Subgroups — #13179674

Order 1728.

4116 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#88857	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×C₃⋊(C₃⋊D₃)	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	#11723913	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6308	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596017	—	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	complemented
19	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
20	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
21	C₂×#6308	C₂²	1	1	yes	characteristic
22	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃³×C₄²	C₂²	1	1	yes	characteristic
25	#88864	C₂²	1	2	yes	—
26	#88859	C₂²	1	1	yes	characteristic
27	#88859	C₂²	1	1	yes	characteristic
28	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
29	#88857	C₂²	1	2	yes	—
30	#88857	C₂²	1	1	yes	characteristic
31	#88857	C₂²	1	1	yes	characteristic
32	#88857	C₂²	1	2	yes	—
33	#88857	C₂²	1	2	yes	—
34	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
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×#3750	—	3	26	no	—
42	C₄⋊C₄×C₃⋊D₃	—	3	13	no	—
43	#64096	—	3	13	no	—
44	#64090	—	3	26	no	—
45	#64086	—	3	26	no	—
46	#64081	—	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	—	4	2	no	—
52	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
53	C₂×C₃³×C₄	C₂³	1	2	yes	—
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)	—	4	2	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	1	yes	characteristic
61	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
62	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
63	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
64	C₄×C₃⋊(C₃⋊D₃)	—	4	2	no	—
65	C₄×C₃⋊(C₃⋊D₃)	C₂×C₄	1	4	yes	complemented
66	#6308	C₂×C₄	1	4	yes	complemented
67	#5546	—	9	13	no	—
68	C₂×C₃²×D₄	C₂×D₃	1	13	yes	—
69	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	—
70	C₂×#638	—	3	26	no	—
71	C₂²×C₃⋊Dic₃	—	3	26	no	—
72	C₂²×C₃⋊Dic₃	—	3	26	no	—
73	#3830	—	3	104	no	—
74	C₂×C₄×C₃⋊D₃	—	3	13	no	—
75	C₂×C₄×C₃⋊D₃	—	3	26	no	—
76	C₂×#634	—	3	13	no	—
77	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
78	C₃²×C₂²⋊C₄	C₂×D₃	1	26	yes	—
79	C₃²×C₄²	C₂×D₃	1	13	yes	—
80	#3757	—	3	26	no	—
81	#3752	—	3	13	no	—
82	#3752	—	3	13	no	—
83	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
84	#3750	—	3	26	no	—
85	#3750	—	3	13	no	—
86	#3750	—	3	13	no	—
87	#3750	—	3	26	no	—
88	#3750	—	3	26	no	—
89	C₄×C₃⋊Dic₃	—	3	26	no	—
90	C₄×C₃⋊Dic₃	—	3	13	no	—
91	C₄×C₃⋊Dic₃	—	3	26	no	—
92	C₂²×C₃³	—	2	2	no	—
93	C₂²×C₃³	—	2	2	no	—
94	C₂²×C₃³	C₂²×C₄	1	4	yes	—
95	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
96	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
97	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
98	C₂×C₃⋊(C₃⋊D₃)	C₂²×C₄	1	2	yes	—
99	C₃³×C₄	—	2	2	no	—
100	C₃³×C₄	—	2	1	no	—
101	C₃³×C₄	—	2	1	no	—
102	C₃³×C₄	C₂²×C₄	1	2	yes	—
103	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
104	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
105	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
106	C₃⋊(C₃⋊Dic₃)	C₂²×C₄	1	4	yes	—
107	C₃⋊(C₃⋊Dic₃)	C₂²×C₄	1	2	yes	—
108	C₂×#237	—	9	13	no	—
109	C₃×C₄×D₄	C₃⋊D₃	1	13	yes	complemented
110	D₄×Dic₃	—	9	13	no	—
111	C₄×#67	—	9	26	no	—
112	C₂×Dic₃⋊C₄	—	9	26	no	—
