Subgroups — C₂×Q₈×D₈₈ #15634188

Order 2816.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₄₄	C₂	1	2	yes	maximal, complemented
2	Q₈×D₈₈	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10622295	C₂	1	3	yes	maximal, complemented
4	C₂×#10622459	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₈₈	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×D₁₁	—	2	2	no	—
8	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂²×D₈₈	C₂²	1	1	yes	characteristic
10	Q₈×D₄₄	—	2	8	no	—
11	Q₈×D₄₄	C₂²	1	16	yes	complemented
12	C₂×#66950	—	2	6	no	—
13	C₂×C₄×D₄₄	C₂²	1	6	yes	—
14	C₂×#66922	C₂²	1	6	yes	—
15	C₈×Q₈×C₁₁	C₂²	1	8	yes	complemented
16	C₂×C₁₁×C₄⋊C₈	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
18	C₂×#66885	C₂²	1	6	yes	—
19	C₂×#66882	C₂²	1	3	yes	—
20	C₂×#66882	C₂²	1	6	yes	—
21	#10622459	C₂²	1	48	yes	complemented
22	C₄×D₈₈	C₂²	1	24	yes	—
23	#10622295	C₂²	1	24	yes	complemented
24	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
25	C₂×Q₈×D₁₁	—	2	16	no	—
26	C₂×Q₈×D₁₁	—	4	16	no	—
27	C₂²×D₄₄	C₂³	1	2	yes	—
28	C₂²×C₄×D₁₁	—	2	6	no	—
29	C₂²×Dic₂₂	—	2	6	no	—
30	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₁₁	D₄	1	4	yes	—
33	C₄×Q₈×C₁₁	C₂³	1	4	yes	—
34	C₂×C₁₁×C₄⋊C₄	D₄	1	3	yes	—
35	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
36	C₂×(C₂×D₁₁)⋊C₄	—	2	6	no	—
37	C₂×C₄⋊Dic₁₁	—	2	6	no	—
38	C₂×C₄⋊Dic₁₁	C₂³	1	6	yes	—
39	C₂×D₈₈	Q₈	1	8	yes	direct product factor
40	C₂×D₈₈	C₂³	1	4	yes	—
41	#66950	—	2	48	no	—
42	C₄×D₄₄	—	2	12	no	—
43	C₄×D₄₄	C₂³	1	24	yes	—
44	#66922	—	2	12	no	—
45	#66922	C₂³	1	24	yes	—
46	C₁₁×C₄⋊C₈	C₂³	1	12	yes	—
47	C₄×C₈×C₁₁	C₂³	1	12	yes	—
48	#66885	C₂³	1	24	yes	—
49	#66882	C₂³	1	12	yes	—
50	#66882	C₂³	1	24	yes	—
51	C₂×Q₈×D₈	—	11	1	no	maximal, Sylow
52	C₂³×D₁₁	—	2	2	no	—
53	C₂×Q₈×C₁₁	D₈	1	8	yes	complemented
54	C₂×Q₈×C₁₁	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₁₁	—	2	6	no	—
59	Q₈×D₁₁	—	4	64	no	—
60	C₂×D₄₄	—	2	2	no	—
61	C₂×D₄₄	—	2	4	no	—
62	C₂×D₄₄	C₂×Q₈	1	8	yes	complemented
63	C₂×D₄₄	C₂⁴	1	4	yes	—
64	C₂×C₄×D₁₁	—	2	24	no	—
65	C₂×C₄×D₁₁	—	4	24	no	—
66	C₂×Dic₂₂	—	2	24	no	—
67	C₂×Dic₂₂	—	4	24	no	—
68	C₂×C₈×C₁₁	—	2	6	no	—
69	C₂×C₈×C₁₁	C₂×Q₈	1	4	yes	complemented
70	C₂×C₈×C₁₁	C₂⁴	1	6	yes	—
71	C₂×C₈×C₁₁	C₂⁴	1	2	yes	—
72	C₁₁×C₄⋊C₄	C₂×D₄	1	12	yes	—
73	C₄²×C₁₁	C₂×D₄	1	6	yes	—
74	C₄²×C₁₁	C₂⁴	1	6	yes	—
75	(C₂×D₁₁)⋊C₄	—	2	24	no	—
76	C₄⋊Dic₁₁	—	2	6	no	—
77	C₄⋊Dic₁₁	—	2	24	no	—
78	C₄⋊Dic₁₁	C₂⁴	1	12	yes	—
79	D₈₈	C₂×Q₈	1	16	yes	direct product factor
80	C₂×D₄×Q₈	—	11	2	no	—
81	Q₈×D₈	—	11	16	no	—
82	C₂×#500	—	11	3	no	—
83	C₂×#474	—	11	6	no	—
84	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic, complemented
85	C₂×C₄×D₈	—	11	3	no	—
86	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
87	C₂²×D₁₁	—	2	8	no	—
88	C₂²×D₁₁	—	2	4	no	—
89	C₂²×D₁₁	—	4	4	no	—
90	C₂²×D₁₁	—	4	4	no	—
91	Q₈×C₁₁	C₂×D₈	1	16	yes	complemented
92	C₂×C₄×C₁₁	—	2	6	no	—
