Subgroups — Q₈×#13093959 #26235168

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20244163	C₂	1	3	yes	maximal
4	#20244161	C₂	1	3	yes	maximal
5	#20243559	C₂	1	3	yes	maximal
6	C₄×#13093959	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689254	—	2	3	no	—
9	Q₈×C₄₇⋊C₈	—	2	1	no	—
10	Q₈×C₄₇⋊C₈	C₄	1	6	yes	—
11	Q₈×C₄₇⋊C₈	C₄	1	2	yes	—
12	C₂×#13093959	C₂²	1	3	yes	—
13	C₂×#13093959	C₂²	1	6	yes	—
14	C₂×#13093959	C₂²	1	1	yes	characteristic
15	#15689147	C₂²	1	3	yes	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
17	C₄×#10625329	C₂²	1	3	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
19	#15688887	C₂²	1	3	yes	—
20	#15688887	C₂²	1	3	yes	—
21	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#446	—	2	3	no	—
23	C₈×Q₈×C₄₇	—	2	1	no	—
24	C₄₇×#423	C₂²	1	3	yes	—
25	C₄×M₁₆×C₄₇	C₂²	1	3	yes	—
26	C₂×#10625329	C₂³	1	3	yes	—
27	C₂×#10625329	C₂³	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
30	#13093959	—	2	6	no	—
31	#13093959	—	4	6	no	—
32	#13093959	Q₈	1	4	yes	direct product factor
33	C₄⋊(C₄₇⋊C₈)	—	2	3	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	3	no	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	6	yes	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	6	yes	—
37	#13093935	—	2	3	no	—
38	C₄×C₄₇⋊C₈	—	2	3	no	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	6	yes	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	6	yes	—
41	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	C₂³	1	3	yes	—
43	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₄₇	C₂×C₄	1	3	yes	—
45	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
47	C₄×Q₈×C₄₇	D₄	1	3	yes	—
48	C₂×C₄⋊C₄×C₄₇	Q₈	1	3	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
50	C₄⋊C₈×C₄₇	—	2	3	no	—
51	C₄⋊C₈×C₄₇	—	2	3	no	—
52	#97×C₄₇	—	2	3	no	—
53	C₄×C₈×C₄₇	—	2	3	no	—
54	C₂×Q₈×C₄₇	—	2	4	no	—
55	C₂×Q₈×C₄₇	C₄⋊C₄	1	3	yes	—
56	C₂×Q₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₄₇	—	2	1	no	—
61	M₁₆×C₄₇	—	2	3	no	—
62	M₁₆×C₄₇	—	2	3	no	—
63	M₁₆×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₄₇	—	2	1	no	—
65	C₂×C₈×C₄₇	—	2	3	no	—
66	C₄⋊C₄×C₄₇	C₄⋊C₄	1	6	yes	—
67	C₄⋊C₄×C₄₇	C₄⋊C₄	1	3	yes	—
68	C₄⋊C₄×C₄₇	C₄⋊C₄	1	3	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
71	C₄²×C₄₇	C₂×D₄	1	3	yes	—
72	C₄²×C₄₇	C₂×D₄	1	3	yes	—
73	#10625329	—	2	1	no	—
74	#10625329	—	2	3	no	—
75	#10625329	—	2	3	no	—
76	#10625329	C₂×Q₈	1	2	yes	—
77	C₂×C₄₇⋊C₈	—	2	1	no	—
78	C₂×C₄₇⋊C₈	—	2	3	no	—
79	C₂×C₄₇⋊C₈	—	2	3	no	—
80	C₂×C₄₇⋊C₈	—	2	3	no	—
81	C₂×C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
82	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
84	C₂×C₄₇⋊C₈	#41	1	2	yes	—
85	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
86	Q₈×C₄₇	—	2	6	no	—
87	Q₈×C₄₇	#108	1	4	yes	complemented
88	C₂×C₄×C₄₇	—	2	6	no	—
89	C₂×C₄×C₄₇	—	4	3	no	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
100	C₈×C₄₇	—	2	2	no	—
101	C₈×C₄₇	—	4	3	no	—
102	C₄₇⋊C₈	—	2	6	no	—
103	C₄₇⋊C₈	—	2	2	no	—
104	C₄₇⋊C₈	—	4	3	no	—
105	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
106	Q₈×#108	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄₇	#557	1	1	yes	characteristic
111	C₂²×C₄₇	Q₈²	1	1	yes	characteristic
112	C₄×C₄₇	—	2	3	no	—
113	C₄×C₄₇	—	2	3	no	—
114	C₄×C₄₇	—	2	3	no	—
115	C₄×C₄₇	C₂×#108	1	6	yes	—
116	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
117	C₄×C₄₇	#520	1	1	yes	characteristic
118	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
119	C₄×C₄₇	#550	1	1	yes	characteristic
120	Q₈×M₁₆	—	47	1	no	—
121	Q₈×M₁₆	D₄₇	1	1	yes	characteristic
122	C₂×C₈×Q₈	—	47	1	no	—
123	#1950	—	47	3	no	—
124	#1950	—	47	3	no	—
125	#1848	—	47	3	no	—
126	C₄×#108	—	47	3	no	—
127	C₂×C₄₇	—	2	1	no	—
128	C₂×C₄₇	—	2	1	no	—
129	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄₇	C₂²×#108	1	1	yes	characteristic
131	C₂×C₄₇	#2916	1	1	yes	characteristic
132	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
133	#446	—	2	3	no	—
134	#446	—	94	3	no	—
