Subgroups — Q₈×C₄⋊#10596842 #34034625

Order 18688.

896 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10596842	C₂	1	2	yes	maximal
2	#22627080	C₂	1	6	yes	maximal
3	#22627075	C₂	1	3	yes	maximal
4	C₄×C₄⋊#10596842	C₂	1	3	yes	maximal
5	C₄×Q₈×#64408	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596842	—	2	8	no	—
7	C₂×C₄⋊#10596842	C₂²	1	3	yes	—
8	C₂×C₄⋊#10596842	C₂²	1	6	yes	—
9	C₂×C₄⋊#10596842	C₂²	1	6	yes	—
10	C₂×C₄⋊#10596842	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596842	C₂²	1	6	yes	—
12	(C₂×#10596842)⋊C₄	C₂²	1	6	yes	—
13	C₂×Q₈×#64408	C₄	1	1	yes	characteristic
14	C₂×Q₈×#64408	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×#64408	C₄	1	3	yes	—
16	C₄⋊C₄×#64408	C₂²	1	3	yes	—
17	C₄×C₄⋊#64408	C₄	1	3	yes	—
18	C₄²×#64408	C₂²	1	3	yes	—
19	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596842	C₂³	1	6	yes	—
21	C₂²×#10596842	C₂³	1	2	yes	—
22	C₄⋊#10596842	—	2	12	no	—
23	C₄⋊#10596842	—	2	6	no	—
24	C₄⋊#10596842	—	4	6	no	—
25	C₄⋊#10596842	Q₈	1	4	yes	direct product factor
26	C₄×#10596842	—	2	12	no	—
27	Q₈×#64408	—	2	4	no	—
28	Q₈×#64408	D₄	1	4	yes	—
29	Q₈×#64408	Q₈	1	4	yes	—
30	C₂×C₄⋊#64408	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊#64408	C₂×C₄	1	3	yes	—
32	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
33	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
34	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
35	C₂×C₄×#64408	C₂³	1	3	yes	—
36	C₂×C₄×#64408	C₂³	1	3	yes	—
37	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
38	(C₂×#64408)⋊C₄	C₂×C₄	1	6	yes	—
39	C₂×Q₈×D₇₃	C₂×C₄	1	1	yes	characteristic
40	Q₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
41	C₄×Dic₁₄₆	C₈	1	3	yes	complemented
42	#14556556	C₈	1	3	yes	complemented
43	C₄⋊C₄×D₇₃	C₂×C₄	1	3	yes	—
44	C₄²×D₇₃	C₂×C₄	1	3	yes	—
45	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic, complemented
46	C₂×#10596842	—	2	2	no	—
47	C₂×#10596842	—	2	2	no	—
48	C₂×#10596842	—	2	6	no	—
49	C₂×#10596842	—	2	6	no	—
50	C₂×#10596842	—	2	6	no	—
51	C₂×#10596842	C₂×Q₈	1	4	yes	—
52	C₂²×#64408	C₂²×C₄	1	3	yes	—
53	C₂²×#64408	C₂²×C₄	1	3	yes	—
54	C₂²×#64408	C₂²×C₄	1	1	yes	characteristic
55	C₂²×#64408	C₂⁴	1	1	yes	characteristic
56	C₄⋊#64408	—	2	6	no	—
57	C₄⋊#64408	C₄⋊C₄	1	12	yes	—
58	C₄×#64408	—	2	3	no	—
59	C₄×#64408	—	2	3	no	—
60	C₄×#64408	—	2	6	no	—
61	C₄×#64408	—	2	6	no	—
62	C₄×#64408	—	4	3	no	—
63	C₄×#64408	C₂×D₄	1	6	yes	—
64	C₄×#64408	C₂×Q₈	1	6	yes	—
65	C₄×#64408	C₂×Q₈	1	2	yes	—
66	C₄×#64408	#41	1	2	yes	—
67	C₂×Dic₁₄₆	C₂×C₈	1	3	yes	—
68	Q₈×D₇₃	C₄⋊C₄	1	4	yes	—
69	Q₈×D₇₃	M₁₆	1	4	yes	—
70	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
71	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
72	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
74	#11924007	C₂×C₈	1	6	yes	—
75	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
76	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
77	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
78	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
79	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
80	C₄²×C₇₃	C₂×C₈	1	3	yes	—
81	C₂²×D₇₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×#64408	—	2	1	no	—
83	C₂×#64408	—	2	3	no	—
84	C₂×#64408	—	2	1	no	—
85	C₂×#64408	—	2	3	no	—
