Subgroups — Q₈×#88048 #15881743

Order 3328.

838 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13115599	C₂	1	3	yes	maximal
4	#13115597	C₂	1	3	yes	maximal
5	#13114992	C₂	1	3	yes	maximal
6	C₄×#88048	C₂	1	3	yes	maximal
7	Q₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	—	2	1	no	—
11	C₁₃×#389	—	2	3	no	—
12	C₁₃×#384	C₂²	1	3	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
14	Q₈×#1055	C₄	1	8	yes	complemented
15	#11718646	—	2	3	no	—
16	C₂×#88048	C₂²	1	3	yes	—
17	C₂×#88048	C₂²	1	6	yes	—
18	C₂×#88048	C₂²	1	1	yes	characteristic
19	#11718596	C₂²	1	3	yes	—
20	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
21	C₂×C₄⋊#1055	C₂²	1	3	yes	—
22	C₂×C₄×#1055	C₂²	1	3	yes	—
23	#11718121	C₂²	1	3	yes	—
24	#11718121	C₂²	1	3	yes	—
25	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	—	4	4	no	—
27	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
31	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	—	2	3	no	—
35	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
36	Q₈×Dic₁₃	—	4	4	no	—
37	C₂×C₄⋊Dic₁₃	—	2	3	no	—
38	C₂×C₄⋊Dic₁₃	—	2	3	no	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	—	2	3	no	—
42	C₂²×#1055	C₂³	1	3	yes	—
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	C₁₃×#95	—	2	3	no	—
45	#88071	—	2	3	no	—
46	#88048	—	2	6	no	—
47	#88048	—	4	6	no	—
48	#88048	Q₈	1	4	yes	direct product factor
49	C₄⋊#1055	C₂×C₄	1	12	yes	—
50	C₄×#1055	C₂×C₄	1	12	yes	—
51	Q₈×#106	—	13	1	no	maximal, Sylow
52	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
53	C₂×Q₈×C₁₃	SD₁₆	1	4	yes	—
54	C₂×Q₈×C₁₃	SD₁₆	1	4	yes	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	2	3	no	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	2	1	no	—
61	C₂²×Dic₁₃	—	2	3	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	3	no	—
64	C₁₃×C₄⋊C₄	—	2	3	no	—
65	C₁₃×C₄⋊C₄	—	4	6	no	—
66	C₁₃×C₄⋊C₄	—	8	3	no	—
67	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
68	C₁₃×C₄⋊C₄	C₂×Q₈	1	2	yes	—
69	C₄²×C₁₃	—	4	6	no	—
70	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
71	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
72	C₄⋊Dic₁₃	—	2	1	no	—
73	C₄⋊Dic₁₃	—	2	3	no	—
74	C₄⋊Dic₁₃	—	2	3	no	—
75	C₄⋊Dic₁₃	—	4	6	no	—
76	C₄⋊Dic₁₃	—	8	3	no	—
77	C₄⋊Dic₁₃	C₂×Q₈	1	2	yes	—
78	C₄×Dic₁₃	—	4	6	no	—
79	C₂×#1055	—	2	3	no	—
80	C₂×#1055	—	2	3	no	—
81	C₂×#1055	C₂²×C₄	1	6	yes	—
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₂×#1055	C₂×Q₈	1	2	yes	—
84	C₂×#1055	#41	1	2	yes	—
85	C₂×C₈×Q₈	—	13	1	no	—
86	Q₈×C₄⋊C₄	—	13	1	no	—
87	Q₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
88	#1948	—	13	3	no	—
89	#1948	—	13	3	no	—
90	#1844	—	13	3	no	—
91	C₄×#106	—	13	3	no	—
92	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₁₃	#106	1	16	yes	complemented
94	C₂×C₄×C₁₃	—	2	3	no	—
95	C₂×C₄×C₁₃	—	2	3	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	4	1	no	—
