Subgroups — C₄⋊Q₈×C₃⋊D₉ #13154278

Order 1728.

1450 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×C₃⋊D₉	C₂	1	2	yes	maximal
2	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11721854	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₄×C₃⋊D₉	C₂	1	4	yes	maximal
5	#11721744	C₂	1	4	yes	maximal, complemented
6	C₄²×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
7	#11721706	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃⋊D₃×C₄⋊Q₈	—	3	1	no	maximal
9	D₉×C₄⋊Q₈	—	3	3	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
11	Q₈×C₃⋊D₉	—	2	8	no	—
12	C₂×C₄×C₃⋊D₉	C₂²	1	4	yes	—
13	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃⋊D₉	C₂²	1	2	yes	—
15	C₂×#6226	C₂²	1	4	yes	—
16	C₂×#6226	C₂²	1	2	yes	—
17	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
18	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊Dic₉)	C₂²	1	4	yes	—
20	#88534	C₂²	1	8	yes	—
21	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
22	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₃⋊D₃	—	3	2	no	—
24	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
25	#64153	—	3	2	no	—
26	C₄⋊C₄×C₃⋊D₃	—	3	4	no	—
27	#64097	—	3	4	no	—
28	C₄²×C₃⋊D₃	—	3	1	no	—
29	#64078	—	3	1	no	—
30	C₂×Q₈×D₉	—	3	6	no	—
31	C₉×C₄⋊Q₈	D₃	1	3	yes	complemented
32	#63506	—	3	6	no	—
33	C₄⋊C₄×D₉	—	3	12	no	—
34	#63450	—	3	12	no	—
35	C₄²×D₉	—	3	3	no	—
36	#63431	—	3	3	no	—
37	C₂²×C₃⋊D₉	C₂³	1	1	yes	characteristic
38	C₃×Q₈×C₉	—	2	4	no	—
39	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
42	C₂×C₃⋊Dic₉	C₂³	1	4	yes	—
43	C₂×C₃⋊Dic₉	C₂³	1	1	yes	characteristic
44	C₂×C₃⋊Dic₉	C₂³	1	2	yes	—
45	C₄×C₃⋊D₉	—	2	8	no	—
46	C₄×C₃⋊D₉	D₄	1	4	yes	—
47	C₄×C₃⋊D₉	Q₈	1	8	yes	complemented
48	#6226	—	2	4	no	—
49	#6226	—	2	8	no	—
50	D₃×C₄⋊Q₈	—	9	3	no	—
51	D₃×C₄⋊Q₈	—	9	1	no	—
52	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
53	Q₈×C₃⋊D₃	—	6	8	no	—
54	C₂×C₄×C₃⋊D₃	—	3	2	no	—
55	C₂×C₄×C₃⋊D₃	—	3	1	no	—
56	C₂×C₄×C₃⋊D₃	—	3	4	no	—
57	C₂×#634	—	3	2	no	—
58	C₂×#634	—	3	4	no	—
59	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
60	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
61	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
62	#3750	—	3	8	no	—
63	C₄×C₃⋊Dic₃	—	3	2	no	—
64	C₄×C₃⋊Dic₃	—	3	1	no	—
65	C₂×Q₈×C₉	C₂×D₃	1	6	yes	—
66	Q₈×D₉	—	6	24	no	—
67	C₂×C₄×D₉	—	3	6	no	—
68	C₂×C₄×D₉	—	3	3	no	—
