Subgroups — C₈×Q₈×Dic₈₉ #36553035

Order 22784.

1070 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×#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26020117	C₂	1	3	yes	maximal
4	C₈×C₄⋊Dic₈₉	C₂	1	3	yes	maximal
5	C₄⋊C₈×Dic₈₉	C₂	1	3	yes	maximal
6	C₄×C₈×Dic₈₉	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₉	C₄	1	2	yes	—
9	Q₈×#10623494	C₄	1	8	yes	complemented
10	C₂×C₄⋊#10623494	C₂²	1	3	yes	—
11	C₂×C₄×#10623494	C₂²	1	3	yes	—
12	C₂×C₈×Dic₈₉	C₂²	1	3	yes	—
13	C₂×C₈×Dic₈₉	C₂²	1	3	yes	—
14	C₂×C₈×Dic₈₉	C₂²	1	3	yes	—
15	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×Dic₈₉	C₄	1	6	yes	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	3	yes	—
18	C₄²×Dic₈₉	C₂²	1	3	yes	—
19	#19151352	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₈₉	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
24	C₂²×#10623494	C₂³	1	3	yes	—
25	C₂²×#10623494	C₂³	1	1	yes	characteristic
26	Q₈×Dic₈₉	C₈	1	16	yes	direct product factor
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	6	yes	—
28	C₂×C₄×Dic₈₉	C₂×C₄	1	6	yes	—
29	C₂×C₄×Dic₈₉	C₂×C₄	1	6	yes	—
30	C₂×C₄×Dic₈₉	C₂³	1	3	yes	—
31	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
32	C₄⋊#10623494	C₂×C₄	1	12	yes	—
33	C₄×#10623494	C₂×C₄	1	12	yes	—
34	C₈×Dic₈₉	—	2	6	no	—
35	C₈×Dic₈₉	—	2	6	no	—
36	C₈×Dic₈₉	—	2	6	no	—
37	C₈×Dic₈₉	Q₈	1	4	yes	direct product factor
38	#15667919	C₂×C₄	1	6	yes	—
39	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
45	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
46	C₄⋊C₈×C₈₉	C₂×C₄	1	12	yes	—
47	C₄×C₈×C₈₉	C₂×C₄	1	12	yes	—
48	C₂²×Dic₈₉	C₂²×C₄	1	6	yes	—
49	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
50	C₂×#10623494	—	2	3	no	—
51	C₂×#10623494	—	2	3	no	—
52	C₂×#10623494	C₂²×C₄	1	6	yes	—
53	C₂×#10623494	C₂²×C₄	1	2	yes	—
54	C₂×#10623494	C₂×Q₈	1	2	yes	—
55	C₂×#10623494	#41	1	2	yes	—
56	C₄⋊Dic₈₉	C₂×C₈	1	24	yes	—
57	C₄×Dic₈₉	—	2	3	no	—
58	C₄×Dic₈₉	—	2	3	no	—
59	C₄×Dic₈₉	—	2	12	no	—
60	C₄×Dic₈₉	C₂×C₈	1	24	yes	—
61	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
62	C₄×Dic₈₉	#41	1	2	yes	—
63	C₂×Q₈×C₈₉	C₄²	1	4	yes	—
64	C₂×Q₈×C₈₉	C₂×C₈	1	8	yes	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₈₉	—	2	3	no	—
69	C₂×C₈×C₈₉	—	2	3	no	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₈₉	#41	1	2	yes	—
74	C₄⋊C₄×C₈₉	C₄²	1	12	yes	—
75	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
76	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
77	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
78	Q₈×C₈₉	C₄×C₈	1	16	yes	complemented
79	C₂×C₄×C₈₉	—	2	6	no	—
80	C₂×C₄×C₈₉	C₂×C₄²	1	6	yes	—
81	C₂×C₄×C₈₉	C₂×C₄²	1	6	yes	—
82	C₂×C₄×C₈₉	C₂²×C₈	1	12	yes	—
83	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
89	C₂×Dic₈₉	—	2	6	no	—
90	C₂×Dic₈₉	—	2	6	no	—
91	C₂×Dic₈₉	C₄×Q₈	1	4	yes	—
