Subgroups — Q₈×#88240 #16808442

Order 3328.

1882 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6118	C₂	1	3	yes	maximal, complemented
2	#13131632	C₂	1	9	yes	maximal, complemented
3	#13131625	C₂	1	9	yes	maximal, complemented
4	Q₈×D₁₃⋊C₈	C₂	1	3	yes	maximal, complemented
5	#13131482	C₂	1	3	yes	maximal
6	C₄×#88240	C₂	1	3	yes	maximal
7	Q₈×#6131	C₂	1	1	yes	maximal, characteristic
8	C₂×#88240	C₂²	1	3	yes	—
9	C₂×#88240	C₂²	1	1	yes	characteristic
10	Q₈²×C₁₃	C₄	1	1	yes	characteristic
11	#11719435	C₄	1	9	yes	—
12	#11719433	C₂²	1	9	yes	—
13	Q₈×D₅₂	C₄	1	3	yes	—
14	C₄×#6131	C₂²	1	3	yes	—
15	C₄×Q₈×D₁₃	C₂²	1	3	yes	—
16	#11719275	C₄	1	3	yes	—
17	#11719158	C₂²	1	18	yes	—
18	Q₈×#1057	C₂²	1	1	yes	characteristic
19	Q₈×#1057	C₂²	1	3	yes	—
20	Q₈×#1057	C₂²	1	6	yes	—
21	#11719080	C₂²	1	18	yes	—
22	#11719079	C₂²	1	18	yes	—
23	#11719074	C₂²	1	9	yes	—
24	#11719074	C₂²	1	9	yes	—
25	#11719072	C₂²	1	9	yes	—
26	#11719072	C₂²	1	9	yes	—
27	#11719052	C₂²	1	9	yes	—
28	D₁₃⋊(C₄⋊C₈)	C₂²	1	9	yes	—
29	C₄×#6118	C₂²	1	9	yes	—
30	C₄×D₁₃⋊C₈	C₂²	1	9	yes	—
31	C₂×#6131	C₂³	1	1	yes	characteristic
32	C₂×Q₈×D₁₃	C₂×C₄	1	3	yes	—
33	#88240	—	2	12	no	—
34	#88240	Q₈	1	8	yes	direct product factor
35	C₂×#6118	C₂³	1	9	yes	—
36	C₂×#6118	C₂³	1	3	yes	—
37	C₂×D₁₃⋊C₈	C₂³	1	9	yes	—
38	C₂×D₁₃⋊C₈	C₂³	1	3	yes	—
39	C₁₃×C₄⋊Q₈	C₂×C₄	1	9	yes	—
40	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
41	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
42	#88200	C₂×C₄	1	18	yes	—
43	Q₈×Dic₁₃	C₂×C₄	1	3	yes	—
44	Q₈×Dic₁₃	C₂³	1	6	yes	—
45	Q₈×Dic₁₃	C₂³	1	1	yes	characteristic
46	#88151	C₂×C₄	1	18	yes	—
47	#88147	C₂×C₄	1	9	yes	—
48	#88146	C₂³	1	9	yes	—
49	C₄⋊C₄×D₁₃	C₂×C₄	1	9	yes	—
50	#88144	C₂×C₄	1	18	yes	—
51	#88141	C₂³	1	9	yes	—
52	C₄×D₅₂	C₂×C₄	1	9	yes	—
53	C₄²×D₁₃	C₂³	1	9	yes	—
54	#88123	C₂×C₄	1	9	yes	—
55	C₄×Dic₂₆	C₂×C₄	1	9	yes	—
56	#88112	C₂³	1	18	yes	—
57	#88110	C₂³	1	18	yes	—
58	C₄⋊#1057	C₂³	1	3	yes	—
59	C₄⋊#1057	C₂³	1	9	yes	—
60	C₄⋊#1057	C₂³	1	18	yes	—
61	C₄×#1057	C₂³	1	3	yes	—
62	C₄×#1057	C₂³	1	9	yes	—
