Subgroups — Q₈×#6129 #13128413

Order 1664.

1068 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
3	C₂×Q₈×Dic₁₃	C₂	1	2	yes	maximal, complemented
4	#11719433	C₂	1	3	yes	maximal, complemented
5	#11719426	C₂	1	3	yes	maximal, complemented
6	#11719340	C₂	1	6	yes	maximal, complemented
7	#11719327	C₂	1	6	yes	maximal, complemented
8	C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719248	C₂	1	3	yes	maximal, complemented
11	C₄×#6129	C₂	1	3	yes	maximal
12	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×#6129	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
16	C₁₃×C₂²⋊Q₈	C₂²	1	6	yes	—
17	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	3	yes	—
19	#88200	C₂²	1	6	yes	—
20	Q₈×Dic₁₃	C₂²	1	6	yes	—
21	Q₈×Dic₁₃	C₂²	1	6	yes	—
22	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₃	C₂²	1	2	yes	—
24	#88198	C₂²	1	6	yes	—
25	D₄×Dic₁₃	C₂²	1	3	yes	—
26	C₄×C₂²⋊D₁₃	C₂²	1	6	yes	—
27	C₂×C₄⋊Dic₁₃	C₂²	1	6	yes	—
28	#88178	C₂²	1	6	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	6	yes	—
30	#88151	C₂²	1	6	yes	—
31	#88146	C₂²	1	3	yes	—
32	C₄⋊C₄×D₁₃	C₂²	1	3	yes	—
33	#88144	C₂²	1	6	yes	—
34	#88143	C₂²	1	12	yes	—
35	#88142	C₂²	1	6	yes	—
36	#88141	C₂²	1	3	yes	—
37	#88141	C₂²	1	6	yes	—
38	#88141	C₂²	1	3	yes	—
39	#88135	C₂²	1	6	yes	—
40	#88132	C₂²	1	12	yes	—
41	#88131	C₂²	1	6	yes	—
42	C₄²×D₁₃	C₂²	1	3	yes	—
43	#88123	C₂²	1	3	yes	—
44	C₄×Dic₂₆	C₂²	1	3	yes	—
45	C₄×Dic₂₆	C₂²	1	3	yes	—
46	C₂×Q₈×C₁₃	—	2	8	no	—
47	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
51	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
52	C₂²×Dic₁₃	C₂³	1	6	yes	—
53	C₂²×Dic₁₃	C₂³	1	2	yes	—
54	Q₈×D₁₃	—	2	4	no	—
55	#6129	Q₈	1	8	yes	direct product factor
56	C₂×C₄×D₁₃	C₂³	1	3	yes	—
57	C₂×C₄×D₁₃	C₂³	1	3	yes	—
58	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
59	C₂×Dic₂₆	C₂³	1	3	yes	—
60	C₂×Dic₂₆	C₂³	1	6	yes	—
61	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
65	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
66	C₄²×C₁₃	C₂³	1	3	yes	—
67	C₂²⋊Dic₁₃	C₂³	1	6	yes	—
68	#6103	C₂³	1	6	yes	—
69	C₄⋊Dic₁₃	C₂³	1	3	yes	—
70	C₄⋊Dic₁₃	C₂³	1	3	yes	—
71	C₄⋊Dic₁₃	C₂³	1	6	yes	—
72	C₄⋊Dic₁₃	C₂³	1	12	yes	—
73	#6101	C₂³	1	6	yes	—
74	#6101	C₂³	1	6	yes	—
75	#6101	C₂³	1	6	yes	—
76	#6101	C₂³	1	12	yes	—
77	C₄×Dic₁₃	C₂³	1	3	yes	—
78	C₄×Dic₁₃	C₂³	1	6	yes	—
79	C₄×Dic₁₃	C₂³	1	3	yes	—
80	C₄×Dic₁₃	C₂³	1	3	yes	—
81	C₄×Dic₁₃	C₂³	1	6	yes	—
82	C₄×Dic₁₃	C₂³	1	6	yes	—
83	Q₈×#41	—	13	1	no	maximal, Sylow
84	C₂³×C₁₃	C₂⁴	1	2	yes	—
85	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
86	Q₈×C₁₃	—	2	12	no	—
87	Q₈×C₁₃	#41	1	4	yes	complemented
88	D₄×C₁₃	C₂×Q₈	1	4	yes	complemented
89	C₂×C₄×C₁₃	—	2	12	no	—
90	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
91	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
92	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
