Subgroups — Q₈×#14559003 #34082687

Order 19328.

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×Dic₃₀₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752937	C₂	1	3	yes	maximal, complemented
3	#23752920	C₂	1	6	yes	maximal, complemented
4	#23752919	C₂	1	3	yes	maximal, complemented
5	Q₈×C₂²⋊D₁₅₁	C₂	1	2	yes	maximal, complemented
6	#23752767	C₂	1	3	yes	maximal, complemented
7	#23752764	C₂	1	6	yes	maximal, complemented
8	C₄×#14559003	C₂	1	3	yes	maximal
9	C₂×Q₈×Dic₁₅₁	C₂	1	2	yes	maximal, complemented
10	C₄×Q₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14559003	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₅₁	C₂²	1	1	yes	characteristic
14	#18012739	C₂²	1	3	yes	—
15	#18012738	C₂²	1	6	yes	—
16	#18012737	C₂²	1	6	yes	—
17	#18012730	C₂²	1	6	yes	—
18	#18012729	C₂²	1	12	yes	—
19	#18012719	C₂²	1	6	yes	—
20	#18012718	C₂²	1	6	yes	—
21	#18012716	C₂²	1	12	yes	—
22	#18012715	C₂²	1	6	yes	—
23	Q₈×Dic₁₅₁	C₂²	1	6	yes	—
24	Q₈×Dic₁₅₁	C₂²	1	2	yes	—
25	Q₈×Dic₁₅₁	C₂²	1	6	yes	—
26	Q₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₀₂	C₂²	1	3	yes	—
28	C₄×Dic₃₀₂	C₂²	1	3	yes	—
29	#18012704	C₂²	1	6	yes	—
30	#18012704	C₂²	1	3	yes	—
31	#18012704	C₂²	1	3	yes	—
32	D₄×Dic₁₅₁	C₂²	1	3	yes	—
33	C₄×C₂²⋊D₁₅₁	C₂²	1	6	yes	—
34	(C₂²⋊D₁₅₁)⋊C₄	C₂²	1	6	yes	—
35	#18012697	C₂²	1	3	yes	—
36	#18012695	C₂²	1	6	yes	—
37	C₂×C₄⋊Dic₁₅₁	C₂²	1	6	yes	—
38	C₄⋊C₄×D₁₅₁	C₂²	1	3	yes	—
39	C₂×C₄×Dic₁₅₁	C₂²	1	6	yes	—
40	C₄²×D₁₅₁	C₂²	1	3	yes	—
41	C₂²×Q₈×C₁₅₁	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₁₅₁	C₂²	1	3	yes	—
43	C₂²⋊Q₈×C₁₅₁	C₂²	1	6	yes	—
44	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
45	C₄×D₄×C₁₅₁	C₂²	1	3	yes	—
46	#14559003	Q₈	1	8	yes	direct product factor
47	C₂×Dic₃₀₂	C₂³	1	3	yes	—
48	C₂×Dic₃₀₂	C₂³	1	6	yes	—
49	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
50	Q₈×D₁₅₁	—	2	4	no	—
51	C₂×C₂²⋊D₁₅₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₅₁	C₂³	1	6	yes	—
53	C₂²×Dic₁₅₁	C₂³	1	2	yes	—
54	C₂×C₄×D₁₅₁	C₂³	1	3	yes	—
55	C₂×C₄×D₁₅₁	C₂³	1	3	yes	—
56	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₁₅₁	C₂³	1	3	yes	—
58	C₄⋊Dic₁₅₁	C₂³	1	6	yes	—
59	C₄⋊Dic₁₅₁	C₂³	1	12	yes	—
60	C₄⋊Dic₁₅₁	C₂³	1	3	yes	—