69	C₂×C₄×D₉	—	3	12	no	—
70	C₂×Dic₁₈	—	3	6	no	—
71	C₂×Dic₁₈	—	3	12	no	—
72	C₉×C₄⋊C₄	C₂×D₃	1	12	yes	—
73	C₄²×C₉	C₂×D₃	1	3	yes	—
74	C₄⋊Dic₉	—	3	12	no	—
75	Dic₉⋊C₄	—	3	24	no	—
76	C₄×Dic₉	—	3	3	no	—
77	C₄×Dic₉	—	3	6	no	—
78	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
79	C₂×C₃⋊D₉	C₂×D₄	1	2	yes	—
80	C₂×C₃⋊D₉	C₂×Q₈	1	4	yes	—
81	C₃×C₄×C₉	—	2	4	no	—
82	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₃×C₄×C₉	C₂×Q₈	1	4	yes	complemented
84	C₃⋊Dic₉	—	2	4	no	—
85	C₃⋊Dic₉	C₂×D₄	1	2	yes	—
86	C₃⋊Dic₉	C₂×Q₈	1	4	yes	complemented
87	C₂×D₃×Q₈	—	9	6	no	—
88	C₂×D₃×Q₈	—	9	2	no	—
89	C₃×C₄⋊Q₈	D₉	1	3	yes	complemented
90	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
91	#944	—	9	6	no	—
92	#944	—	9	2	no	—
93	D₃×C₄⋊C₄	—	9	12	no	—
94	D₃×C₄⋊C₄	—	9	4	no	—
95	#888	—	9	12	no	—
96	#888	—	9	4	no	—
97	C₄²×D₃	—	9	3	no	—
98	C₄²×D₃	—	9	1	no	—
99	#869	—	9	3	no	—
100	#869	—	9	1	no	—
101	C₂²×C₃⋊D₃	—	3	1	no	—
102	C₃²×Q₈	—	2	4	no	—
103	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
106	C₂×C₃⋊Dic₃	—	3	2	no	—
107	C₂×C₃⋊Dic₃	—	3	1	no	—
108	C₂×C₃⋊Dic₃	—	3	4	no	—
109	C₄×C₃⋊D₃	—	3	4	no	—
110	C₄×C₃⋊D₃	—	3	8	no	—
111	C₄×C₃⋊D₃	—	6	8	no	—
112	#634	—	6	8	no	—
113	#634	—	6	4	no	—
114	C₂²×D₉	—	3	3	no	—
115	Q₈×C₉	—	2	12	no	—
116	C₂×C₄×C₉	C₂²×D₃	1	12	yes	—
117	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
118	C₂×C₄×C₉	C₂²×D₃	1	6	yes	—
119	C₂×Dic₉	—	3	6	no	—
120	C₂×Dic₉	—	3	3	no	—
121	C₂×Dic₉	—	3	12	no	—
122	C₄×D₉	—	3	12	no	—
123	C₄×D₉	—	3	24	no	—
124	C₄×D₉	—	6	24	no	—
125	Dic₁₈	—	6	24	no	—
126	Dic₁₈	—	6	12	no	—
127	C₂×C₄⋊Q₈	—	27	1	no	Sylow
128	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₃×C₉	C₂²×Q₈	1	2	yes	—
130	C₃⋊D₉	C₄⋊Q₈	1	4	yes	direct product factor
131	C₂×C₃×Q₈	C₂×D₉	1	6	yes	—
132	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
133	D₃×Q₈	—	18	8	no	—
134	D₃×Q₈	—	18	24	no	—
135	C₂×C₄×D₃	—	9	6	no	—
136	C₂×C₄×D₃	—	9	3	no	—
137	C₂×C₄×D₃	—	9	2	no	—
138	C₂×C₄×D₃	—	9	1	no	—
139	C₂×C₄×D₃	—	9	12	no	—
140	C₂×C₄×D₃	—	9	4	no	—
141	C₂×Dic₆	—	9	6	no	—
142	C₂×Dic₆	—	9	12	no	—
143	C₂×Dic₆	—	9	2	no	—
144	C₂×Dic₆	—	9	4	no	—
145	C₃×C₄⋊C₄	C₂×D₉	1	12	yes	—
146	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
147	C₃×C₄²	C₂×D₉	1	3	yes	—
148	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
149	C₄⋊Dic₃	—	9	12	no	—
150	C₄⋊Dic₃	—	9	4	no	—
151	Dic₃⋊C₄	—	9	24	no	—