113	D₃×C₄⋊C₄	—	9	13	no	—
114	#887	—	9	13	no	—
115	#67⋊C₄	—	9	26	no	—
116	#877	—	9	26	no	—
117	#872	—	9	13	no	—
118	C₄×Dic₆	—	9	13	no	—
119	C₂³×C₃²	C₂²×D₃	1	26	yes	—
120	C₂²×C₃⋊D₃	—	3	13	no	—
121	C₃²×D₄	C₄×D₃	1	52	yes	complemented
122	C₂×C₃²×C₄	—	4	26	no	—
123	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
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	#638	—	3	104	no	—
128	C₂×C₃⋊Dic₃	—	3	104	no	—
129	C₂×C₃⋊Dic₃	—	3	26	no	—
130	C₂×C₃⋊Dic₃	—	3	13	no	—
131	C₂×C₃⋊Dic₃	—	3	13	no	—
132	C₂×C₃⋊Dic₃	—	3	13	no	—
133	C₂×C₃⋊Dic₃	—	3	26	no	—
134	C₂×C₃⋊Dic₃	—	12	26	no	—
135	C₄×C₃⋊D₃	—	3	52	no	—
136	C₄×C₃⋊D₃	—	12	26	no	—
137	#634	—	3	52	no	—
138	#520	—	27	1	no	Sylow
139	C₂×C₃³	—	2	4	no	—
140	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
143	C₃⋊(C₃⋊D₃)	—	2	2	no	—
144	C₂×C₃×D₄	C₂×C₃⋊D₃	1	13	yes	—
145	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
146	C₂×#67	—	9	26	no	—
147	C₂²×Dic₃	—	9	26	no	—
148	C₂²×Dic₃	—	9	26	no	—
149	#237	—	9	104	no	—
150	C₂×C₄×D₃	—	9	13	no	—
151	C₂×C₄×D₃	—	9	26	no	—
152	C₂×Dic₆	—	9	13	no	—
153	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
154	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
155	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
156	#217	—	9	26	no	—
157	(C₂×D₃)⋊C₄	—	9	13	no	—
158	(C₂×D₃)⋊C₄	—	9	13	no	—
159	C₄⋊Dic₃	—	9	13	no	—
160	Dic₃⋊C₄	—	9	26	no	—
161	Dic₃⋊C₄	—	9	13	no	—
162	Dic₃⋊C₄	—	9	13	no	—
163	Dic₃⋊C₄	—	9	26	no	—
164	Dic₃⋊C₄	—	9	26	no	—
165	C₄×Dic₃	—	9	26	no	—
166	C₄×Dic₃	—	9	13	no	—
167	C₄×Dic₃	—	9	26	no	—
168	C₂²×C₃²	—	2	26	no	—
169	C₂²×C₃²	—	2	26	no	—
170	C₂²×C₃²	C₂×C₄×D₃	1	52	yes	—
171	C₂²×C₃²	C₂³×D₃	1	13	yes	—
172	C₂×C₃⋊D₃	—	3	26	no	—
173	C₂×C₃⋊D₃	—	6	13	no	—
174	C₂×C₃⋊D₃	—	6	13	no	—
175	C₃²×C₄	—	2	26	no	—
176	C₃²×C₄	—	2	13	no	—
177	C₃²×C₄	—	2	13	no	—
178	C₃²×C₄	C₂×C₄×D₃	1	26	yes	—
179	C₃⋊Dic₃	—	3	52	no	—
180	C₃⋊Dic₃	—	3	26	no	—
181	C₃⋊Dic₃	—	6	26	no	—
182	C₃⋊Dic₃	—	6	13	no	—
183	C₃⋊Dic₃	—	6	13	no	—
184	C₂×#41	—	27	1	no	—
185	C₄×Q₈	—	27	1	no	—
186	C₄×Q₈	—	27	1	no	—
187	C₄×D₄	—	27	1	no	—
188	C₄×D₄	—	27	2	no	—
189	C₄×D₄	—	27	2	no	—
190	C₄×D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
191	#117	—	27	2	no	—
192	#117	—	27	1	no	—
193	C₂×C₄⋊C₄	—	27	2	no	—
194	C₂×C₄⋊C₄	—	27	1	no	—
195	C₃³	#520	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₃	C₂²×C₃⋊D₃	1	26	yes	—
197	C₂²×D₃	—	9	13	no	—
198	C₃×D₄	C₄×C₃⋊D₃	1	52	yes	complemented
199	C₂×C₃×C₄	—	4	26	no	—
200	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
201	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
202	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
203	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
204	#67	—	9	104	no	—
205	C₂×Dic₃	—	9	26	no	—
206	C₂×Dic₃	—	9	13	no	—
207	C₂×Dic₃	—	9	104	no	—
208	C₂×Dic₃	—	9	13	no	—
209	C₂×Dic₃	—	9	13	no	—
210	C₂×Dic₃	—	9	26	no	—
211	C₂×Dic₃	—	36	26	no	—
212	C₄×D₃	—	9	52	no	—
213	C₄×D₃	—	36	26	no	—
214	Dic₆	—	9	52	no	—
215	C₂×C₃²	—	2	52	no	—
216	C₂×C₃²	C₂²×C₄×D₃	1	13	yes	—
217	C₂×C₃²	#1004	1	13	yes	—
218	C₂×C₃²	#1010	1	13	yes	—
219	C₃⋊D₃	—	6	26	no	—
220	#41	—	27	8	no	—
221	C₂×Q₈	—	27	1	no	—
222	C₂×D₄	—	27	2	no	—
223	C₂×D₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
224	C₂²×C₄	—	27	2	no	—
225	C₂²×C₄	—	27	1	no	—
226	C₂²×C₄	—	27	2	no	—
227	C₂²×C₄	—	27	2	no	—