63	C₄×#1057	C₂³	1	9	yes	—
64	C₄×#1057	C₂³	1	9	yes	—
65	Q₈×#131	—	13	1	no	maximal, Sylow
66	C₂×Q₈×C₁₃	C₂²×C₄	1	6	yes	—
67	C₂×Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	#6131	C₂×Q₈	1	4	yes	—
70	#6131	#41	1	4	yes	—
71	Q₈×D₁₃	—	2	12	no	—
72	C₂×D₅₂	C₂²×C₄	1	3	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	9	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	9	yes	—
75	C₂×C₄×D₁₃	C₂⁴	1	3	yes	—
76	C₂×Dic₂₆	C₂²×C₄	1	9	yes	—
77	C₂×#1057	C₂⁴	1	3	yes	—
78	C₂×#1057	C₂⁴	1	9	yes	—
79	C₂×#1057	C₂⁴	1	1	yes	characteristic
80	C₂×#1057	C₂⁴	1	3	yes	—
81	#6118	—	2	18	no	—
82	#6118	C₂×Q₈	1	12	yes	complemented
83	D₁₃⋊C₈	—	2	18	no	—
84	D₁₃⋊C₈	C₂×Q₈	1	12	yes	complemented
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	9	yes	—
86	C₁₃×C₄⋊C₄	C₂²×C₄	1	9	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	9	yes	—
88	#6103	C₂²×C₄	1	18	yes	—
89	C₄⋊Dic₁₃	C₂²×C₄	1	3	yes	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	9	yes	—
91	C₄⋊Dic₁₃	C₂²×C₄	1	18	yes	—
92	#6101	C₂²×C₄	1	18	yes	—
93	C₄×Dic₁₃	C₂²×C₄	1	9	yes	—
94	C₄×Dic₁₃	C₂⁴	1	3	yes	—
95	C₄×Dic₁₃	C₂⁴	1	9	yes	—
96	Q₈×#41	—	13	1	no	—
97	Q₈×M₁₆	—	13	3	no	—
98	#2962	—	13	9	no	—
99	#2960	—	13	9	no	—
100	C₂×C₈×Q₈	—	13	3	no	—
101	#2906	—	13	3	no	—
102	C₄×#131	—	13	3	no	—
103	C₂²×D₁₃	C₂³×C₄	1	3	yes	—
104	Q₈×C₁₃	—	4	6	no	—
105	Q₈×C₁₃	C₄×Q₈	1	4	yes	—
106	Q₈×C₁₃	#131	1	4	yes	complemented
107	C₂×C₄×C₁₃	C₂³×C₄	1	9	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
110	C₂×Dic₁₃	C₂³×C₄	1	3	yes	—
111	C₂×Dic₁₃	C₂³×C₄	1	9	yes	—
112	C₂×Dic₁₃	C₂⁵	1	1	yes	characteristic
113	D₅₂	C₄×Q₈	1	12	yes	—
114	C₄×D₁₃	—	2	18	no	—
115	C₄×D₁₃	—	4	9	no	—
116	C₄×D₁₃	C₂²×Q₈	1	6	yes	—
117	C₄×D₁₃	C₂×#41	1	6	yes	—
118	Dic₂₆	—	2	18	no	—
119	#1057	—	2	3	no	—
120	#1057	—	2	9	no	—
121	#1057	C₂²×Q₈	1	2	yes	—
122	#1057	C₂²×Q₈	1	6	yes	—
123	C₂×#131	—	13	1	no	—
124	C₂×#131	—	13	3	no	—
125	Q₈²	#261	1	1	yes	characteristic
126	#557	—	13	3	no	—
127	#554	—	13	9	no	—
128	D₄×Q₈	—	13	3	no	—
129	#533	—	13	9	no	—
130	C₄×#41	—	13	3	no	—
131	C₂×C₄×Q₈	—	13	3	no	—
