Subgroups — Q₈×#15667943 #36573252

Order 22784.

838 conjugacy classes of subgroups.

The subgroup lattice has 274 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	Q₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021015	C₂	1	3	yes	maximal
4	#26021013	C₂	1	3	yes	maximal
5	#26020408	C₂	1	3	yes	maximal
6	C₄×#15667943	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₉	—	2	1	no	—
9	Q₈×#10623494	C₄	1	8	yes	complemented
10	C₂×#15667943	C₂²	1	3	yes	—
11	C₂×#15667943	C₂²	1	6	yes	—
12	C₂×#15667943	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#10623494	C₂²	1	3	yes	—
14	C₂×C₄×#10623494	C₂²	1	3	yes	—
15	#19151497	—	2	3	no	—
16	#19151468	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	3	yes	—
18	#19151360	C₂²	1	3	yes	—
19	#19151360	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
21	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	#389×C₈₉	—	2	3	no	—
23	#384×C₈₉	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
25	C₂²×#10623494	C₂³	1	3	yes	—
26	C₂²×#10623494	C₂³	1	1	yes	characteristic
27	Q₈×Dic₈₉	—	4	4	no	—
28	C₂×C₄⋊Dic₈₉	—	2	3	no	—
29	C₂×C₄⋊Dic₈₉	—	2	3	no	—
30	C₂×C₄⋊Dic₈₉	C₂³	1	3	yes	—
31	C₂×C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₈₉	—	2	3	no	—
33	#15667943	—	2	6	no	—
34	#15667943	—	4	6	no	—
35	#15667943	Q₈	1	4	yes	direct product factor
36	C₄⋊#10623494	C₂×C₄	1	12	yes	—
37	C₄×#10623494	C₂×C₄	1	12	yes	—
38	#15667919	—	2	3	no	—
39	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₈₉	—	4	4	no	—
41	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
42	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
45	C₂×C₄⋊C₄×C₈₉	D₄	1	3	yes	—
46	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
47	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	—	2	3	no	—
49	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
50	#95×C₈₉	—	2	3	no	—
51	C₂²×Dic₈₉	—	2	1	no	—
52	C₂²×Dic₈₉	—	2	3	no	—
53	C₂×#10623494	—	2	3	no	—
54	C₂×#10623494	—	2	3	no	—
55	C₂×#10623494	C₂²×C₄	1	6	yes	—
56	C₂×#10623494	C₂²×C₄	1	2	yes	—
57	C₂×#10623494	C₂×Q₈	1	2	yes	—
58	C₂×#10623494	#41	1	2	yes	—
59	C₄⋊Dic₈₉	—	2	1	no	—
60	C₄⋊Dic₈₉	—	2	3	no	—
61	C₄⋊Dic₈₉	—	2	3	no	—
62	C₄⋊Dic₈₉	—	4	6	no	—
63	C₄⋊Dic₈₉	—	8	3	no	—
64	C₄⋊Dic₈₉	C₂×Q₈	1	2	yes	—
65	C₄×Dic₈₉	—	4	6	no	—
66	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
67	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	—
68	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	—
69	C₂²×C₄×C₈₉	—	2	1	no	—
70	C₂²×C₄×C₈₉	—	2	3	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₄⋊C₄×C₈₉	—	2	3	no	—
76	C₄⋊C₄×C₈₉	—	2	3	no	—
77	C₄⋊C₄×C₈₉	—	4	6	no	—
78	C₄⋊C₄×C₈₉	—	8	3	no	—
79	C₄⋊C₄×C₈₉	C₄⋊C₄	1	12	yes	—
80	C₄⋊C₄×C₈₉	C₂×Q₈	1	2	yes	—
81	C₄²×C₈₉	—	4	6	no	—
82	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
83	C₄²×C₈₉	C₂×Q₈	1	6	yes	—
84	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₈₉	#106	1	16	yes	complemented
86	C₂×C₄×C₈₉	—	2	3	no	—
87	C₂×C₄×C₈₉	—	2	3	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	4	1	no	—
90	C₂×C₄×C₈₉	—	4	1	no	—
