Subgroups — C₄×#88425 #13137254

Order 1728.

410 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88425	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720297	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720266	C₂	1	1	yes	maximal, characteristic
7	#11720265	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#3715	—	3	1	no	maximal
9	C₄×#3675	—	3	1	no	maximal
10	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×D₉	—	2	1	no	—
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃×(C₂×D₉)⋊C₄	—	2	1	no	—
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×Dic₃⋊C₄	—	2	1	no	—
19	C₄×C₉×Dic₃	—	2	1	no	—
20	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
21	#88425	C₄	1	4	yes	direct product factor
22	C₃×C₄×D₁₂	—	3	1	no	—
23	C₃×C₄×Dic₆	D₃	1	1	yes	characteristic
24	C₂×#3715	—	3	1	no	—
25	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
26	#63574	—	3	1	no	—
27	#63565	—	3	1	no	—
28	#63562	—	3	1	no	—
29	C₄×Q₈×C₉	—	3	1	no	—
30	C₂×#3675	—	3	1	no	—
31	C₄×D₃₆	D₃	1	1	yes	characteristic
32	#63395	—	3	1	no	—
33	#63369	—	3	1	no	—
34	#63367	—	3	1	no	—
35	C₄×C₉⋊C₈	—	3	1	no	—
36	C₄×C₉⋊C₈	—	6	1	no	—
37	C₂²×C₃×D₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₂×C₉×Dic₃	—	2	1	no	—
43	C₂×C₃×Dic₉	—	2	1	no	—
44	C₃×D₃₆	—	2	1	no	—
45	C₃×D₃₆	C₂×C₄	1	2	yes	—
46	C₃×C₄×D₉	—	4	2	no	—
47	C₉×Dic₆	—	2	1	no	—
48	C₉×Dic₆	C₂×C₄	1	2	yes	complemented
49	C₃⋊(C₉⋊C₈)	—	2	1	no	—
50	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	complemented
51	C₄×#215	—	9	1	no	—
52	C₄×#214	—	9	1	no	—
53	C₂×C₃×D₁₂	—	3	1	no	—
54	C₂×C₃×C₄×D₃	—	6	1	no	—
55	C₂×C₃×Dic₆	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
59	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
60	C₃×C₄⋊Dic₃	—	3	1	no	—
61	C₃×C₄⋊Dic₃	C₂×D₃	1	1	yes	characteristic
62	C₃×Dic₃⋊C₄	—	2	1	no	—
63	C₃×C₄×Dic₃	—	2	1	no	—
64	#3715	—	3	4	no	—
65	C₂×Q₈×C₉	—	3	1	no	—
66	C₂×D₃₆	C₂×D₃	1	1	yes	characteristic
67	C₂×C₄×D₉	—	2	1	no	—
68	C₉×C₄⋊C₄	—	3	1	no	—
69	C₉×C₄⋊C₄	—	6	1	no	—
70	C₄²×C₉	—	2	1	no	—
71	C₄²×C₉	—	6	1	no	—
72	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
73	#3675	—	3	4	no	—
74	(C₂×D₉)⋊C₄	—	2	1	no	—
75	C₄⋊Dic₉	C₂×D₃	1	1	yes	characteristic
76	C₂×C₉⋊C₈	—	3	1	no	—
77	C₂×C₉⋊C₈	—	3	1	no	—
78	C₂×C₉⋊C₈	—	6	1	no	—
79	C₂×C₉⋊C₈	—	6	1	no	—
80	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×D₉	—	2	2	no	—
82	C₂×C₃×D₉	—	4	1	no	—
83	C₂×C₃×D₉	—	4	1	no	—
84	C₃×C₄×C₉	—	2	1	no	—
85	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
86	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₄×C₉	#41	1	1	yes	characteristic