132	#446	—	13	18	no	—
133	#446	—	13	18	no	—
134	C₈×Q₈	—	13	1	no	—
135	C₈×Q₈	—	13	9	no	—
136	C₈×Q₈	—	13	9	no	—
137	C₈×Q₈	—	13	3	no	—
138	C₈×Q₈	—	13	6	no	—
139	#424	—	13	18	no	—
140	#423	—	13	9	no	—
141	C₂×C₄⋊C₈	—	13	9	no	—
142	#405	—	13	9	no	—
143	#405	—	13	9	no	—
144	C₄×M₁₆	—	13	9	no	—
145	C₂×C₄×C₈	—	13	9	no	—
146	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
147	C₂×D₁₃	—	2	3	no	—
148	C₂×D₁₃	—	2	3	no	—
149	C₂×D₁₃	C₂×C₄×Q₈	1	6	yes	—
150	C₄×C₁₃	—	2	9	no	—
151	C₄×C₁₃	C₂×C₄×Q₈	1	6	yes	—
152	C₄×C₁₃	C₂×#131	1	6	yes	—
153	Dic₁₃	—	2	9	no	—
154	Dic₁₃	—	2	3	no	—
155	Dic₁₃	C₂³×Q₈	1	1	yes	characteristic
156	Dic₁₃	C₂²×#41	1	1	yes	characteristic
157	C₂×#41	—	13	1	no	—
158	C₂²×Q₈	—	13	3	no	—
159	#131	—	13	8	no	—
160	#131	—	26	12	no	—
161	C₂×M₁₆	—	13	3	no	—
162	C₂×M₁₆	—	13	9	no	—
163	C₂²×C₈	—	13	3	no	—
164	C₂²×C₈	—	13	9	no	—
165	C₄⋊Q₈	—	13	9	no	—
166	C₄⋊Q₈	C₂×#261	1	9	yes	—
167	#125	—	13	18	no	—
168	C₂²⋊Q₈	—	13	18	no	—
169	C₂²⋊Q₈	—	13	18	no	—
170	C₄×Q₈	—	13	1	no	—
171	C₄×Q₈	—	13	3	no	—
172	C₄×Q₈	—	13	6	no	—
173	C₄×Q₈	—	13	9	no	—
174	C₄×Q₈	—	13	9	no	—
175	C₄×Q₈	C₂×#261	1	3	yes	—
176	C₄×Q₈	C₂×#261	1	3	yes	—
177	C₄×D₄	—	13	9	no	—
178	C₄×D₄	—	13	9	no	—
179	#117	—	13	9	no	—
180	C₂×C₄⋊C₄	—	13	9	no	—
181	C₂×C₄²	—	13	9	no	—
182	C₄⋊C₈	—	13	3	no	—
183	C₄⋊C₈	—	13	9	no	—
184	C₄⋊C₈	—	13	18	no	—
185	C₄⋊C₈	—	13	18	no	—
186	#97	—	13	18	no	—
187	C₄×C₈	—	13	3	no	—
188	C₄×C₈	—	13	9	no	—
189	C₄×C₈	—	13	9	no	—
190	C₄×C₈	—	13	9	no	—
191	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
192	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
193	C₂×C₁₃	#3573	1	1	yes	characteristic
194	D₁₃	—	2	6	no	—
195	#41	—	13	4	no	—
196	#41	—	13	4	no	—
197	C₂×Q₈	—	13	9	no	—
198	C₂×Q₈	—	26	12	no	—
199	C₂×Q₈	C₂²×#261	1	6	yes	—
200	C₂×Q₈	C₂²×#261	1	1	yes	characteristic
201	C₂×Q₈	C₂²×#261	1	1	yes	characteristic
202	C₂×D₄	—	13	3	no	—
203	C₂²×C₄	—	13	3	no	—
204	C₂²×C₄	—	13	9	no	—
205	C₂²×C₄	—	13	9	no	—
206	M₁₆	—	13	12	no	—
207	M₁₆	—	26	18	no	—
208	C₂×C₈	—	13	1	no	—
209	C₂×C₈	—	13	3	no	—
210	C₂×C₈	—	13	12	no	—
