Subgroups — D₄×C₄⋊(C₃⋊(C₃⋊D₃)) #13179677

Order 1728.

5992 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	2	yes	maximal, complemented
3	C₃³×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723971	C₂	1	2	yes	maximal, complemented
5	#11723936	C₂	1	2	yes	maximal, complemented
6	#11723923	C₂	1	4	yes	maximal, complemented
7	#11723915	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄⋊(C₃⋊D₃)	—	3	13	no	maximal
10	C₂³×C₃⋊(C₃⋊D₃)	C₂²	1	4	yes	complemented
11	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃³×C₄	C₂²	1	2	yes	complemented
13	C₂×#6312	C₂²	1	4	yes	complemented
14	D₄×C₃⋊(C₃⋊D₃)	—	2	8	no	—
15	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	—	2	8	no	—
16	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	2	yes	complemented
18	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	4	yes	complemented
19	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	2	yes	complemented
20	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	2	yes	complemented
21	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
22	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
23	C₃³×C₂²⋊C₄	C₂²	1	2	yes	complemented
24	C₃³×C₄²	C₂²	1	1	yes	characteristic, complemented
25	#88859	C₂²	1	2	yes	complemented
26	#88859	C₂²	1	4	yes	complemented
27	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic, complemented
28	C₂×D₄×C₃⋊D₃	—	3	26	no	—
29	C₂²×C₄⋊(C₃⋊D₃)	—	3	26	no	—
30	C₃²×C₄×D₄	D₃	1	13	yes	complemented
31	#64139	—	3	26	no	—
32	#64104	—	3	26	no	—
33	#64091	—	3	52	no	—
34	#64083	—	3	13	no	—
35	C₄×C₄⋊(C₃⋊D₃)	—	3	13	no	—
36	C₂³×C₃³	C₂³	1	2	yes	—
37	C₂²×C₃⋊(C₃⋊D₃)	—	2	8	no	—
38	C₂²×C₃⋊(C₃⋊D₃)	—	2	8	no	—
39	C₂²×C₃⋊(C₃⋊D₃)	—	4	4	no	—
40	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	4	yes	—
41	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	2	yes	—
42	C₃³×D₄	D₄	1	4	yes	complemented
43	C₂×C₃³×C₄	—	2	4	no	—
44	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃³×C₄	C₂³	1	2	yes	—
46	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
47	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
48	#6312	—	2	8	no	—
49	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
50	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	8	no	—
51	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	4	no	—
52	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	4	no	—
53	C₄⋊(C₃⋊(C₃⋊D₃))	—	4	2	no	—
54	C₄⋊(C₃⋊(C₃⋊D₃))	D₄	1	4	yes	direct product factor
55	C₄×C₃⋊(C₃⋊D₃)	—	2	4	no	—
56	D₄×D₁₂	—	9	13	no	—
57	C₂³×C₃⋊D₃	—	3	52	no	—
58	C₂×C₃²×D₄	C₂×D₃	1	13	yes	—
59	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	complemented
60	C₂×#638	—	3	52	no	—
61	D₄×C₃⋊D₃	—	6	104	no	—
62	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
63	C₂×C₄⋊(C₃⋊D₃)	—	3	26	no	—
64	C₂×C₄⋊(C₃⋊D₃)	—	3	26	no	—
65	C₂×C₄⋊(C₃⋊D₃)	—	3	52	no	—
66	C₂×C₄⋊(C₃⋊D₃)	—	3	26	no	—
67	C₂×C₄⋊(C₃⋊D₃)	—	6	104	no	—
68	C₂×C₄×C₃⋊D₃	—	3	26	no	—
69	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	complemented
70	C₃²×C₂²⋊C₄	C₂×D₃	1	26	yes	complemented
71	C₃²×C₄²	C₂×D₃	1	13	yes	complemented
72	#3752	—	3	26	no	—
73	#3752	—	3	52	no	—