94	C₂²⋊D₁₃	C₂×Q₈	1	8	yes	complemented
95	C₂×Dic₁₃	—	4	6	no	—
96	C₂×Dic₁₃	C₂×Q₈	1	8	yes	complemented
97	C₂×Dic₁₃	C₂⁴	1	3	yes	—
98	C₂×Dic₁₃	C₂⁴	1	3	yes	—
99	C₂×Dic₁₃	C₂⁴	1	6	yes	—
100	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
101	C₂×Dic₁₃	C₂⁴	1	2	yes	—
102	C₄×D₁₃	—	2	6	no	—
103	C₄×D₁₃	—	4	3	no	—
104	C₄×D₁₃	C₂×Q₈	1	4	yes	complemented
105	Dic₂₆	—	2	6	no	—
106	Dic₂₆	—	4	6	no	—
107	Dic₂₆	C₂×Q₈	1	4	yes	complemented
108	Q₈²	—	13	1	no	—
109	#557	—	13	3	no	—
110	#554	—	13	6	no	—
111	#554	—	13	3	no	—
112	D₄×Q₈	—	13	2	no	—
113	D₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
114	#533	—	13	3	no	—
115	#533	—	13	6	no	—
116	C₄×#41	—	13	3	no	—
117	C₂×C₄×Q₈	—	13	2	no	—
118	C₂×C₄×Q₈	—	13	1	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	C₂²×Q₈	1	4	yes	—
122	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
123	C₂×D₁₃	—	2	1	no	—
124	C₂×D₁₃	—	2	1	no	—
125	C₂×D₁₃	C₂²×Q₈	1	2	yes	—
126	C₄×C₁₃	—	2	6	no	—
127	C₄×C₁₃	—	2	3	no	—
128	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
129	C₄×C₁₃	C₂×#41	1	6	yes	—
130	Dic₁₃	—	2	3	no	—
131	Dic₁₃	—	2	6	no	—
132	Dic₁₃	—	2	3	no	—
133	Dic₁₃	C₂²×Q₈	1	4	yes	—
134	Dic₁₃	C₂²×Q₈	1	2	yes	—
135	C₂×#41	—	13	1	no	—
136	C₂²×Q₈	—	13	1	no	—
137	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
138	C₄⋊Q₈	—	13	6	no	—
139	C₄⋊Q₈	—	13	6	no	—
140	C₄⋊Q₈	—	13	3	no	—
141	C₄⋊Q₈	C₂×D₁₃	1	3	yes	—
142	#125	—	13	12	no	—
143	#125	—	13	6	no	—
144	C₂²⋊Q₈	—	13	6	no	—
145	C₂²⋊Q₈	—	13	6	no	—
146	C₂²⋊Q₈	—	13	6	no	—
147	C₂²⋊Q₈	—	13	12	no	—
148	C₂²⋊Q₈	C₂×D₁₃	1	6	yes	—
149	C₄×Q₈	—	13	1	no	—
150	C₄×Q₈	—	13	6	no	—
151	C₄×Q₈	—	13	2	no	—
152	C₄×Q₈	—	13	6	no	—
153	C₄×Q₈	—	13	3	no	—
154	C₄×Q₈	—	13	3	no	—
155	C₄×Q₈	—	13	6	no	—
156	C₄×Q₈	—	13	3	no	—
157	C₄×Q₈	—	13	3	no	—
158	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
159	C₄×D₄	—	13	3	no	—
160	C₄×D₄	—	13	6	no	—
161	C₄×D₄	—	13	6	no	—
162	C₄×D₄	C₂×D₁₃	1	3	yes	—
163	#117	—	13	3	no	—
164	#117	—	13	6	no	—
165	C₂×C₄⋊C₄	—	13	6	no	—
166	C₂×C₄⋊C₄	—	13	3	no	—
167	C₂×C₄²	—	13	6	no	—
168	C₂×C₄²	—	13	3	no	—
169	C₂×C₁₃	—	2	4	no	—
170	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
171	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
172	C₂×C₁₃	#585	1	1	yes	characteristic
173	D₁₃	—	2	2	no	—
174	#41	—	13	8	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	13	1	no	—
177	C₂×Q₈	—	13	3	no	—
178	C₂×Q₈	—	13	6	no	—
179	C₂×Q₈	—	26	4	no	—
180	C₂×Q₈	C₂²×D₁₃	1	6	yes	—
181	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×D₄	—	13	2	no	—
183	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₄	—	13	2	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	6	no	—
187	C₂²×C₄	—	13	3	no	—
188	C₂²×C₄	—	13	3	no	—
189	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
190	C₄⋊C₄	—	13	3	no	—
191	C₄⋊C₄	—	13	6	no	—
192	C₄⋊C₄	—	13	12	no	—
193	C₄⋊C₄	—	13	6	no	—
