Subgroups — C₃×D₉×C₄⋊Q₈ #13153293

Order 1728.

938 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle	C₂²×C₃²	C₂³×D₃
prime radical	C₂²×C₃×D₉	C₂³
hypercenter	C₃×C₄⋊Q₈	D₉
hypocenter	C₉	C₂×C₃×C₄⋊Q₈
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×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. C₃×#63506	C₂	semidirect
5. C₃×#63450	C₂	semidirect
7. C₃×#63431	C₂	semidirect
9. D₉×C₄⋊Q₈	C₃	direct (N × H)
32. C₉×C₄⋊Q₈	C₂×C₃	semidirect
33. #63506	C₂×C₃	semidirect
35. #63450	C₂×C₃	semidirect
37. #63431	C₂×C₃	semidirect
48. C₃×C₄×D₉	Q₈	semidirect
87. C₃×C₄×C₉	C₂×Q₈	semidirect
90. C₃×Dic₉	C₂×Q₈	semidirect
97. C₃×C₄⋊Q₈	D₉	direct (N × H)
133. C₄×D₉	C₃×Q₈	semidirect
139. C₃×D₉	C₄⋊Q₈	direct (N × H)
187. C₄×C₉	C₂×C₃×Q₈	semidirect
190. Dic₉	C₂×C₃×Q₈	semidirect
195. C₄⋊Q₈	C₃×D₉	direct (N × H)
198. C₃×C₉	C₂×C₄⋊Q₈	semidirect
236. D₉	C₃×C₄⋊Q₈	direct (N × H)
266. C₃×C₄	Q₈×D₉	semidirect
274. C₉	C₂×C₃×C₄⋊Q₈	semidirect
304. C₄	C₃×Q₈×D₉	semidirect
306. C₃	D₉×C₄⋊Q₈	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×C₃×Q₈×D₉	2
C₃×C₉×C₄⋊Q₈	2
C₃×#63506	2
C₃×C₄⋊C₄×D₉	2
C₃×#63450	2
C₃×C₄²×D₉	2
C₃×#63431	2
C₃×D₃×C₄⋊Q₈	3
D₉×C₄⋊Q₈	3

Subgroups — C3×D9×C4⋊Q8 #13153293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₉×C₄⋊Q₈ #13153293