91	C₂×C₄×C₈₉	—	4	3	no	—
92	C₂×C₄×C₈₉	—	4	3	no	—
93	C₂×C₄×C₈₉	—	4	3	no	—
94	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
96	C₂×C₄×C₈₉	C₂×SD₁₆	1	6	yes	—
97	C₂×C₄×C₈₉	C₂×SD₁₆	1	6	yes	—
98	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂×Dic₈₉	—	2	2	no	—
105	C₂×Dic₈₉	—	4	1	no	—
106	C₂×Dic₈₉	—	4	1	no	—
107	C₂×Dic₈₉	—	4	3	no	—
108	C₂×Dic₈₉	—	4	3	no	—
109	C₂×Dic₈₉	—	4	3	no	—
110	#10623494	—	2	6	no	—
111	#10623494	C₄×Q₈	1	4	yes	complemented
112	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
114	C₂²×C₈₉	#550	1	1	yes	characteristic
115	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
116	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
117	C₂²×C₈₉	#577	1	1	yes	characteristic
118	C₂²×C₈₉	#577	1	1	yes	characteristic
119	C₄×C₈₉	—	2	6	no	—
120	C₄×C₈₉	—	4	2	no	—
121	C₄×C₈₉	—	8	3	no	—
122	C₄×C₈₉	C₂×#106	1	12	yes	—
123	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₄×C₈₉	#520	1	1	yes	characteristic
125	C₄×C₈₉	#557	1	1	yes	characteristic
126	C₄×C₈₉	Q₈²	1	1	yes	characteristic
127	Dic₈₉	—	4	2	no	—
128	Dic₈₉	—	8	3	no	—
129	Q₈×#106	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈₉	C₂²×#106	1	1	yes	characteristic
132	C₂×C₈₉	#2912	1	1	yes	characteristic
133	C₂×C₈₉	Q₈×SD₁₆	1	1	yes	characteristic
134	C₂×C₈₉	Q₈×SD₁₆	1	1	yes	characteristic
135	C₂×C₈₉	#3392	1	1	yes	characteristic
136	C₂×C₈₉	#3392	1	1	yes	characteristic
137	C₂×C₈×Q₈	—	89	1	no	—
138	Q₈×C₄⋊C₄	—	89	1	no	—
139	Q₈×C₄⋊C₄	D₈₉	1	1	yes	characteristic
140	#1948	—	89	3	no	—
141	#1948	—	89	3	no	—
142	#1844	—	89	3	no	—
143	C₄×#106	—	89	3	no	—
144	C₈₉	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄×Q₈	—	2	1	no	—
146	C₂×C₄×Q₈	—	178	1	no	—
147	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
148	C₈×Q₈	—	89	8	no	—
149	C₂×#106	—	89	1	no	—
150	C₂×#106	—	89	6	no	—
151	C₂×#106	—	89	3	no	—
152	C₂×C₄⋊C₈	—	89	3	no	—
153	C₂×C₄×C₈	—	89	3	no	—
154	#389	—	2	3	no	—
155	#389	—	178	3	no	—
156	#384	—	89	3	no	—
157	#384	C₂×D₈₉	1	3	yes	—
158	C₄×C₄⋊C₄	—	89	3	no	—
159	C₄×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
160	#340	—	89	3	no	—
161	#340	—	89	3	no	—
162	C₂²×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂²×C₈	—	89	1	no	—
164	C₂²×C₈	—	89	3	no	—
165	C₄×Q₈	—	4	4	no	—
166	C₄×Q₈	—	356	4	no	—
167	C₄×Q₈	Dic₁₇₈	1	4	yes	—
168	C₄×Q₈	C₄×D₈₉	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	—	2	3	no	—
171	C₂×C₄⋊C₄	—	89	1	no	—
172	C₂×C₄⋊C₄	—	89	3	no	—
173	C₂×C₄⋊C₄	—	178	3	no	—
174	C₂×C₄⋊C₄	—	178	3	no	—
175	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	3	yes	—
176	C₂×C₄⋊C₄	C₂²×D₈₉	1	3	yes	—
177	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
178	C₂×C₄²	—	2	3	no	—
179	C₂×C₄²	—	178	3	no	—
180	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
181	#106	—	89	4	no	—
182	#106	—	356	6	no	—
183	#106	—	178	6	no	—
184	C₄⋊C₈	—	89	12	no	—
185	C₄×C₈	—	89	12	no	—
186	#95	—	2	3	no	—
187	#95	—	178	3	no	—
188	C₂×Q₈	#13083553	1	4	yes	—
189	C₂×Q₈	#13083561	1	4	yes	—
190	C₂×Q₈	#13083562	1	4	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	3	no	—
