Subgroups — C₂×Q₈×#10625335 #27096700

Order 12032.

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×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#10625335	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689519	C₂	1	3	yes	maximal, complemented
4	C₂×#15689405	C₂	1	3	yes	maximal, complemented
5	C₂×#15689413	C₂	1	3	yes	maximal
6	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#10625335	C₂	1	3	yes	maximal
8	C₂×D₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625335	C₂²	1	1	yes	characteristic
10	Q₈×Dic₉₄	—	2	4	no	—
11	Q₈×Dic₉₄	C₂²	1	8	yes	complemented
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	#15689519	C₂²	1	24	yes	complemented
17	#15689413	C₂²	1	24	yes	—
18	#15689405	C₂²	1	24	yes	complemented
19	Q₈×C₄₇⋊C₈	C₂²	1	8	yes	complemented
20	C₄×#10625335	C₂²	1	24	yes	—
21	C₂×#13093955	C₂²	1	3	yes	—
22	C₂×#13093955	C₂²	1	6	yes	—
23	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
24	C₂×#13093947	C₂²	1	3	yes	—
25	C₂×#13093945	C₂²	1	3	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
27	C₂³×Q₈×C₄₇	—	2	1	no	—
28	D₄×Q₈×C₄₇	—	2	4	no	—
29	D₄×Q₈×C₄₇	C₂²	1	8	yes	—
30	C₂×C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
31	C₂×C₂²⋊Q₈×C₄₇	—	2	3	no	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₄₇	C₂²	1	3	yes	—
34	C₂²×Dic₉₄	—	2	3	no	—
35	C₂²×Dic₉₄	C₂³	1	1	yes	characteristic
36	C₂×#10625335	Q₈	1	8	yes	direct product factor
37	C₂×#10625335	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	#13093955	C₂³	1	12	yes	—
51	#13093955	C₂³	1	24	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	12	yes	—
53	#13093947	C₂³	1	12	yes	—
54	#13093945	C₂³	1	12	yes	—
55	C₄×C₄₇⋊C₈	C₂³	1	12	yes	—
56	C₂²×Q₈×C₄₇	—	2	8	no	—
57	C₂²×Q₈×C₄₇	—	2	3	no	—
58	C₂²×Q₈×C₄₇	—	4	8	no	—
59	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₇	—	2	3	no	—
62	C₄⋊Q₈×C₄₇	—	2	6	no	—
63	C₄⋊Q₈×C₄₇	C₂³	1	12	yes	—
64	C₂²⋊Q₈×C₄₇	—	2	24	no	—
65	C₄×Q₈×C₄₇	D₄	1	4	yes	—
66	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
67	C₄×D₄×C₄₇	—	2	6	no	—
68	C₄×D₄×C₄₇	C₂³	1	12	yes	—
69	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
70	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
71	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
72	C₂×C₂²⋊C₄×C₄₇	—	2	3	no	—
73	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
74	C₂⁴×C₄₇	—	2	1	no	—
75	C₂×Q₈×C₄₇	—	2	12	no	—
76	C₂×Q₈×C₄₇	—	4	32	no	—
77	C₂×Q₈×C₄₇	—	4	12	no	—
78	C₂×Q₈×C₄₇	SD₁₆	1	8	yes	complemented
79	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
80	C₂×D₄×C₄₇	—	2	1	no	—
81	C₂×D₄×C₄₇	—	2	2	no	—
82	C₂×D₄×C₄₇	C₂×Q₈	1	4	yes	—
83	C₂×D₄×C₄₇	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₄₇	—	2	3	no	—
85	C₂²×C₄×C₄₇	—	2	12	no	—
86	C₂²×C₄×C₄₇	—	4	12	no	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₄₇	—	2	1	no	—
91	C₂²×Dic₄₇	—	2	3	no	—