93	C₂×C₄×C₁₁	C₂×D₈	1	12	yes	—
94	C₂×C₄×C₁₁	C₂²×D₄	1	6	yes	—
95	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₁	C₂²×D₄	1	6	yes	—
97	C₂×C₄×C₁₁	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
99	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
100	C₂×Dic₁₁	—	2	12	no	—
101	C₂×Dic₁₁	—	4	12	no	—
102	D₄₄	—	2	8	no	—
103	D₄₄	—	2	4	no	—
104	D₄₄	C₂²×Q₈	1	8	yes	complemented
105	C₄×D₁₁	—	4	48	no	—
106	Dic₂₂	—	4	48	no	—
107	C₈×C₁₁	—	2	6	no	—
108	C₈×C₁₁	C₂²×Q₈	1	4	yes	complemented
109	C₂³×Q₈	—	22	2	no	—
110	C₂²×D₈	—	11	1	no	—
111	D₄×Q₈	—	11	16	no	—
112	D₄×Q₈	—	22	8	no	—
113	C₂×C₄⋊Q₈	—	11	6	no	—
114	C₂×C₂²⋊Q₈	—	22	6	no	—
115	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
116	C₂×C₄×D₄	—	11	6	no	—
117	#500	—	11	24	no	—
118	#474	—	11	48	no	—
119	C₈×Q₈	C₂×D₁₁	1	8	yes	complemented
120	C₄×D₈	—	11	24	no	—
121	C₂×#107	—	11	3	no	—
122	C₂×#107	—	11	6	no	—
123	C₂×C₄⋊C₈	C₂×D₁₁	1	3	yes	—
124	C₂×#102	—	11	6	no	—
125	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
126	C₂²×C₁₁	D₄×Q₈	1	2	yes	—
127	C₂²×C₁₁	C₂²×D₈	1	2	yes	—
128	C₂²×C₁₁	#578	1	2	yes	—
129	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
130	C₂×D₁₁	—	2	8	no	—
131	C₂×D₁₁	—	4	16	no	—
132	C₂×D₁₁	—	4	4	no	—
133	C₂×D₁₁	—	4	4	no	—
134	C₄×C₁₁	—	2	6	no	—
135	C₄×C₁₁	D₄×Q₈	1	2	yes	—
136	C₄×C₁₁	C₂²×D₈	1	12	yes	—
137	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
138	C₄×C₁₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
139	Dic₁₁	—	4	12	no	—
140	C₂²×Q₈	—	22	16	no	—
141	C₂²×Q₈	—	22	6	no	—
142	C₂²×Q₈	—	44	16	no	—
143	C₂²×Q₈	D₄₄	1	1	yes	characteristic
144	C₂²×D₄	—	11	2	no	—
145	C₂³×C₄	—	22	6	no	—
146	C₂×D₈	—	11	8	no	—
147	C₂×D₈	—	11	4	no	—
148	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
149	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
150	C₄⋊Q₈	—	11	24	no	—
151	C₄⋊Q₈	—	22	12	no	—
152	C₂²⋊Q₈	—	22	48	no	—
153	C₄×Q₈	D₄₄	1	4	yes	—
154	C₄×Q₈	C₂²×D₁₁	1	4	yes	—
155	C₄×D₄	—	11	24	no	—
156	C₄×D₄	—	22	12	no	—
157	C₂×C₄⋊C₄	—	11	6	no	—
158	C₂×C₄⋊C₄	—	22	6	no	—
159	C₂×C₄⋊C₄	D₄₄	1	3	yes	—
160	C₂×C₂²⋊C₄	—	22	6	no	—
161	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
162	#107	—	11	12	no	—
163	#107	—	11	24	no	—
164	C₄⋊C₈	C₂²×D₁₁	1	12	yes	—
165	#102	—	11	24	no	—
166	C₄×C₈	C₂²×D₁₁	1	12	yes	—
167	C₂×C₁₁	Q₈×D₈	1	4	yes	complemented
168	C₂×C₁₁	C₂×D₄×Q₈	1	1	yes	characteristic
169	C₂×C₁₁	C₂³×D₈	1	1	yes	characteristic
170	C₂×C₁₁	C₂×#578	1	1	yes	characteristic
171	D₁₁	—	4	8	no	—
172	C₂⁴	—	22	2	no	—
173	C₂×Q₈	—	22	24	no	—
174	C₂×Q₈	—	44	64	no	—
175	C₂×Q₈	—	44	24	no	—
176	C₂×Q₈	D₈₈	1	8	yes	direct product factor
177	C₂×Q₈	C₂×D₄₄	1	4	yes	—
178	C₂×D₄	—	11	8	no	—
179	C₂×D₄	—	11	4	no	—
180	C₂×D₄	—	22	2	no	—
181	C₂×D₄	—	22	4	no	—
182	C₂²×C₄	—	22	6	no	—
183	C₂²×C₄	—	22	24	no	—
184	C₂²×C₄	—	44	24	no	—
185	C₂²×C₄	C₂×D₄₄	1	3	yes	—
186	C₂²×C₄	C₂×D₄₄	1	3	yes	—
187	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
