Subgroups — C₂×Q₈⋊#13083549 #36464227

Order 22784.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#13083549	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	—	2	1	no	—
6	C₂²×#13083549	—	2	1	no	—
7	C₄⋊#13083549	—	2	1	no	—
8	C₄⋊#13083549	C₂²	1	2	yes	—
9	C₄×#13083549	—	2	1	no	—
10	C₄×#13083549	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₈₉	—	2	1	no	—
13	C₈×Q₈×C₈₉	—	2	1	no	—
14	C₈×Q₈×C₈₉	C₄	1	2	yes	—
15	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083549	—	2	4	no	—
21	C₂×#13083549	—	2	2	no	—
22	C₂×#13083549	—	2	2	no	—
23	C₂×#13083549	—	4	2	no	—
24	C₂²×Q₈×C₈₉	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	—	2	1	no	—
26	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₈₉	—	2	1	no	—
29	C₄×Q₈×C₈₉	—	2	1	no	—
30	C₄×Q₈×C₈₉	C₈	1	2	yes	—
31	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	—	2	1	no	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₈₉	—	2	2	no	—
37	C₄⋊C₈×C₈₉	—	2	2	no	—
38	C₄⋊C₈×C₈₉	—	2	1	no	—
39	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₉	—	2	2	no	—
41	C₄×C₈×C₈₉	—	2	2	no	—
42	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	#13083549	—	2	4	no	—
47	#13083549	—	4	2	no	—
48	C₂×Q₈×C₈₉	—	2	1	no	—
49	C₂×Q₈×C₈₉	—	2	2	no	—
50	C₂×Q₈×C₈₉	C₁₆	1	4	yes	complemented
51	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₈₉	—	2	1	no	—
53	C₂²×C₄×C₈₉	—	2	1	no	—
54	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	—	4	2	no	—
60	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
63	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
64	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₈₉	—	2	2	no	—
67	C₄⋊C₄×C₈₉	—	2	1	no	—
68	C₄⋊C₄×C₈₉	—	2	2	no	—
69	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
70	C₄²×C₈₉	—	2	2	no	—
71	C₄²×C₈₉	—	2	2	no	—
72	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
75	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
77	Q₈×C₈₉	—	2	4	no	—
78	Q₈×C₈₉	—	2	2	no	—
79	Q₈×C₈₉	C₂×C₁₆	1	4	yes	complemented
80	C₂×C₄×C₈₉	—	2	4	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	4	2	no	—
85	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#103	1	2	yes	—
89	C₂×C₄×C₈₉	#104	1	2	yes	—
90	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
91	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
95	C₈×C₈₉	—	2	1	no	—
96	C₈×C₈₉	—	2	1	no	—
97	C₈×C₈₉	—	4	2	no	—
98	C₈×C₈₉	#103	1	2	yes	—
99	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₈₉	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₈₉	C₂²⋊C₁₆	1	2	yes	—
103	C₂²×C₈₉	#350	1	2	yes	—
104	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₈₉	—	2	4	no	—
106	C₄×C₈₉	—	2	1	no	—
107	C₄×C₈₉	—	2	1	no	—
108	C₄×C₈₉	—	4	2	no	—
109	C₄×C₈₉	Q₈⋊C₈	1	2	yes	—
110	C₄×C₈₉	C₂²⋊C₁₆	1	2	yes	—
111	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
112	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
113	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
114	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
115	C₂×Q₈⋊C₁₆	—	89	1	no	maximal, Sylow
116	C₂×C₈₉	Q₈⋊C₁₆	1	4	yes	complemented
117	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
118	C₂×C₈₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₈₉	C₂×#350	1	1	yes	characteristic
120	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
121	C₂×C₄⋊C₁₆	—	89	1	no	—