74	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
75	C₂²×C₃³	—	2	2	no	—
76	C₂²×C₃³	—	2	2	no	—
77	C₂²×C₃³	C₂×D₄	1	4	yes	complemented
78	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
79	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
80	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
81	C₂×C₃⋊(C₃⋊D₃)	—	2	4	no	—
82	C₂×C₃⋊(C₃⋊D₃)	—	2	4	no	—
83	C₂×C₃⋊(C₃⋊D₃)	—	2	4	no	—
84	C₂×C₃⋊(C₃⋊D₃)	—	4	16	no	—
85	C₂×C₃⋊(C₃⋊D₃)	C₂×D₄	1	4	yes	complemented
86	C₃³×C₄	—	2	2	no	—
87	C₃³×C₄	—	2	1	no	—
88	C₃³×C₄	C₂×D₄	1	2	yes	complemented
89	C₃³×C₄	C₂×D₄	1	2	yes	complemented
90	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
91	C₂×D₃×D₄	—	9	26	no	—
92	C₂²×D₁₂	—	9	26	no	—
93	C₃×C₄×D₄	C₃⋊D₃	1	13	yes	complemented
94	#930	—	9	26	no	—
95	#895	—	9	26	no	—
96	#882	—	9	52	no	—
97	#874	—	9	13	no	—
98	C₄×D₁₂	—	9	13	no	—
99	C₂³×C₃²	C₂²×D₃	1	26	yes	—
100	C₂²×C₃⋊D₃	—	3	26	no	—
101	C₂²×C₃⋊D₃	—	3	52	no	—
102	C₂²×C₃⋊D₃	—	6	104	no	—
103	C₂²×C₃⋊D₃	—	6	104	no	—
104	C₂²×C₃⋊D₃	—	12	52	no	—
105	C₃²×D₄	D₁₂	1	52	yes	complemented
106	C₂×C₃²×C₄	—	2	52	no	—
107	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
109	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
110	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
111	#638	—	6	104	no	—
112	C₂×C₃⋊Dic₃	—	3	26	no	—
113	C₄⋊(C₃⋊D₃)	—	3	52	no	—
114	C₄⋊(C₃⋊D₃)	—	6	104	no	—
115	C₄⋊(C₃⋊D₃)	—	6	52	no	—
116	C₄⋊(C₃⋊D₃)	—	6	52	no	—
117	C₄⋊(C₃⋊D₃)	—	12	26	no	—
118	C₄×C₃⋊D₃	—	6	52	no	—
119	D₄²	—	27	1	no	Sylow
120	C₂×C₃³	—	2	4	no	—
121	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
124	C₃⋊(C₃⋊D₃)	—	2	4	no	—
125	C₃⋊(C₃⋊D₃)	—	4	4	no	—
126	C₂³×D₃	—	9	52	no	—
127	C₂×C₃×D₄	C₂×C₃⋊D₃	1	13	yes	—
128	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	complemented
129	C₂×#67	—	9	52	no	—
130	D₃×D₄	—	18	104	no	—
131	C₂×D₁₂	—	9	13	no	—
132	C₂×D₁₂	—	9	26	no	—
133	C₂×D₁₂	—	9	26	no	—
134	C₂×D₁₂	—	9	52	no	—
135	C₂×D₁₂	—	9	26	no	—
136	C₂×D₁₂	—	18	104	no	—
137	C₂×C₄×D₃	—	9	26	no	—
138	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	complemented
139	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	26	yes	complemented
140	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	complemented
141	(C₂×D₃)⋊C₄	—	9	26	no	—
142	(C₂×D₃)⋊C₄	—	9	52	no	—
143	C₄⋊Dic₃	—	9	13	no	—
144	C₂²×C₃²	—	2	26	no	—
145	C₂²×C₃²	—	2	26	no	—
146	C₂²×C₃²	C₂×D₁₂	1	52	yes	complemented
147	C₂²×C₃²	C₂³×D₃	1	13	yes	—
148	C₂×C₃⋊D₃	—	3	52	no	—
149	C₂×C₃⋊D₃	—	6	26	no	—
150	C₂×C₃⋊D₃	—	6	26	no	—
151	C₂×C₃⋊D₃	—	6	52	no	—
152	C₂×C₃⋊D₃	—	6	52	no	—
153	C₂×C₃⋊D₃	—	6	52	no	—
154	C₂×C₃⋊D₃	—	12	208	no	—
155	C₃²×C₄	—	2	26	no	—
156	C₃²×C₄	—	2	13	no	—
157	C₃²×C₄	C₂×D₁₂	1	26	yes	complemented
158	C₃²×C₄	D₃×D₄	1	26	yes	complemented
159	C₃⋊Dic₃	—	6	26	no	—
160	C₂²×D₄	—	27	2	no	—
161	C₂²×D₄	—	27	2	no	—
162	#127	—	27	1	no	—
163	#121	—	27	2	no	—
164	#121	—	27	2	no	—
165	#120	—	27	4	no	—
166	C₄×D₄	—	27	1	no	—
167	C₄×D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
168	C₃³	D₄²	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₃	C₂²×C₃⋊D₃	1	26	yes	—
170	C₂²×D₃	—	9	26	no	—
171	C₂²×D₃	—	9	52	no	—
172	C₂²×D₃	—	18	104	no	—