152	Dic₃⋊C₄	—	9	8	no	—
153	C₄×Dic₃	—	9	6	no	—
154	C₄×Dic₃	—	9	3	no	—
155	C₄×Dic₃	—	9	2	no	—
156	C₄×Dic₃	—	9	1	no	—
157	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
158	C₂×C₃⋊D₃	—	3	2	no	—
159	C₂×C₃⋊D₃	—	3	4	no	—
160	C₃²×C₄	—	2	4	no	—
161	C₃²×C₄	D₃×D₄	1	2	yes	—
162	C₃²×C₄	D₃×Q₈	1	4	yes	—
163	C₃⋊Dic₃	—	3	2	no	—
164	C₃⋊Dic₃	—	3	4	no	—
165	C₃⋊Dic₃	—	6	4	no	—
166	C₂²×C₉	C₂³×D₃	1	3	yes	—
167	C₂×D₉	—	3	6	no	—
168	C₂×D₉	—	3	12	no	—
169	C₄×C₉	—	2	12	no	—
170	C₄×C₉	D₃×D₄	1	6	yes	—
171	C₄×C₉	D₃×Q₈	1	12	yes	complemented
172	Dic₉	—	3	6	no	—
173	Dic₉	—	3	12	no	—
174	Dic₉	—	6	12	no	—
175	C₂²×Q₈	—	27	2	no	—
176	C₄⋊Q₈	—	27	2	no	—
177	C₄⋊Q₈	—	27	4	no	—
178	C₄⋊Q₈	—	27	1	no	—
179	C₄⋊Q₈	C₃⋊D₉	1	1	yes	characteristic, direct product factor
180	C₂×C₄⋊C₄	—	27	4	no	—
181	C₂×C₄²	—	27	1	no	—
182	C₃×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₃	—	9	1	no	—
184	C₂²×D₃	—	9	3	no	—
185	C₃×Q₈	—	2	12	no	—
186	C₃×Q₈	—	2	4	no	—
187	C₂×C₃×C₄	C₂²×D₉	1	12	yes	—
188	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
189	C₂×C₃×C₄	C₂²×D₉	1	6	yes	—
190	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
191	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
193	C₂×Dic₃	—	9	2	no	—
194	C₂×Dic₃	—	9	1	no	—
195	C₂×Dic₃	—	9	6	no	—
196	C₂×Dic₃	—	9	3	no	—
197	C₂×Dic₃	—	9	4	no	—
198	C₂×Dic₃	—	9	12	no	—
199	C₄×D₃	—	9	12	no	—
200	C₄×D₃	—	9	24	no	—
201	C₄×D₃	—	9	4	no	—
202	C₄×D₃	—	9	8	no	—
203	C₄×D₃	—	18	8	no	—
204	C₄×D₃	—	18	24	no	—
205	Dic₆	—	18	8	no	—
206	Dic₆	—	18	24	no	—
207	Dic₆	—	18	12	no	—
208	Dic₆	—	18	4	no	—
209	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
210	C₂×C₃²	C₂×D₃×Q₈	1	2	yes	—
211	C₃⋊D₃	—	3	4	no	—
212	C₂×C₉	C₂×D₃×D₄	1	3	yes	—
213	C₂×C₉	C₂×D₃×Q₈	1	6	yes	—
214	D₉	—	3	12	no	—
215	C₂×Q₈	—	27	4	no	—
216	C₂×Q₈	—	27	2	no	—
217	C₂×Q₈	—	54	8	no	—
218	C₂×Q₈	C₂×C₃⋊D₉	1	2	yes	—
219	C₂²×C₄	—	27	2	no	—
220	C₂²×C₄	—	27	1	no	—
221	C₂²×C₄	—	27	4	no	—
222	C₄⋊C₄	—	27	8	no	—
223	C₄⋊C₄	—	27	4	no	—
224	C₄⋊C₄	C₂×C₃⋊D₉	1	4	yes	—
225	C₄²	—	27	1	no	—
226	C₄²	—	27	2	no	—
227	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
228	C₂²×C₃	C₂³×D₉	1	3	yes	—
229	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
230	C₂×D₃	—	9	2	no	—
231	C₂×D₃	—	9	4	no	—
232	C₂×D₃	—	9	6	no	—
233	C₂×D₃	—	9	12	no	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	2	12	no	—
