Subgroups — C₄×#89078 #13179673

Order 1728.

4340 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×#89078	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6312	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal, complemented
6	#11723933	C₂	1	1	yes	maximal, characteristic, complemented
7	#11723928	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723922	C₂	1	2	yes	maximal, complemented
9	#11723918	C₂	1	2	yes	maximal, complemented
10	C₄²×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6308	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#3830	—	3	13	no	maximal
13	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃³×C₄	C₂²	1	2	yes	—
15	C₂×#6312	C₂²	1	2	yes	—
16	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
17	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
18	#89078	C₄	1	8	yes	direct product factor
19	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
22	C₂×#6308	C₂²	1	1	yes	characteristic
23	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃³×C₄²	C₂²	1	1	yes	characteristic
26	#88864	C₂²	1	2	yes	—
27	#88859	C₂²	1	2	yes	—
28	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
29	#88857	C₂²	1	2	yes	—
30	#88857	C₂²	1	2	yes	—
31	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
32	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
33	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
34	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
36	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
37	C₂×#3830	—	3	13	no	—
38	C₃²×C₄×D₄	D₃	1	13	yes	complemented
39	D₄×C₃⋊Dic₃	—	3	13	no	—
40	C₄×#638	—	3	26	no	—
41	C₂×C₄×C₃⋊Dic₃	—	3	26	no	—
42	#64101	—	3	13	no	—
43	#64096	—	3	13	no	—
44	#64090	—	3	26	no	—
45	#64086	—	3	26	no	—
46	C₄²×C₃⋊D₃	—	3	13	no	—
47	C₄×#634	—	3	13	no	—
48	C₂³×C₃³	C₂³	1	2	yes	—
49	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
50	C₃³×D₄	C₂×C₄	1	4	yes	complemented
51	C₂×C₃³×C₄	—	2	4	no	—
52	C₂×C₃³×C₄	C₂³	1	2	yes	—
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
55	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
56	#6312	C₂×C₄	1	8	yes	complemented
57	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
58	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	8	yes	complemented
59	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
60	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
61	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
62	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
63	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
64	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
65	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
66	C₄×C₃⋊(C₃⋊D₃)	C₂×C₄	1	4	yes	complemented
67	#6308	C₂×C₄	1	4	yes	complemented
68	C₄×#237	—	9	13	no	—
69	C₂×C₃²×D₄	C₂×D₃	1	13	yes	—
70	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	—
71	C₂×#638	—	3	26	no	—
72	C₂²×C₃⋊Dic₃	—	3	26	no	—
73	C₂²×C₃⋊Dic₃	—	3	26	no	—
