Subgroups — Q₈×#6649 #14418834

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×D₅	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11747007	C₂	1	3	yes	maximal
5	C₄×#6649	C₂	1	3	yes	maximal
6	#11746862	C₂	1	3	yes	maximal, complemented
7	#11746861	C₂	1	3	yes	maximal, complemented
8	Q₈×#676	—	3	1	no	maximal, Hall
9	C₂×C₃×Q₈×D₅	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₃×Q₈×Dic₅	C₄	1	1	yes	characteristic
12	C₃×D₅×C₄⋊C₄	C₄	1	3	yes	—
13	C₃×#4033	C₄	1	3	yes	—
14	C₃×C₄²×D₅	C₂²	1	3	yes	—
15	C₃×C₄×Dic₁₀	C₂²	1	3	yes	—
16	C₂×#6649	C₂²	1	3	yes	—
17	C₂×#6649	C₂²	1	1	yes	characteristic
18	C₄⋊(C₃⋊#168)	C₂²	1	3	yes	—
19	C₄×C₃⋊#168	C₂²	1	3	yes	—
20	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	3	yes	—
21	C₄×C₅×C₃⋊C₈	C₂²	1	3	yes	—
22	#90982	C₂²	1	6	yes	—
23	#90952	C₂²	1	3	yes	—
24	#90952	C₂²	1	3	yes	—
25	Q₈×#208	—	5	1	no	maximal, Hall
26	C₄×Q₈×D₅	D₃	1	1	yes	characteristic
27	C₅×C₈×Q₈	—	3	1	no	—
28	Q₈×#168	—	3	1	no	—
29	#65126	—	3	3	no	—
30	#65120	—	3	3	no	—
31	C₄×#676	—	3	3	no	—
32	#64968	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₃×Q₈×D₅	—	2	4	no	—
35	C₂×C₃×C₄×D₅	C₂×C₄	1	3	yes	—
36	C₂×C₃×C₄×D₅	C₂×C₄	1	3	yes	—
37	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	C₂×C₄	1	3	yes	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
40	C₃×C₄²×C₅	C₂³	1	3	yes	—
41	C₂×C₃⋊#168	C₂³	1	3	yes	—
42	C₂×C₃⋊#168	C₂³	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	C₂³	1	3	yes	—
44	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊Dic₅	C₂×C₄	1	3	yes	—
46	C₃×#683	C₂×C₄	1	6	yes	—
47	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	—
48	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	—
49	C₃×C₄×Dic₅	C₂³	1	3	yes	—
50	#6649	—	2	6	no	—
51	#6649	Q₈	1	4	yes	direct product factor
52	C₂×C₃×C₄×Q₈	—	5	1	no	—
53	#5024	—	5	3	no	—
54	#5024	—	5	3	no	—
55	Q₈×C₃⋊C₈	—	5	1	no	—
56	Q₈×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
57	#4924	—	5	3	no	—
58	C₄×#208	—	5	3	no	—
59	C₂×Q₈×D₅	Dic₃	1	1	yes	characteristic
60	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
61	Q₈×Dic₅	Dic₃	1	1	yes	characteristic
62	C₂×#676	—	3	1	no	—
63	C₂×#676	—	3	3	no	—
64	D₅×C₄⋊C₄	Dic₃	1	3	yes	—
65	#4033	Dic₃	1	3	yes	—
66	C₄²×D₅	C₂×D₃	1	3	yes	—
67	C₄×Dic₁₀	C₂×D₃	1	3	yes	—
68	C₅×C₄⋊C₈	—	3	3	no	—
69	C₄×C₅×C₈	—	3	3	no	—
70	#3947	—	3	3	no	—
71	#3947	—	3	3	no	—
72	#3946	—	3	6	no	—
73	C₄⋊#168	—	3	3	no	—
74	C₄×#168	—	3	3	no	—
75	Q₈×M₁₆	—	15	1	no	Sylow
