Subgroups — Q₈×#14581725 #35423241

Order 21248.

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×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24908221	C₂	1	3	yes	maximal
4	#24908219	C₂	1	3	yes	maximal
5	#24907622	C₂	1	3	yes	maximal
6	C₄×#14581725	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₈₃	—	2	1	no	—
9	Q₈×C₈₃⋊C₈	C₄	1	8	yes	complemented
10	C₂×#14581725	C₂²	1	3	yes	—
11	C₂×#14581725	C₂²	1	6	yes	—
12	C₂×#14581725	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	3	yes	—
14	C₂×C₄×C₈₃⋊C₈	C₂²	1	3	yes	—
15	#19127600	—	2	3	no	—
16	#19127571	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₈₃	C₂²	1	3	yes	—
18	#19127463	C₂²	1	3	yes	—
19	#19127463	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₈₃	—	2	1	no	—
21	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
22	#389×C₈₃	—	2	3	no	—
23	#384×C₈₃	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₈₃	C₂²	1	3	yes	—
25	C₂²×C₈₃⋊C₈	C₂³	1	3	yes	—
26	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
27	Q₈×Dic₈₃	—	4	4	no	—
28	C₂×C₄⋊Dic₈₃	—	2	3	no	—
29	C₂×C₄⋊Dic₈₃	—	2	3	no	—
30	C₂×C₄⋊Dic₈₃	C₂³	1	3	yes	—
31	C₂×C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₈₃	—	2	3	no	—
33	#14581725	—	2	6	no	—
34	#14581725	—	4	6	no	—
35	#14581725	Q₈	1	4	yes	direct product factor
36	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	12	yes	—
37	C₄×C₈₃⋊C₈	C₂×C₄	1	12	yes	—
38	C₈₃⋊#95	—	2	3	no	—
39	C₂²×Q₈×C₈₃	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₈₃	—	4	4	no	—
41	C₄×Q₈×C₈₃	C₂×C₄	1	4	yes	—
42	C₄×Q₈×C₈₃	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄×C₈₃	—	2	3	no	—
44	C₂×C₄⋊C₄×C₈₃	—	2	3	no	—
45	C₂×C₄⋊C₄×C₈₃	D₄	1	3	yes	—
46	C₂×C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
47	C₂×C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₃	—	2	3	no	—
49	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
50	#95×C₈₃	—	2	3	no	—
51	C₂²×Dic₈₃	—	2	1	no	—
52	C₂²×Dic₈₃	—	2	3	no	—
53	C₂×C₈₃⋊C₈	—	2	3	no	—
54	C₂×C₈₃⋊C₈	—	2	3	no	—
55	C₂×C₈₃⋊C₈	C₂²×C₄	1	6	yes	—
56	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
57	C₂×C₈₃⋊C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₈₃⋊C₈	#41	1	2	yes	—
59	C₄⋊Dic₈₃	—	2	1	no	—
60	C₄⋊Dic₈₃	—	2	3	no	—
61	C₄⋊Dic₈₃	—	2	3	no	—
62	C₄⋊Dic₈₃	—	4	6	no	—
63	C₄⋊Dic₈₃	—	8	3	no	—
64	C₄⋊Dic₈₃	C₂×Q₈	1	2	yes	—
65	C₄×Dic₈₃	—	4	6	no	—
66	C₂×Q₈×C₈₃	C₄⋊C₄	1	4	yes	—
67	C₂×Q₈×C₈₃	D₈	1	4	yes	—
68	C₂×Q₈×C₈₃	Q₁₆	1	4	yes	—
69	C₂²×C₄×C₈₃	—	2	1	no	—
70	C₂²×C₄×C₈₃	—	2	3	no	—
71	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₈₃	—	2	1	no	—
75	C₄⋊C₄×C₈₃	—	2	3	no	—
76	C₄⋊C₄×C₈₃	—	2	3	no	—
77	C₄⋊C₄×C₈₃	—	4	6	no	—
78	C₄⋊C₄×C₈₃	—	8	3	no	—
79	C₄⋊C₄×C₈₃	C₄⋊C₄	1	12	yes	—
80	C₄⋊C₄×C₈₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₈₃	—	4	6	no	—
82	C₄²×C₈₃	C₂²×C₄	1	6	yes	—
83	C₄²×C₈₃	C₂×Q₈	1	6	yes	—
84	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₈₃	#107	1	16	yes	complemented
86	C₂×C₄×C₈₃	—	2	3	no	—
87	C₂×C₄×C₈₃	—	2	3	no	—
88	C₂×C₄×C₈₃	—	2	2	no	—
89	C₂×C₄×C₈₃	—	4	1	no	—
90	C₂×C₄×C₈₃	—	4	1	no	—
91	C₂×C₄×C₈₃	—	4	3	no	—