88	C₉×Dic₃	—	2	2	no	—
89	C₉×Dic₃	—	4	1	no	—
90	C₃×Dic₉	—	4	1	no	—
91	C₃×C₄×Q₈	—	3	1	no	—
92	C₃×C₄×D₄	—	9	1	no	—
93	C₂×#215	—	9	1	no	—
94	C₂×#214	—	9	1	no	—
95	C₄×D₁₂	—	3	1	no	—
96	C₄×Dic₆	D₉	1	1	yes	characteristic, complemented
97	#835	—	9	1	no	—
98	#832	—	9	1	no	—
99	#810	—	9	1	no	—
100	#809	—	9	1	no	—
101	#808	—	9	1	no	—
102	#808	—	9	1	no	—
103	C₄×C₃⋊C₈	—	9	1	no	—
104	C₄×C₃⋊C₈	—	9	1	no	—
105	C₄×C₃⋊C₈	—	18	1	no	—
106	C₂²×C₃×D₃	—	6	1	no	—
107	C₂×C₃²×C₄	#67	1	1	yes	characteristic
108	C₂×C₃²×C₄	#67	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₃×Dic₃	—	2	1	no	—
111	C₂×C₃×Dic₃	—	2	1	no	—
112	C₂×C₃×Dic₃	—	6	1	no	—
113	C₃×D₁₂	—	3	2	no	—
114	C₃×D₁₂	—	6	1	no	—
115	C₃×C₄×D₃	—	12	2	no	—
116	C₃×Dic₆	—	2	1	no	—
117	C₃×Dic₆	C₄×D₃	1	2	yes	—
118	C₂²×D₉	—	2	1	no	—
119	C₃⋊(C₃⋊C₈)	—	3	2	no	—
120	C₃⋊(C₃⋊C₈)	—	6	1	no	—
121	Q₈×C₉	—	3	2	no	—
122	Q₈×C₉	—	6	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	6	1	no	—
127	C₂×C₄×C₉	—	6	1	no	—
128	C₂×C₄×C₉	#67	1	1	yes	characteristic
129	C₂×C₄×C₉	#67	1	1	yes	characteristic
130	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
131	C₂×Dic₉	—	2	1	no	—
132	D₃₆	—	2	1	no	—
133	D₃₆	C₄×D₃	1	2	yes	—
134	C₄×D₉	—	4	2	no	—
135	C₉⋊C₈	—	3	2	no	—
136	C₉⋊C₈	—	6	2	no	—
137	C₉⋊C₈	—	6	2	no	—
138	C₉⋊C₈	—	6	1	no	—
139	C₄×SD₁₆	—	27	1	no	Sylow
140	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₃×C₉	#135	1	1	yes	characteristic
143	C₃×D₉	—	4	2	no	—
144	C₂×C₃×Q₈	—	3	1	no	—
145	C₂×C₃×D₄	—	9	1	no	—
146	C₂²×C₃×C₄	—	18	1	no	—
147	C₂×D₁₂	—	3	1	no	—
148	C₂×C₄×D₃	—	6	1	no	—
149	C₂×Dic₆	C₂×D₉	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	3	1	no	—
151	C₃×C₄⋊C₄	—	6	1	no	—
152	C₃×C₄⋊C₄	—	9	1	no	—
153	C₃×C₂²⋊C₄	—	18	1	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	6	1	no	—
156	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
157	C₃×C₄²	D₃²	1	1	yes	characteristic
158	#215	—	9	4	no	—
159	#214	—	9	4	no	—
160	(C₂×D₃)⋊C₄	—	6	1	no	—
161	C₄⋊Dic₃	—	3	1	no	—
162	C₄⋊Dic₃	C₂×D₉	1	1	yes	characteristic
163	Dic₃⋊C₄	—	2	1	no	—
164	C₄×Dic₃	—	2	1	no	—
165	C₂×C₃⋊C₈	—	9	1	no	—
166	C₂×C₃⋊C₈	—	9	1	no	—
167	C₂×C₃⋊C₈	—	9	1	no	—
168	C₂×C₃⋊C₈	—	9	1	no	—
169	C₂×C₃⋊C₈	—	18	1	no	—
170	C₂×C₃⋊C₈	—	18	1	no	—
171	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
172	C₂×C₃×D₃	—	6	2	no	—
173	C₂×C₃×D₃	—	12	1	no	—
174	C₂×C₃×D₃	—	12	1	no	—
175	C₃²×C₄	—	2	1	no	—
176	C₃²×C₄	#215	1	2	yes	—
177	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₃²×C₄	#235	1	1	yes	characteristic
179	C₃×Dic₃	—	2	2	no	—
180	C₃×Dic₃	—	4	1	no	—
181	C₃×Dic₃	—	12	1	no	—
