Subgroups — C₂²×Q₈×#41 #10585936

Order 512.

36450 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#41	C₂	1	96	yes	maximal, direct product factor
2	C₂²×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#557	C₂	1	3	yes	maximal, complemented
4	C₂²×#554	C₂	1	9	yes	maximal, complemented
5	C₂²×D₄×Q₈	C₂	1	3	yes	maximal, complemented
6	C₂²×#533	C₂	1	9	yes	maximal, complemented
7	C₂²×C₄×#41	C₂	1	3	yes	maximal
8	C₂³×C₄×Q₈	C₂	1	3	yes	maximal, complemented
9	C₂³×#41	C₂²	1	1	yes	characteristic
10	C₂⁴×Q₈	C₂²	1	3	yes	—
11	Q₈×#41	C₂²	1	1024	yes	direct product factor
12	C₂×Q₈²	C₂²	1	48	yes	complemented
13	C₂×#557	C₂²	1	144	yes	complemented
14	C₂×#554	C₂²	1	432	yes	complemented
15	C₂×D₄×Q₈	C₂²	1	144	yes	complemented
16	C₂×#533	C₂²	1	432	yes	complemented
17	C₂²×C₄⋊Q₈	C₂²	1	9	yes	—
18	C₂²×C₄⋊Q₈	C₂²	1	9	yes	—
19	C₂²×#125	C₂²	1	18	yes	—
20	C₂²×C₂²⋊Q₈	C₂²	1	18	yes	—
21	C₂²×C₂²⋊Q₈	C₂²	1	18	yes	—
22	C₂×C₄×#41	C₂²	1	144	yes	—
23	C₂²×C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₂²×C₄×Q₈	C₂²	1	9	yes	—
25	C₂²×C₄×Q₈	C₂²	1	3	yes	—
26	C₂²×C₄×Q₈	C₂²	1	9	yes	—
27	C₂²×C₄×Q₈	C₂²	1	3	yes	—
28	C₂²×C₄×Q₈	C₂²	1	3	yes	—
29	C₂²×C₄×Q₈	C₂²	1	6	yes	—
30	C₂²×C₄×Q₈	C₂²	1	144	yes	complemented
31	C₂²×C₄×D₄	C₂²	1	9	yes	—
32	C₂²×C₄×D₄	C₂²	1	9	yes	—
33	C₂²×#117	C₂²	1	9	yes	—
34	C₂³×C₄⋊C₄	C₂²	1	9	yes	—
35	C₂³×C₄²	C₂²	1	9	yes	—
36	C₂²×#41	Q₈	1	32	yes	direct product factor
37	C₂²×#41	C₂³	1	24	yes	—
38	C₂³×Q₈	—	2	48	no	—
39	C₂³×Q₈	C₂³	1	9	yes	—
40	C₂³×Q₈	C₂³	1	72	yes	—
41	C₂³×Q₈	C₂³	1	6	yes	—
42	C₂³×Q₈	C₂³	1	1	yes	characteristic
43	C₂³×Q₈	C₂³	1	1	yes	characteristic
44	C₂³×D₄	C₂³	1	3	yes	—
45	C₂⁴×C₄	C₂³	1	9	yes	—
46	C₂⁴×C₄	C₂³	1	9	yes	—
47	C₂⁴×C₄	C₂³	1	3	yes	—
48	Q₈²	C₂³	1	256	yes	complemented
49	#557	C₂³	1	768	yes	complemented
50	#554	C₂³	1	2304	yes	complemented
51	D₄×Q₈	C₂³	1	768	yes	complemented
52	#533	C₂³	1	2304	yes	complemented
53	C₂×C₄⋊Q₈	C₂³	1	216	yes	—
54	C₂×C₄⋊Q₈	C₂³	1	216	yes	—
55	C₂×#125	C₂³	1	432	yes	—
56	C₂×C₂²⋊Q₈	C₂³	1	432	yes	—
57	C₂×C₂²⋊Q₈	C₂³	1	432	yes	—
58	C₄×#41	C₂³	1	768	yes	—
59	C₂×C₄×Q₈	C₂³	1	24	yes	—