193	C₂²×C₄	—	178	3	no	—
194	C₂²×C₄	—	178	1	no	—
195	C₂²×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
197	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
198	C₂×C₈	—	89	2	no	—
199	C₂×C₈	—	89	2	no	—
200	C₂×C₈	—	89	2	no	—
201	C₂×C₈	—	89	6	no	—
202	C₂×C₈	—	178	3	no	—
203	C₂×C₈	—	178	3	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	3	no	—
206	C₄⋊C₄	—	2	3	no	—
207	C₄⋊C₄	—	4	6	no	—
208	C₄⋊C₄	—	8	3	no	—
209	C₄⋊C₄	—	89	2	no	—
210	C₄⋊C₄	—	356	6	no	—
211	C₄⋊C₄	—	178	3	no	—
212	C₄⋊C₄	—	178	3	no	—
213	C₄⋊C₄	—	178	1	no	—
214	C₄⋊C₄	—	712	3	no	—
215	C₄⋊C₄	#13083553	1	12	yes	—
216	C₄⋊C₄	Q₈×D₈₉	1	2	yes	—
217	C₄²	—	4	6	no	—
218	C₄²	—	356	6	no	—
219	C₄²	C₂×C₄×D₈₉	1	6	yes	—
220	C₄²	C₂×Dic₁₇₈	1	6	yes	—
221	C₂³	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
222	Q₈	#15667943	1	16	yes	direct product factor
223	C₂×C₄	—	2	3	no	—
224	C₂×C₄	—	2	3	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	4	3	no	—
230	C₂×C₄	—	4	3	no	—
231	C₂×C₄	—	356	1	no	—
232	C₂×C₄	—	356	1	no	—
233	C₂×C₄	—	356	3	no	—
234	C₂×C₄	—	356	3	no	—
235	C₂×C₄	—	356	3	no	—
236	C₂×C₄	—	178	2	no	—
237	C₂×C₄	C₂×#13083553	1	6	yes	—
238	C₂×C₄	C₂×#13083553	1	6	yes	—
239	C₂×C₄	C₂×#13083561	1	6	yes	—
240	C₂×C₄	C₂×#13083562	1	6	yes	—
241	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
244	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
245	C₂×C₄	#15668058	1	1	yes	characteristic
246	C₂×C₄	#15668060	1	1	yes	characteristic
247	C₈	—	89	4	no	—
248	C₈	—	178	6	no	—
249	C₂²	C₂²×#13083553	1	1	yes	characteristic
250	C₂²	Q₈×C₂²⋊D₈₉	1	1	yes	characteristic
251	C₂²	#19152267	1	1	yes	characteristic
252	C₂²	C₂²×#13083561	1	1	yes	characteristic
253	C₂²	C₂²×#13083562	1	1	yes	characteristic
254	C₂²	#19152406	1	1	yes	characteristic
255	C₂²	#19152409	1	1	yes	characteristic
256	C₄	—	2	6	no	—
257	C₄	—	4	2	no	—
258	C₄	—	8	3	no	—
259	C₄	—	356	2	no	—
260	C₄	—	712	3	no	—
261	C₄	C₂×#15667943	1	12	yes	—
262	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
263	C₄	#19152080	1	1	yes	characteristic
264	C₄	#19152324	1	1	yes	characteristic
265	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
266	C₂	Q₈×#13083553	1	1	yes	characteristic
267	C₂	C₂²×#15667943	1	1	yes	characteristic
268	C₂	#26028845	1	1	yes	characteristic
269	C₂	Q₈×#13083561	1	1	yes	characteristic
270	C₂	Q₈×#13083562	1	1	yes	characteristic
271	C₂	#26033229	1	1	yes	characteristic
272	C₂	#26033233	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	Q₈×C₄⋊C₄×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₈₉
hypocenter	C₈₉	Q₈×#106
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Q₈×#10623494	C₄	semidirect
35. #15667943	Q₈	direct (N × H)
85. Q₈×C₈₉	#106	semidirect
111. #10623494	C₄×Q₈	semidirect
144. C₈₉	Q₈×#106	semidirect
222. Q₈	#15667943	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
Q₈×C₄⋊Dic₈₉	2
#26021015	2
#26021013	2
#26020408	2
C₄×#15667943	2
Q₈×C₄⋊C₄×C₈₉	2
Q₈×#106	89

Subgroups — Q8×#15667943 #36573252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#15667943 #36573252