182	C₂²×C₉	—	2	1	no	—
183	C₂²×C₉	C₂×#67	1	1	yes	characteristic
184	C₂×D₉	—	2	2	no	—
185	C₂×D₉	—	4	1	no	—
186	C₂×D₉	—	4	1	no	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	2	1	no	—
189	C₄×C₉	—	2	2	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	6	2	no	—
193	C₄×C₉	—	12	1	no	—
194	C₄×C₉	#214	1	2	yes	complemented
195	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
196	C₄×C₉	#235	1	1	yes	characteristic
197	Dic₉	—	4	1	no	—
198	C₂×SD₁₆	—	27	1	no	—
199	C₄×Q₈	—	3	1	no	—
200	C₄×D₄	—	9	1	no	—
201	#106	—	27	1	no	—
202	#103	—	27	1	no	—
203	#102	—	27	1	no	—
204	C₄×C₈	—	27	1	no	—
205	C₃×C₉	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	18	1	no	—
207	C₂²×D₃	—	6	1	no	—
208	C₃×Q₈	—	3	2	no	—
209	C₃×Q₈	—	6	1	no	—
210	C₃×D₄	—	9	2	no	—
211	C₃×D₄	—	18	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	6	1	no	—
216	C₂×C₃×C₄	—	6	1	no	—
217	C₂×C₃×C₄	—	18	1	no	—
218	C₂×C₃×C₄	—	36	2	no	—
219	C₂×C₃×C₄	#611	1	1	yes	characteristic
220	C₂×C₃×C₄	#611	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
222	C₂×C₃×C₄	#625	1	1	yes	characteristic
223	C₂×C₃×C₄	#625	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
225	C₂×Dic₃	—	2	1	no	—
226	C₂×Dic₃	—	2	1	no	—
227	C₂×Dic₃	—	6	1	no	—
228	D₁₂	—	3	2	no	—
229	D₁₂	—	6	1	no	—
230	C₄×D₃	—	12	2	no	—
231	Dic₆	—	2	1	no	—
232	Dic₆	C₄×D₉	1	2	yes	complemented
233	C₃⋊C₈	—	9	2	no	—
234	C₃⋊C₈	—	9	2	no	—
235	C₃⋊C₈	—	18	2	no	—
236	C₃⋊C₈	—	18	2	no	—
237	C₃⋊C₈	—	18	1	no	—
238	C₃⋊C₈	—	18	1	no	—
239	C₂×C₃²	C₄×#67	1	1	yes	characteristic
240	C₂×C₃²	C₂×#215	1	1	yes	characteristic
241	C₂×C₃²	#950	1	1	yes	characteristic
242	C₃×D₃	—	12	2	no	—
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	C₄×#67	1	1	yes	characteristic
247	C₂×C₉	C₂×#214	1	1	yes	characteristic
248	C₂×C₉	#950	1	1	yes	characteristic
249	D₉	—	4	2	no	—
250	C₂×Q₈	—	3	1	no	—
251	C₂×D₄	—	9	1	no	—
252	C₂²×C₄	—	18	1	no	—
253	SD₁₆	—	27	4	no	—
254	C₂×C₈	—	27	1	no	—
255	C₂×C₈	—	27	1	no	—
256	C₄⋊C₄	—	3	1	no	—
257	C₄⋊C₄	—	6	1	no	—
258	C₄⋊C₄	—	9	1	no	—
259	C₂²⋊C₄	—	18	1	no	—
260	C₄²	—	6	1	no	—
261	C₄²	D₃×D₉	1	1	yes	characteristic
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	18	2	no	—
264	C₂²×C₃	—	36	1	no	—
265	C₂²×C₃	—	36	1	no	—
266	C₂²×C₃	C₂×#611	1	1	yes	characteristic
267	C₂²×C₃	C₂×#625	1	1	yes	characteristic
268	C₂×D₃	—	6	2	no	—
269	C₂×D₃	—	12	1	no	—
270	C₂×D₃	—	12	1	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	6	2	no	—
278	C₃×C₄	—	12	1	no	—
279	C₃×C₄	—	36	1	no	—
280	C₃×C₄	#3675	1	2	yes	complemented
281	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
