Subgroups — Q₈×#88110 #16797478

Order 3328.

954 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1057	C₂	1	2	yes	maximal
2	#13130632	C₂	1	6	yes	maximal
3	#13130622	C₂	1	3	yes	maximal
4	C₄×#88110	C₂	1	3	yes	maximal
5	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×Q₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
9	Q₈×#1057	C₄	1	16	yes	complemented
10	C₂×#88110	C₂²	1	3	yes	—
11	C₂×#88110	C₂²	1	6	yes	—
12	C₂×#88110	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#1057	C₂²	1	6	yes	—
14	C₂×C₄×#1057	C₂²	1	6	yes	—
15	C₄⋊C₄×Dic₁₃	C₄	1	3	yes	—
16	C₄⋊C₄×Dic₁₃	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₁₃	C₄	1	3	yes	—
18	C₄²×Dic₁₃	C₂²	1	3	yes	—
19	#11718290	C₂²	1	6	yes	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1057	C₂³	1	6	yes	—
22	C₂²×#1057	C₂³	1	2	yes	—
23	C₄×Q₈×C₁₃	—	2	4	no	—
24	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
26	Q₈×Dic₁₃	—	2	4	no	—
27	Q₈×Dic₁₃	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
29	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
34	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
36	#88110	—	2	12	no	—
37	#88110	—	2	6	no	—
38	#88110	Q₈	1	4	yes	direct product factor
39	C₄⋊#1057	C₂×C₄	1	24	yes	—
40	C₄×#1057	C₂×C₄	1	24	yes	—
41	#88071	C₂×C₄	1	6	yes	—
42	Q₈×#97	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
44	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
45	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
49	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
51	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
53	C₂×#1057	—	2	6	no	—
54	C₂×#1057	—	2	6	no	—
55	C₂×#1057	C₂²×C₄	1	12	yes	—
56	C₂×#1057	C₂²×C₄	1	4	yes	—
57	C₂×#1057	C₂×Q₈	1	4	yes	—
58	C₂×#1057	#41	1	4	yes	—
59	C₁₃×C₄⋊C₄	—	2	6	no	—
60	C₄²×C₁₃	—	2	6	no	—
61	C₄⋊Dic₁₃	—	2	6	no	—
62	C₄⋊Dic₁₃	C₄²	1	12	yes	—
63	C₄×Dic₁₃	—	2	6	no	—
64	C₄×Dic₁₃	—	2	6	no	—
65	C₄×Dic₁₃	—	4	3	no	—
66	C₄×Dic₁₃	—	4	3	no	—
67	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
68	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
69	C₄×Dic₁₃	#41	1	2	yes	—
70	C₂×C₈×Q₈	—	13	2	no	—
71	C₄²×Q₈	—	13	1	no	—
72	#1770	—	13	6	no	—
73	#1767	—	13	3	no	—
74	C₄×#97	—	13	3	no	—
75	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
76	Q₈×C₁₃	#97	1	16	yes	complemented
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	3	no	—
80	C₂×C₄×C₁₃	—	2	3	no	—
81	C₂×C₄×C₁₃	—	2	3	no	—
82	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
83	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
85	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	3	no	—
89	C₂×Dic₁₃	—	2	3	no	—
90	C₂×Dic₁₃	—	2	3	no	—
91	C₂×Dic₁₃	—	2	3	no	—
92	C₂×Dic₁₃	—	2	3	no	—
93	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
94	C₂×Dic₁₃	C₄×Q₈	1	2	yes	—
95	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
96	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
97	C₂×Dic₁₃	C₂×#41	1	2	yes	—
98	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
99	#1057	—	2	12	no	—
100	#1057	C₄×Q₈	1	8	yes	complemented
101	C₂×C₄×Q₈	—	13	1	no	—
102	C₂×C₄×Q₈	—	13	1	no	—
103	C₂×C₄×Q₈	#261	1	1	yes	characteristic
104	C₈×Q₈	—	13	16	no	—
105	C₂×C₄⋊C₈	—	13	6	no	—
106	C₂×#97	—	13	1	no	—
107	C₂×#97	—	13	3	no	—
108	C₂×#97	—	13	6	no	—
109	C₂×C₄×C₈	—	13	6	no	—
110	C₄×C₄⋊C₄	—	13	3	no	—
111	C₄×C₄⋊C₄	—	13	3	no	—
112	C₄×C₄⋊C₄	—	13	3	no	—
113	C₄³	—	13	3	no	—
114	#336	—	13	6	no	—
115	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
117	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
118	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
119	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
120	C₂²×C₁₃	#568	1	1	yes	characteristic
121	C₂²×C₁₃	#568	1	1	yes	characteristic