92	C₂×C₄×C₈₃	—	4	3	no	—
93	C₂×C₄×C₈₃	—	4	3	no	—
94	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	6	yes	—
95	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	6	yes	—
96	C₂×C₄×C₈₃	C₂×D₈	1	6	yes	—
97	C₂×C₄×C₈₃	C₂×Q₁₆	1	6	yes	—
98	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂×Dic₈₃	—	2	2	no	—
105	C₂×Dic₈₃	—	4	1	no	—
106	C₂×Dic₈₃	—	4	1	no	—
107	C₂×Dic₈₃	—	4	3	no	—
108	C₂×Dic₈₃	—	4	3	no	—
109	C₂×Dic₈₃	—	4	3	no	—
110	C₈₃⋊C₈	—	2	6	no	—
111	C₈₃⋊C₈	C₄×Q₈	1	4	yes	complemented
112	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₈₃	D₄×Q₈	1	1	yes	characteristic
114	C₂²×C₈₃	#550	1	1	yes	characteristic
115	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
116	C₂²×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
117	C₂²×C₈₃	#576	1	1	yes	characteristic
118	C₂²×C₈₃	#578	1	1	yes	characteristic
119	C₄×C₈₃	—	2	6	no	—
120	C₄×C₈₃	—	4	2	no	—
121	C₄×C₈₃	—	8	3	no	—
122	C₄×C₈₃	C₂×#107	1	12	yes	—
123	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₄×C₈₃	#520	1	1	yes	characteristic
125	C₄×C₈₃	#557	1	1	yes	characteristic
126	C₄×C₈₃	Q₈²	1	1	yes	characteristic
127	Dic₈₃	—	4	2	no	—
128	Dic₈₃	—	8	3	no	—
129	Q₈×#107	—	83	1	no	maximal, Sylow
130	C₂×C₈₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈₃	C₂²×#107	1	1	yes	characteristic
132	C₂×C₈₃	#2913	1	1	yes	characteristic
133	C₂×C₈₃	Q₈×D₈	1	1	yes	characteristic
134	C₂×C₈₃	Q₈×Q₁₆	1	1	yes	characteristic
135	C₂×C₈₃	#3391	1	1	yes	characteristic
136	C₂×C₈₃	#3395	1	1	yes	characteristic
137	C₂×C₈×Q₈	—	83	1	no	—
138	Q₈×C₄⋊C₄	—	83	1	no	—
139	Q₈×C₄⋊C₄	D₈₃	1	1	yes	characteristic
140	#1947	—	83	3	no	—
141	#1947	—	83	3	no	—
142	#1845	—	83	3	no	—
143	C₄×#107	—	83	3	no	—
144	C₈₃	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄×Q₈	—	2	1	no	—
146	C₂×C₄×Q₈	—	166	1	no	—
147	C₂×C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
148	C₈×Q₈	—	83	8	no	—
149	C₂×#107	—	83	6	no	—
150	C₂×#107	—	83	1	no	—
151	C₂×#107	—	83	3	no	—
152	C₂×C₄⋊C₈	—	83	3	no	—
153	C₂×C₄×C₈	—	83	3	no	—
154	#389	—	2	3	no	—
155	#389	—	166	3	no	—
156	#384	—	83	3	no	—
157	#384	C₂×D₈₃	1	3	yes	—
158	C₄×C₄⋊C₄	—	83	3	no	—
159	C₄×C₄⋊C₄	C₂×D₈₃	1	3	yes	—
160	#340	—	83	3	no	—
161	#340	—	83	3	no	—
162	C₂²×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
163	C₂²×C₈	—	83	1	no	—
164	C₂²×C₈	—	83	3	no	—
165	C₄×Q₈	—	4	4	no	—
166	C₄×Q₈	—	332	4	no	—
167	C₄×Q₈	Dic₁₆₆	1	4	yes	—
168	C₄×Q₈	C₄×D₈₃	1	4	yes	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	—	2	3	no	—
171	C₂×C₄⋊C₄	—	83	1	no	—
172	C₂×C₄⋊C₄	—	83	3	no	—
173	C₂×C₄⋊C₄	—	166	3	no	—
174	C₂×C₄⋊C₄	—	166	3	no	—
175	C₂×C₄⋊C₄	C₂²⋊D₈₃	1	3	yes	—
176	C₂×C₄⋊C₄	C₂²×D₈₃	1	3	yes	—
177	C₂×C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
178	C₂×C₄²	—	2	3	no	—
179	C₂×C₄²	—	166	3	no	—
180	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
181	#107	—	332	6	no	—
182	#107	—	83	4	no	—
183	#107	—	166	6	no	—
184	C₄⋊C₈	—	83	12	no	—
185	C₄×C₈	—	83	12	no	—
186	#95	—	2	3	no	—
187	#95	—	166	3	no	—
188	C₂×Q₈	Dic₈₃⋊C₄	1	4	yes	—
189	C₂×Q₈	#13050575	1	4	yes	—
