Subgroups — Q₈×#64541 #14577216

Order 2432.

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	Q₈×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927272	C₂	1	3	yes	maximal, complemented
3	#11927254	C₂	1	3	yes	maximal, complemented
4	#11927249	C₂	1	6	yes	maximal, complemented
5	Q₈×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₂²⋊D₁₉	C₂	1	2	yes	maximal, complemented
7	#11927101	C₂	1	3	yes	maximal, complemented
8	#11927098	C₂	1	6	yes	maximal, complemented
9	C₄×#64541	C₂	1	3	yes	maximal
10	C₄×Q₈×D₁₉	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₉	C₂²	1	2	yes	—
14	C₂×#64541	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
16	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
20	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
21	#10597392	C₂²	1	6	yes	—
22	Q₈×Dic₁₉	C₂²	1	2	yes	—
23	#10597390	C₂²	1	6	yes	—
24	C₄×C₂²⋊D₁₉	C₂²	1	6	yes	—
25	C₄×C₂²⋊D₁₉	C₂²	1	6	yes	—
26	#10597372	C₂²	1	3	yes	—
27	#10597370	C₂²	1	6	yes	—
28	#10597370	C₂²	1	6	yes	—
29	#10597343	C₂²	1	6	yes	—
30	#10597342	C₂²	1	12	yes	—
31	C₄⋊C₄×D₁₉	C₂²	1	6	yes	—
32	#10597335	C₂²	1	12	yes	—
33	#10597334	C₂²	1	6	yes	—
34	#10597333	C₂²	1	6	yes	—
35	C₄×D₇₆	C₂²	1	3	yes	—
36	C₄×D₇₆	C₂²	1	3	yes	—
37	#10597318	C₂²	1	6	yes	—
38	C₄²×D₁₉	C₂²	1	6	yes	—
39	#10597316	C₂²	1	6	yes	—
40	#10597315	C₂²	1	3	yes	—
41	#10597315	C₂²	1	3	yes	—
42	C₄×Dic₃₈	C₂²	1	6	yes	—
43	C₄×Dic₃₈	C₂²	1	6	yes	—
44	C₄×Dic₃₈	C₂²	1	3	yes	—
45	C₄×Dic₃₈	C₂²	1	3	yes	—
46	C₂×Q₈×C₁₉	—	2	4	no	—
47	C₂×Q₈×C₁₉	C₂³	1	3	yes	—
48	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
50	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₉	C₂³	1	2	yes	—
53	Q₈×D₁₉	—	2	8	no	—
54	#64541	Q₈	1	8	yes	direct product factor
55	C₂×D₇₆	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₁₉	C₂³	1	6	yes	—
57	C₂×C₄×D₁₉	C₂³	1	6	yes	—
58	C₂×C₄×D₁₉	C₂³	1	2	yes	—
59	C₂×Dic₃₈	C₂³	1	6	yes	—
60	C₂×Dic₃₈	C₂³	1	6	yes	—
61	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
63	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
64	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
65	C₄²×C₁₉	C₂³	1	3	yes	—
66	C₄²×C₁₉	C₂³	1	3	yes	—
67	C₄²×C₁₉	C₂³	1	3	yes	—
68	C₄²×C₁₉	C₂³	1	3	yes	—
69	C₂²⋊Dic₁₉	C₂³	1	6	yes	—
70	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
71	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
72	C₄⋊Dic₁₉	C₂³	1	6	yes	—
73	C₄⋊Dic₁₉	C₂³	1	3	yes	—
74	C₄⋊Dic₁₉	C₂³	1	3	yes	—
75	C₄⋊Dic₁₉	C₂³	1	6	yes	—
76	Dic₁₉⋊C₄	C₂³	1	6	yes	—
77	Dic₁₉⋊C₄	C₂³	1	12	yes	—
78	Dic₁₉⋊C₄	C₂³	1	12	yes	—
79	Dic₁₉⋊C₄	C₂³	1	6	yes	—
80	C₄×Dic₁₉	C₂³	1	6	yes	—
81	C₄×Dic₁₉	C₂³	1	6	yes	—
82	C₄×Dic₁₉	C₂³	1	6	yes	—
83	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₉	C₂⁴	1	2	yes	—
85	Q₈×C₁₉	—	2	6	no	—
86	Q₈×C₁₉	#41	1	4	yes	complemented
87	C₂×C₄×C₁₉	—	2	6	no	—
88	C₂×C₄×C₁₉	—	4	3	no	—
89	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	complemented