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	4	3	no	—
124	C₄×C₁₃	C₂×#97	1	12	yes	—
125	Dic₁₃	—	2	6	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	4	3	no	—
128	Dic₁₃	C₂×C₄×Q₈	1	2	yes	—
129	Dic₁₃	C₄×#41	1	2	yes	—
130	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
131	C₂²×C₈	—	13	2	no	—
132	C₂²×C₈	—	13	6	no	—
133	C₄×Q₈	—	2	4	no	—
134	C₄×Q₈	—	13	8	no	—
135	C₄×Q₈	—	26	4	no	—
136	C₂×C₄⋊C₄	—	13	3	no	—
137	C₂×C₄⋊C₄	—	13	3	no	—
138	C₂×C₄⋊C₄	C₂×#261	1	3	yes	—
139	C₂×C₄²	—	13	3	no	—
140	C₂×C₄²	—	13	1	no	—
141	C₂×C₄²	—	13	3	no	—
142	C₂×C₄²	—	13	3	no	—
143	C₂×C₄²	—	13	3	no	—
144	C₂×C₄²	—	13	3	no	—
145	C₂×C₄²	C₂×#261	1	3	yes	—
146	C₄⋊C₈	—	13	24	no	—
147	#97	—	13	4	no	—
148	#97	—	26	6	no	—
149	#97	—	26	12	no	—
150	C₄×C₈	—	13	24	no	—
151	#95	—	13	6	no	—
152	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
153	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
154	C₂×C₁₃	#2875	1	1	yes	characteristic
155	C₂×C₁₃	Q₈×M₁₆	1	1	yes	characteristic
156	C₂×C₁₃	Q₈×M₁₆	1	1	yes	characteristic
157	C₂×C₁₃	#2971	1	1	yes	characteristic
158	C₂×C₁₃	#2971	1	1	yes	characteristic
159	C₂×Q₈	#6118	1	4	yes	—
160	C₂×Q₈	C₄×#261	1	4	yes	—
161	C₂×Q₈	#6122	1	4	yes	—
162	C₂²×C₄	—	13	1	no	—
163	C₂²×C₄	—	13	3	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	3	no	—
166	C₂²×C₄	C₂²×#261	1	3	yes	—
167	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
168	C₂²×C₄	C₂²×#261	1	3	yes	—
169	C₂×C₈	—	13	4	no	—
170	C₂×C₈	—	13	4	no	—
171	C₂×C₈	—	13	4	no	—
172	C₂×C₈	—	13	12	no	—
173	C₂×C₈	—	26	6	no	—
174	C₂×C₈	—	26	6	no	—
175	C₄⋊C₄	—	2	6	no	—
176	C₄⋊C₄	—	13	12	no	—
177	C₄⋊C₄	—	26	6	no	—
178	C₄²	—	2	6	no	—
179	C₄²	—	13	12	no	—
180	C₄²	—	13	2	no	—
181	C₄²	—	13	2	no	—
182	C₄²	—	26	6	no	—
183	C₄²	—	26	6	no	—
184	C₄²	—	52	3	no	—
185	C₄²	—	52	3	no	—
186	C₁₃	Q₈×#97	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂³×#261	1	1	yes	characteristic
188	Q₈	#88110	1	16	yes	direct product factor
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	13	2	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	6	no	—
199	C₂×C₄	—	13	2	no	—
200	C₂×C₄	—	26	1	no	—
201	C₂×C₄	—	26	3	no	—
202	C₂×C₄	—	26	1	no	—
203	C₂×C₄	—	26	3	no	—
204	C₂×C₄	—	26	3	no	—
205	C₂×C₄	—	26	3	no	—
206	C₂×C₄	—	26	3	no	—
207	C₂×C₄	C₂×#6118	1	6	yes	—
208	C₂×C₄	C₂×C₄×#261	1	6	yes	—
209	C₂×C₄	Q₈×#261	1	2	yes	—
210	C₂×C₄	C₂×#6122	1	6	yes	—
211	C₈	—	13	8	no	—
212	C₈	—	26	12	no	—
213	C₂²	C₂²×#6118	1	1	yes	characteristic
214	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
215	C₂²	#11719625	1	1	yes	characteristic
216	C₂²	C₂×Q₈×#261	1	1	yes	characteristic
217	C₂²	#11719629	1	1	yes	characteristic
218	C₂²	#41×#261	1	1	yes	characteristic
219	C₂²	C₂²×#6122	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	3	no	—
222	C₄	—	13	2	no	—
223	C₄	—	13	2	no	—
224	C₄	—	26	2	no	—
225	C₄	—	26	6	no	—
226	C₄	—	52	3	no	—
227	C₄	C₂×#88110	1	12	yes	—
228	C₂	C₄×Q₈×#261	1	1	yes	characteristic
229	C₂	C₂²×#88110	1	1	yes	characteristic
230	C₂	#13131408	1	1	yes	characteristic
231	C₂	Q₈×#6122	1	1	yes	characteristic
232	C₂	Q₈×#6118	1	1	yes	characteristic
233	C₂	#13131680	1	1	yes	characteristic
234	C₂	#13131682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×#261
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×Dic₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	Q₈×#97
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×#1057	C₄	semidirect
38. #88110	Q₈	direct (N × H)
76. Q₈×C₁₃	#97	semidirect
100. #1057	C₄×Q₈	semidirect
186. C₁₃	Q₈×#97	semidirect
188. Q₈	#88110	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×#1057	2
#13130632	2
#13130622	2
C₄×#88110	2
C₄×Q₈×Dic₁₃	2
Q₈×#97	13

Subgroups — Q8×#88110 #16797478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88110 #16797478