—
63	#13063250	C₂³	1	12	yes	—
64	#13063247	C₂³	1	12	yes	—
65	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
67	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₄×Q₈×C₄₃	D₄	1	4	yes	—
69	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
71	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
72	C₄⋊C₈×C₄₃	C₂³	1	12	yes	—
73	C₄×C₈×C₄₃	C₂³	1	12	yes	—
74	C₂³×D₄₃	—	2	1	no	—
75	C₂×Q₈×C₄₃	SD₁₆	1	8	yes	complemented
76	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₃	—	2	1	no	—
81	C₂²×Dic₄₃	—	2	3	no	—
82	C₂²×Dic₄₃	—	2	3	no	—
83	Q₈×D₄₃	—	4	32	no	—
84	C₂×D₁₇₂	—	2	1	no	—
85	C₂×D₁₇₂	—	2	2	no	—
86	C₂×D₁₇₂	C₂×Q₈	1	4	yes	—
87	C₂×D₁₇₂	C₂⁴	1	2	yes	—
88	C₂×C₄×D₄₃	—	2	12	no	—
89	C₂×C₄×D₄₃	—	4	12	no	—
90	C₂×Dic₈₆	—	2	1	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	—	2	12	no	—
93	C₂×Dic₈₆	—	2	12	no	—
94	C₂×Dic₈₆	—	4	12	no	—
95	C₂×Dic₈₆	—	8	6	no	—
96	C₂×Dic₈₆	C₂×Q₈	1	4	yes	complemented
97	C₂×Dic₈₆	C₂⁴	1	2	yes	—
98	C₂×C₈×C₄₃	—	2	6	no	—
99	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	complemented
100	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
102	C₄⋊C₄×C₄₃	C₂×D₄	1	12	yes	—
103	C₄²×C₄₃	C₂×D₄	1	6	yes	—
104	C₄²×C₄₃	C₂⁴	1	6	yes	—
105	(C₂×D₄₃)⋊C₄	—	2	12	no	—
106	C₄⋊Dic₄₃	—	2	12	no	—
107	C₄⋊Dic₄₃	—	2	12	no	—
108	C₄⋊Dic₄₃	—	2	3	no	—
109	C₄⋊Dic₄₃	—	2	3	no	—
110	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
111	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
112	Dic₄₃⋊C₄	—	2	12	no	—
113	C₄×Dic₄₃	—	2	12	no	—
114	#10621873	C₂×Q₈	1	16	yes	direct product factor
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	C₂×SD₁₆	1	16	yes	complemented
121	C₂×C₄×C₄₃	—	2	6	no	—
122	C₂×C₄×C₄₃	C₂×SD₁₆	1	12	yes	—
123	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
126	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
128	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
129	C₂×Dic₄₃	—	2	4	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	2	6	no	—
132	C₂×Dic₄₃	—	2	6	no	—
133	C₂×Dic₄₃	—	4	6	no	—
134	C₂×Dic₄₃	—	4	6	no	—
135	D₁₇₂	—	2	4	no	—
136	D₁₇₂	—	2	2	no	—
137	D₁₇₂	C₂²×Q₈	1	4	yes	—
138	C₄×D₄₃	—	4	24	no	—
139	Dic₈₆	—	2	4	no	—
140	Dic₈₆	—	2	2	no	—
141	Dic₈₆	—	4	24	no	—
142	Dic₈₆	—	8	12	no	—
143	Dic₈₆	C₂²×Q₈	1	4	yes	complemented
144	C₈×C₄₃	—	2	6	no	—
145	C₈×C₄₃	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	6	no	—
156	C₄×C₄₃	D₄×Q₈	1	2	yes	—
157	C₄×C₄₃	C₂²×SD₁₆	1	12	yes	—
158	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
159	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₄₃	—	2	4	no	—
161	Dic₄₃	—	4	6	no	—
162	Dic₄₃	—	4	6	no	—
163	C₂×Q₈²	—	43	1	no	—
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	D₄₃	1	1	yes	characteristic, complemented