135	C₈×Q₈	—	2	1	no	—
136	C₈×Q₈	—	47	6	no	—
137	C₈×Q₈	—	47	2	no	—
138	C₈×Q₈	—	94	1	no	—
139	C₂×#108	—	47	1	no	—
140	C₂×#108	—	47	3	no	—
141	C₂×#108	—	47	6	no	—
142	#423	—	47	3	no	—
143	#423	C₂×D₄₇	1	3	yes	—
144	C₂×C₄⋊C₈	—	47	3	no	—
145	C₄×M₁₆	—	47	3	no	—
146	C₄×M₁₆	C₂×D₄₇	1	3	yes	—
147	C₂×C₄×C₈	—	47	3	no	—
148	#341	—	47	3	no	—
149	#341	—	47	3	no	—
150	C₄₇	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
152	C₂×M₁₆	—	47	1	no	—
153	C₂×M₁₆	—	47	3	no	—
154	C₂×M₁₆	C₂²×D₄₇	1	3	yes	—
155	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
156	C₂²×C₈	—	47	1	no	—
157	C₂²×C₈	—	47	3	no	—
158	C₄×Q₈	C₄×D₄₇	1	3	yes	—
159	C₄×Q₈	C₄×D₄₇	1	1	yes	characteristic
160	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₄×Q₈	C₂²⋊D₄₇	1	3	yes	—
162	C₂×C₄⋊C₄	Dic₉₄	1	3	yes	—
163	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
164	#108	—	47	4	no	—
165	#108	—	94	6	no	—
166	#108	—	188	6	no	—
167	C₄⋊C₈	—	2	3	no	—
168	C₄⋊C₈	—	2	3	no	—
169	C₄⋊C₈	—	47	6	no	—
170	C₄⋊C₈	—	47	6	no	—
171	C₄⋊C₈	—	94	3	no	—
172	C₄⋊C₈	—	94	3	no	—
173	#97	—	2	3	no	—
174	#97	—	94	3	no	—
175	C₄×C₈	—	2	3	no	—
176	C₄×C₈	—	47	6	no	—
177	C₄×C₈	—	47	6	no	—
178	C₄×C₈	—	94	3	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	Dic₄₇⋊C₄	1	3	yes	—
181	C₂×Q₈	Dic₄₇⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
183	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
184	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
185	M₁₆	—	2	1	no	—
186	M₁₆	—	2	3	no	—
187	M₁₆	—	2	3	no	—
188	M₁₆	—	47	2	no	—
189	M₁₆	—	94	1	no	—
190	M₁₆	—	94	3	no	—
191	M₁₆	—	94	3	no	—
192	M₁₆	Q₈×D₄₇	1	2	yes	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	3	no	—
195	C₂×C₈	—	47	2	no	—
196	C₂×C₈	—	47	2	no	—
197	C₂×C₈	—	47	2	no	—
198	C₂×C₈	—	47	6	no	—
199	C₂×C₈	—	94	1	no	—
200	C₂×C₈	—	94	3	no	—
201	C₂×C₈	—	94	3	no	—
202	C₂×C₈	—	94	3	no	—
203	C₄⋊C₄	Dic₄₇⋊C₄	1	6	yes	—
204	C₄⋊C₄	Dic₄₇⋊C₄	1	3	yes	—
205	C₄⋊C₄	Dic₄₇⋊C₄	1	3	yes	—
206	C₄²	C₂×C₄×D₄₇	1	3	yes	—
207	C₄²	C₂×C₄×D₄₇	1	3	yes	—
208	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
209	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
210	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
211	Q₈	—	2	6	no	—
212	Q₈	#13093959	1	4	yes	direct product factor
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	4	3	no	—
215	C₂×C₄	C₂×Dic₄₇⋊C₄	1	3	yes	—
216	C₂×C₄	C₂×Dic₄₇⋊C₄	1	3	yes	—
217	C₂×C₄	C₂×Dic₄₇⋊C₄	1	3	yes	—
218	C₂×C₄	C₂×Dic₄₇⋊C₄	1	3	yes	—
219	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
222	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
223	C₂×C₄	#13094070	1	1	yes	characteristic
224	C₂×C₄	#13094072	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	4	3	no	—
227	C₈	—	47	4	no	—
228	C₈	—	94	2	no	—
229	C₈	—	94	6	no	—
230	C₈	—	188	3	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
234	C₂²	#15689849	1	1	yes	characteristic
235	C₂²	Q₈×Dic₉₄	1	1	yes	characteristic
236	C₄	—	2	3	no	—
237	C₄	—	2	3	no	—
238	C₄	—	2	3	no	—
239	C₄	C₂×#13093959	1	6	yes	—
240	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
241	C₄	#15689605	1	1	yes	characteristic
242	C₄	Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₄	#15689792	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	Q₈×Dic₄₇⋊C₄	1	1	yes	characteristic
247	C₂	C₂²×#13093959	1	1	yes	characteristic
248	C₂	#20251997	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	Q₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₄₇	C₂²×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	Q₈×M₁₆	D₄₇
hypocenter	C₄₇	Q₈×#108
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
32. #13093959	Q₈	direct (N × H)
87. Q₈×C₄₇	#108	semidirect
150. C₄₇	Q₈×#108	semidirect
212. Q₈	#13093959	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#10625329	2
C₂×Q₈×C₄₇⋊C₈	2
#20244163	2
#20244161	2
#20243559	2
C₄×#13093959	2
Q₈×M₁₆×C₄₇	2
Q₈×#108	47

Subgroups — Q8×#13093959 #26235168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13093959 #26235168