74	#3830	—	3	104	no	—
75	C₂×C₄×C₃⋊D₃	—	3	13	no	—
76	C₂×C₄×C₃⋊D₃	—	3	13	no	—
77	C₂×C₄×C₃⋊D₃	—	3	13	no	—
78	C₂×#634	—	3	13	no	—
79	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
80	C₃²×C₂²⋊C₄	C₂×D₃	1	26	yes	—
81	C₃²×C₄²	C₂×D₃	1	13	yes	—
82	#3757	—	3	26	no	—
83	#3752	—	3	26	no	—
84	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
85	#3750	—	3	26	no	—
86	#3750	—	3	26	no	—
87	C₄×C₃⋊Dic₃	—	3	26	no	—
88	C₄×C₃⋊Dic₃	—	3	13	no	—
89	C₄×C₃⋊Dic₃	—	3	13	no	—
90	C₄×C₃⋊Dic₃	—	3	13	no	—
91	C₄×C₃⋊Dic₃	—	3	26	no	—
92	C₄×C₃⋊Dic₃	—	3	26	no	—
93	C₂²×C₃³	—	2	2	no	—
94	C₂²×C₃³	—	2	2	no	—
95	C₂²×C₃³	C₂²×C₄	1	4	yes	—
96	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
97	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
98	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
99	C₂×C₃⋊(C₃⋊D₃)	C₂²×C₄	1	2	yes	—
100	C₃³×C₄	—	2	1	no	—
101	C₃³×C₄	—	2	2	no	—
102	C₃³×C₄	C₂²×C₄	1	2	yes	—
103	C₃³×C₄	#41	1	2	yes	complemented
104	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
105	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
106	C₃⋊(C₃⋊Dic₃)	C₂²×C₄	1	4	yes	—
107	C₃⋊(C₃⋊Dic₃)	C₂²×C₄	1	2	yes	—
108	C₃⋊(C₃⋊Dic₃)	#41	1	2	yes	complemented
109	C₂×#237	—	9	13	no	—
110	C₃×C₄×D₄	C₃⋊D₃	1	13	yes	complemented
111	D₄×Dic₃	—	9	13	no	—
112	C₄×#67	—	9	26	no	—
113	C₂×C₄×Dic₃	—	9	26	no	—
114	#892	—	9	13	no	—
115	#887	—	9	13	no	—
116	#67⋊C₄	—	9	26	no	—
117	#877	—	9	26	no	—
118	C₄²×D₃	—	9	13	no	—
119	C₄×Dic₆	—	9	13	no	—
120	C₂³×C₃²	C₂²×D₃	1	26	yes	—
121	C₂²×C₃⋊D₃	—	3	13	no	—
122	C₃²×D₄	C₄×D₃	1	52	yes	complemented
123	C₂×C₃²×C₄	—	2	52	no	—
124	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
125	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
126	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
127	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
128	#638	—	3	104	no	—
129	C₂×C₃⋊Dic₃	—	3	104	no	—
130	C₂×C₃⋊Dic₃	—	3	13	no	—
131	C₂×C₃⋊Dic₃	—	3	13	no	—
132	C₂×C₃⋊Dic₃	—	3	26	no	—
133	C₂×C₃⋊Dic₃	—	3	13	no	—
134	C₂×C₃⋊Dic₃	—	3	26	no	—
135	C₂×C₃⋊Dic₃	—	6	52	no	—
136	C₄×C₃⋊D₃	—	3	52	no	—
137	C₄×C₃⋊D₃	—	6	26	no	—
138	C₄×C₃⋊D₃	—	6	26	no	—
139	#634	—	3	52	no	—
140	C₄×#41	—	27	1	no	Sylow
141	C₂×C₃³	—	2	4	no	—
142	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₃³	C₂×#41	1	1	yes	characteristic
144	C₂×C₃³	C₂×#41	1	1	yes	characteristic
145	C₃⋊(C₃⋊D₃)	—	2	2	no	—
146	C₂×C₃×D₄	C₂×C₃⋊D₃	1	13	yes	—
147	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
148	C₂×#67	—	9	26	no	—
149	C₂²×Dic₃	—	9	26	no	—
150	C₂²×Dic₃	—	9	26	no	—
151	#237	—	9	104	no	—
152	C₂×C₄×D₃	—	9	13	no	—
153	C₂×C₄×D₃	—	9	13	no	—
154	C₂×C₄×D₃	—	9	13	no	—
155	C₂×Dic₆	—	9	13	no	—
156	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
157	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
158	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
159	#217	—	9	26	no	—
160	(C₂×D₃)⋊C₄	—	9	26	no	—
161	C₄⋊Dic₃	—	9	13	no	—
162	Dic₃⋊C₄	—	9	26	no	—
163	Dic₃⋊C₄	—	9	26	no	—