90	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
91	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
92	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
93	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
95	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₁₉	C₂×Q₈	1	8	yes	complemented
97	C₂×Dic₁₉	C₂⁴	1	6	yes	—
98	C₂×Dic₁₉	C₂⁴	1	6	yes	—
99	C₂×Dic₁₉	C₂⁴	1	2	yes	—
100	D₇₆	C₂×Q₈	1	4	yes	complemented
101	C₄×D₁₉	—	2	12	no	—
102	C₄×D₁₉	—	4	6	no	—
103	C₄×D₁₉	C₂×Q₈	1	8	yes	complemented
104	Dic₃₈	—	2	12	no	—
105	Dic₃₈	—	4	6	no	—
106	Dic₃₈	C₂×Q₈	1	4	yes	complemented
107	Q₈×#41	—	19	1	no	maximal, Sylow
108	C₂²×C₁₉	—	2	1	no	—
109	C₂²×C₁₉	—	2	1	no	—
110	C₂²×C₁₉	C₂²×Q₈	1	2	yes	—
111	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
112	C₂×D₁₉	—	2	2	no	—
113	C₂×D₁₉	—	2	2	no	—
114	C₂×D₁₉	C₂²×Q₈	1	4	yes	—
115	C₄×C₁₉	—	2	3	no	—
116	C₄×C₁₉	—	2	3	no	—
117	C₄×C₁₉	—	2	3	no	—
118	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
119	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
120	C₄×C₁₉	C₂×#41	1	6	yes	—
121	Dic₁₉	—	2	6	no	—
122	Dic₁₉	—	2	6	no	—
123	Dic₁₉	C₂²×Q₈	1	4	yes	—
124	Q₈²	—	19	1	no	—
125	#557	—	19	3	no	—
126	#554	—	19	3	no	—
127	#554	—	19	6	no	—
128	D₄×Q₈	—	19	2	no	—
129	D₄×Q₈	—	19	1	no	—
130	#533	—	19	6	no	—
131	#533	—	19	3	no	—
132	C₄×#41	—	19	3	no	—
133	C₂×C₄×Q₈	—	19	2	no	—
134	C₂×C₄×Q₈	D₁₉	1	1	yes	characteristic, complemented
135	C₂×C₁₉	—	2	2	no	—
136	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
137	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
138	C₂×C₁₉	#585	1	1	yes	characteristic
139	D₁₉	—	2	4	no	—
140	C₂×#41	—	19	1	no	—
141	C₂²×Q₈	—	19	2	no	—
142	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	19	6	no	—
144	C₄⋊Q₈	—	19	6	no	—
145	C₄⋊Q₈	—	19	3	no	—
146	C₄⋊Q₈	—	19	3	no	—
147	#125	—	19	12	no	—
148	#125	—	19	6	no	—
149	C₂²⋊Q₈	—	19	6	no	—
150	C₂²⋊Q₈	—	19	6	no	—
151	C₂²⋊Q₈	—	19	6	no	—
152	C₂²⋊Q₈	—	19	6	no	—
153	C₂²⋊Q₈	—	19	12	no	—
154	C₄×Q₈	—	19	2	no	—
155	C₄×Q₈	—	19	6	no	—
156	C₄×Q₈	—	19	6	no	—
157	C₄×Q₈	—	19	3	no	—
158	C₄×Q₈	—	19	3	no	—
159	C₄×Q₈	—	19	6	no	—
160	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
161	C₄×Q₈	C₂×D₁₉	1	3	yes	—
162	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₁₉	1	3	yes	—
164	C₄×D₄	—	19	6	no	—
165	C₄×D₄	—	19	6	no	—
166	C₄×D₄	—	19	3	no	—
167	C₄×D₄	—	19	3	no	—
168	#117	—	19	3	no	—
169	#117	—	19	6	no	—
170	C₂×C₄⋊C₄	—	19	6	no	—
171	C₂×C₄⋊C₄	C₂×D₁₉	1	3	yes	—
172	C₂×C₄²	—	19	6	no	—
173	C₂×C₄²	C₂×D₁₉	1	3	yes	—
174	C₁₉	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
175	#41	—	19	8	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	19	1	no	—
178	C₂×Q₈	—	19	6	no	—
179	C₂×Q₈	—	19	6	no	—
180	C₂×Q₈	—	38	8	no	—
181	C₂×Q₈	C₂²×D₁₉	1	3	yes	—
182	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
183	C₂×D₄	—	19	2	no	—
184	C₂×D₄	—	19	1	no	—
185	C₂²×C₄	—	19	2	no	—
186	C₂²×C₄	—	19	6	no	—
187	C₂²×C₄	—	19	6	no	—