76	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₅×Q₈	M₁₆	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
82	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
83	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₄×D₅	—	2	6	no	—
85	C₃×C₄×D₅	—	4	3	no	—
86	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
87	C₃×C₄×D₅	#41	1	2	yes	—
88	C₃×Dic₁₀	—	2	6	no	—
89	C₃⋊#168	—	2	3	no	—
90	C₃⋊#168	C₂×Q₈	1	2	yes	complemented
91	C₅×C₃⋊C₈	—	2	3	no	—
92	C₅×C₃⋊C₈	C₂×Q₈	1	2	yes	complemented
93	C₂²×C₃×Q₈	—	5	1	no	—
94	C₃×C₄×Q₈	—	5	1	no	—
95	C₃×C₄×Q₈	—	5	3	no	—
96	C₃×C₄×Q₈	—	5	3	no	—
97	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
98	C₂×C₃×C₄⋊C₄	—	5	3	no	—
99	C₂×C₃×C₄²	—	5	3	no	—
100	C₂×#208	—	5	1	no	—
101	C₂×#208	—	5	3	no	—
102	C₄⋊(C₃⋊C₈)	—	5	3	no	—
103	C₄⋊(C₃⋊C₈)	—	5	6	no	—
104	C₄⋊(C₃⋊C₈)	C₂×D₅	1	3	yes	—
105	#803	—	5	3	no	—
106	#803	—	5	3	no	—
107	C₄×C₃⋊C₈	—	5	3	no	—
108	C₄×C₃⋊C₈	C₂×D₅	1	3	yes	—
109	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
110	Q₈×D₅	—	2	4	no	—
111	C₂×C₄×D₅	C₂×Dic₃	1	3	yes	—
112	C₂×C₄×D₅	C₂×Dic₃	1	3	yes	—
113	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
114	C₂×Dic₁₀	C₂×Dic₃	1	3	yes	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	—	3	3	no	—
117	C₅×C₄⋊C₄	C₂×Dic₃	1	3	yes	—
118	C₄²×C₅	C₂²×D₃	1	3	yes	—
119	C₄⋊Dic₅	C₂×Dic₃	1	3	yes	—
120	#683	C₂×Dic₃	1	6	yes	—
121	C₄×Dic₅	C₂×Dic₃	1	3	yes	—
122	C₄×Dic₅	C₂×Dic₃	1	3	yes	—
123	C₄×Dic₅	C₂²×D₃	1	3	yes	—
124	C₂×#168	—	3	1	no	—
125	C₂×#168	—	3	3	no	—
126	#676	—	3	4	no	—
127	#676	—	6	6	no	—
128	C₂×C₄×Q₈	—	5	1	no	—
129	#446	—	15	3	no	—
130	#446	—	15	3	no	—
131	C₈×Q₈	—	3	1	no	—
132	C₈×Q₈	—	15	1	no	—
133	#423	—	15	3	no	—
134	C₄×M₁₆	—	15	3	no	—
135	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₃×D₅	—	2	1	no	—
137	C₂×C₃×D₅	—	2	1	no	—
138	C₂×C₃×D₅	C₄×Q₈	1	2	yes	—
139	C₃×C₄×C₅	—	2	3	no	—
140	C₃×C₄×C₅	C₂×M₁₆	1	6	yes	—
141	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₃×Dic₅	—	2	3	no	—
144	C₃×Dic₅	—	2	3	no	—
145	C₃×Dic₅	C₄×Q₈	1	2	yes	—
146	C₂×C₃×Q₈	—	5	3	no	—
147	C₂×C₃×Q₈	—	10	4	no	—
148	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
149	C₂²×C₃×C₄	—	5	1	no	—
150	C₂²×C₃×C₄	—	5	3	no	—
151	C₂²×C₃×C₄	—	5	3	no	—
152	C₃×C₄⋊C₄	—	5	6	no	—
153	C₃×C₄⋊C₄	—	5	3	no	—
154	C₃×C₄⋊C₄	C₄×D₅	1	3	yes	—
155	C₃×C₄²	—	5	3	no	—
156	C₃×C₄²	—	5	3	no	—
157	C₃×C₄²	—	5	3	no	—
158	C₃×C₄²	C₂²×D₅	1	3	yes	—
159	#208	—	5	4	no	—
160	#208	—	10	6	no	—
161	C₂×C₃⋊C₈	—	5	1	no	—
162	C₂×C₃⋊C₈	—	5	3	no	—
163	C₂×C₃⋊C₈	C₂²×D₅	1	3	yes	—
164	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
