Subgroups — #13179703

Order 1728.

3108 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11723987	C₂	1	1	yes	maximal, characteristic
3	#11723985	C₂	1	1	yes	maximal, characteristic, complemented
4	#11723938	C₂	1	1	yes	maximal, characteristic
5	#11723937	C₂	1	2	yes	maximal
6	#11723930	C₂	1	2	yes	maximal, complemented
7	#11723929	C₂	1	1	yes	maximal, characteristic, complemented
8	#11723913	C₂	1	2	yes	maximal
9	C₄²×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
10	C₄×#6308	C₂	1	2	yes	maximal, complemented
11	C₄×#6308	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596049	—	3	13	no	maximal
13	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
17	C₂×#6308	C₂²	1	2	yes	—
18	C₂×#6308	C₂²	1	1	yes	characteristic
19	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃³×C₄²	C₂²	1	1	yes	characteristic
22	C₃³×C₄²	C₂²	1	2	yes	—
23	#88859	C₂²	1	2	yes	—
24	#88859	C₂²	1	2	yes	—
25	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
26	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
27	#88857	C₂²	1	2	yes	—
28	#88857	C₂²	1	2	yes	—
29	#88857	C₂²	1	2	yes	—
30	#88857	C₂²	1	2	yes	—
31	#88857	C₂²	1	2	yes	—
32	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
33	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
34	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
36	C₃²×C₄×Q₈	D₃	1	13	yes	complemented
37	#64155	—	3	13	no	—
38	#64153	—	3	13	no	—
39	#64106	—	3	13	no	—
40	#64105	—	3	26	no	—
41	#64098	—	3	26	no	—
42	#64097	—	3	13	no	—
43	#64081	—	3	26	no	—
44	C₄²×C₃⋊D₃	—	3	13	no	—
45	C₄×#634	—	3	26	no	—
46	C₄×#634	—	3	13	no	—
47	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
48	C₃³×Q₈	—	2	2	no	—
49	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃³×C₄	C₂³	1	2	yes	—
51	C₂×C₃³×C₄	C₂³	1	2	yes	—
52	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
53	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
54	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
55	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
56	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
57	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
58	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
59	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
60	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
61	C₄×C₃⋊(C₃⋊D₃)	Q₈	1	4	yes	complemented
62	#6308	—	2	2	no	—
63	#6308	—	2	4	no	—
64	#5578	—	9	13	no	—
65	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
66	C₂×C₄×C₃⋊D₃	—	3	13	no	—
67	C₂×C₄×C₃⋊D₃	—	3	13	no	—
68	C₂×C₄×C₃⋊D₃	—	3	13	no	—
69	C₂×#634	—	3	26	no	—
70	C₂×#634	—	3	13	no	—
71	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
72	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
73	C₃²×C₄²	C₂×D₃	1	13	yes	—
74	C₃²×C₄²	C₂×D₃	1	26	yes	—
75	#3752	—	3	26	no	—
76	#3752	—	3	26	no	—
77	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
78	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
79	#3750	—	3	26	no	—
80	#3750	—	3	26	no	—
81	#3750	—	3	26	no	—
82	#3750	—	3	26	no	—
83	#3750	—	3	26	no	—
84	C₄×C₃⋊Dic₃	—	3	13	no	—
85	C₄×C₃⋊Dic₃	—	3	26	no	—
86	C₄×C₃⋊Dic₃	—	3	13	no	—
87	C₄×C₃⋊Dic₃	—	3	13	no	—
88	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
89	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