188	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
189	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
190	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
191	C₄⋊C₄	—	19	12	no	—
192	C₄⋊C₄	—	19	6	no	—
193	C₄⋊C₄	—	19	6	no	—
194	C₄⋊C₄	—	19	12	no	—
195	C₄⋊C₄	—	19	6	no	—
196	C₄⋊C₄	—	19	3	no	—
197	C₄⋊C₄	—	19	3	no	—
198	C₄⋊C₄	—	19	6	no	—
199	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
200	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
201	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
202	C₂²⋊C₄	—	19	6	no	—
203	C₂²⋊C₄	—	19	6	no	—
204	C₂²⋊C₄	—	19	6	no	—
205	C₄²	—	19	6	no	—
206	C₄²	—	19	6	no	—
207	C₄²	—	19	6	no	—
208	C₄²	C₂²×D₁₉	1	3	yes	—
209	C₄²	C₂²×D₁₉	1	3	yes	—
210	C₄²	C₂²×D₁₉	1	3	yes	—
211	C₄²	C₂²×D₁₉	1	3	yes	—
212	C₂³	—	19	2	no	—
213	C₂³	C₂³×D₁₉	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	19	4	no	—
216	Q₈	—	38	12	no	—
217	Q₈	—	76	6	no	—
218	Q₈	#64541	1	4	yes	direct product factor
219	D₄	—	19	8	no	—
220	D₄	—	19	4	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	4	3	no	—
223	C₂×C₄	—	19	2	no	—
224	C₂×C₄	—	19	8	no	—
225	C₂×C₄	—	19	6	no	—
226	C₂×C₄	—	19	6	no	—
227	C₂×C₄	—	38	12	no	—
228	C₂×C₄	—	76	6	no	—
229	C₂×C₄	Q₈×D₁₉	1	4	yes	complemented
230	C₂×C₄	C₂³×D₁₉	1	3	yes	—
231	C₂×C₄	C₂³×D₁₉	1	3	yes	—
232	C₂×C₄	C₂³×D₁₉	1	3	yes	—
233	C₂×C₄	C₂³×D₁₉	1	3	yes	—
234	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	19	4	no	—
239	C₂²	—	38	2	no	—
240	C₂²	—	38	2	no	—
241	C₂²	C₂×Q₈×D₁₉	1	2	yes	—
242	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	19	4	no	—
247	C₄	—	38	6	no	—
248	C₄	—	38	6	no	—
249	C₄	C₂×#64541	1	6	yes	—
250	C₄	C₂×Q₈×D₁₉	1	2	yes	—
251	C₄	C₂×Q₈×D₁₉	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	38	4	no	—
254	C₂	C₂²×Q₈×D₁₉	1	1	yes	characteristic
255	C₂	C₂²×#64541	1	1	yes	characteristic
256	C₂	#11927380	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₄×Q₈×C₁₉	C₂
prime radical	C₂×#64541	C₂²
hypercenter	C₂×C₄×Q₈	D₁₉
hypocenter	C₁₉	Q₈×#41
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
2. #11927272	C₂	semidirect
3. #11927254	C₂	semidirect
4. #11927249	C₂	semidirect
5. Q₈×D₇₆	C₂	semidirect
6. Q₈×C₂²⋊D₁₉	C₂	semidirect
7. #11927101	C₂	semidirect
8. #11927098	C₂	semidirect
10. C₄×Q₈×D₁₉	C₂	semidirect
11. C₂×C₄×Q₈×C₁₉	C₂	semidirect
54. #64541	Q₈	direct (N × H)
86. Q₈×C₁₉	#41	semidirect
89. C₂×C₄×C₁₉	C₂×Q₈	semidirect
96. C₂²⋊D₁₉	C₂×Q₈	semidirect
100. D₇₆	C₂×Q₈	semidirect
103. C₄×D₁₉	C₂×Q₈	semidirect
106. Dic₃₈	C₂×Q₈	semidirect
134. C₂×C₄×Q₈	D₁₉	semidirect
174. C₁₉	Q₈×#41	semidirect
218. Q₈	#64541	direct (N × H)
229. C₂×C₄	Q₈×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₈	2
#11927272	2
#11927254	2
#11927249	2
Q₈×D₇₆	2
Q₈×C₂²⋊D₁₉	2
#11927101	2
#11927098	2
C₄×#64541	2
C₄×Q₈×D₁₉	2
C₂×C₄×Q₈×C₁₉	2
Q₈×#41	19

Subgroups — Q8×#64541 #14577216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#64541 #14577216