173	C₂²×D₃	—	18	104	no	—
174	C₂²×D₃	—	36	52	no	—
175	C₃×D₄	C₄⋊(C₃⋊D₃)	1	52	yes	complemented
176	C₂×C₃×C₄	—	2	52	no	—
177	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
178	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
179	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
180	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
181	#67	—	18	104	no	—
182	C₂×Dic₃	—	9	26	no	—
183	D₁₂	—	9	52	no	—
184	D₁₂	—	18	104	no	—
185	D₁₂	—	18	52	no	—
186	D₁₂	—	18	52	no	—
187	D₁₂	—	36	26	no	—
188	C₄×D₃	—	18	52	no	—
189	C₂×C₃²	—	2	52	no	—
190	C₂×C₃²	C₂²×D₁₂	1	13	yes	—
191	C₂×C₃²	C₂×D₃×D₄	1	13	yes	—
192	C₂×C₃²	#1009	1	13	yes	—
193	C₃⋊D₃	—	6	52	no	—
194	C₃⋊D₃	—	12	52	no	—
195	C₂⁴	—	27	4	no	—
196	C₂×D₄	—	27	4	no	—
197	C₂×D₄	—	27	1	no	—
198	C₂×D₄	—	27	4	no	—
199	C₂×D₄	—	27	2	no	—
200	C₂×D₄	—	27	2	no	—
201	C₂×D₄	—	27	2	no	—
202	C₂×D₄	—	54	8	no	—
203	C₂×D₄	—	54	8	no	—
204	C₂×D₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
205	C₂²×C₄	—	27	2	no	—
206	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
207	C₄⋊C₄	—	27	1	no	—
208	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
209	C₂²⋊C₄	—	27	2	no	—
210	C₂²⋊C₄	—	27	4	no	—
211	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
212	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
213	C₂²×C₃	—	2	26	no	—
214	C₂²×C₃	—	2	26	no	—
215	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	52	yes	complemented
216	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
217	C₂×D₃	—	9	52	no	—
218	C₂×D₃	—	18	26	no	—
219	C₂×D₃	—	18	26	no	—
220	C₂×D₃	—	18	52	no	—
221	C₂×D₃	—	18	52	no	—
222	C₂×D₃	—	18	52	no	—
223	C₂×D₃	—	36	208	no	—
224	C₃×C₄	—	2	26	no	—
225	C₃×C₄	—	2	13	no	—
226	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	26	yes	complemented
227	C₃×C₄	D₄×C₃⋊D₃	1	26	yes	complemented
228	Dic₃	—	18	26	no	—
229	C₃²	D₄×D₁₂	1	13	yes	complemented
230	C₂³	—	27	2	no	—
231	C₂³	—	27	4	no	—
232	C₂³	—	54	8	no	—
233	C₂³	—	54	8	no	—
234	C₂³	—	108	4	no	—
235	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
236	D₄	—	27	4	no	—
237	D₄	—	54	8	no	—
238	D₄	—	54	8	no	—
239	D₄	—	54	4	no	—
240	D₄	—	54	4	no	—
241	D₄	—	108	2	no	—
242	D₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	4	yes	direct product factor
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	27	2	no	—
245	C₂×C₄	—	54	4	no	—
246	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
248	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
250	C₂×C₃	—	2	52	no	—
251	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	1	13	yes	—
252	C₂×C₃	C₂×D₄×C₃⋊D₃	1	13	yes	—
253	C₂×C₃	#64366	1	13	yes	—
254	D₃	—	18	52	no	—
255	D₃	—	36	52	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	27	4	no	—
259	C₂²	—	54	2	no	—
260	C₂²	—	54	2	no	—
261	C₂²	—	54	4	no	—
262	C₂²	—	54	4	no	—
263	C₂²	—	54	4	no	—
264	C₂²	—	108	16	no	—
265	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	4	yes	complemented
266	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	54	2	no	—
270	C₄	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	2	yes	complemented
271	C₄	D₄×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
