Subgroups — C₂×Q₈×#10625337 #27096725

Order 12032.

1830 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	Q₈×#10625337	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689526	C₂	1	3	yes	maximal
4	C₂×#15689418	C₂	1	3	yes	maximal
5	C₂×#15689415	C₂	1	3	yes	maximal
6	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#10625337	C₂	1	3	yes	maximal
8	C₂×Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625337	C₂²	1	1	yes	characteristic
10	Q₈×Dic₉₄	—	2	4	no	—
11	Q₈×Dic₉₄	C₂²	1	8	yes	—
12	C₂×#13094024	C₂²	1	3	yes	—
13	C₂×#13094023	—	2	3	no	—
14	C₂×Q₈×Dic₄₇	—	2	1	no	—
15	C₂×C₄×Dic₉₄	C₂²	1	3	yes	—
16	#15689526	C₂²	1	24	yes	—
17	#15689418	C₂²	1	24	yes	—
18	#15689415	C₂²	1	24	yes	—
19	Q₈×C₄₇⋊C₈	C₂²	1	8	yes	—
20	C₄×#10625337	C₂²	1	24	yes	—
21	C₂×#13093957	C₂²	1	3	yes	—
22	C₂×#13093957	C₂²	1	6	yes	—
23	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
24	C₂×#13093948	C₂²	1	3	yes	—
25	C₂×#13093947	C₂²	1	3	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
27	Q₈²×C₄₇	—	2	4	no	—
28	Q₈²×C₄₇	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₄₇	—	2	3	no	—
30	C₂×C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
31	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
33	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂²×Dic₉₄	—	2	3	no	—
35	C₂²×Dic₉₄	C₂³	1	1	yes	characteristic
36	C₂×#10625337	Q₈	1	8	yes	direct product factor
37	C₂×#10625337	C₂³	1	4	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	#13094024	—	2	6	no	—
41	#13094024	C₂³	1	12	yes	—
42	#13094023	—	2	24	no	—
43	Q₈×Dic₄₇	—	2	8	no	—
44	C₄×Dic₉₄	—	2	6	no	—
45	C₄×Dic₉₄	C₂³	1	12	yes	—
46	C₂×C₄⋊Dic₄₇	—	2	3	no	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	3	yes	—
48	C₂×Dic₄₇⋊C₄	—	2	3	no	—
49	C₂×C₄×Dic₄₇	—	2	3	no	—
50	#13093957	C₂³	1	12	yes	—
51	#13093957	C₂³	1	24	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	12	yes	—
53	#13093948	C₂³	1	12	yes	—
54	#13093947	C₂³	1	12	yes	—
55	C₄×C₄₇⋊C₈	C₂³	1	12	yes	—
56	C₂²×Q₈×C₄₇	—	2	3	no	—
57	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₄₇	—	2	24	no	—
60	C₄⋊Q₈×C₄₇	—	2	6	no	—
61	C₄⋊Q₈×C₄₇	C₂³	1	12	yes	—
62	C₄×Q₈×C₄₇	—	2	8	no	—
63	C₄×Q₈×C₄₇	—	2	6	no	—
64	C₄×Q₈×C₄₇	D₄	1	4	yes	—
65	C₄×Q₈×C₄₇	C₂³	1	12	yes	—
66	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
68	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
69	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
70	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
71	C₂×C₄²×C₄₇	—	2	3	no	—
72	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
73	C₂×Q₈×C₄₇	—	2	1	no	—
74	C₂×Q₈×C₄₇	—	2	12	no	—
75	C₂×Q₈×C₄₇	—	2	2	no	—
76	C₂×Q₈×C₄₇	—	8	6	no	—
77	C₂×Q₈×C₄₇	Q₁₆	1	8	yes	complemented
78	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
79	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
80	C₂×Q₈×C₄₇	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₄₇	—	2	1	no	—
82	C₂²×C₄×C₄₇	—	2	3	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₇	—	2	1	no	—
87	C₂²×Dic₄₇	—	2	3	no	—
88	C₂×Dic₉₄	—	2	1	no	—
89	C₂×Dic₉₄	—	2	2	no	—
90	C₂×Dic₉₄	—	2	12	no	—
91	C₂×Dic₉₄	—	8	6	no	—
92	C₂×Dic₉₄	C₂×Q₈	1	4	yes	—
93	C₂×Dic₉₄	C₂⁴	1	2	yes	—
94	C₄⋊C₄×C₄₇	—	2	12	no	—
95	C₄⋊C₄×C₄₇	—	2	12	no	—
96	C₄⋊C₄×C₄₇	—	2	3	no	—
97	C₄⋊C₄×C₄₇	C₂×D₄	1	12	yes	—
98	C₄⋊C₄×C₄₇	C₂⁴	1	6	yes	—
99	C₄²×C₄₇	—	2	12	no	—
100	C₄²×C₄₇	C₂×D₄	1	6	yes	—
101	C₄²×C₄₇	C₂⁴	1	6	yes	—
102	#10625337	C₂×Q₈	1	16	yes	direct product factor
103	C₄⋊Dic₄₇	—	2	12	no	—
104	C₄⋊Dic₄₇	—	2	3	no	—