92	C₂×Dic₈₉	C₂²×C₈	1	12	yes	—
93	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
94	C₂×Dic₈₉	#131	1	4	yes	—
95	C₈×C₈₉	—	2	6	no	—
96	C₈×C₈₉	C₄×Q₈	1	4	yes	complemented
97	#10623494	—	2	6	no	—
98	#10623494	C₄×Q₈	1	4	yes	complemented
99	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
100	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
102	C₂²×C₈₉	C₂³×C₈	1	2	yes	—
103	C₂²×C₈₉	C₂×#131	1	2	yes	—
104	C₄×C₈₉	—	2	6	no	—
105	C₄×C₈₉	C₂×C₄×C₈	1	12	yes	—
106	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
109	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
110	Dic₈₉	—	2	12	no	—
111	Dic₈₉	C₈×Q₈	1	8	yes	direct product factor
112	C₄×C₈×Q₈	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	C₄²×Q₈	1	1	yes	characteristic
114	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
115	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
116	C₂×C₈₉	C₂×C₈×Q₈	1	2	yes	—
117	C₂×C₈₉	C₈×#41	1	2	yes	—
118	C₂×C₈×Q₈	—	89	1	no	—
119	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
120	C₄²×Q₈	—	89	1	no	—
121	C₈×C₄⋊C₄	—	89	3	no	—
122	C₈×C₄⋊C₄	—	89	3	no	—
123	C₄×C₄⋊C₈	—	89	3	no	—
124	C₄²×C₈	—	89	3	no	—
125	C₈₉	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
126	C₂×C₄×Q₈	—	89	2	no	—
127	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
128	C₈×Q₈	—	89	8	no	—
129	C₈×Q₈	Dic₈₉	1	8	yes	direct product factor
130	C₂×C₄⋊C₈	—	89	3	no	—
131	C₂×C₄⋊C₈	C₂×D₈₉	1	3	yes	—
132	C₂×C₄×C₈	—	89	1	no	—
133	C₂×C₄×C₈	—	89	3	no	—
134	C₂×C₄×C₈	—	89	3	no	—
135	C₂×C₄×C₈	—	89	3	no	—
136	C₂×C₄×C₈	—	89	3	no	—
137	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
138	C₄×C₄⋊C₄	—	89	6	no	—
139	C₄×C₄⋊C₄	—	89	3	no	—
140	C₄³	—	89	3	no	—
141	#336	—	89	6	no	—
142	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
143	C₂²×C₈	—	89	1	no	—
144	C₂²×C₈	—	89	3	no	—
145	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
146	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
147	C₄×Q₈	—	89	16	no	—
148	C₄×Q₈	C₄×D₈₉	1	4	yes	—
149	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
150	C₂×C₄⋊C₄	—	89	6	no	—
151	C₂×C₄⋊C₄	C₄×D₈₉	1	3	yes	—
152	C₂×C₄²	—	89	6	no	—
153	C₂×C₄²	—	89	1	no	—
154	C₂×C₄²	—	89	6	no	—
155	C₂×C₄²	—	89	3	no	—
156	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
157	C₄⋊C₈	—	89	12	no	—
158	C₄⋊C₈	C₂×Dic₈₉	1	12	yes	—
159	C₄×C₈	—	89	4	no	—
160	C₄×C₈	—	89	12	no	—
161	C₄×C₈	—	178	6	no	—
162	C₄×C₈	—	178	6	no	—
163	C₄×C₈	—	178	6	no	—
164	C₄×C₈	C₂×Dic₈₉	1	12	yes	—
165	#95	—	89	6	no	—
166	C₂×Q₈	C₄×Dic₈₉	1	4	yes	—
167	C₂×Q₈	C₈×D₈₉	1	8	yes	—
168	C₂²×C₄	—	89	2	no	—
169	C₂²×C₄	—	89	6	no	—
170	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
171	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
172	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	89	2	no	—
176	C₂×C₈	—	89	2	no	—