92	C₂×Dic₉₄	—	2	1	no	—
93	C₂×Dic₉₄	—	2	2	no	—
94	C₂×Dic₉₄	—	2	12	no	—
95	C₂×Dic₉₄	—	8	6	no	—
96	C₂×Dic₉₄	C₂×Q₈	1	4	yes	complemented
97	C₂×Dic₉₄	C₂⁴	1	2	yes	—
98	C₄⋊C₄×C₄₇	—	2	12	no	—
99	C₄⋊C₄×C₄₇	—	2	3	no	—
100	C₄⋊C₄×C₄₇	C₂×D₄	1	12	yes	—
101	C₄⋊C₄×C₄₇	C₂⁴	1	6	yes	—
102	C₂²⋊C₄×C₄₇	—	2	12	no	—
103	C₄²×C₄₇	C₂×D₄	1	6	yes	—
104	C₄²×C₄₇	C₂⁴	1	6	yes	—
105	#10625335	C₂×Q₈	1	16	yes	direct product factor
106	C₄⋊Dic₄₇	—	2	12	no	—
107	C₄⋊Dic₄₇	—	2	3	no	—
108	C₄⋊Dic₄₇	C₂⁴	1	6	yes	—
109	Dic₄₇⋊C₄	—	2	12	no	—
110	C₄×Dic₄₇	—	2	12	no	—
111	C₂×C₄₇⋊C₈	—	2	6	no	—
112	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	complemented
113	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
114	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
115	C₂³×C₄₇	—	2	4	no	—
116	C₂³×C₄₇	—	2	2	no	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂³×C₄₇	—	4	2	no	—
119	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	Q₈×C₄₇	—	4	24	no	—
121	Q₈×C₄₇	C₂×SD₁₆	1	16	yes	complemented
122	D₄×C₄₇	—	2	4	no	—
123	D₄×C₄₇	—	2	2	no	—
124	D₄×C₄₇	C₂²×Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	—	2	6	no	—
126	C₂×C₄×C₄₇	—	2	6	no	—
127	C₂×C₄×C₄₇	—	4	6	no	—
128	C₂×C₄×C₄₇	—	4	24	no	—
129	C₂×C₄×C₄₇	C₂×SD₁₆	1	12	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
133	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
134	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
135	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
136	C₂×Dic₄₇	—	2	4	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	2	6	no	—
139	C₂×Dic₄₇	—	4	6	no	—
140	Dic₉₄	—	2	4	no	—
141	Dic₉₄	—	2	2	no	—
142	Dic₉₄	—	8	12	no	—
143	Dic₉₄	C₂²×Q₈	1	4	yes	complemented
144	C₄₇⋊C₈	—	2	6	no	—
145	C₄₇⋊C₈	C₂²×Q₈	1	4	yes	complemented
146	C₂×Q₈×SD₁₆	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	4	no	—
148	C₂²×C₄₇	—	4	8	no	—
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	—	4	2	no	—
151	C₂²×C₄₇	D₄×Q₈	1	2	yes	—
152	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
153	C₂²×C₄₇	#577	1	2	yes	—
154	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
155	C₄×C₄₇	—	2	6	no	—
156	C₄×C₄₇	—	4	6	no	—
157	C₄×C₄₇	D₄×Q₈	1	2	yes	—
158	C₄×C₄₇	C₂²×SD₁₆	1	12	yes	—
159	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
160	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₄₇	—	2	4	no	—
162	Dic₄₇	—	4	6	no	—
163	C₂×Q₈²	—	47	1	no	—
164	C₂×D₄×Q₈	D₄₇	1	1	yes	characteristic
165	Q₈×SD₁₆	—	47	16	no	—
166	C₂×#498	—	47	3	no	—
167	C₂×#475	—	47	3	no	—
168	C₂×#476	—	47	3	no	—
169	C₂×C₈×Q₈	—	47	1	no	—
170	C₂×C₄×SD₁₆	—	47	3	no	—
171	C₂×C₄₇	—	4	4	no	—
172	C₂×C₄₇	Q₈×SD₁₆	1	4	yes	complemented
173	C₂×C₄₇	C₂×D₄×Q₈	1	1	yes	characteristic
174	C₂×C₄₇	C₂³×SD₁₆	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#577	1	1	yes	characteristic
176	C₂³×Q₈	—	2	1	no	—
177	C₂²×SD₁₆	—	47	1	no	—