98	C₂×C₄×C₁₃	—	4	1	no	—
99	C₂×C₄×C₁₃	—	4	3	no	—
100	C₂×C₄×C₁₃	—	4	3	no	—
101	C₂×C₄×C₁₃	—	4	3	no	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
104	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
105	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	2	no	—
113	C₂×Dic₁₃	—	4	1	no	—
114	C₂×Dic₁₃	—	4	1	no	—
115	C₂×Dic₁₃	—	4	3	no	—
116	C₂×Dic₁₃	—	4	3	no	—
117	C₂×Dic₁₃	—	4	3	no	—
118	#1055	—	2	6	no	—
119	#1055	C₄×Q₈	1	4	yes	complemented
120	C₂×C₄×Q₈	—	2	1	no	—
121	C₂×C₄×Q₈	—	26	1	no	—
122	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
123	C₈×Q₈	—	13	8	no	—
124	C₂×#106	—	13	1	no	—
125	C₂×#106	—	13	3	no	—
126	C₂×#106	—	13	6	no	—
127	C₂×C₄⋊C₈	—	13	3	no	—
128	C₂×C₄×C₈	—	13	3	no	—
129	#389	—	2	3	no	—
130	#389	—	26	3	no	—
131	#384	—	13	3	no	—
132	#384	C₂×D₁₃	1	3	yes	—
133	C₄×C₄⋊C₄	—	13	3	no	—
134	C₄×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
135	#340	—	13	3	no	—
136	#340	—	13	3	no	—
137	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
139	C₂²×C₁₃	#550	1	1	yes	characteristic
140	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
141	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
142	C₂²×C₁₃	#577	1	1	yes	characteristic
143	C₂²×C₁₃	#577	1	1	yes	characteristic
144	C₄×C₁₃	—	2	6	no	—
145	C₄×C₁₃	—	4	2	no	—
146	C₄×C₁₃	—	8	3	no	—
147	C₄×C₁₃	C₂×#106	1	12	yes	—
148	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₁₃	#520	1	1	yes	characteristic
150	C₄×C₁₃	#557	1	1	yes	characteristic
151	C₄×C₁₃	Q₈²	1	1	yes	characteristic
152	Dic₁₃	—	4	2	no	—
153	Dic₁₃	—	8	3	no	—
154	C₂²×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
155	C₂²×C₈	—	13	1	no	—
156	C₂²×C₈	—	13	3	no	—
157	C₄×Q₈	—	4	4	no	—
158	C₄×Q₈	—	52	4	no	—
159	C₄×Q₈	Dic₂₆	1	4	yes	—
160	C₄×Q₈	C₄×D₁₃	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	3	no	—
162	C₂×C₄⋊C₄	—	2	3	no	—
163	C₂×C₄⋊C₄	—	13	1	no	—
164	C₂×C₄⋊C₄	—	13	3	no	—
165	C₂×C₄⋊C₄	—	26	3	no	—
166	C₂×C₄⋊C₄	—	26	3	no	—
167	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
168	C₂×C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
169	C₂×C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×C₄²	—	2	3	no	—
171	C₂×C₄²	—	26	3	no	—
172	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
173	#106	—	13	4	no	—
174	#106	—	26	6	no	—
175	#106	—	52	6	no	—
176	C₄⋊C₈	—	13	12	no	—
177	C₄×C₈	—	13	12	no	—
178	#95	—	2	3	no	—
179	#95	—	26	3	no	—
180	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
182	C₂×C₁₃	#2912	1	1	yes	characteristic
183	C₂×C₁₃	Q₈×SD₁₆	1	1	yes	characteristic
184	C₂×C₁₃	Q₈×SD₁₆	1	1	yes	characteristic
185	C₂×C₁₃	#3392	1	1	yes	characteristic
186	C₂×C₁₃	#3392	1	1	yes	characteristic
187	C₂×Q₈	#6101	1	4	yes	—
188	C₂×Q₈	#6105	1	4	yes	—
189	C₂×Q₈	#6106	1	4	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	3	no	—
192	C₂²×C₄	—	26	1	no	—
193	C₂²×C₄	—	26	3	no	—