177	C₂×C₈	—	89	2	no	—
178	C₂×C₈	—	89	6	no	—
179	C₂×C₈	—	178	3	no	—
180	C₂×C₈	—	178	3	no	—
181	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
182	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
183	C₂×C₈	Q₈×D₈₉	1	2	yes	—
184	C₂×C₈	#13083575	1	2	yes	—
185	C₄⋊C₄	—	89	24	no	—
186	C₄⋊C₄	C₄×Dic₈₉	1	12	yes	—
187	C₄²	—	89	24	no	—
188	C₄²	—	89	2	no	—
189	C₄²	—	89	2	no	—
190	C₄²	—	178	12	no	—
191	C₄²	—	178	3	no	—
192	C₄²	—	178	3	no	—
193	C₄²	C₂×C₄×D₈₉	1	6	yes	—
194	C₄²	C₂²×Dic₈₉	1	6	yes	—
195	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
196	Q₈	C₈×Dic₈₉	1	16	yes	direct product factor
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	89	4	no	—
199	C₂×C₄	—	89	4	no	—
200	C₂×C₄	—	89	4	no	—
201	C₂×C₄	—	89	12	no	—
202	C₂×C₄	—	178	6	no	—
203	C₂×C₄	—	178	6	no	—
204	C₂×C₄	C₂×C₄×Dic₈₉	1	6	yes	—
205	C₂×C₄	C₂×C₄×Dic₈₉	1	6	yes	—
206	C₂×C₄	C₂×C₈×D₈₉	1	12	yes	—
207	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
208	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
209	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
210	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
211	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
212	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
213	C₈	—	2	6	no	—
214	C₈	—	89	4	no	—
215	C₈	—	178	6	no	—
216	C₈	Q₈×Dic₈₉	1	4	yes	direct product factor
217	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
218	C₂²	C₄×Q₈×D₈₉	1	1	yes	characteristic
219	C₂²	C₄×#13083575	1	1	yes	characteristic
220	C₂²	C₂²×C₈×D₈₉	1	2	yes	—
221	C₂²	#131×D₈₉	1	2	yes	—
222	C₄	—	2	6	no	—
223	C₄	—	89	8	no	—
224	C₄	—	178	12	no	—
225	C₄	C₂×C₈×Dic₈₉	1	12	yes	—
226	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
227	C₄	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
228	C₄	C₄×#13083575	1	1	yes	characteristic
229	C₄	#41×Dic₈₉	1	1	yes	characteristic
230	C₂	C₄×Q₈×Dic₈₉	1	1	yes	characteristic
231	C₂	C₂²×C₈×Dic₈₉	1	1	yes	characteristic
232	C₂	#131×Dic₈₉	1	1	yes	characteristic
233	C₂	C₈×Q₈×D₈₉	1	2	yes	—
234	C₂	C₈×#13083575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	C₄×C₈×Q₈
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×#10623494	C₄	semidirect
21. C₈×Q₈×C₈₉	C₄	semidirect
26. Q₈×Dic₈₉	C₈	direct (N × H)
37. C₈×Dic₈₉	Q₈	direct (N × H)
78. Q₈×C₈₉	C₄×C₈	semidirect
96. C₈×C₈₉	C₄×Q₈	semidirect
98. #10623494	C₄×Q₈	semidirect
111. Dic₈₉	C₈×Q₈	direct (N × H)
125. C₈₉	C₄×C₈×Q₈	semidirect
129. C₈×Q₈	Dic₈₉	direct (N × H)
196. Q₈	C₈×Dic₈₉	direct (N × H)
216. C₈	Q₈×Dic₈₉	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×#10623494	2
C₄×Q₈×Dic₈₉	2
#26020117	2
C₈×C₄⋊Dic₈₉	2
C₄⋊C₈×Dic₈₉	2
C₄×C₈×Dic₈₉	2
C₂×C₈×Q₈×C₈₉	2
C₄×C₈×Q₈	89

Subgroups — C8×Q8×Dic89 #36553035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Q₈×Dic₈₉ #36553035