122	C₂×C₄×C₁₆	—	89	1	no	—
123	Q₈⋊C₁₆	—	89	4	no	—
124	C₈₉	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄×Q₈	Dic₈₉	1	1	yes	characteristic
126	C₂²×C₁₆	—	178	1	no	—
127	C₂²×C₁₆	—	178	1	no	—
128	C₈×Q₈	—	2	1	no	—
129	C₈×Q₈	—	2	1	no	—
130	C₈×Q₈	Dic₈₉	1	2	yes	—
131	C₈×Q₈	C₂×D₈₉	1	2	yes	—
132	C₂×C₄⋊C₈	—	2	1	no	—
133	C₂×C₄⋊C₈	Dic₈₉	1	1	yes	characteristic
134	C₂×C₄×C₈	—	2	1	no	—
135	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
136	C₄⋊C₁₆	—	89	2	no	—
137	C₄⋊C₁₆	—	178	1	no	—
138	C₄×C₁₆	—	89	2	no	—
139	C₄×C₁₆	—	178	1	no	—
140	C₂²×Q₈	#10623494	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	#10623494	1	2	yes	—
147	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	#10623494	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
152	C₂×C₁₆	—	356	2	no	—
153	C₂×C₁₆	—	178	2	no	—
154	C₂×C₁₆	—	178	4	no	—
155	C₂×C₁₆	—	178	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	#13083549	1	4	yes	complemented
169	C₂×Q₈	C₂×#10623494	1	2	yes	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
174	C₂²×C₄	#13083558	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
180	C₂×C₈	#13083561	1	2	yes	—
181	C₂×C₈	#13083564	1	2	yes	—
182	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	C₂×#10623494	1	2	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
191	C₄²	C₂×#10623494	1	1	yes	characteristic
192	C₄²	#13083558	1	1	yes	characteristic
193	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
194	C₁₆	—	356	2	no	—
195	C₁₆	—	178	4	no	—
196	C₂³	C₂²⋊#10623494	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	2	2	no	—
199	Q₈	C₂×#13083549	1	4	yes	complemented
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
206	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
207	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
208	C₂×C₄	#15667936	1	2	yes	—
209	C₂×C₄	#15667939	1	2	yes	—
210	C₂×C₄	C₂×#13083549	1	2	yes	—
211	C₂×C₄	#15667952	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
213	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
214	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	4	2	no	—
218	C₈	#15667936	1	2	yes	—
219	C₈	C₂×#13083561	1	1	yes	characteristic
220	C₈	C₂×#13083564	1	1	yes	characteristic
221	C₂²	Q₈⋊#10623494	1	2	yes	—
222	C₂²	C₂²⋊#13083549	1	2	yes	—
223	C₂²	#19151382	1	2	yes	—
224	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	2	no	—
229	C₄	Q₈⋊#10623494	1	2	yes	—
230	C₄	C₂²⋊#13083549	1	2	yes	—
231	C₄	C₂×#15667936	1	1	yes	characteristic
232	C₄	C₂×#15667939	1	1	yes	characteristic
233	C₄	C₂²×#13083549	1	1	yes	characteristic
234	C₄	C₂×#15667952	1	1	yes	characteristic
235	C₂	Q₈⋊#13083549	1	4	yes	direct product factor
236	C₂	C₂×Q₈⋊#10623494	1	1	yes	characteristic
237	C₂	C₂×C₂²⋊#13083549	1	1	yes	characteristic
238	C₂	C₂×#19151382	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#13083549	C₂	direct (N × H)
50. C₂×Q₈×C₈₉	C₁₆	semidirect
79. Q₈×C₈₉	C₂×C₁₆	semidirect
116. C₂×C₈₉	Q₈⋊C₁₆	semidirect
124. C₈₉	C₂×Q₈⋊C₁₆	semidirect
168. C₂×Q₈	#13083549	semidirect
199. Q₈	C₂×#13083549	semidirect
235. C₂	Q₈⋊#13083549	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×C₄⋊#13083549	2
C₂×C₄×#13083549	2
Q₈⋊#13083549	2
C₂×C₈×Q₈×C₈₉	2
C₂×Q₈⋊C₁₆	89

Subgroups — C2×Q8⋊#13083549 #36464227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈⋊#13083549 #36464227