105	C₄⋊Dic₄₇	C₂⁴	1	6	yes	—
106	Dic₄₇⋊C₄	—	2	12	no	—
107	C₄×Dic₄₇	—	2	12	no	—
108	C₂×C₄₇⋊C₈	—	2	6	no	—
109	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
110	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
111	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₄₇	—	2	4	no	—
114	Q₈×C₄₇	—	2	2	no	—
115	Q₈×C₄₇	—	8	12	no	—
116	Q₈×C₄₇	C₂×Q₁₆	1	16	yes	complemented
117	Q₈×C₄₇	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	6	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	6	no	—
122	C₂×C₄×C₄₇	—	4	6	no	—
123	C₂×C₄×C₄₇	C₂×Q₁₆	1	12	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
127	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
129	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	6	no	—
133	C₂×Dic₄₇	—	4	6	no	—
134	Dic₉₄	—	2	4	no	—
135	Dic₉₄	—	2	2	no	—
136	Dic₉₄	—	8	12	no	—
137	Dic₉₄	C₂²×Q₈	1	4	yes	—
138	C₄₇⋊C₈	—	2	6	no	—
139	C₄₇⋊C₈	C₂²×Q₈	1	4	yes	—
140	C₂×Q₈×Q₁₆	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	D₄×Q₈	1	2	yes	—
142	C₂²×C₄₇	C₂²×Q₁₆	1	2	yes	—
143	C₂²×C₄₇	#576	1	2	yes	—
144	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
145	C₄×C₄₇	—	2	4	no	—
146	C₄×C₄₇	—	2	6	no	—
147	C₄×C₄₇	—	4	6	no	—
148	C₄×C₄₇	D₄×Q₈	1	2	yes	—
149	C₄×C₄₇	C₂²×Q₁₆	1	12	yes	—
150	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
151	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₄₇	—	2	4	no	—
153	Dic₄₇	—	4	6	no	—
154	C₂×Q₈²	—	47	1	no	—
155	C₂×Q₈²	D₄₇	1	1	yes	characteristic
156	Q₈×Q₁₆	—	47	16	no	—
157	C₂×#500	—	47	3	no	—
158	C₂×#477	—	47	3	no	—
159	C₂×#477	—	47	3	no	—
160	C₂×C₈×Q₈	—	47	1	no	—
161	C₂×C₄×Q₁₆	—	47	3	no	—
162	C₂×C₄₇	Q₈×Q₁₆	1	4	yes	complemented
163	C₂×C₄₇	C₂×D₄×Q₈	1	1	yes	characteristic
164	C₂×C₄₇	C₂³×Q₁₆	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#576	1	1	yes	characteristic
166	C₂²×Q₁₆	—	47	1	no	—
167	Q₈²	—	2	4	no	—
168	Q₈²	—	47	8	no	—
169	Q₈²	—	94	4	no	—
170	Q₈²	C₂×D₄₇	1	8	yes	—
171	C₂×C₄⋊Q₈	—	2	3	no	—
172	C₂×C₄⋊Q₈	—	47	3	no	—
173	C₂×C₄⋊Q₈	—	94	3	no	—
174	C₂×C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
175	C₂×C₄×Q₈	—	2	1	no	—
176	C₂×C₄×Q₈	—	47	3	no	—
177	C₂×C₄×Q₈	—	94	1	no	—
178	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
179	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
180	#500	—	47	24	no	—
181	#477	—	47	24	no	—
182	#477	—	47	24	no	—
183	C₈×Q₈	—	47	8	no	—
184	C₄×Q₁₆	—	47	24	no	—
185	C₂×#107	—	47	6	no	—
186	C₂×#107	—	47	3	no	—
187	C₂×C₄⋊C₈	—	47	3	no	—
188	C₂×#103	—	47	3	no	—
189	C₂×#103	—	47	3	no	—
190	C₂×C₄×C₈	—	47	3	no	—
191	C₄₇	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	2	3	no	—
193	C₂²×Q₈	—	47	1	no	—
194	C₂²×Q₈	—	94	3	no	—
195	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×Q₁₆	—	47	8	no	—
198	C₂×Q₁₆	—	47	4	no	—
199	C₂²×C₈	—	47	1	no	—
200	C₂²×C₈	—	47	3	no	—
201	C₄⋊Q₈	—	2	24	no	—
202	C₄⋊Q₈	—	2	6	no	—
203	C₄⋊Q₈	—	47	12	no	—
204	C₄⋊Q₈	—	94	24	no	—
205	C₄⋊Q₈	—	94	6	no	—
206	C₄⋊Q₈	C₂²×D₄₇	1	12	yes	—
207	C₄×Q₈	—	2	8	no	—
208	C₄×Q₈	—	2	6	no	—
209	C₄×Q₈	—	47	12	no	—
210	C₄×Q₈	—	94	8	no	—
211	C₄×Q₈	—	94	6	no	—
212	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
213	C₄×Q₈	C₂²×D₄₇	1	12	yes	—
214	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
215	C₂×C₄⋊C₄	—	2	3	no	—
216	C₂×C₄⋊C₄	—	2	3	no	—
217	C₂×C₄⋊C₄	—	47	3	no	—
218	C₂×C₄⋊C₄	—	94	3	no	—
219	C₂×C₄⋊C₄	—	94	3	no	—
220	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
221	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
222	C₂×C₄²	—	2	3	no	—
223	C₂×C₄²	—	94	3	no	—
