Subgroups — Q₈×#88055 #15861587

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×#1055	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13114706	C₂	1	3	yes	maximal
4	#13114703	C₂	1	3	yes	maximal
5	#13114691	C₂	1	3	yes	maximal
6	C₄×#88055	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₃	C₄	1	2	yes	—
10	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	C₂×#88055	C₂²	1	3	yes	—
14	C₂×#88055	C₂²	1	3	yes	—
15	C₂×#88055	C₂²	1	3	yes	—
16	C₂×#88055	C₂²	1	1	yes	characteristic
17	Q₈×#1055	C₄	1	8	yes	complemented
18	C₄⋊C₄×Dic₁₃	C₄	1	6	yes	—
19	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
20	C₄²×Dic₁₃	C₂²	1	3	yes	—
21	C₂×C₄⋊#1055	C₂²	1	3	yes	—
22	C₂×C₄×#1055	C₂²	1	3	yes	—
23	#11718142	C₂²	1	6	yes	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
31	Q₈×Dic₁₃	—	2	8	no	—
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	6	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
34	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
36	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
37	C₂²×#1055	C₂³	1	3	yes	—
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
41	#88071	C₂×C₄	1	6	yes	—
42	#88055	—	2	6	no	—
43	#88055	—	2	6	no	—
44	#88055	—	2	6	no	—
45	#88055	Q₈	1	4	yes	direct product factor
46	C₄⋊#1055	C₂×C₄	1	12	yes	—
47	C₄×#1055	C₂×C₄	1	12	yes	—
48	Q₈×#97	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
50	C₂×Q₈×C₁₃	M₁₆	1	8	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
55	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₁₃	—	2	3	no	—
57	C₂×C₈×C₁₃	—	2	3	no	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₃	#41	1	2	yes	—
62	C₁₃×C₄⋊C₄	C₄²	1	12	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
64	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
65	C₄⋊Dic₁₃	—	2	12	no	—
66	C₄×Dic₁₃	—	2	12	no	—
67	C₄×Dic₁₃	—	2	3	no	—
68	C₄×Dic₁₃	—	2	3	no	—
69	C₄×Dic₁₃	—	4	6	no	—
70	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
71	C₄×Dic₁₃	#41	1	2	yes	—
72	C₂×#1055	—	2	3	no	—
73	C₂×#1055	—	2	3	no	—
74	C₂×#1055	C₂²×C₄	1	6	yes	—
75	C₂×#1055	C₂²×C₄	1	2	yes	—
76	C₂×#1055	C₂×Q₈	1	2	yes	—
77	C₂×#1055	#41	1	2	yes	—
78	C₂×C₈×Q₈	—	13	1	no	—
79	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
80	C₄²×Q₈	—	13	1	no	—
81	#1770	—	13	3	no	—
82	#1770	—	13	3	no	—
83	#1767	—	13	3	no	—
84	C₄×#97	—	13	3	no	—
85	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
86	Q₈×C₁₃	#97	1	16	yes	complemented
87	C₂×C₄×C₁₃	—	2	6	no	—
88	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
89	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
90	C₂×C₄×C₁₃	C₂×M₁₆	1	12	yes	—
91	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₁₃	—	2	2	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	2	6	no	—
100	C₂×Dic₁₃	—	2	6	no	—
101	C₂×Dic₁₃	—	2	6	no	—
102	C₂×Dic₁₃	C₄×Q₈	1	4	yes	—
103	C₈×C₁₃	—	2	6	no	—
104	C₈×C₁₃	C₄×Q₈	1	4	yes	complemented
105	#1055	—	2	6	no	—
106	#1055	C₄×Q₈	1	4	yes	complemented
107	C₂×C₄×Q₈	—	13	2	no	—
108	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
109	C₈×Q₈	—	13	8	no	—
110	C₈×Q₈	Dic₁₃	1	8	yes	complemented
111	C₂×C₄⋊C₈	—	13	3	no	—
112	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
113	C₂×#97	—	13	1	no	—
114	C₂×#97	—	13	3	no	—
115	C₂×#97	—	13	3	no	—
116	C₂×#97	—	13	3	no	—
117	C₂×C₄×C₈	—	13	3	no	—
118	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
119	C₄×C₄⋊C₄	—	13	6	no	—
120	C₄×C₄⋊C₄	—	13	3	no	—
121	C₄³	—	13	3	no	—
122	#336	—	13	6	no	—
123	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
126	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
127	C₂²×C₁₃	#568	1	2	yes	—
128	C₄×C₁₃	—	2	6	no	—
129	C₄×C₁₃	C₂×#97	1	12	yes	—
130	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
133	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
134	Dic₁₃	—	2	4	no	—
135	Dic₁₃	—	4	6	no	—
136	C₂²×Q₈	C₄×D₁₃	1	1	yes	characteristic