86	C₂×#64408	—	2	3	no	—
87	C₂×#64408	—	2	3	no	—
88	C₂×#64408	—	2	3	no	—
89	C₂×#64408	C₂×C₄⋊C₄	1	6	yes	—
90	C₂×#64408	C₄×Q₈	1	2	yes	—
91	C₂×#64408	C₂²×D₄	1	1	yes	characteristic
92	C₂×#64408	C₂²×Q₈	1	1	yes	characteristic
93	C₂×#64408	C₂²×Q₈	1	1	yes	characteristic
94	C₂×#64408	C₂×#41	1	1	yes	characteristic
95	C₂×#64408	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂×#64408	2₋¹⁺⁴	1	1	yes	characteristic
97	#10596842	—	2	4	no	—
98	#10596842	—	4	6	no	—
99	Q₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
100	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
101	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
102	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
104	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
105	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
106	C₄×D₇₃	—	2	3	no	—
107	C₄×D₇₃	—	2	3	no	—
108	C₄×D₇₃	C₂×C₄⋊C₄	1	6	yes	—
109	C₄×D₇₃	C₄×Q₈	1	2	yes	—
110	C₄×D₇₃	C₂×M₁₆	1	6	yes	—
111	C₄×D₇₃	#131	1	2	yes	—
112	Dic₁₄₆	C₄⋊C₈	1	12	yes	complemented
113	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
114	C₂×D₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂×D₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₂×D₇₃	#520	1	1	yes	characteristic
117	C₂×D₇₃	C₂²×M₁₆	1	1	yes	characteristic
118	C₂×D₇₃	C₂×#131	1	1	yes	characteristic
119	C₂×D₇₃	#568	1	1	yes	characteristic
120	#64408	—	2	6	no	—
121	#64408	—	2	2	no	—
122	#64408	—	4	3	no	—
123	#64408	D₄×Q₈	1	1	yes	characteristic
124	#64408	#550	1	1	yes	characteristic
125	#64408	#557	1	1	yes	characteristic
126	#64408	Q₈²	1	1	yes	characteristic
127	C₄×C₇₃	—	2	3	no	—
128	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
129	C₄×C₇₃	C₈×Q₈	1	2	yes	complemented
130	Dic₇₃	—	2	3	no	—
131	Dic₇₃	C₂×C₄⋊C₈	1	6	yes	—
132	Dic₇₃	C₈×Q₈	1	2	yes	complemented
133	Q₈×C₄⋊C₈	—	73	1	no	maximal, Sylow
134	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
136	C₂×C₇₃	#2988	1	1	yes	characteristic
137	D₇₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
138	D₇₃	#2906	1	1	yes	characteristic
139	D₇₃	Q₈×M₁₆	1	1	yes	characteristic
140	D₇₃	#2991	1	1	yes	characteristic
141	C₂×C₈×Q₈	—	73	2	no	—
142	C₄²×Q₈	—	73	1	no	—
143	#1939	—	73	6	no	—
144	#1842	—	73	3	no	—
145	C₄×C₄⋊C₈	—	73	3	no	—
146	C₇₃	Q₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄×Q₈	—	73	1	no	—
148	C₂×C₄×Q₈	—	73	1	no	—
149	C₂×C₄×Q₈	—	73	1	no	—
150	C₈×Q₈	—	146	8	no	—
151	C₂×C₄⋊C₈	—	73	6	no	—
152	C₂×C₄⋊C₈	—	73	1	no	—
153	C₂×C₄⋊C₈	—	73	3	no	—
154	C₂×C₄⋊C₈	—	73	6	no	—
155	C₂×C₄×C₈	—	73	6	no	—
156	C₄×C₄⋊C₄	—	73	3	no	—
157	C₄×C₄⋊C₄	—	73	3	no	—
158	C₄×C₄⋊C₄	—	73	3	no	—
159	C₄³	—	73	3	no	—
160	#336	—	73	6	no	—
161	C₂²×Q₈	—	73	1	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	—	73	6	no	—
164	C₄×Q₈	—	73	4	no	—
165	C₄×Q₈	—	73	4	no	—
166	C₄×Q₈	—	73	1	no	—
167	C₄×Q₈	—	73	3	no	—
168	C₄×Q₈	—	73	3	no	—
169	C₄×Q₈	—	146	4	no	—
170	C₄×Q₈	#10596842	1	1	yes	characteristic, complemented
171	C₂×C₄⋊C₄	—	73	3	no	—
172	C₂×C₄⋊C₄	—	73	3	no	—
173	C₂×C₄⋊C₄	—	73	3	no	—
174	C₂×C₄²	—	73	3	no	—
175	C₂×C₄²	—	73	3	no	—
176	C₂×C₄²	—	73	1	no	—
177	C₂×C₄²	—	73	3	no	—
178	C₂×C₄²	—	73	3	no	—
179	C₂×C₄²	—	73	3	no	—
180	C₂×C₄²	—	73	3	no	—
181	C₄⋊C₈	—	292	6	no	—
182	C₄⋊C₈	—	73	4	no	—
183	C₄⋊C₈	—	146	6	no	—