60	C₂×C₄×Q₈	C₂³	1	216	yes	—
61	C₂×C₄×Q₈	C₂³	1	72	yes	—
62	C₂×C₄×Q₈	C₂³	1	216	yes	—
63	C₂×C₄×Q₈	C₂³	1	72	yes	—
64	C₂×C₄×Q₈	C₂³	1	72	yes	—
65	C₂×C₄×Q₈	C₂³	1	144	yes	—
66	C₂×C₄×Q₈	C₂³	1	768	yes	complemented
67	C₂×C₄×D₄	C₂³	1	216	yes	—
68	C₂×C₄×D₄	C₂³	1	216	yes	—
69	C₂×#117	C₂³	1	216	yes	—
70	C₂²×C₄⋊C₄	C₂³	1	3	yes	—
71	C₂²×C₄⋊C₄	C₂³	1	18	yes	—
72	C₂²×C₄⋊C₄	C₂³	1	9	yes	—
73	C₂²×C₄⋊C₄	C₂³	1	216	yes	—
74	C₂²×C₄⋊C₄	C₂³	1	9	yes	—
75	C₂²×C₄⋊C₄	C₂³	1	9	yes	—
76	C₂²×C₄⋊C₄	C₂³	1	18	yes	—
77	C₂²×C₂²⋊C₄	C₂³	1	18	yes	—
78	C₂²×C₄²	C₂³	1	9	yes	—
79	C₂²×C₄²	C₂³	1	9	yes	—
80	C₂²×C₄²	C₂³	1	216	yes	—
81	C₂²×C₄²	C₂³	1	3	yes	—
82	C₂²×C₄²	C₂³	1	9	yes	—
83	C₂⁵	C₂⁴	1	3	yes	—
84	C₂×#41	C₂×Q₈	1	384	yes	direct product factor
85	C₂×#41	C₂⁴	1	64	yes	—
86	C₂²×Q₈	—	2	576	no	—
87	C₂²×Q₈	—	2	72	no	—
88	C₂²×Q₈	—	4	24	no	—
89	C₂²×Q₈	C₂×Q₈	1	16	yes	complemented
90	C₂²×Q₈	#41	1	16	yes	direct product factor
91	C₂²×Q₈	C₂⁴	1	108	yes	—
92	C₂²×Q₈	C₂⁴	1	192	yes	—
93	C₂²×Q₈	C₂⁴	1	72	yes	—
94	C₂²×Q₈	C₂⁴	1	12	yes	—
95	C₂²×Q₈	C₂⁴	1	12	yes	—
96	C₂²×D₄	C₂×Q₈	1	48	yes	complemented
97	C₂²×D₄	C₂⁴	1	36	yes	—
98	C₂³×C₄	—	2	72	no	—
99	C₂³×C₄	—	4	36	no	—
100	C₂³×C₄	C₂×Q₈	1	48	yes	complemented
101	C₂³×C₄	C₂⁴	1	9	yes	—
102	C₂³×C₄	C₂⁴	1	9	yes	—
103	C₂³×C₄	C₂⁴	1	3	yes	—
104	C₂³×C₄	C₂⁴	1	108	yes	—
105	C₂³×C₄	C₂⁴	1	108	yes	—
106	C₂³×C₄	C₂⁴	1	3	yes	—
107	C₂³×C₄	C₂⁴	1	36	yes	—
108	C₂³×C₄	C₂⁴	1	3	yes	—
109	C₂³×C₄	C₂⁴	1	1	yes	characteristic
110	C₄⋊Q₈	C₂⁴	1	576	yes	—
111	C₄⋊Q₈	C₂⁴	1	576	yes	—
112	#125	C₂⁴	1	1152	yes	—
113	C₂²⋊Q₈	C₂⁴	1	1152	yes	—
114	C₂²⋊Q₈	C₂⁴	1	1152	yes	—
115	C₄×Q₈	C₂⁴	1	64	yes	—
116	C₄×Q₈	C₂⁴	1	576	yes	—
117	C₄×Q₈	C₂⁴	1	192	yes	—
118	C₄×Q₈	C₂⁴	1	576	yes	—
119	C₄×Q₈	C₂⁴	1	192	yes	—
120	C₄×Q₈	C₂⁴	1	192	yes	—
121	C₄×Q₈	C₂⁴	1	384	yes	—
122	C₄×D₄	C₂⁴	1	576	yes	—
123	C₄×D₄	C₂⁴	1	576	yes	—
124	#117	C₂⁴	1	576	yes	—
125	C₂×C₄⋊C₄	C₂⁴	1	36	yes	—
126	C₂×C₄⋊C₄	C₂⁴	1	216	yes	—
127	C₂×C₄⋊C₄	C₂⁴	1	108	yes	—
128	C₂×C₄⋊C₄	C₂⁴	1	576	yes	—
129	C₂×C₄⋊C₄	C₂⁴	1	108	yes	—
130	C₂×C₄⋊C₄	C₂⁴	1	108	yes	—
131	C₂×C₄⋊C₄	C₂⁴	1	216	yes	—
132	C₂×C₂²⋊C₄	C₂⁴	1	216	yes	—