236	C₃×C₄	D₄×D₉	1	6	yes	—
237	C₃×C₄	Q₈×D₉	1	12	yes	complemented
238	C₃×C₄	D₄×C₃⋊D₃	1	2	yes	—
239	C₃×C₄	Q₈×C₃⋊D₃	1	4	yes	—
240	Dic₃	—	9	2	no	—
241	Dic₃	—	9	4	no	—
242	Dic₃	—	9	6	no	—
243	Dic₃	—	9	12	no	—
244	Dic₃	—	18	4	no	—
245	Dic₃	—	18	12	no	—
246	C₃²	D₃×C₄⋊Q₈	1	1	yes	characteristic
247	C₉	D₃×C₄⋊Q₈	1	3	yes	complemented
248	C₂³	—	27	1	no	—
249	Q₈	—	2	4	no	—
250	Q₈	—	54	8	no	—
251	Q₈	—	54	4	no	—
252	C₂×C₄	—	27	2	no	—
253	C₂×C₄	—	27	1	no	—
254	C₂×C₄	—	27	4	no	—
255	C₂×C₄	—	27	8	no	—
256	C₂×C₄	—	27	4	no	—
257	C₂×C₄	—	54	8	no	—
258	C₂×C₄	C₂²×C₃⋊D₉	1	4	yes	—
259	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₃⋊D₉	1	2	yes	—
261	C₂×C₃	C₂×D₄×D₉	1	3	yes	—
262	C₂×C₃	C₂×Q₈×D₉	1	6	yes	—
263	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
264	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	2	yes	—
265	D₃	—	9	4	no	—
266	D₃	—	9	12	no	—
267	C₂²	—	27	2	no	—
268	C₂²	—	27	4	no	—
269	C₂²	C₂³×C₃⋊D₉	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	27	2	no	—
272	C₄	—	27	4	no	—
273	C₄	—	54	4	no	—
274	C₄	D₄×C₃⋊D₉	1	2	yes	—
275	C₄	Q₈×C₃⋊D₉	1	4	yes	complemented
276	C₃	D₉×C₄⋊Q₈	1	3	yes	complemented
277	C₃	C₃⋊D₃×C₄⋊Q₈	1	1	yes	characteristic
278	C₂	—	27	4	no	—
279	C₂	C₂×D₄×C₃⋊D₉	1	1	yes	characteristic
280	C₂	C₂×Q₈×C₃⋊D₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×C₃⋊D₉
commutator, cosocle	C₂²×C₃×C₉	C₂⁴
Frattini	C₂²×C₃	C₂³×C₃⋊D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle	C₂²×C₃²	C₂³×D₃
prime radical	C₂²×C₃⋊D₉	C₂³
hypercenter	C₄⋊Q₈	C₃⋊D₉
hypocenter	C₃×C₉	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₄⋊Q₈	C₂	semidirect
3. #11721854	C₂	semidirect
5. #11721744	C₂	semidirect
7. #11721706	C₂	semidirect
31. C₉×C₄⋊Q₈	D₃	semidirect
47. C₄×C₃⋊D₉	Q₈	semidirect
83. C₃×C₄×C₉	C₂×Q₈	semidirect
86. C₃⋊Dic₉	C₂×Q₈	semidirect
89. C₃×C₄⋊Q₈	D₉	semidirect
130. C₃⋊D₉	C₄⋊Q₈	direct (N × H)
171. C₄×C₉	D₃×Q₈	semidirect
179. C₄⋊Q₈	C₃⋊D₉	direct (N × H)
182. C₃×C₉	C₂×C₄⋊Q₈	semidirect
237. C₃×C₄	Q₈×D₉	semidirect
247. C₉	D₃×C₄⋊Q₈	semidirect
275. C₄	Q₈×C₃⋊D₉	semidirect
276. C₃	D₉×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₃⋊D₉	2
C₃×C₉×C₄⋊Q₈	2
#11721854	2
C₄⋊C₄×C₃⋊D₉	2
#11721744	2
C₄²×C₃⋊D₉	2
#11721706	2
C₃⋊D₃×C₄⋊Q₈	3
D₉×C₄⋊Q₈	3

Subgroups — C4⋊Q8×C3⋊D9 #13154278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊Q₈×C₃⋊D₉ #13154278