61	Dic₁₅₁⋊C₄	C₂³	1	6	yes	—
62	Dic₁₅₁⋊C₄	C₂³	1	6	yes	—
63	Dic₁₅₁⋊C₄	C₂³	1	6	yes	—
64	Dic₁₅₁⋊C₄	C₂³	1	12	yes	—
65	C₂²⋊Dic₁₅₁	C₂³	1	6	yes	—
66	(C₂×D₁₅₁)⋊C₄	C₂³	1	6	yes	—
67	C₄×Dic₁₅₁	C₂³	1	3	yes	—
68	C₄×Dic₁₅₁	C₂³	1	3	yes	—
69	C₄×Dic₁₅₁	C₂³	1	3	yes	—
70	C₄×Dic₁₅₁	C₂³	1	6	yes	—
71	C₄×Dic₁₅₁	C₂³	1	6	yes	—
72	C₄×Dic₁₅₁	C₂³	1	6	yes	—
73	C₂×Q₈×C₁₅₁	—	2	8	no	—
74	C₂×Q₈×C₁₅₁	C₂³	1	6	yes	—
75	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₅₁	C₂³	1	6	yes	—
78	C₄⋊C₄×C₁₅₁	C₂³	1	3	yes	—
79	C₄⋊C₄×C₁₅₁	C₂³	1	3	yes	—
80	C₄⋊C₄×C₁₅₁	C₂³	1	6	yes	—
81	C₂²⋊C₄×C₁₅₁	C₂³	1	6	yes	—
82	C₄²×C₁₅₁	C₂³	1	3	yes	—
83	C₂²×D₁₅₁	C₂⁴	1	1	yes	characteristic
84	Dic₃₀₂	—	2	6	no	—
85	Dic₃₀₂	—	4	6	no	—
86	Dic₃₀₂	C₂×Q₈	1	4	yes	complemented
87	C₂²⋊D₁₅₁	C₂×Q₈	1	8	yes	complemented
88	C₂×Dic₁₅₁	—	4	6	no	—
89	C₂×Dic₁₅₁	C₂×Q₈	1	8	yes	complemented
90	C₂×Dic₁₅₁	C₂⁴	1	3	yes	—
91	C₂×Dic₁₅₁	C₂⁴	1	6	yes	—
92	C₂×Dic₁₅₁	C₂⁴	1	3	yes	—
93	C₂×Dic₁₅₁	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₁₅₁	C₂⁴	1	2	yes	—
95	C₄×D₁₅₁	—	2	6	no	—
96	C₄×D₁₅₁	—	4	3	no	—
97	C₄×D₁₅₁	C₂×Q₈	1	4	yes	complemented
98	C₂³×C₁₅₁	C₂⁴	1	2	yes	—
99	Q₈×C₁₅₁	—	2	12	no	—
100	Q₈×C₁₅₁	#41	1	4	yes	complemented
101	D₄×C₁₅₁	C₂×Q₈	1	4	yes	complemented
102	C₂×C₄×C₁₅₁	—	2	12	no	—
103	C₂×C₄×C₁₅₁	C₂⁴	1	6	yes	—
104	C₂×C₄×C₁₅₁	C₂⁴	1	3	yes	—
105	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
106	C₂×C₄×C₁₅₁	C₂⁴	1	3	yes	—
107	C₂²×C₁₅₁	—	2	2	no	—
108	C₂²×C₁₅₁	—	2	2	no	—
109	C₂²×C₁₅₁	C₂²×Q₈	1	4	yes	—
110	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
111	C₂×D₁₅₁	—	2	1	no	—
112	C₂×D₁₅₁	—	2	1	no	—
113	C₂×D₁₅₁	C₂²×Q₈	1	2	yes	—
114	C₄×C₁₅₁	—	2	6	no	—
115	C₄×C₁₅₁	—	2	3	no	—
116	C₄×C₁₅₁	C₂²×Q₈	1	2	yes	—
117	C₄×C₁₅₁	C₂×#41	1	6	yes	—
118	Dic₁₅₁	—	2	3	no	—
119	Dic₁₅₁	—	2	3	no	—
120	Dic₁₅₁	—	2	6	no	—
121	Dic₁₅₁	C₂²×Q₈	1	2	yes	—
122	Dic₁₅₁	C₂²×Q₈	1	4	yes	—
123	C₂×C₁₅₁	—	2	4	no	—
124	C₂×C₁₅₁	C₂³×Q₈	1	1	yes	characteristic
125	C₂×C₁₅₁	C₂²×#41	1	1	yes	characteristic
126	C₂×C₁₅₁	#585	1	1	yes	characteristic
127	D₁₅₁	—	2	2	no	—
128	C₁₅₁	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
129	Q₈×#41	—	151	1	no	maximal, Sylow
130	Q₈²	—	151	1	no	—
131	#557	—	151	3	no	—
132	#554	—	151	6	no	—