165	C₂²×D₅	C₂²×Dic₃	1	1	yes	characteristic
166	C₅×Q₈	#208	1	4	yes	complemented
167	C₂×C₄×C₅	C₂²×Dic₃	1	3	yes	—
168	C₂×C₄×C₅	C₂²×Dic₃	1	3	yes	—
169	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
170	C₂×Dic₅	C₂²×Dic₃	1	3	yes	—
171	C₂×Dic₅	C₂²×Dic₃	1	3	yes	—
172	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
173	C₄×D₅	—	2	6	no	—
174	C₄×D₅	—	4	3	no	—
175	C₄×D₅	D₃×Q₈	1	2	yes	—
176	C₄×D₅	#239	1	2	yes	—
177	Dic₁₀	—	2	6	no	—
178	C₅×C₈	—	3	2	no	—
179	C₅×C₈	—	6	3	no	—
180	#168	—	3	2	no	—
181	#168	—	6	3	no	—
182	C₂²×Q₈	—	5	1	no	—
183	C₂×M₁₆	—	15	1	no	—
184	C₂×M₁₆	—	15	3	no	—
185	C₄×Q₈	—	5	1	no	—
186	C₄×Q₈	—	5	3	no	—
187	C₄×Q₈	—	5	3	no	—
188	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	5	3	no	—
190	C₂×C₄²	—	5	3	no	—
191	C₄⋊C₈	—	3	3	no	—
192	C₄⋊C₈	—	15	6	no	—
193	C₄⋊C₈	—	15	3	no	—
194	#97	—	15	3	no	—
195	#97	—	15	3	no	—
196	C₄×C₈	—	3	3	no	—
197	C₄×C₈	—	15	3	no	—
198	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂²×M₁₆	1	1	yes	characteristic
200	C₂×C₃×C₅	#568	1	1	yes	characteristic
201	C₃×D₅	—	2	2	no	—
202	C₂³×C₃	—	5	1	no	—
203	C₃×Q₈	—	10	6	no	—
204	C₃×Q₈	#676	1	4	yes	complemented
205	C₂×C₃×C₄	—	5	1	no	—
206	C₂×C₃×C₄	—	5	2	no	—
207	C₂×C₃×C₄	—	5	2	no	—
208	C₂×C₃×C₄	—	5	3	no	—
209	C₂×C₃×C₄	—	5	3	no	—
210	C₂×C₃×C₄	—	10	6	no	—
211	C₂×C₃×C₄	—	20	3	no	—
212	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
213	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
214	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
215	C₃⋊C₈	—	2	3	no	—
216	C₃⋊C₈	—	5	2	no	—
217	C₃⋊C₈	—	10	3	no	—
218	C₃⋊C₈	Q₈×D₅	1	2	yes	complemented
219	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
220	C₂×D₅	—	2	1	no	—
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	Q₈×Dic₃	1	2	yes	—
223	C₄×C₅	—	2	3	no	—
224	C₄×C₅	C₂×#208	1	6	yes	—
225	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
226	C₄×C₅	C₂×#239	1	1	yes	characteristic
227	Dic₅	—	2	3	no	—
228	Dic₅	—	2	3	no	—
229	Dic₅	Q₈×Dic₃	1	2	yes	—
230	C₂×Q₈	—	5	3	no	—
231	C₂×Q₈	—	10	4	no	—
232	C₂×Q₈	D₅×Dic₃	1	1	yes	characteristic
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	3	no	—
235	C₂²×C₄	—	5	3	no	—
236	M₁₆	—	15	4	no	—
237	M₁₆	—	30	6	no	—
238	C₂×C₈	—	3	1	no	—
239	C₂×C₈	—	3	3	no	—
240	C₂×C₈	—	15	1	no	—
241	C₂×C₈	—	15	3	no	—
242	C₄⋊C₄	—	5	6	no	—
243	C₄⋊C₄	—	5	3	no	—
244	C₄⋊C₄	D₅×Dic₃	1	3	yes	—
245	C₄²	—	5	3	no	—
246	C₄²	—	5	3	no	—
247	C₄²	—	5	3	no	—
248	C₄²	C₂×D₃×D₅	1	3	yes	—
249	C₃×C₅	Q₈×M₁₆	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	5	2	no	—