194	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
196	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	13	6	no	—
201	C₂×C₈	—	26	3	no	—
202	C₂×C₈	—	26	3	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	3	no	—
205	C₄⋊C₄	—	2	3	no	—
206	C₄⋊C₄	—	4	6	no	—
207	C₄⋊C₄	—	8	3	no	—
208	C₄⋊C₄	—	13	2	no	—
209	C₄⋊C₄	—	26	1	no	—
210	C₄⋊C₄	—	26	3	no	—
211	C₄⋊C₄	—	26	3	no	—
212	C₄⋊C₄	—	52	6	no	—
213	C₄⋊C₄	—	104	3	no	—
214	C₄⋊C₄	#6101	1	12	yes	—
215	C₄⋊C₄	Q₈×D₁₃	1	2	yes	—
216	C₄²	—	4	6	no	—
217	C₄²	—	52	6	no	—
218	C₄²	C₂×Dic₂₆	1	6	yes	—
219	C₄²	C₂×C₄×D₁₃	1	6	yes	—
220	C₁₃	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
222	Q₈	#88048	1	16	yes	direct product factor
223	C₂×C₄	—	2	3	no	—
224	C₂×C₄	—	2	3	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	4	3	no	—
230	C₂×C₄	—	4	3	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	52	1	no	—
233	C₂×C₄	—	52	1	no	—
234	C₂×C₄	—	52	3	no	—
235	C₂×C₄	—	52	3	no	—
236	C₂×C₄	—	52	3	no	—
237	C₂×C₄	C₂×#6101	1	6	yes	—
238	C₂×C₄	C₂×#6101	1	6	yes	—
239	C₂×C₄	C₂×#6105	1	6	yes	—
240	C₂×C₄	C₂×#6106	1	6	yes	—
241	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
244	C₂×C₄	#88254	1	1	yes	characteristic
245	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
246	C₂×C₄	#88256	1	1	yes	characteristic
247	C₈	—	13	4	no	—
248	C₈	—	26	6	no	—
249	C₂²	C₂²×#6101	1	1	yes	characteristic
250	C₂²	C₂²×#6105	1	1	yes	characteristic
251	C₂²	C₂²×#6106	1	1	yes	characteristic
252	C₂²	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
253	C₂²	#11719539	1	1	yes	characteristic
254	C₂²	#11719541	1	1	yes	characteristic
255	C₂²	#11719542	1	1	yes	characteristic
256	C₄	—	2	6	no	—
257	C₄	—	4	2	no	—
258	C₄	—	8	3	no	—
259	C₄	—	52	2	no	—
260	C₄	—	104	3	no	—
261	C₄	C₂×#88048	1	12	yes	—
262	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
263	C₄	#11719274	1	1	yes	characteristic
264	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
265	C₄	#11719279	1	1	yes	characteristic
266	C₂	Q₈×#6101	1	1	yes	characteristic
267	C₂	C₂²×#88048	1	1	yes	characteristic
268	C₂	#13123429	1	1	yes	characteristic
269	C₂	Q₈×#6106	1	1	yes	characteristic
270	C₂	Q₈×#6105	1	1	yes	characteristic
271	C₂	#13127813	1	1	yes	characteristic
272	C₂	#13127817	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	Q₈×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	Q₈×#106
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. Q₈×#1055	C₄	semidirect
48. #88048	Q₈	direct (N × H)
93. Q₈×C₁₃	#106	semidirect
119. #1055	C₄×Q₈	semidirect
220. C₁₃	Q₈×#106	semidirect
222. Q₈	#88048	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×#1055	2
Q₈×C₄⋊Dic₁₃	2
#13115599	2
#13115597	2
#13114992	2
C₄×#88048	2
Q₈×C₁₃×C₄⋊C₄	2
Q₈×#106	13

Subgroups — Q8×#88048 #15881743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88048 #15881743