164	C₄×Dic₃	—	9	26	no	—
165	C₄×Dic₃	—	9	13	no	—
166	C₄×Dic₃	—	9	13	no	—
167	C₄×Dic₃	—	9	13	no	—
168	C₄×Dic₃	—	9	26	no	—
169	C₄×Dic₃	—	9	26	no	—
170	C₂²×C₃²	—	2	26	no	—
171	C₂²×C₃²	—	2	26	no	—
172	C₂²×C₃²	C₂×C₄×D₃	1	52	yes	—
173	C₂²×C₃²	C₂³×D₃	1	13	yes	—
174	C₂×C₃⋊D₃	—	3	26	no	—
175	C₂×C₃⋊D₃	—	6	13	no	—
176	C₂×C₃⋊D₃	—	6	13	no	—
177	C₃²×C₄	—	2	26	no	—
178	C₃²×C₄	—	2	13	no	—
179	C₃²×C₄	C₂×C₄×D₃	1	26	yes	—
180	C₃²×C₄	#237	1	26	yes	complemented
181	C₃⋊Dic₃	—	3	52	no	—
182	C₃⋊Dic₃	—	3	26	no	—
183	C₃⋊Dic₃	—	3	26	no	—
184	C₃⋊Dic₃	—	6	13	no	—
185	C₃⋊Dic₃	—	6	26	no	—
186	C₂×#41	—	27	1	no	—
187	C₄×Q₈	—	27	1	no	—
188	C₄×Q₈	—	27	1	no	—
189	C₄×D₄	—	27	1	no	—
190	C₄×D₄	—	27	2	no	—
191	C₄×D₄	—	27	2	no	—
192	C₄×D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
193	#117	—	27	1	no	—
194	#117	—	27	2	no	—
195	C₂×C₄²	—	27	2	no	—
196	C₂×C₄²	—	27	1	no	—
197	C₃³	C₄×#41	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₃	C₂²×C₃⋊D₃	1	26	yes	—
199	C₂²×D₃	—	9	13	no	—
200	C₃×D₄	C₄×C₃⋊D₃	1	52	yes	complemented
201	C₂×C₃×C₄	—	2	52	no	—
202	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
203	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
204	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
205	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
206	#67	—	9	104	no	—
207	C₂×Dic₃	—	9	13	no	—
208	C₂×Dic₃	—	9	13	no	—
209	C₂×Dic₃	—	9	26	no	—
210	C₂×Dic₃	—	9	104	no	—
211	C₂×Dic₃	—	9	13	no	—
212	C₂×Dic₃	—	9	26	no	—
213	C₂×Dic₃	—	18	52	no	—
214	C₄×D₃	—	9	52	no	—
215	C₄×D₃	—	18	26	no	—
216	C₄×D₃	—	18	26	no	—
217	Dic₆	—	9	52	no	—
218	C₂×C₃²	—	2	52	no	—
219	C₂×C₃²	C₂²×C₄×D₃	1	13	yes	—
220	C₂×C₃²	C₂×#237	1	13	yes	—
221	C₂×C₃²	D₃×#41	1	13	yes	—
222	C₃⋊D₃	—	6	26	no	—
223	#41	—	27	8	no	—
224	C₂×Q₈	—	27	1	no	—
225	C₂×D₄	—	27	2	no	—
226	C₂×D₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
227	C₂²×C₄	—	27	2	no	—
228	C₂²×C₄	—	27	1	no	—
229	C₂²×C₄	—	27	2	no	—
230	C₂²×C₄	—	27	1	no	—
231	C₂²×C₄	—	27	1	no	—
232	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
233	C₄⋊C₄	—	27	2	no	—
234	C₄⋊C₄	—	27	2	no	—
235	C₄⋊C₄	—	27	1	no	—
236	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
237	C₂²⋊C₄	—	27	2	no	—
238	C₂²⋊C₄	—	27	2	no	—
239	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
240	C₄²	—	27	2	no	—
241	C₄²	—	27	1	no	—
242	C₄²	—	27	2	no	—
243	C₄²	—	27	1	no	—
244	C₄²	—	27	1	no	—
245	C₄²	—	27	2	no	—
246	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
247	C₂²×C₃	—	2	26	no	—
248	C₂²×C₃	—	2	26	no	—
249	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	52	yes	—
250	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
251	C₂×D₃	—	9	26	no	—
252	C₂×D₃	—	18	13	no	—
253	C₂×D₃	—	18	13	no	—
254	C₃×C₄	—	2	26	no	—
255	C₃×C₄	—	2	13	no	—
256	C₃×C₄	C₂×C₄×C₃⋊D₃	1	26	yes	—
257	C₃×C₄	#3830	1	26	yes	complemented
258	Dic₃	—	9	52	no	—
259	Dic₃	—	9	26	no	—
260	Dic₃	—	9	26	no	—
261	Dic₃	—	18	13	no	—
262	Dic₃	—	18	26	no	—
263	C₃²	C₄×#237	1	13	yes	complemented