184	C₄⋊C₈	—	146	12	no	—
185	C₄×C₈	—	146	12	no	—
186	#95	—	73	6	no	—
187	C₂×Q₈	—	73	4	no	—
188	C₂×Q₈	—	73	4	no	—
189	C₂×Q₈	—	73	3	no	—
190	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
191	C₂²×C₄	—	73	1	no	—
192	C₂²×C₄	—	73	1	no	—
193	C₂²×C₄	—	73	3	no	—
194	C₂²×C₄	—	73	3	no	—
195	C₂²×C₄	—	73	1	no	—
196	C₂²×C₄	—	73	3	no	—
197	C₂²×C₄	—	73	3	no	—
198	C₂×C₈	—	73	4	no	—
199	C₂×C₈	—	146	6	no	—
200	C₂×C₈	—	146	6	no	—
201	C₂×C₈	—	146	6	no	—
202	C₂×C₈	—	146	2	no	—
203	C₂×C₈	—	146	2	no	—
204	C₄⋊C₄	—	73	12	no	—
205	C₄⋊C₄	—	73	6	no	—
206	C₄⋊C₄	—	73	3	no	—
207	C₄⋊C₄	—	146	6	no	—
208	C₄⋊C₄	C₂×#10596842	1	3	yes	—
209	C₄²	—	292	3	no	—
210	C₄²	—	73	6	no	—
211	C₄²	—	73	2	no	—
212	C₄²	—	73	6	no	—
213	C₄²	—	73	2	no	—
214	C₄²	—	73	3	no	—
215	C₄²	—	73	3	no	—
216	C₄²	—	73	3	no	—
217	C₄²	—	146	3	no	—
218	C₄²	—	146	3	no	—
219	C₄²	—	146	6	no	—
220	C₄²	—	146	6	no	—
221	C₄²	C₂×#10596842	1	3	yes	—
222	C₂³	—	73	1	no	—
223	Q₈	—	73	12	no	—
224	Q₈	C₄⋊#10596842	1	4	yes	direct product factor
225	C₂×C₄	—	73	1	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	73	1	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	73	3	no	—
233	C₂×C₄	—	73	6	no	—
234	C₂×C₄	—	73	6	no	—
235	C₂×C₄	—	73	2	no	—
236	C₂×C₄	—	73	2	no	—
237	C₂×C₄	—	73	6	no	—
238	C₂×C₄	—	73	2	no	—
239	C₂×C₄	—	73	3	no	—
240	C₂×C₄	—	146	3	no	—
241	C₂×C₄	—	146	3	no	—
242	C₂×C₄	—	146	3	no	—
243	C₂×C₄	—	146	3	no	—
244	C₂×C₄	—	146	3	no	—
245	C₂×C₄	—	146	1	no	—
246	C₂×C₄	—	146	3	no	—
247	C₂×C₄	—	146	1	no	—
248	C₂×C₄	—	146	3	no	—
249	C₂×C₄	C₂²×#10596842	1	3	yes	—
250	C₂×C₄	C₂²×#10596842	1	3	yes	—
251	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
252	C₈	—	292	6	no	—
253	C₈	—	146	4	no	—
254	C₂²	—	73	1	no	—
255	C₂²	—	73	1	no	—
256	C₂²	—	73	1	no	—
257	C₂²	—	73	1	no	—
258	C₂²	—	73	1	no	—
259	C₂²	—	73	1	no	—
260	C₂²	C₂³×#10596842	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	292	3	no	—
263	C₄	—	73	1	no	—
264	C₄	—	73	1	no	—
265	C₄	—	73	1	no	—
266	C₄	—	73	1	no	—
267	C₄	—	73	6	no	—
268	C₄	—	73	2	no	—
269	C₄	—	146	6	no	—
270	C₄	—	146	2	no	—
271	C₄	—	146	3	no	—
272	C₄	C₂×C₄⋊#10596842	1	6	yes	—
273	C₄	Q₈×#10596842	1	2	yes	complemented
274	C₂	—	73	1	no	—
275	C₂	—	73	1	no	—
276	C₂	—	73	1	no	—
277	C₂	—	73	1	no	—
278	C₂	C₂²×C₄⋊#10596842	1	1	yes	characteristic
279	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
280	C₂	#22627175	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×Q₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂³×C₄
cosocle	C₂²×#64408	C₂⁴
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	Q₈×C₄⋊C₈
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
25. C₄⋊#10596842	Q₈	direct (N × H)
40. Q₈×Dic₇₃	C₈	semidirect
41. C₄×Dic₁₄₆	C₈	semidirect
42. #14556556	C₈	semidirect
45. C₄×Q₈×C₇₃	C₈	semidirect
99. Q₈×C₇₃	C₄⋊C₈	semidirect
112. Dic₁₄₆	C₄⋊C₈	semidirect
129. C₄×C₇₃	C₈×Q₈	semidirect
132. Dic₇₃	C₈×Q₈	semidirect
146. C₇₃	Q₈×C₄⋊C₈	semidirect
170. C₄×Q₈	#10596842	semidirect
224. Q₈	C₄⋊#10596842	direct (N × H)
273. C₄	Q₈×#10596842	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10596842	2
#22627080	2
#22627075	2
C₄×C₄⋊#10596842	2
C₄×Q₈×#64408	2
Q₈×C₄⋊C₈	73

Subgroups — Q8×C4⋊#10596842 #34034625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₄⋊#10596842 #34034625