228	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
229	C₄⋊C₄	—	27	2	no	—
230	C₄⋊C₄	—	27	1	no	—
231	C₄⋊C₄	—	27	2	no	—
232	C₄⋊C₄	—	27	2	no	—
233	C₄⋊C₄	—	27	1	no	—
234	C₄⋊C₄	—	27	1	no	—
235	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
236	C₂²⋊C₄	—	27	2	no	—
237	C₂²⋊C₄	—	27	1	no	—
238	C₂²⋊C₄	—	27	1	no	—
239	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
240	C₄²	—	27	2	no	—
241	C₄²	—	27	2	no	—
242	C₄²	—	27	1	no	—
243	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
244	C₂²×C₃	—	2	26	no	—
245	C₂²×C₃	—	2	26	no	—
246	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	52	yes	—
247	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
248	C₂×D₃	—	9	26	no	—
249	C₂×D₃	—	18	13	no	—
250	C₂×D₃	—	18	13	no	—
251	C₃×C₄	—	2	26	no	—
252	C₃×C₄	—	2	13	no	—
253	C₃×C₄	—	2	13	no	—
254	C₃×C₄	C₂×C₄×C₃⋊D₃	1	26	yes	—
255	Dic₃	—	9	52	no	—
256	Dic₃	—	9	26	no	—
257	Dic₃	—	18	26	no	—
258	Dic₃	—	18	13	no	—
259	Dic₃	—	18	13	no	—
260	C₃²	#5546	1	13	yes	complemented
261	C₂³	—	27	1	no	—
262	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
263	Q₈	—	27	4	no	—
264	D₄	—	27	8	no	—
265	D₄	C₄×C₃⋊(C₃⋊D₃)	1	4	yes	complemented
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	27	2	no	—
268	C₂×C₄	—	27	1	no	—
269	C₂×C₄	—	27	8	no	—
270	C₂×C₄	—	27	4	no	—
271	C₂×C₄	—	27	1	no	—
272	C₂×C₄	—	27	2	no	—
273	C₂×C₄	—	27	1	no	—
274	C₂×C₄	—	108	2	no	—
275	C₂×C₄	—	108	2	no	—
276	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
278	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
280	C₂×C₃	—	2	52	no	—
281	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	13	yes	—
282	C₂×C₃	#64361	1	13	yes	—
283	C₂×C₃	#64367	1	13	yes	—
284	D₃	—	18	26	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	27	2	no	—
288	C₂²	—	54	1	no	—
289	C₂²	—	54	1	no	—
290	C₂²	C₂×C₄×C₃⋊(C₃⋊D₃)	1	4	yes	—
291	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	27	2	no	—
296	C₄	—	27	4	no	—
297	C₄	—	54	2	no	—
298	C₄	—	54	1	no	—
299	C₄	—	54	1	no	—
300	C₄	C₂×C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
301	C₃	#10596017	1	13	yes	complemented
302	C₂	—	2	4	no	—
303	C₂	—	54	2	no	—
304	C₂	C₂²×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
305	C₂	#11724679	1	1	yes	characteristic
306	C₂	#11724685	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂³×C₄
Fitting	C₃³×C₄×D₄	C₂
socle, cosocle	C₂²×C₃³	C₂⁴
prime radical	C₂×#89078	C₂
hypercenter	C₄×D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#520
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ 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×#88857	C₂	semidirect
6. C₄⋊C₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
7. #11723928	C₂	semidirect
8. #11723922	C₂	semidirect
9. #11723918	C₂	semidirect
10. #11723913	C₂	semidirect
11. C₄×#6308	C₂	semidirect
18. #89078	C₄	semidirect
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
65. C₄×C₃⋊(C₃⋊D₃)	C₂×C₄	semidirect
66. #6308	C₂×C₄	semidirect
109. C₃×C₄×D₄	C₃⋊D₃	semidirect
121. C₃²×D₄	C₄×D₃	semidirect
190. C₄×D₄	C₃⋊(C₃⋊D₃)	semidirect
195. C₃³	#520	semidirect
198. C₃×D₄	C₄×C₃⋊D₃	semidirect
260. C₃²	#5546	semidirect
265. D₄	C₄×C₃⋊(C₃⋊D₃)	semidirect
301. C₃	#10596017	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×#88857	2
C₄⋊C₄×C₃⋊(C₃⋊D₃)	2
#11723928	2
#11723922	2
#11723918	2
#11723913	2
C₄×#6308	2
#10596017	3