264	C₂³	—	27	1	no	—
265	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
266	Q₈	—	27	4	no	—
267	D₄	—	27	8	no	—
268	D₄	C₄×C₃⋊(C₃⋊D₃)	1	4	yes	complemented
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	27	1	no	—
271	C₂×C₄	—	27	1	no	—
272	C₂×C₄	—	27	2	no	—
273	C₂×C₄	—	27	8	no	—
274	C₂×C₄	—	27	4	no	—
275	C₂×C₄	—	27	1	no	—
276	C₂×C₄	—	27	2	no	—
277	C₂×C₄	—	54	2	no	—
278	C₂×C₄	—	54	4	no	—
279	C₂×C₄	—	54	2	no	—
280	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
281	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
284	C₂×C₃	—	2	52	no	—
285	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	13	yes	—
286	C₂×C₃	C₂×#3830	1	13	yes	—
287	C₂×C₃	#41×C₃⋊D₃	1	13	yes	—
288	D₃	—	18	26	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	27	2	no	—
292	C₂²	—	54	1	no	—
293	C₂²	—	54	1	no	—
294	C₂²	C₂×C₄×C₃⋊(C₃⋊D₃)	1	4	yes	—
295	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	27	2	no	—
299	C₄	—	27	4	no	—
300	C₄	—	27	2	no	—
301	C₄	—	54	1	no	—
302	C₄	—	54	2	no	—
303	C₄	C₂×C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
304	C₄	#89078	1	2	yes	direct product factor
305	C₃	C₄×#3830	1	13	yes	complemented
306	C₂	—	2	4	no	—
307	C₂	—	54	2	no	—
308	C₂	C₂²×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
309	C₂	C₂×#89078	1	1	yes	characteristic
310	C₂	#41×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂³×C₄
Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄×D₄	C₂
socle, cosocle	C₂²×C₃³	C₂⁴
prime radical	C₂×#89078	C₂
hypercenter	C₄×D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₄×#41
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄×D₄	C₂	semidirect
3. D₄×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
4. C₄×#6312	C₂	semidirect
5. C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
6. #11723933	C₂	semidirect
7. #11723928	C₂	semidirect
8. #11723922	C₂	semidirect
9. #11723918	C₂	semidirect
10. C₄²×C₃⋊(C₃⋊D₃)	C₂	semidirect
11. C₄×#6308	C₂	semidirect
18. #89078	C₄	direct (N × H)
38. C₃²×C₄×D₄	D₃	semidirect
50. C₃³×D₄	C₂×C₄	semidirect
56. #6312	C₂×C₄	semidirect
58. C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	semidirect
66. C₄×C₃⋊(C₃⋊D₃)	C₂×C₄	semidirect
67. #6308	C₂×C₄	semidirect
103. C₃³×C₄	#41	semidirect
108. C₃⋊(C₃⋊Dic₃)	#41	semidirect
110. C₃×C₄×D₄	C₃⋊D₃	semidirect
122. C₃²×D₄	C₄×D₃	semidirect
180. C₃²×C₄	#237	semidirect
192. C₄×D₄	C₃⋊(C₃⋊D₃)	semidirect
197. C₃³	C₄×#41	semidirect
200. C₃×D₄	C₄×C₃⋊D₃	semidirect
257. C₃×C₄	#3830	semidirect
263. C₃²	C₄×#237	semidirect
268. D₄	C₄×C₃⋊(C₃⋊D₃)	semidirect
304. C₄	#89078	direct (N × H)
305. C₃	C₄×#3830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89078	2
C₃³×C₄×D₄	2
D₄×C₃⋊(C₃⋊Dic₃)	2
C₄×#6312	2
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
#11723933	2
#11723928	2
#11723922	2
#11723918	2
C₄²×C₃⋊(C₃⋊D₃)	2
C₄×#6308	2
C₄×#3830	3

Subgroups — C4×#89078 #13179673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#89078 #13179673