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	3	no	—
139	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
140	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
141	C₄×Q₈	—	26	8	no	—
142	C₄×Q₈	C₄×D₁₃	1	4	yes	—
143	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
144	C₂×C₄⋊C₄	—	13	6	no	—
145	C₂×C₄⋊C₄	C₄×D₁₃	1	3	yes	—
146	C₂×C₄²	—	13	1	no	—
147	C₂×C₄²	—	13	6	no	—
148	C₂×C₄²	—	13	3	no	—
149	C₂×C₄²	—	13	6	no	—
150	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
151	C₄⋊C₈	—	13	12	no	—
152	C₄⋊C₈	C₂×Dic₁₃	1	12	yes	—
153	#97	—	13	4	no	—
154	#97	—	26	6	no	—
155	#97	—	26	6	no	—
156	#97	—	26	6	no	—
157	C₄×C₈	—	13	12	no	—
158	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
159	#95	—	13	6	no	—
160	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
161	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
162	C₂×C₁₃	#2875	1	1	yes	characteristic
163	C₂×C₁₃	Q₈×M₁₆	1	2	yes	—
164	C₂×C₁₃	#2971	1	2	yes	—
165	C₂×Q₈	#6094	1	8	yes	—
166	C₂×Q₈	C₄×Dic₁₃	1	4	yes	—
167	C₂²×C₄	—	13	2	no	—
168	C₂²×C₄	—	13	6	no	—
169	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
170	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
171	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
172	C₂×C₈	—	2	3	no	—
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	13	6	no	—
178	C₂×C₈	—	26	3	no	—
179	C₂×C₈	—	26	3	no	—
180	C₂×C₈	Q₈×D₁₃	1	2	yes	—
181	C₂×C₈	#6131	1	2	yes	—
182	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
183	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
184	C₄⋊C₄	—	26	12	no	—
185	C₄⋊C₄	C₄×Dic₁₃	1	12	yes	—
186	C₄²	—	13	2	no	—
187	C₄²	—	13	2	no	—
188	C₄²	—	26	12	no	—
189	C₄²	—	26	3	no	—
190	C₄²	—	26	3	no	—
191	C₄²	—	52	6	no	—
192	C₄²	C₂×C₄×D₁₃	1	6	yes	—
193	C₄²	C₂²×Dic₁₃	1	6	yes	—
194	C₁₃	Q₈×#97	1	1	yes	characteristic, complemented, Sylow
195	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
196	Q₈	#88055	1	16	yes	direct product factor
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	13	4	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	26	2	no	—
201	C₂×C₄	—	26	6	no	—
202	C₂×C₄	—	26	6	no	—
203	C₂×C₄	—	26	6	no	—
204	C₂×C₄	C₂×#6094	1	12	yes	—
205	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
206	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
207	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂×#6131	1	1	yes	characteristic
210	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
211	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
213	C₈	—	2	6	no	—
214	C₈	—	13	4	no	—
215	C₈	—	26	6	no	—
216	C₈	Q₈×Dic₁₃	1	4	yes	complemented
217	C₂²	C₄×Q₈×D₁₃	1	1	yes	characteristic
218	C₂²	C₄×#6131	1	1	yes	characteristic
219	C₂²	C₂²×#6094	1	2	yes	—
220	C₂²	#11719455	1	2	yes	—
221	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
222	C₄	—	2	6	no	—
223	C₄	—	26	4	no	—
224	C₄	—	52	6	no	—
225	C₄	C₂×#88055	1	12	yes	—
226	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
227	C₄	C₄×#6131	1	1	yes	characteristic
228	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
229	C₄	#41×Dic₁₃	1	1	yes	characteristic
230	C₂	C₄×Q₈×Dic₁₃	1	1	yes	characteristic
231	C₂	C₂²×#88055	1	1	yes	characteristic
232	C₂	#13123262	1	1	yes	characteristic
233	C₂	Q₈×#6094	1	2	yes	—
234	C₂	#13123855	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	Q₈×#97
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₁₃	C₄	semidirect
17. Q₈×#1055	C₄	semidirect
45. #88055	Q₈	direct (N × H)
86. Q₈×C₁₃	#97	semidirect
104. C₈×C₁₃	C₄×Q₈	semidirect
106. #1055	C₄×Q₈	semidirect
110. C₈×Q₈	Dic₁₃	semidirect
194. C₁₃	Q₈×#97	semidirect
196. Q₈	#88055	direct (N × H)
216. C₈	Q₈×Dic₁₃	semidirect

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
C₄×Q₈×Dic₁₃	2
#13114706	2
#13114703	2
#13114691	2
C₄×#88055	2
C₂×C₈×Q₈×C₁₃	2
Q₈×#97	13

Subgroups — Q8×#88055 #15861587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#88055 #15861587