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	—	43	8	no	—
179	Q₈²	—	86	4	no	—
180	D₄×Q₈	—	43	8	no	—
181	D₄×Q₈	—	86	4	no	—
182	C₂×C₄⋊Q₈	—	43	3	no	—
183	C₂×C₄⋊Q₈	—	43	3	no	—
184	C₂×C₄⋊Q₈	—	86	3	no	—
185	C₂×C₂²⋊Q₈	—	86	3	no	—
186	C₂×C₄×Q₈	—	43	3	no	—
187	C₂×C₄×Q₈	—	86	1	no	—
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	C₂×D₄₃	1	8	yes	complemented
194	C₄×SD₁₆	—	43	24	no	—
195	C₂×#106	—	43	6	no	—
196	C₂×#106	—	43	3	no	—
197	C₂×C₄⋊C₈	C₂×D₄₃	1	3	yes	—
198	C₂×#103	—	43	3	no	—
199	C₂×#102	—	43	3	no	—
200	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
201	C₄₃	C₂×Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	43	1	no	—
203	C₂²×Q₈	—	86	8	no	—
204	C₂²×Q₈	—	86	3	no	—
205	C₂²×Q₈	—	86	3	no	—
206	C₂²×Q₈	—	172	8	no	—
207	C₂²×Q₈	D₁₇₂	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	C₂²×D₄₃	1	3	yes	—
213	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
214	C₄⋊Q₈	—	43	12	no	—
215	C₄⋊Q₈	—	43	12	no	—
216	C₄⋊Q₈	—	86	24	no	—
217	C₄⋊Q₈	—	86	6	no	—
218	C₄⋊Q₈	—	86	6	no	—
219	C₂²⋊Q₈	—	86	24	no	—
220	C₄×Q₈	—	43	12	no	—
221	C₄×Q₈	—	86	8	no	—
222	C₄×Q₈	—	86	6	no	—
223	C₄×Q₈	D₁₇₂	1	4	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	—	43	3	no	—
228	C₂×C₄⋊C₄	—	43	3	no	—
229	C₂×C₄⋊C₄	—	86	3	no	—
230	C₂×C₄⋊C₄	—	86	3	no	—
231	C₂×C₄⋊C₄	—	86	3	no	—
232	C₂×C₄⋊C₄	D₁₇₂	1	3	yes	—
233	C₂×C₂²⋊C₄	—	86	3	no	—
234	C₂×C₄²	—	86	3	no	—
235	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
236	#106	—	43	12	no	—
237	#106	—	43	24	no	—
238	C₄⋊C₈	C₂²×D₄₃	1	12	yes	—
239	#103	—	43	12	no	—
240	#102	—	43	12	no	—
241	C₄×C₈	C₂²×D₄₃	1	12	yes	—
242	C₂⁴	—	86	1	no	—
243	C₂×Q₈	—	43	4	no	—
244	C₂×Q₈	—	43	2	no	—
245	C₂×Q₈	—	86	1	no	—
246	C₂×Q₈	—	86	12	no	—
247	C₂×Q₈	—	86	12	no	—
248	C₂×Q₈	—	86	2	no	—
249	C₂×Q₈	—	344	6	no	—
250	C₂×Q₈	—	172	32	no	—
251	C₂×Q₈	—	172	12	no	—
252	C₂×Q₈	#10621873	1	8	yes	direct product factor
253	C₂×Q₈	C₂×D₁₇₂	1	4	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	—	86	1	no	—
259	C₂²×C₄	—	86	3	no	—
260	C₂²×C₄	—	86	3	no	—
261	C₂²×C₄	—	86	12	no	—
262	C₂²×C₄	—	172	12	no	—
263	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
264	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
265	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
266	SD₁₆	—	43	16	no	—
267	C₂×C₈	—	2	6	no	—
268	C₂×C₈	Q₈×D₄₃	1	4	yes	complemented
269	C₂×C₈	C₂³×D₄₃	1	6	yes	—
270	C₂×C₈	C₂³×D₄₃	1	2	yes	—
271	C₄⋊C₄	—	43	6	no	—
272	C₄⋊C₄	—	43	6	no	—
273	C₄⋊C₄	—	86	3	no	—
274	C₄⋊C₄	—	86	3	no	—
275	C₄⋊C₄	—	86	12	no	—
276	C₄⋊C₄	—	86	12	no	—
277	C₄⋊C₄	—	86	12	no	—
278	C₄⋊C₄	C₂×D₁₇₂	1	12	yes	—
279	C₂²⋊C₄	—	86	12	no	—
280	C₄²	—	86	12	no	—
281	C₄²	C₂×D₁₇₂	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×D₁₇₂	1	1	yes	characteristic