90	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
91	C₂×C₃⋊(C₃⋊D₃)	C₂×Q₈	1	2	yes	—
92	C₃³×C₄	—	2	2	no	—
93	C₃³×C₄	—	2	1	no	—
94	C₃³×C₄	—	2	2	no	—
95	C₃³×C₄	C₂×Q₈	1	2	yes	—
96	C₃³×C₄	#41	1	2	yes	—
97	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
98	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
99	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
100	C₃⋊(C₃⋊Dic₃)	C₂×Q₈	1	2	yes	—
101	C₃⋊(C₃⋊Dic₃)	#41	1	2	yes	—
102	C₃×C₄×Q₈	C₃⋊D₃	1	13	yes	complemented
103	#946	—	9	13	no	—
104	#944	—	9	13	no	—
105	#897	—	9	13	no	—
106	#896	—	9	26	no	—
107	#889	—	9	26	no	—
108	#888	—	9	13	no	—
109	#872	—	9	26	no	—
110	C₄²×D₃	—	9	13	no	—
111	C₄×Dic₆	—	9	26	no	—
112	C₄×Dic₆	—	9	13	no	—
113	C₂²×C₃⋊D₃	—	3	13	no	—
114	C₃²×Q₈	—	2	26	no	—
115	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
116	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
117	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
118	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
119	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
120	C₂×C₃⋊Dic₃	—	3	13	no	—
121	C₂×C₃⋊Dic₃	—	3	13	no	—
122	C₂×C₃⋊Dic₃	—	3	13	no	—
123	C₂×C₃⋊Dic₃	—	3	26	no	—
124	C₂×C₃⋊Dic₃	—	3	26	no	—
125	C₄×C₃⋊D₃	—	3	52	no	—
126	C₄×C₃⋊D₃	—	6	26	no	—
127	C₄×C₃⋊D₃	—	6	26	no	—
128	#634	—	6	52	no	—
129	#634	—	6	26	no	—
130	#533	—	27	1	no	Sylow
131	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₃³	C₂×#41	1	1	yes	characteristic
133	C₂×C₃³	C₂×#41	1	1	yes	characteristic
134	C₃⋊(C₃⋊D₃)	—	2	2	no	—
135	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
136	C₂×C₄×D₃	—	9	13	no	—
137	C₂×C₄×D₃	—	9	13	no	—
138	C₂×C₄×D₃	—	9	13	no	—
139	C₂×Dic₆	—	9	26	no	—
140	C₂×Dic₆	—	9	13	no	—
141	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
142	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
143	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
144	C₃×C₄²	C₂×C₃⋊D₃	1	26	yes	—
145	(C₂×D₃)⋊C₄	—	9	26	no	—
146	(C₂×D₃)⋊C₄	—	9	26	no	—
147	C₄⋊Dic₃	—	9	13	no	—
148	C₄⋊Dic₃	—	9	26	no	—
149	Dic₃⋊C₄	—	9	26	no	—
150	Dic₃⋊C₄	—	9	26	no	—
151	Dic₃⋊C₄	—	9	26	no	—
152	Dic₃⋊C₄	—	9	26	no	—
153	Dic₃⋊C₄	—	9	26	no	—
154	C₄×Dic₃	—	9	13	no	—
155	C₄×Dic₃	—	9	26	no	—
156	C₄×Dic₃	—	9	13	no	—
157	C₄×Dic₃	—	9	13	no	—
158	C₂²×C₃²	C₂³×D₃	1	13	yes	—
159	C₂×C₃⋊D₃	—	3	26	no	—
160	C₂×C₃⋊D₃	—	6	13	no	—
161	C₂×C₃⋊D₃	—	6	13	no	—
162	C₃²×C₄	—	2	26	no	—
163	C₃²×C₄	—	2	26	no	—
164	C₃²×C₄	—	2	13	no	—
165	C₃²×C₄	#235	1	26	yes	—
166	C₃²×C₄	D₃×Q₈	1	26	yes	—
167	C₃⋊Dic₃	—	3	26	no	—
168	C₃⋊Dic₃	—	3	26	no	—
169	C₃⋊Dic₃	—	6	26	no	—
170	C₃⋊Dic₃	—	6	13	no	—
171	C₃⋊Dic₃	—	6	26	no	—
172	C₄⋊Q₈	—	27	1	no	—
173	C₄⋊Q₈	—	27	1	no	—
174	#125	—	27	2	no	—
175	C₂²⋊Q₈	—	27	1	no	—
176	C₂²⋊Q₈	—	27	2	no	—
177	C₂²⋊Q₈	—	27	1	no	—
178	C₄×Q₈	—	27	2	no	—
179	C₄×Q₈	—	27	1	no	—
180	C₄×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
181	#117	—	27	2	no	—
182	C₂×C₄²	—	27	1	no	—
183	C₃³	#533	1	1	yes	characteristic, complemented, Sylow
184	C₂²×D₃	—	9	13	no	—
185	C₃×Q₈	—	2	26	no	—
186	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
187	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
188	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
189	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
190	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