188	D₈	—	11	16	no	—
189	C₂×C₈	—	2	6	no	—
190	C₂×C₈	Q₈×D₁₁	1	4	yes	complemented
191	C₂×C₈	C₂³×D₁₁	1	6	yes	—
192	C₂×C₈	C₂³×D₁₁	1	2	yes	—
193	C₄⋊C₄	—	11	12	no	—
194	C₄⋊C₄	—	22	6	no	—
195	C₄⋊C₄	—	22	24	no	—
196	C₄⋊C₄	C₂×D₄₄	1	12	yes	—
197	C₂²⋊C₄	—	22	24	no	—
198	C₄²	C₂×D₄₄	1	6	yes	—
199	C₄²	C₂³×D₁₁	1	6	yes	—
200	C₁₁	C₂×Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	22	8	no	—
202	C₂³	—	22	4	no	—
203	C₂³	—	44	4	no	—
204	C₂³	—	44	4	no	—
205	C₂³	C₂²×D₄₄	1	1	yes	characteristic
206	Q₈	—	44	48	no	—
207	Q₈	C₂×D₈₈	1	16	yes	direct product factor
208	D₄	—	11	8	no	—
209	D₄	—	22	8	no	—
210	D₄	—	22	4	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	22	12	no	—
213	C₂×C₄	—	44	12	no	—
214	C₂×C₄	—	44	48	no	—
215	C₂×C₄	C₂×D₈₈	1	12	yes	—
216	C₂×C₄	C₂²×D₄₄	1	6	yes	—
217	C₂×C₄	C₂²×D₄₄	1	6	yes	—
218	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
219	C₂×C₄	C₂×Q₈×D₁₁	1	2	yes	—
220	C₂×C₄	#67043	1	2	yes	—
221	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂×Q₈×D₁₁	1	4	yes	complemented
224	C₂²	—	22	8	no	—
225	C₂²	—	44	16	no	—
226	C₂²	—	44	4	no	—
227	C₂²	—	44	4	no	—
228	C₂²	Q₈×D₄₄	1	2	yes	—
229	C₂²	C₂²×D₈₈	1	2	yes	—
230	C₂²	#10623284	1	2	yes	—
231	C₂²	C₂³×D₄₄	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	44	12	no	—
234	C₄	Q₈×D₄₄	1	2	yes	—
235	C₄	C₂²×D₈₈	1	12	yes	—
236	C₄	C₂²×Q₈×D₁₁	1	1	yes	characteristic
237	C₄	C₂×#67043	1	1	yes	characteristic
238	C₂	—	44	8	no	—
239	C₂	Q₈×D₈₈	1	4	yes	direct product factor
240	C₂	C₂×Q₈×D₄₄	1	1	yes	characteristic
241	C₂	C₂³×D₈₈	1	1	yes	characteristic
242	C₂	C₂×#10623284	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₄₄
commutator, cosocle	C₂×C₄×C₁₁	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂×C₈×Q₈×C₁₁	C₂
socle	C₂³×C₁₁	C₂²×D₄
prime radical	C₂²×D₈₈	C₂²
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	C₂×Q₈×D₈
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₄₄	C₂	semidirect
2. Q₈×D₈₈	C₂	direct (N × H)
3. C₂×#10622295	C₂	semidirect
4. C₂×#10622459	C₂	semidirect
6. C₂×C₈×Q₈×C₁₁	C₂	semidirect
11. Q₈×D₄₄	C₂²	semidirect
15. C₈×Q₈×C₁₁	C₂²	semidirect
21. #10622459	C₂²	semidirect
23. #10622295	C₂²	semidirect
39. C₂×D₈₈	Q₈	direct (N × H)
53. C₂×Q₈×C₁₁	D₈	semidirect
62. C₂×D₄₄	C₂×Q₈	semidirect
69. C₂×C₈×C₁₁	C₂×Q₈	semidirect
79. D₈₈	C₂×Q₈	direct (N × H)
84. C₂×C₈×Q₈	D₁₁	semidirect
91. Q₈×C₁₁	C₂×D₈	semidirect
104. D₄₄	C₂²×Q₈	semidirect
108. C₈×C₁₁	C₂²×Q₈	semidirect
119. C₈×Q₈	C₂×D₁₁	semidirect
167. C₂×C₁₁	Q₈×D₈	semidirect
176. C₂×Q₈	D₈₈	direct (N × H)
190. C₂×C₈	Q₈×D₁₁	semidirect
200. C₁₁	C₂×Q₈×D₈	semidirect
207. Q₈	C₂×D₈₈	direct (N × H)
223. C₈	C₂×Q₈×D₁₁	semidirect
239. C₂	Q₈×D₈₈	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×D₄₄	2
Q₈×D₈₈	2
C₂×#10622295	2
C₂×#10622459	2
C₂×C₄×D₈₈	2
C₂×C₈×Q₈×C₁₁	2
C₂×Q₈×D₈	11

Subgroups — C2×Q8×D88 #15634188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₈₈ #15634188