288	Q₈	—	43	4	no	—
289	Q₈	—	86	4	no	—
290	Q₈	—	86	2	no	—
291	Q₈	—	344	12	no	—
292	Q₈	—	172	24	no	—
293	Q₈	C₂×#10621873	1	16	yes	direct product factor
294	D₄	—	43	4	no	—
295	D₄	—	86	4	no	—
296	D₄	—	86	2	no	—
297	C₂×C₄	—	2	6	no	—
298	C₂×C₄	—	86	4	no	—
299	C₂×C₄	—	86	2	no	—
300	C₂×C₄	—	86	6	no	—
301	C₂×C₄	—	86	6	no	—
302	C₂×C₄	—	172	6	no	—
303	C₂×C₄	—	172	6	no	—
304	C₂×C₄	—	172	24	no	—
305	C₂×C₄	C₂×#10621873	1	12	yes	—
306	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
307	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
309	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
310	C₂×C₄	#13063372	1	2	yes	—
311	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
312	C₈	—	2	6	no	—
313	C₈	C₂×Q₈×D₄₃	1	4	yes	complemented
314	C₂²	—	86	4	no	—
315	C₂²	—	172	2	no	—
316	C₂²	—	172	2	no	—
317	C₂²	—	172	8	no	—
318	C₂²	Q₈×D₁₇₂	1	2	yes	—
319	C₂²	C₂²×#10621873	1	2	yes	—
320	C₂²	#14583711	1	2	yes	—
321	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
322	C₄	—	2	6	no	—
323	C₄	—	86	4	no	—
324	C₄	—	172	6	no	—
325	C₄	—	172	6	no	—
326	C₄	Q₈×D₁₇₂	1	2	yes	—
327	C₄	C₂²×#10621873	1	12	yes	—
328	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
329	C₄	C₂×#13063372	1	1	yes	characteristic
330	C₂	—	172	4	no	—
331	C₂	Q₈×#10621873	1	4	yes	direct product factor
332	C₂	C₂×Q₈×D₁₇₂	1	1	yes	characteristic
333	C₂	C₂³×#10621873	1	1	yes	characteristic
334	C₂	C₂×#14583711	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₁₇₂	C₂³
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	C₂×Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₈₆	C₂	semidirect
3. Q₈×#10621873	C₂	direct (N × H)
4. C₂×#14583297	C₂	semidirect
5. C₂×#14583185	C₂	semidirect
8. C₂×C₈×Q₈×C₄₃	C₂	semidirect
12. Q₈×Dic₈₆	C₂²	semidirect
22. #14583297	C₂²	semidirect
24. #14583185	C₂²	semidirect
31. C₈×Q₈×C₄₃	C₂²	semidirect
41. C₂×#10621873	Q₈	direct (N × H)
75. C₂×Q₈×C₄₃	SD₁₆	semidirect
96. C₂×Dic₈₆	C₂×Q₈	semidirect
99. C₂×C₈×C₄₃	C₂×Q₈	semidirect
114. #10621873	C₂×Q₈	direct (N × H)
120. Q₈×C₄₃	C₂×SD₁₆	semidirect
143. Dic₈₆	C₂²×Q₈	semidirect
145. C₈×C₄₃	C₂²×Q₈	semidirect
169. C₂×C₈×Q₈	D₄₃	semidirect
171. C₂×C₄₃	Q₈×SD₁₆	semidirect
193. C₈×Q₈	C₂×D₄₃	semidirect
201. C₄₃	C₂×Q₈×SD₁₆	semidirect
252. C₂×Q₈	#10621873	direct (N × H)
268. C₂×C₈	Q₈×D₄₃	semidirect
293. Q₈	C₂×#10621873	direct (N × H)
313. C₈	C₂×Q₈×D₄₃	semidirect
331. C₂	Q₈×#10621873	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×Dic₈₆	2
C₂×Q₈×D₁₇₂	2
Q₈×#10621873	2
C₂×#14583297	2
C₂×#14583185	2
C₂×#14583190	2
C₂×C₄×#10621873	2
C₂×C₈×Q₈×C₄₃	2
C₂×Q₈×SD₁₆	43

Subgroups — C2×Q8×#10621873 #25977020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#10621873 #25977020