191	C₂×Dic₃	—	9	13	no	—
192	C₂×Dic₃	—	9	13	no	—
193	C₂×Dic₃	—	9	13	no	—
194	C₂×Dic₃	—	9	26	no	—
195	C₂×Dic₃	—	9	26	no	—
196	C₄×D₃	—	9	52	no	—
197	C₄×D₃	—	18	26	no	—
198	C₄×D₃	—	18	26	no	—
199	Dic₆	—	18	52	no	—
200	Dic₆	—	18	26	no	—
201	C₂×C₃²	C₂×#235	1	13	yes	—
202	C₂×C₃²	C₂×D₃×Q₈	1	13	yes	—
203	C₂×C₃²	D₃×#41	1	13	yes	—
204	C₃⋊D₃	—	6	26	no	—
205	C₂×Q₈	—	27	2	no	—
206	C₂×Q₈	—	27	1	no	—
207	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
208	C₂²×C₄	—	27	1	no	—
209	C₂²×C₄	—	27	1	no	—
210	C₂²×C₄	—	27	1	no	—
211	C₄⋊C₄	—	27	2	no	—
212	C₄⋊C₄	—	27	2	no	—
213	C₄⋊C₄	—	27	2	no	—
214	C₄⋊C₄	—	27	2	no	—
215	C₄⋊C₄	—	27	2	no	—
216	C₄⋊C₄	—	27	2	no	—
217	C₄⋊C₄	—	27	1	no	—
218	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
219	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
220	C₂²⋊C₄	—	27	2	no	—
221	C₂²⋊C₄	—	27	2	no	—
222	C₄²	—	27	1	no	—
223	C₄²	—	27	1	no	—
224	C₄²	—	27	1	no	—
225	C₄²	—	27	2	no	—
226	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
227	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
228	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
229	C₂×D₃	—	9	26	no	—
230	C₂×D₃	—	18	13	no	—
231	C₂×D₃	—	18	13	no	—
232	C₃×C₄	—	2	26	no	—
233	C₃×C₄	—	2	26	no	—
234	C₃×C₄	—	2	13	no	—
235	C₃×C₄	#3828	1	26	yes	—
236	C₃×C₄	Q₈×C₃⋊D₃	1	26	yes	—
237	Dic₃	—	9	26	no	—
238	Dic₃	—	9	26	no	—
239	Dic₃	—	18	26	no	—
240	Dic₃	—	18	13	no	—
241	Dic₃	—	18	26	no	—
242	C₃²	#5578	1	13	yes	complemented
243	C₂³	—	27	1	no	—
244	Q₈	—	2	2	no	—
245	Q₈	—	54	4	no	—
246	Q₈	—	54	2	no	—
247	C₂×C₄	—	27	1	no	—
248	C₂×C₄	—	27	1	no	—
249	C₂×C₄	—	27	4	no	—
250	C₂×C₄	—	27	1	no	—
251	C₂×C₄	—	27	2	no	—
252	C₂×C₄	—	27	2	no	—
253	C₂×C₄	—	54	2	no	—
254	C₂×C₄	—	54	2	no	—
255	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
257	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
258	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
260	C₂×C₃	C₂×#3828	1	13	yes	—
261	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	13	yes	—
262	C₂×C₃	#41×C₃⋊D₃	1	13	yes	—
263	D₃	—	18	26	no	—
264	C₂²	—	27	2	no	—
265	C₂²	—	54	1	no	—
266	C₂²	—	54	1	no	—
267	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	27	2	no	—
272	C₄	—	27	2	no	—
273	C₄	—	54	1	no	—
274	C₄	—	54	2	no	—
275	C₄	—	54	2	no	—
276	C₄	#89076	1	2	yes	—
277	C₄	Q₈×C₃⋊(C₃⋊D₃)	1	2	yes	—
278	C₃	#10596049	1	13	yes	complemented
279	C₂	—	54	2	no	—
280	C₂	C₂×#89076	1	1	yes	characteristic
281	C₂	C₂×Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
282	C₂	#41×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄×Q₈	C₂
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₂³
hypercenter	C₄×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#533
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄×Q₈	C₂	semidirect
3. #11723985	C₂	semidirect
6. #11723930	C₂	semidirect
7. #11723929	C₂	semidirect
10. C₄×#6308	C₂	semidirect
11. C₄×#6308	C₂	semidirect
36. C₃²×C₄×Q₈	D₃	semidirect
61. C₄×C₃⋊(C₃⋊D₃)	Q₈	semidirect
102. C₃×C₄×Q₈	C₃⋊D₃	semidirect
180. C₄×Q₈	C₃⋊(C₃⋊D₃)	semidirect
183. C₃³	#533	semidirect
242. C₃²	#5578	semidirect
278. C₃	#10596049	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
#11723987	2
#11723985	2
#11723938	2
#11723937	2
#11723930	2
#11723929	2
#11723913	2
C₄²×C₃⋊(C₃⋊D₃)	2
C₄×#6308	2
#10596049	3