133	#554	—	151	3	no	—
134	D₄×Q₈	—	151	2	no	—
135	D₄×Q₈	D₁₅₁	1	1	yes	characteristic, complemented
136	#533	—	151	3	no	—
137	#533	—	151	6	no	—
138	C₄×#41	—	151	3	no	—
139	C₂×C₄×Q₈	—	151	2	no	—
140	C₂×C₄×Q₈	—	151	1	no	—
141	C₂×#41	—	151	1	no	—
142	C₂²×Q₈	—	151	1	no	—
143	C₂²×Q₈	C₂×D₁₅₁	1	2	yes	—
144	C₄⋊Q₈	—	151	6	no	—
145	C₄⋊Q₈	—	151	6	no	—
146	C₄⋊Q₈	—	151	3	no	—
147	C₄⋊Q₈	C₂×D₁₅₁	1	3	yes	—
148	#125	—	151	12	no	—
149	#125	—	151	6	no	—
150	C₂²⋊Q₈	—	151	6	no	—
151	C₂²⋊Q₈	—	151	6	no	—
152	C₂²⋊Q₈	—	151	6	no	—
153	C₂²⋊Q₈	—	151	12	no	—
154	C₂²⋊Q₈	C₂×D₁₅₁	1	6	yes	—
155	C₄×Q₈	—	151	1	no	—
156	C₄×Q₈	—	151	6	no	—
157	C₄×Q₈	—	151	2	no	—
158	C₄×Q₈	—	151	6	no	—
159	C₄×Q₈	—	151	3	no	—
160	C₄×Q₈	—	151	3	no	—
161	C₄×Q₈	—	151	6	no	—
162	C₄×Q₈	—	151	3	no	—
163	C₄×Q₈	—	151	3	no	—
164	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
165	C₄×D₄	—	151	3	no	—
166	C₄×D₄	—	151	6	no	—
167	C₄×D₄	—	151	6	no	—
168	C₄×D₄	C₂×D₁₅₁	1	3	yes	—
169	#117	—	151	3	no	—
170	#117	—	151	6	no	—
171	C₂×C₄⋊C₄	—	151	6	no	—
172	C₂×C₄⋊C₄	—	151	3	no	—
173	C₂×C₄²	—	151	6	no	—
174	C₂×C₄²	—	151	3	no	—
175	#41	—	151	8	no	—
176	C₂×Q₈	—	2	8	no	—
177	C₂×Q₈	—	302	4	no	—
178	C₂×Q₈	—	151	3	no	—
179	C₂×Q₈	—	151	6	no	—
180	C₂×Q₈	—	151	1	no	—
181	C₂×Q₈	C₂²×D₁₅₁	1	6	yes	—
182	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
183	C₂×D₄	—	151	2	no	—
184	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
185	C₂²×C₄	—	151	3	no	—
186	C₂²×C₄	—	151	6	no	—
187	C₂²×C₄	—	151	3	no	—
188	C₂²×C₄	—	151	2	no	—
189	C₂²×C₄	—	151	1	no	—
190	C₂²×C₄	C₂²×D₁₅₁	1	6	yes	—
191	C₄⋊C₄	—	151	6	no	—
192	C₄⋊C₄	—	151	6	no	—
193	C₄⋊C₄	—	151	6	no	—
194	C₄⋊C₄	—	151	12	no	—
195	C₄⋊C₄	—	151	3	no	—
196	C₄⋊C₄	—	151	12	no	—
197	C₄⋊C₄	—	151	6	no	—
198	C₄⋊C₄	—	151	3	no	—
199	C₄⋊C₄	C₂²×D₁₅₁	1	3	yes	—
200	C₄⋊C₄	C₂²×D₁₅₁	1	3	yes	—
201	C₄⋊C₄	C₂²×D₁₅₁	1	6	yes	—
202	C₂²⋊C₄	—	151	6	no	—
203	C₂²⋊C₄	—	151	6	no	—
204	C₂²⋊C₄	C₂²×D₁₅₁	1	6	yes	—
205	C₄²	—	151	3	no	—
206	C₄²	—	151	6	no	—
207	C₄²	—	151	3	no	—
208	C₄²	—	151	3	no	—
209	C₄²	—	151	6	no	—
210	C₄²	—	151	6	no	—
211	C₄²	C₂²×D₁₅₁	1	3	yes	—
212	C₂³	—	151	1	no	—
213	C₂³	C₂³×D₁₅₁	1	2	yes	—
214	Q₈	—	2	12	no	—
215	Q₈	—	302	6	no	—
216	Q₈	—	604	6	no	—