133	C₂×C₄²	C₂⁴	1	108	yes	—
134	C₂×C₄²	C₂⁴	1	108	yes	—
135	C₂×C₄²	C₂⁴	1	108	yes	—
136	C₂×C₄²	C₂⁴	1	36	yes	—
137	C₂×C₄²	C₂⁴	1	576	yes	—
138	C₂⁴	—	2	12	no	—
139	C₂⁴	—	2	12	no	—
140	C₂⁴	C₂²×Q₈	1	24	yes	—
141	C₂⁴	C₂⁵	1	18	yes	—
142	C₂⁴	C₂⁵	1	1	yes	characteristic
143	#41	C₂²×Q₈	1	512	yes	direct product factor
144	C₂×Q₈	—	2	768	no	—
145	C₂×Q₈	—	2	432	no	—
146	C₂×Q₈	—	4	144	no	—
147	C₂×Q₈	C₂²×Q₈	1	96	yes	complemented
148	C₂×Q₈	C₂×#41	1	96	yes	direct product factor
149	C₂×Q₈	C₂⁵	1	144	yes	—
150	C₂×Q₈	C₂⁵	1	96	yes	—
151	C₂×Q₈	C₂⁵	1	16	yes	—
152	C₂×Q₈	C₂⁵	1	16	yes	—
153	C₂×D₄	C₂²×Q₈	1	288	yes	complemented
154	C₂×D₄	C₂⁵	1	48	yes	—
155	C₂²×C₄	—	2	36	no	—
156	C₂²×C₄	—	2	12	no	—
157	C₂²×C₄	—	2	36	no	—
158	C₂²×C₄	—	2	432	no	—
159	C₂²×C₄	—	4	216	no	—
160	C₂²×C₄	C₂²×Q₈	1	8	yes	—
161	C₂²×C₄	C₂²×Q₈	1	24	yes	—
162	C₂²×C₄	C₂²×Q₈	1	288	yes	complemented
163	C₂²×C₄	C₂×#41	1	24	yes	—
164	C₂²×C₄	C₂⁵	1	18	yes	—
165	C₂²×C₄	C₂⁵	1	54	yes	—
166	C₂²×C₄	C₂⁵	1	54	yes	—
167	C₂²×C₄	C₂⁵	1	144	yes	—
168	C₂²×C₄	C₂⁵	1	144	yes	—
169	C₂²×C₄	C₂⁵	1	6	yes	—
170	C₂²×C₄	C₂⁵	1	18	yes	—
171	C₂²×C₄	C₂⁵	1	48	yes	—
172	C₂²×C₄	C₂⁵	1	18	yes	—
173	C₄⋊C₄	C₂⁵	1	48	yes	—
174	C₄⋊C₄	C₂⁵	1	288	yes	—
175	C₄⋊C₄	C₂⁵	1	144	yes	—
176	C₄⋊C₄	C₂⁵	1	144	yes	—
177	C₄⋊C₄	C₂⁵	1	144	yes	—
178	C₄⋊C₄	C₂⁵	1	288	yes	—
179	C₂²⋊C₄	C₂⁵	1	288	yes	—
180	C₄²	C₂⁵	1	48	yes	—
181	C₄²	C₂⁵	1	144	yes	—
182	C₄²	C₂⁵	1	144	yes	—
183	C₄²	C₂⁵	1	144	yes	—
184	C₂³	—	2	36	no	—
185	C₂³	—	2	36	no	—
186	C₂³	—	2	96	no	—
187	C₂³	C₂³×Q₈	1	72	yes	—
188	C₂³	C₂³×Q₈	1	4	yes	—
189	C₂³	C₂²×#41	1	4	yes	—
190	C₂³	#585	1	4	yes	—
191	C₂³	C₂⁶	1	12	yes	—
192	C₂³	C₂⁶	1	3	yes	—
193	Q₈	—	2	288	no	—
194	Q₈	—	4	96	no	—
195	Q₈	C₂³×Q₈	1	64	yes	complemented
196	Q₈	C₂²×#41	1	64	yes	direct product factor
197	D₄	C₂³×Q₈	1	192	yes	complemented
198	C₂×C₄	—	2	108	no	—
199	C₂×C₄	—	2	36	no	—
200	C₂×C₄	—	2	108	no	—
201	C₂×C₄	—	2	288	no	—
202	C₂×C₄	—	4	144	no	—
203	C₂×C₄	C₂³×Q₈	1	24	yes	—
204	C₂×C₄	C₂³×Q₈	1	72	yes	—
205	C₂×C₄	C₂³×Q₈	1	192	yes	complemented
206	C₂×C₄	C₂²×#41	1	72	yes	—
207	C₂×C₄	C₂⁶	1	12	yes	—