190	C₂×Q₈	#13050580	1	4	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	3	no	—
193	C₂²×C₄	—	166	3	no	—
194	C₂²×C₄	—	166	1	no	—
195	C₂²×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
197	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
198	C₂×C₈	—	83	2	no	—
199	C₂×C₈	—	83	2	no	—
200	C₂×C₈	—	83	2	no	—
201	C₂×C₈	—	83	6	no	—
202	C₂×C₈	—	166	3	no	—
203	C₂×C₈	—	166	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	—	332	6	no	—
210	C₄⋊C₄	—	83	2	no	—
211	C₄⋊C₄	—	664	3	no	—
212	C₄⋊C₄	—	166	3	no	—
213	C₄⋊C₄	—	166	3	no	—
214	C₄⋊C₄	—	166	1	no	—
215	C₄⋊C₄	Dic₈₃⋊C₄	1	12	yes	—
216	C₄⋊C₄	Q₈×D₈₃	1	2	yes	—
217	C₄²	—	4	6	no	—
218	C₄²	—	332	6	no	—
219	C₄²	C₂×C₄×D₈₃	1	6	yes	—
220	C₄²	C₂×Dic₁₆₆	1	6	yes	—
221	C₂³	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
222	Q₈	#14581725	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	—	332	1	no	—
232	C₂×C₄	—	332	1	no	—
233	C₂×C₄	—	332	3	no	—
234	C₂×C₄	—	332	3	no	—
235	C₂×C₄	—	332	3	no	—
236	C₂×C₄	—	166	2	no	—
237	C₂×C₄	C₂×Dic₈₃⋊C₄	1	6	yes	—
238	C₂×C₄	C₂×Dic₈₃⋊C₄	1	6	yes	—
239	C₂×C₄	C₂×#13050575	1	6	yes	—
240	C₂×C₄	C₂×#13050580	1	6	yes	—
241	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
242	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
244	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
245	C₂×C₄	#14581838	1	1	yes	characteristic
246	C₂×C₄	#14581840	1	1	yes	characteristic
247	C₈	—	83	4	no	—
248	C₈	—	166	6	no	—
249	C₂²	C₂²×Dic₈₃⋊C₄	1	1	yes	characteristic
250	C₂²	Q₈×C₂²⋊D₈₃	1	1	yes	characteristic
251	C₂²	#19128370	1	1	yes	characteristic
252	C₂²	C₂²×#13050575	1	1	yes	characteristic
253	C₂²	C₂²×#13050580	1	1	yes	characteristic
254	C₂²	#19128504	1	1	yes	characteristic
255	C₂²	#19128516	1	1	yes	characteristic
256	C₄	—	2	6	no	—
257	C₄	—	4	2	no	—
258	C₄	—	8	3	no	—
259	C₄	—	332	2	no	—
260	C₄	—	664	3	no	—
261	C₄	C₂×#14581725	1	12	yes	—
262	C₄	C₄×Q₈×D₈₃	1	1	yes	characteristic
263	C₄	#19128183	1	1	yes	characteristic
264	C₄	#19128427	1	1	yes	characteristic
265	C₄	Q₈×Dic₁₆₆	1	1	yes	characteristic
266	C₂	Q₈×Dic₈₃⋊C₄	1	1	yes	characteristic
267	C₂	C₂²×#14581725	1	1	yes	characteristic
268	C₂	#24916061	1	1	yes	characteristic
269	C₂	Q₈×#13050575	1	1	yes	characteristic
270	C₂	Q₈×#13050580	1	1	yes	characteristic
271	C₂	#24920435	1	1	yes	characteristic
272	C₂	#24920466	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	Q₈×C₄⋊C₄×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×D₄
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₈₃
hypocenter	C₈₃	Q₈×#107
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
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×C₈₃⋊C₈	C₄	semidirect
35. #14581725	Q₈	direct (N × H)
85. Q₈×C₈₃	#107	semidirect
111. C₈₃⋊C₈	C₄×Q₈	semidirect
144. C₈₃	Q₈×#107	semidirect
222. Q₈	#14581725	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×C₈₃⋊C₈	2
Q₈×C₄⋊Dic₈₃	2
#24908221	2
#24908219	2
#24907622	2
C₄×#14581725	2
Q₈×C₄⋊C₄×C₈₃	2
Q₈×#107	83

Subgroups — Q8×#14581725 #35423241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14581725 #35423241