224	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
225	#107	—	47	12	no	—
226	#107	—	47	24	no	—
227	C₄⋊C₈	—	47	12	no	—
228	#103	—	47	12	no	—
229	#103	—	47	12	no	—
230	C₄×C₈	—	47	12	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	12	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	8	6	no	—
235	C₂×Q₈	—	47	4	no	—
236	C₂×Q₈	—	47	2	no	—
237	C₂×Q₈	—	94	1	no	—
238	C₂×Q₈	—	94	12	no	—
239	C₂×Q₈	—	94	2	no	—
240	C₂×Q₈	—	376	6	no	—
241	C₂×Q₈	#10625337	1	8	yes	direct product factor
242	C₂×Q₈	Q₈×D₄₇	1	4	yes	—
243	C₂×Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
244	C₂×Q₈	C₂³×D₄₇	1	2	yes	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	3	no	—
247	C₂²×C₄	—	94	1	no	—
248	C₂²×C₄	—	94	3	no	—
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
251	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
252	Q₁₆	—	47	16	no	—
253	C₂×C₈	—	47	4	no	—
254	C₂×C₈	—	47	2	no	—
255	C₂×C₈	—	47	6	no	—
256	C₂×C₈	—	94	6	no	—
257	C₄⋊C₄	—	2	12	no	—
258	C₄⋊C₄	—	2	12	no	—
259	C₄⋊C₄	—	2	3	no	—
260	C₄⋊C₄	—	47	6	no	—
261	C₄⋊C₄	—	94	12	no	—
262	C₄⋊C₄	—	94	3	no	—
263	C₄⋊C₄	—	94	12	no	—
264	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
265	C₄⋊C₄	C₂³×D₄₇	1	6	yes	—
266	C₄²	—	2	12	no	—
267	C₄²	—	94	12	no	—
268	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
269	C₄²	C₂³×D₄₇	1	6	yes	—
270	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	2	2	no	—
273	Q₈	—	8	12	no	—
274	Q₈	—	47	4	no	—
275	Q₈	—	94	4	no	—
276	Q₈	—	94	2	no	—
277	Q₈	—	376	12	no	—
278	Q₈	C₂×#10625337	1	16	yes	direct product factor
279	Q₈	C₂×Q₈×D₄₇	1	4	yes	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	6	no	—
284	C₂×C₄	—	4	6	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	6	no	—
288	C₂×C₄	—	188	6	no	—
289	C₂×C₄	C₂×#10625337	1	12	yes	—
290	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
293	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
294	C₂×C₄	#13094072	1	2	yes	—
295	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
296	C₈	—	47	4	no	—
297	C₈	—	94	6	no	—
298	C₂²	Q₈×C₂²⋊D₄₇	1	2	yes	—
299	C₂²	C₂²×#10625337	1	2	yes	—
300	C₂²	#15689926	1	2	yes	—
301	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
302	C₄	—	2	6	no	—
303	C₄	—	2	4	no	—
304	C₄	—	4	6	no	—
305	C₄	—	94	4	no	—
306	C₄	—	188	6	no	—
307	C₄	Q₈×C₂²⋊D₄₇	1	2	yes	—
308	C₄	C₂²×#10625337	1	12	yes	—
309	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
310	C₄	C₂×#13094072	1	1	yes	characteristic
311	C₂	Q₈×#10625337	1	4	yes	direct product factor
312	C₂	C₂×Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
313	C₂	C₂³×#10625337	1	1	yes	characteristic
314	C₂	C₂×#15689926	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×Q₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×Q₈²	D₄₇
hypocenter	C₄₇	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
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×#10625337	C₂	direct (N × H)
36. C₂×#10625337	Q₈	direct (N × H)
77. C₂×Q₈×C₄₇	Q₁₆	semidirect
102. #10625337	C₂×Q₈	direct (N × H)
116. Q₈×C₄₇	C₂×Q₁₆	semidirect
162. C₂×C₄₇	Q₈×Q₁₆	semidirect
191. C₄₇	C₂×Q₈×Q₁₆	semidirect
241. C₂×Q₈	#10625337	direct (N × H)
278. Q₈	C₂×#10625337	direct (N × H)
311. C₂	Q₈×#10625337	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×Dic₉₄	2
Q₈×#10625337	2
C₂×#15689526	2
C₂×#15689418	2
C₂×#15689415	2
C₂×Q₈×C₄₇⋊C₈	2
C₂×C₄×#10625337	2
C₂×Q₈²×C₄₇	2
C₂×Q₈×Q₁₆	47

Subgroups — C2×Q8×#10625337 #27096725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#10625337 #27096725