272	C₃	D₄×C₄⋊(C₃⋊D₃)	1	13	yes	complemented
273	C₂	—	2	4	no	—
274	C₂	—	54	4	no	—
275	C₂	—	108	4	no	—
276	C₂	C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
277	C₂	C₂×D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
278	C₂	#11724684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₃³×C₄×D₄	C₂
hypercenter	C₄×D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	D₄²
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
2. C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
3. C₃³×C₄×D₄	C₂	semidirect
4. #11723971	C₂	semidirect
5. #11723936	C₂	semidirect
6. #11723923	C₂	semidirect
7. #11723915	C₂	semidirect
8. C₄×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
10. C₂³×C₃⋊(C₃⋊D₃)	C₂²	semidirect
12. C₂²×C₃³×C₄	C₂²	semidirect
13. C₂×#6312	C₂²	semidirect
17. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	semidirect
18. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	semidirect
19. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	semidirect
20. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	semidirect
21. C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	semidirect
22. C₃³×C₄⋊C₄	C₂²	semidirect
23. C₃³×C₂²⋊C₄	C₂²	semidirect
24. C₃³×C₄²	C₂²	semidirect
25. #88859	C₂²	semidirect
26. #88859	C₂²	semidirect
27. C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	semidirect
30. C₃²×C₄×D₄	D₃	semidirect
42. C₃³×D₄	D₄	semidirect
54. C₄⋊(C₃⋊(C₃⋊D₃))	D₄	direct (N × H)
59. C₂²×C₃²×C₄	C₂×D₃	semidirect
69. C₃²×C₄⋊C₄	C₂×D₃	semidirect
70. C₃²×C₂²⋊C₄	C₂×D₃	semidirect
71. C₃²×C₄²	C₂×D₃	semidirect
77. C₂²×C₃³	C₂×D₄	semidirect
85. C₂×C₃⋊(C₃⋊D₃)	C₂×D₄	semidirect
88. C₃³×C₄	C₂×D₄	semidirect
89. C₃³×C₄	C₂×D₄	semidirect
93. C₃×C₄×D₄	C₃⋊D₃	semidirect
105. C₃²×D₄	D₁₂	semidirect
128. C₂²×C₃×C₄	C₂×C₃⋊D₃	semidirect
138. C₃×C₄⋊C₄	C₂×C₃⋊D₃	semidirect
139. C₃×C₂²⋊C₄	C₂×C₃⋊D₃	semidirect
140. C₃×C₄²	C₂×C₃⋊D₃	semidirect
146. C₂²×C₃²	C₂×D₁₂	semidirect
157. C₃²×C₄	C₂×D₁₂	semidirect
158. C₃²×C₄	D₃×D₄	semidirect
167. C₄×D₄	C₃⋊(C₃⋊D₃)	semidirect
168. C₃³	D₄²	semidirect
175. C₃×D₄	C₄⋊(C₃⋊D₃)	semidirect
206. C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	semidirect
208. C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	semidirect
211. C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	semidirect
212. C₄²	C₂×C₃⋊(C₃⋊D₃)	semidirect
215. C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	semidirect
226. C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	semidirect
227. C₃×C₄	D₄×C₃⋊D₃	semidirect
229. C₃²	D₄×D₁₂	semidirect
242. D₄	C₄⋊(C₃⋊(C₃⋊D₃))	direct (N × H)
265. C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	semidirect
270. C₄	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	semidirect
271. C₄	D₄×C₃⋊(C₃⋊D₃)	semidirect
272. C₃	D₄×C₄⋊(C₃⋊D₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₃⋊(C₃⋊D₃)	2
C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	2
C₃³×C₄×D₄	2
#11723971	2
#11723936	2
#11723923	2
#11723915	2
C₄×C₄⋊(C₃⋊(C₃⋊D₃))	2
D₄×C₄⋊(C₃⋊D₃)	3

Subgroups — D4×C4⋊(C3⋊(C3⋊D3)) #13179677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₄⋊(C₃⋊(C₃⋊D₃)) #13179677