282	C₃×C₄	#3697	1	1	yes	characteristic
283	C₃×C₄	#3715	1	2	yes	—
284	C₃×C₄	#3798	1	1	yes	characteristic
285	C₃×C₄	C₄×D₃²	1	1	yes	characteristic
286	Dic₃	—	2	2	no	—
287	Dic₃	—	4	1	no	—
288	Dic₃	—	12	1	no	—
289	C₃²	C₄×#215	1	1	yes	characteristic
290	C₉	—	2	1	no	—
291	C₉	C₄×#214	1	1	yes	characteristic, complemented
292	C₂³	—	18	1	no	—
293	Q₈	—	3	2	no	—
294	Q₈	—	6	1	no	—
295	D₄	—	9	2	no	—
296	D₄	—	18	1	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	18	1	no	—
300	C₂×C₄	—	36	2	no	—
301	C₂×C₄	#6194	1	1	yes	characteristic
302	C₂×C₄	#6194	1	1	yes	characteristic
303	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
304	C₈	—	27	2	no	—
305	C₈	—	54	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	36	2	no	—
310	C₂×C₃	C₄×#611	1	1	yes	characteristic
311	C₂×C₃	C₂×#3675	1	1	yes	characteristic
312	C₂×C₃	#63513	1	1	yes	characteristic
313	C₂×C₃	#63822	1	1	yes	characteristic
314	C₂×C₃	C₂×#3715	1	1	yes	characteristic
315	C₂×C₃	C₄×#625	1	1	yes	characteristic
316	D₃	—	12	2	no	—
317	C₂²	—	18	2	no	—
318	C₂²	—	36	1	no	—
319	C₂²	—	36	1	no	—
320	C₂²	C₂×#6194	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	6	2	no	—
323	C₄	—	12	1	no	—
324	C₄	—	36	1	no	—
325	C₄	#88425	1	2	yes	direct product factor
326	C₄	#88640	1	1	yes	characteristic
327	C₄	C₄×D₃×D₉	1	1	yes	characteristic
328	C₃	—	2	1	no	—
329	C₃	C₄×#3675	1	1	yes	characteristic, complemented
330	C₃	C₄×#3715	1	1	yes	characteristic
331	C₂	—	36	2	no	—
332	C₂	#11720881	1	1	yes	characteristic
333	C₂	C₂×#88425	1	1	yes	characteristic
334	C₂	C₄×#6194	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6194
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₃×D₃₆	C₂²
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	C₄×SD₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×C₉×Dic₆	C₂	semidirect
4. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
5. #11720297	C₂	semidirect
7. #11720265	C₂	semidirect
21. #88425	C₄	direct (N × H)
48. C₉×Dic₆	C₂×C₄	semidirect
50. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
85. C₃×C₄×C₉	SD₁₆	semidirect
96. C₄×Dic₆	D₉	semidirect
194. C₄×C₉	#214	semidirect
205. C₃×C₉	C₄×SD₁₆	semidirect
232. Dic₆	C₄×D₉	semidirect
280. C₃×C₄	#3675	semidirect
291. C₉	C₄×#214	semidirect
325. C₄	#88425	direct (N × H)
329. C₃	C₄×#3675	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₃₆	2
C₄×C₉×Dic₆	2
C₂×#88425	2
C₄×C₃⋊(C₉⋊C₈)	2
#11720297	2
#11720266	2
#11720265	2
C₄×#3715	3
C₄×#3675	3

Subgroups — C4×#88425 #13137254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88425 #13137254