211	C₂×C₈	—	13	3	no	—
212	C₂×C₈	—	13	9	no	—
213	C₂×C₈	—	26	18	no	—
214	C₄⋊C₄	—	13	3	no	—
215	C₄⋊C₄	—	13	18	no	—
216	C₄⋊C₄	—	13	9	no	—
217	C₄⋊C₄	—	13	18	no	—
218	C₄⋊C₄	C₂²×#261	1	9	yes	—
219	C₄⋊C₄	C₂²×#261	1	9	yes	—
220	C₂²⋊C₄	—	13	18	no	—
221	C₄²	—	13	3	no	—
222	C₄²	—	13	9	no	—
223	C₄²	—	13	9	no	—
224	C₄²	C₂²×#261	1	9	yes	—
225	C₁₃	Q₈×#131	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	13	3	no	—
227	Q₈	—	4	6	no	—
228	Q₈	—	26	18	no	—
229	Q₈	#88240	1	4	yes	direct product factor
230	Q₈	Q₈×#261	1	4	yes	—
231	D₄	—	13	12	no	—
232	C₂×C₄	—	13	1	no	—
233	C₂×C₄	—	13	6	no	—
234	C₂×C₄	—	13	6	no	—
235	C₂×C₄	—	13	3	no	—
236	C₂×C₄	—	13	9	no	—
237	C₂×C₄	—	26	18	no	—
238	C₂×C₄	—	52	9	no	—
239	C₂×C₄	C₂³×#261	1	9	yes	—
240	C₂×C₄	C₂³×#261	1	3	yes	—
241	C₂×C₄	C₂³×#261	1	3	yes	—
242	C₈	—	13	2	no	—
243	C₈	—	13	6	no	—
244	C₈	—	26	3	no	—
245	C₈	—	26	9	no	—
246	C₂²	—	13	6	no	—
247	C₂²	—	26	3	no	—
248	C₂²	—	26	3	no	—
249	C₂²	C₂⁴×#261	1	1	yes	characteristic
250	C₄	—	2	9	no	—
251	C₄	—	13	1	no	—
252	C₄	—	13	1	no	—
253	C₄	—	26	9	no	—
254	C₄	—	26	3	no	—
255	C₄	C₂×#88240	1	6	yes	—
256	C₄	C₂×Q₈×#261	1	6	yes	—
257	C₂	—	26	6	no	—
258	C₂	C₂²×Q₈×#261	1	1	yes	characteristic
259	C₂	C₂²×#88240	1	1	yes	characteristic
260	C₂	#13131831	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×#261
commutator, socle	C₂²×C₁₃	C₂⁴×C₄
Fitting	Q₈²×C₁₃	C₄
prime radical	C₂×#6131	C₂³
cosocle	C₂×Dic₁₃	C₂⁵
hypercenter	Q₈²	#261
hypocenter	C₁₃	Q₈×#131
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈×#6118	C₂	semidirect
2. #13131632	C₂	semidirect
3. #13131625	C₂	semidirect
4. Q₈×D₁₃⋊C₈	C₂	semidirect
34. #88240	Q₈	direct (N × H)
82. #6118	C₂×Q₈	semidirect
84. D₁₃⋊C₈	C₂×Q₈	semidirect
106. Q₈×C₁₃	#131	semidirect
225. C₁₃	Q₈×#131	semidirect
229. Q₈	#88240	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×#6118	2
#13131632	2
#13131625	2
Q₈×D₁₃⋊C₈	2
#13131482	2
C₄×#88240	2
Q₈×#6131	2
Q₈×#131	13

Subgroups — Q8×#88240 #16808442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88240 #16808442