208	C₂×C₄	C₂⁶	1	36	yes	—
209	C₂×C₄	C₂⁶	1	36	yes	—
210	C₂×C₄	C₂⁶	1	4	yes	—
211	C₂×C₄	C₂⁶	1	12	yes	—
212	C₂×C₄	C₂⁶	1	12	yes	—
213	C₂²	—	2	12	no	—
214	C₂²	—	2	12	no	—
215	C₂²	—	2	144	no	—
216	C₂²	Q₈×#41	1	16	yes	direct product factor
217	C₂²	C₂⁴×Q₈	1	24	yes	—
218	C₂²	C₂⁴×Q₈	1	6	yes	—
219	C₂²	C₂³×#41	1	6	yes	—
220	C₂²	C₂×#585	1	6	yes	—
221	C₂²	C₂⁷	1	1	yes	characteristic
222	C₄	—	2	36	no	—
223	C₄	—	2	12	no	—
224	C₄	—	2	36	no	—
225	C₄	C₂⁴×Q₈	1	8	yes	—
226	C₄	C₂⁴×Q₈	1	24	yes	—
227	C₄	C₂³×#41	1	24	yes	—
228	C₂	—	2	24	no	—
229	C₂	C₂×Q₈×#41	1	12	yes	direct product factor
230	C₂	C₂⁵×Q₈	1	1	yes	characteristic
231	C₂	C₂⁴×#41	1	1	yes	characteristic
232	C₂	C₂²×#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
socle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×#41	C₂	direct (N × H)
2. C₂²×Q₈²	C₂	semidirect
3. C₂²×#557	C₂	semidirect
4. C₂²×#554	C₂	semidirect
5. C₂²×D₄×Q₈	C₂	semidirect
6. C₂²×#533	C₂	semidirect
8. C₂³×C₄×Q₈	C₂	semidirect
11. Q₈×#41	C₂²	direct (N × H)
12. C₂×Q₈²	C₂²	semidirect
13. C₂×#557	C₂²	semidirect
14. C₂×#554	C₂²	semidirect
15. C₂×D₄×Q₈	C₂²	semidirect
16. C₂×#533	C₂²	semidirect
30. C₂²×C₄×Q₈	C₂²	semidirect
36. C₂²×#41	Q₈	direct (N × H)
48. Q₈²	C₂³	semidirect
49. #557	C₂³	semidirect
50. #554	C₂³	semidirect
51. D₄×Q₈	C₂³	semidirect
52. #533	C₂³	semidirect
66. C₂×C₄×Q₈	C₂³	semidirect
84. C₂×#41	C₂×Q₈	direct (N × H)
89. C₂²×Q₈	C₂×Q₈	semidirect
90. C₂²×Q₈	#41	direct (N × H)
96. C₂²×D₄	C₂×Q₈	semidirect
100. C₂³×C₄	C₂×Q₈	semidirect
143. #41	C₂²×Q₈	direct (N × H)
147. C₂×Q₈	C₂²×Q₈	semidirect
148. C₂×Q₈	C₂×#41	direct (N × H)
153. C₂×D₄	C₂²×Q₈	semidirect
162. C₂²×C₄	C₂²×Q₈	semidirect
195. Q₈	C₂³×Q₈	semidirect
196. Q₈	C₂²×#41	direct (N × H)
197. D₄	C₂³×Q₈	semidirect
205. C₂×C₄	C₂³×Q₈	semidirect
216. C₂²	Q₈×#41	direct (N × H)
229. C₂	C₂×Q₈×#41	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×#41	2
C₂²×Q₈²	2
C₂²×#557	2
C₂²×#554	2
C₂²×D₄×Q₈	2
C₂²×#533	2
C₂²×C₄×#41	2
C₂³×C₄×Q₈	2

Subgroups — C22×Q8×#41 #10585936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×#41 #10585936