178	Q₈²	—	47	8	no	—
179	Q₈²	—	94	4	no	—
180	D₄×Q₈	—	2	4	no	—
181	D₄×Q₈	C₂×D₄₇	1	8	yes	—
182	C₂×C₄⋊Q₈	—	47	3	no	—
183	C₂×C₄⋊Q₈	—	94	3	no	—
184	C₂×C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
185	C₂×C₂²⋊Q₈	—	2	3	no	—
186	C₂×C₄×Q₈	—	47	3	no	—
187	C₂×C₄×Q₈	—	94	1	no	—
188	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
189	C₂×C₄×D₄	C₂×D₄₇	1	3	yes	—
190	#498	—	47	24	no	—
191	#476	—	47	24	no	—
192	#475	—	47	24	no	—
193	C₈×Q₈	—	47	8	no	—
194	C₄×SD₁₆	—	47	24	no	—
195	C₂×#106	—	47	6	no	—
196	C₂×#106	—	47	3	no	—
197	C₂×C₄⋊C₈	—	47	3	no	—
198	C₂×#103	—	47	3	no	—
199	C₂×#102	—	47	3	no	—
200	C₂×C₄×C₈	—	47	3	no	—
201	C₄₇	C₂×Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	2	8	no	—
203	C₂²×Q₈	—	2	3	no	—
204	C₂²×Q₈	—	4	8	no	—
205	C₂²×Q₈	—	47	1	no	—
206	C₂²×Q₈	—	94	3	no	—
207	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂²×D₄	C₂²×D₄₇	1	1	yes	characteristic
209	C₂³×C₄	—	2	3	no	—
210	C₂×SD₁₆	—	47	8	no	—
211	C₂×SD₁₆	—	47	4	no	—
212	C₂²×C₈	—	47	1	no	—
213	C₂²×C₈	—	47	3	no	—
214	C₄⋊Q₈	—	2	6	no	—
215	C₄⋊Q₈	—	47	12	no	—
216	C₄⋊Q₈	—	94	24	no	—
217	C₄⋊Q₈	—	94	6	no	—
218	C₄⋊Q₈	C₂²×D₄₇	1	12	yes	—
219	C₂²⋊Q₈	—	2	24	no	—
220	C₄×Q₈	—	47	12	no	—
221	C₄×Q₈	—	94	8	no	—
222	C₄×Q₈	—	94	6	no	—
223	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
224	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
225	C₄×D₄	—	2	6	no	—
226	C₄×D₄	C₂²×D₄₇	1	12	yes	—
227	C₂×C₄⋊C₄	—	2	3	no	—
228	C₂×C₄⋊C₄	—	47	3	no	—
229	C₂×C₄⋊C₄	—	94	3	no	—
230	C₂×C₄⋊C₄	—	94	3	no	—
231	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
232	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
233	C₂×C₂²⋊C₄	—	2	3	no	—
234	C₂×C₄²	—	94	3	no	—
235	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
236	#106	—	47	12	no	—
237	#106	—	47	24	no	—
238	C₄⋊C₈	—	47	12	no	—
239	#103	—	47	12	no	—
240	#102	—	47	12	no	—
241	C₄×C₈	—	47	12	no	—
242	C₂⁴	—	2	1	no	—
243	C₂×Q₈	—	2	12	no	—
244	C₂×Q₈	—	4	32	no	—
245	C₂×Q₈	—	4	12	no	—
246	C₂×Q₈	—	47	4	no	—
247	C₂×Q₈	—	47	2	no	—
248	C₂×Q₈	—	94	1	no	—
249	C₂×Q₈	—	94	12	no	—
250	C₂×Q₈	—	94	2	no	—
251	C₂×Q₈	—	376	6	no	—
252	C₂×Q₈	#10625335	1	8	yes	direct product factor
253	C₂×Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	Q₈×D₄₇	1	4	yes	—
257	C₂×D₄	C₂³×D₄₇	1	2	yes	—
258	C₂²×C₄	—	2	3	no	—
259	C₂²×C₄	—	2	12	no	—
260	C₂²×C₄	—	4	12	no	—
261	C₂²×C₄	—	94	1	no	—
262	C₂²×C₄	—	94	3	no	—
263	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
265	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	SD₁₆	—	47	16	no	—
267	C₂×C₈	—	47	4	no	—
268	C₂×C₈	—	47	2	no	—
269	C₂×C₈	—	47	6	no	—
270	C₂×C₈	—	94	6	no	—
271	C₄⋊C₄	—	2	12	no	—
272	C₄⋊C₄	—	2	3	no	—
273	C₄⋊C₄	—	47	6	no	—
274	C₄⋊C₄	—	94	12	no	—
275	C₄⋊C₄	—	94	3	no	—
276	C₄⋊C₄	—	94	12	no	—
277	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