251	C₂²×C₃	—	10	1	no	—
252	C₂²×C₃	—	10	1	no	—
253	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
254	C₃×C₄	—	2	3	no	—
255	C₃×C₄	—	5	2	no	—
256	C₃×C₄	—	10	3	no	—
257	C₃×C₄	—	10	3	no	—
258	C₃×C₄	C₂×#676	1	6	yes	—
259	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
260	C₃×C₄	D₅×#41	1	1	yes	characteristic
261	C₂×C₅	C₂²×#208	1	1	yes	characteristic
262	C₂×C₅	C₂×Q₈×Dic₃	1	1	yes	characteristic
263	C₂×C₅	#5819	1	1	yes	characteristic
264	D₅	—	2	2	no	—
265	C₂³	—	5	1	no	—
266	Q₈	—	10	6	no	—
267	Q₈	#6649	1	4	yes	direct product factor
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	2	no	—
270	C₂×C₄	—	5	2	no	—
271	C₂×C₄	—	5	3	no	—
272	C₂×C₄	—	5	3	no	—
273	C₂×C₄	—	10	6	no	—
274	C₂×C₄	—	20	3	no	—
275	C₂×C₄	C₂×D₅×Dic₃	1	3	yes	—
276	C₂×C₄	C₂×D₅×Dic₃	1	3	yes	—
277	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
278	C₈	—	3	2	no	—
279	C₈	—	6	3	no	—
280	C₈	—	15	2	no	—
281	C₈	—	30	3	no	—
282	C₂×C₃	—	10	2	no	—
283	C₂×C₃	C₄×Q₈×D₅	1	1	yes	characteristic
284	C₂×C₃	C₂²×#676	1	1	yes	characteristic
285	C₂×C₃	#66084	1	1	yes	characteristic
286	C₅	Q₈×#208	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	5	2	no	—
288	C₂²	—	10	1	no	—
289	C₂²	—	10	1	no	—
290	C₂²	C₂²×D₅×Dic₃	1	1	yes	characteristic
291	C₄	—	2	3	no	—
292	C₄	—	5	2	no	—
293	C₄	—	10	3	no	—
294	C₄	—	10	3	no	—
295	C₄	C₂×#6649	1	6	yes	—
296	C₄	D₃×Q₈×D₅	1	1	yes	characteristic
297	C₄	D₅×#239	1	1	yes	characteristic
298	C₃	Q₈×#676	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	10	2	no	—
300	C₂	C₂²×#6649	1	1	yes	characteristic
301	C₂	Q₈×D₅×Dic₃	1	1	yes	characteristic
302	C₂	#11755296	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₃×C₄×C₅×Q₈	C₂²
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	Q₈×M₁₆
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊#168	C₂	semidirect
3. C₅×Q₈×C₃⋊C₈	C₂	semidirect
6. #11746862	C₂	semidirect
7. #11746861	C₂	semidirect
51. #6649	Q₈	direct (N × H)
56. Q₈×C₃⋊C₈	D₅	semidirect
77. C₃×C₅×Q₈	M₁₆	semidirect
90. C₃⋊#168	C₂×Q₈	semidirect
92. C₅×C₃⋊C₈	C₂×Q₈	semidirect
166. C₅×Q₈	#208	semidirect
204. C₃×Q₈	#676	semidirect
218. C₃⋊C₈	Q₈×D₅	semidirect
249. C₃×C₅	Q₈×M₁₆	semidirect
267. Q₈	#6649	direct (N × H)
286. C₅	Q₈×#208	semidirect
298. C₃	Q₈×#676	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×D₅	2
Q₈×C₃⋊#168	2
C₅×Q₈×C₃⋊C₈	2
#11747007	2
C₄×#6649	2
#11746862	2
#11746861	2
Q₈×#676	3
Q₈×#208	5

Subgroups — Q8×#6649 #14418834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6649 #14418834