217	Q₈	—	151	4	no	—
218	Q₈	#14559003	1	4	yes	direct product factor
219	D₄	—	151	8	no	—
220	D₄	Q₈×D₁₅₁	1	4	yes	complemented
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	302	6	no	—
223	C₂×C₄	—	604	3	no	—
224	C₂×C₄	—	604	6	no	—
225	C₂×C₄	—	151	3	no	—
226	C₂×C₄	—	151	3	no	—
227	C₂×C₄	—	151	6	no	—
228	C₂×C₄	—	151	8	no	—
229	C₂×C₄	—	151	1	no	—
230	C₂×C₄	—	151	2	no	—
231	C₂×C₄	—	151	4	no	—
232	C₂×C₄	C₂³×D₁₅₁	1	6	yes	—
233	C₂×C₄	C₂³×D₁₅₁	1	3	yes	—
234	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₅₁	1	3	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	302	1	no	—
239	C₂²	—	302	1	no	—
240	C₂²	—	151	2	no	—
241	C₂²	C₂×Q₈×D₁₅₁	1	4	yes	—
242	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
243	C₄	—	2	6	no	—
244	C₄	—	2	3	no	—
245	C₄	—	302	3	no	—
246	C₄	—	302	3	no	—
247	C₄	—	302	6	no	—
248	C₄	—	151	2	no	—
249	C₄	—	151	4	no	—
250	C₄	C₂×Q₈×D₁₅₁	1	2	yes	—
251	C₄	C₂×#14559003	1	6	yes	—
252	C₂	—	2	4	no	—
253	C₂	—	302	2	no	—
254	C₂	C₂²×Q₈×D₁₅₁	1	1	yes	characteristic
255	C₂	C₂²×#14559003	1	1	yes	characteristic
256	C₂	#23753049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁵
Fitting	D₄×Q₈×C₁₅₁	C₂
prime radical	C₂×#14559003	C₂²
hypercenter	D₄×Q₈	D₁₅₁
hypocenter	C₁₅₁	Q₈×#41
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₀₂	C₂	semidirect
2. #23752937	C₂	semidirect
3. #23752920	C₂	semidirect
4. #23752919	C₂	semidirect
5. Q₈×C₂²⋊D₁₅₁	C₂	semidirect
6. #23752767	C₂	semidirect
7. #23752764	C₂	semidirect
9. C₂×Q₈×Dic₁₅₁	C₂	semidirect
10. C₄×Q₈×D₁₅₁	C₂	semidirect
11. D₄×Q₈×C₁₅₁	C₂	semidirect
46. #14559003	Q₈	direct (N × H)
86. Dic₃₀₂	C₂×Q₈	semidirect
87. C₂²⋊D₁₅₁	C₂×Q₈	semidirect
89. C₂×Dic₁₅₁	C₂×Q₈	semidirect
97. C₄×D₁₅₁	C₂×Q₈	semidirect
100. Q₈×C₁₅₁	#41	semidirect
101. D₄×C₁₅₁	C₂×Q₈	semidirect
128. C₁₅₁	Q₈×#41	semidirect
135. D₄×Q₈	D₁₅₁	semidirect
218. Q₈	#14559003	direct (N × H)
220. D₄	Q₈×D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₀₂	2
#23752937	2
#23752920	2
#23752919	2
Q₈×C₂²⋊D₁₅₁	2
#23752767	2
#23752764	2
C₄×#14559003	2
C₂×Q₈×Dic₁₅₁	2
C₄×Q₈×D₁₅₁	2
D₄×Q₈×C₁₅₁	2
Q₈×#41	151

Subgroups — Q8×#14559003 #34082687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14559003 #34082687