278	C₄⋊C₄	C₂³×D₄₇	1	6	yes	—
279	C₂²⋊C₄	—	2	12	no	—
280	C₄²	—	94	12	no	—
281	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
282	C₄²	C₂³×D₄₇	1	6	yes	—
283	C₂³	—	2	4	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
288	Q₈	—	4	24	no	—
289	Q₈	—	47	4	no	—
290	Q₈	—	94	4	no	—
291	Q₈	—	94	2	no	—
292	Q₈	—	376	12	no	—
293	Q₈	C₂×#10625335	1	16	yes	direct product factor
294	D₄	—	2	4	no	—
295	D₄	—	2	2	no	—
296	D₄	C₂×Q₈×D₄₇	1	4	yes	—
297	C₂×C₄	—	2	6	no	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	4	6	no	—
300	C₂×C₄	—	4	24	no	—
301	C₂×C₄	—	94	4	no	—
302	C₂×C₄	—	94	2	no	—
303	C₂×C₄	—	94	6	no	—
304	C₂×C₄	—	188	6	no	—
305	C₂×C₄	C₂×#10625335	1	12	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
307	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
309	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
310	C₂×C₄	#13094072	1	2	yes	—
311	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
312	C₈	—	47	4	no	—
313	C₈	—	94	6	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	8	no	—
318	C₂²	Q₈×C₂²⋊D₄₇	1	2	yes	—
319	C₂²	C₂²×#10625335	1	2	yes	—
320	C₂²	#15689931	1	2	yes	—
321	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
322	C₄	—	2	6	no	—
323	C₄	—	4	6	no	—
324	C₄	—	94	4	no	—
325	C₄	—	188	6	no	—
326	C₄	Q₈×C₂²⋊D₄₇	1	2	yes	—
327	C₄	C₂²×#10625335	1	12	yes	—
328	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
329	C₄	C₂×#13094072	1	1	yes	characteristic
330	C₂	—	4	4	no	—
331	C₂	Q₈×#10625335	1	4	yes	direct product factor
332	C₂	C₂×Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₂	C₂³×#10625335	1	1	yes	characteristic
334	C₂	C₂×#15689931	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×D₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₄×C₄₇	C₂³
hypercenter	C₂×D₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₉₄	C₂	semidirect
2. Q₈×#10625335	C₂	direct (N × H)
3. C₂×#15689519	C₂	semidirect
4. C₂×#15689405	C₂	semidirect
6. C₂×Q₈×C₄₇⋊C₈	C₂	semidirect
11. Q₈×Dic₉₄	C₂²	semidirect
16. #15689519	C₂²	semidirect
18. #15689405	C₂²	semidirect
19. Q₈×C₄₇⋊C₈	C₂²	semidirect
36. C₂×#10625335	Q₈	direct (N × H)
78. C₂×Q₈×C₄₇	SD₁₆	semidirect
96. C₂×Dic₉₄	C₂×Q₈	semidirect
105. #10625335	C₂×Q₈	direct (N × H)
112. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
121. Q₈×C₄₇	C₂×SD₁₆	semidirect
143. Dic₉₄	C₂²×Q₈	semidirect
145. C₄₇⋊C₈	C₂²×Q₈	semidirect
172. C₂×C₄₇	Q₈×SD₁₆	semidirect
201. C₄₇	C₂×Q₈×SD₁₆	semidirect
252. C₂×Q₈	#10625335	direct (N × H)
293. Q₈	C₂×#10625335	direct (N × H)
331. C₂	Q₈×#10625335	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×#10625335	2
C₂×#15689519	2
C₂×#15689405	2
C₂×#15689413	2
C₂×Q₈×C₄₇⋊C₈	2
C₂×C₄×#10625335	2
C₂×D₄×Q₈×C₄₇	2
C₂×Q₈×SD₁₆	47

Subgroups — C2×Q8×#10625335 #27096700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#10625335 #27096700