194	C₄⋊C₄	—	13	6	no	—
195	C₄⋊C₄	—	13	6	no	—
196	C₄⋊C₄	—	13	3	no	—
197	C₄⋊C₄	—	13	12	no	—
198	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
199	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
200	C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
201	C₂²⋊C₄	—	13	6	no	—
202	C₂²⋊C₄	—	13	6	no	—
203	C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
204	C₄²	—	13	3	no	—
205	C₄²	—	13	6	no	—
206	C₄²	—	13	3	no	—
207	C₄²	—	13	6	no	—
208	C₄²	—	13	3	no	—
209	C₄²	—	13	6	no	—
210	C₄²	C₂²×D₁₃	1	3	yes	—
211	C₁₃	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	13	1	no	—
213	C₂³	C₂³×D₁₃	1	2	yes	—
214	Q₈	—	2	12	no	—
215	Q₈	—	13	4	no	—
216	Q₈	—	26	6	no	—
217	Q₈	—	52	6	no	—
218	Q₈	#6129	1	4	yes	direct product factor
219	D₄	—	13	8	no	—
220	D₄	Q₈×D₁₃	1	4	yes	complemented
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	13	8	no	—
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	2	no	—
225	C₂×C₄	—	13	4	no	—
226	C₂×C₄	—	13	3	no	—
227	C₂×C₄	—	13	3	no	—
228	C₂×C₄	—	13	6	no	—
229	C₂×C₄	—	26	6	no	—
230	C₂×C₄	—	52	3	no	—
231	C₂×C₄	—	52	6	no	—
232	C₂×C₄	C₂³×D₁₃	1	6	yes	—
233	C₂×C₄	C₂³×D₁₃	1	3	yes	—
234	C₂×C₄	C₂³×D₁₃	1	3	yes	—
235	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	13	2	no	—
239	C₂²	—	26	1	no	—
240	C₂²	—	26	1	no	—
241	C₂²	C₂×Q₈×D₁₃	1	4	yes	—
242	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
243	C₄	—	2	6	no	—
244	C₄	—	2	3	no	—
245	C₄	—	13	2	no	—
246	C₄	—	13	4	no	—
247	C₄	—	26	3	no	—
248	C₄	—	26	3	no	—
249	C₄	—	26	6	no	—
250	C₄	C₂×#6129	1	6	yes	—
251	C₄	C₂×Q₈×D₁₃	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	—	26	2	no	—
254	C₂	C₂²×#6129	1	1	yes	characteristic
255	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
256	C₂	#11719652	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	D₄×Q₈×C₁₃	C₂
prime radical	C₂×#6129	C₂²
hypercenter	D₄×Q₈	D₁₃
hypocenter	C₁₃	Q₈×#41
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. D₄×Q₈×C₁₃	C₂	semidirect
2. Q₈×C₂²⋊D₁₃	C₂	semidirect
3. C₂×Q₈×Dic₁₃	C₂	semidirect
4. #11719433	C₂	semidirect
5. #11719426	C₂	semidirect
6. #11719340	C₂	semidirect
7. #11719327	C₂	semidirect
8. C₄×Q₈×D₁₃	C₂	semidirect
9. Q₈×Dic₂₆	C₂	semidirect
10. #11719248	C₂	semidirect
55. #6129	Q₈	direct (N × H)
87. Q₈×C₁₃	#41	semidirect
88. D₄×C₁₃	C₂×Q₈	semidirect
94. C₂²⋊D₁₃	C₂×Q₈	semidirect
96. C₂×Dic₁₃	C₂×Q₈	semidirect
104. C₄×D₁₃	C₂×Q₈	semidirect
107. Dic₂₆	C₂×Q₈	semidirect
113. D₄×Q₈	D₁₃	semidirect
211. C₁₃	Q₈×#41	semidirect
218. Q₈	#6129	direct (N × H)
220. D₄	Q₈×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
D₄×Q₈×C₁₃	2
Q₈×C₂²⋊D₁₃	2
C₂×Q₈×Dic₁₃	2
#11719433	2
#11719426	2
#11719340	2
#11719327	2
C₄×Q₈×D₁₃	2
Q₈×Dic₂₆	2
#11719248	2
C₄×#6129	2
Q₈×#41	13

Subgroups — Q8×#6129 #13128413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6129 #13128413