Subgroups — #13179636

Order 1728.

3192 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#6312	C₂	1	1	yes	maximal, characteristic
3	C₄×#6312	C₂	1	2	yes	maximal
4	#11723965	C₂	1	2	yes	maximal, complemented
5	#11723965	C₂	1	1	yes	maximal, characteristic, complemented
6	#11723938	C₂	1	1	yes	maximal, characteristic, complemented
7	#11723937	C₂	1	2	yes	maximal, complemented
8	C₄⋊C₄×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
9	#11723930	C₂	1	2	yes	maximal, complemented
10	#11723929	C₂	1	1	yes	maximal, characteristic, complemented
11	#11723928	C₂	1	1	yes	maximal, characteristic, complemented
12	#10595982	—	3	13	no	maximal
13	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃³×C₄	C₂²	1	2	yes	—
15	C₂×#6312	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
18	C₂×#6308	C₂²	1	1	yes	characteristic
19	C₂×#6308	C₂²	1	2	yes	—
20	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
23	#88864	C₂²	1	2	yes	—
24	#88864	C₂²	1	1	yes	characteristic
25	#88859	C₂²	1	1	yes	characteristic
26	#88859	C₂²	1	2	yes	—
27	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
28	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
29	#88857	C₂²	1	2	yes	—
30	#88857	C₂²	1	2	yes	—
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	1	yes	characteristic
35	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
36	C₂×C₃²×C₄⋊C₄	D₃	1	13	yes	complemented
37	C₄×#638	—	3	13	no	—
38	C₄×#638	—	3	26	no	—
39	#64133	—	3	26	no	—
40	#64133	—	3	13	no	—
41	#64106	—	3	13	no	—
42	#64105	—	3	26	no	—
43	C₄⋊C₄×C₃⋊D₃	—	3	13	no	—
44	#64098	—	3	26	no	—
45	#64097	—	3	13	no	—
46	#64096	—	3	13	no	—
47	C₂³×C₃³	C₂³	1	1	yes	characteristic
48	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
49	C₂×C₃³×C₄	—	2	2	no	—
50	C₂×C₃³×C₄	—	4	2	no	—
51	C₂×C₃³×C₄	C₂³	1	2	yes	—
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	#6312	Q₈	1	4	yes	complemented
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	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	2	yes	—
60	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	1	yes	characteristic
61	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
62	C₄×C₃⋊(C₃⋊D₃)	—	4	2	no	—
63	#6308	—	2	2	no	—
64	#6308	—	4	2	no	—
65	#5511	—	9	13	no	—
66	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
67	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	—
68	C₂×#638	—	3	13	no	—
69	C₂×C₄×C₃⋊D₃	—	3	13	no	—
70	C₂×C₄×C₃⋊D₃	—	3	26	no	—
71	C₂×#634	—	3	13	no	—
72	C₂×#634	—	3	26	no	—
73	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
74	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
75	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
76	#3757	—	3	26	no	—
77	#3757	—	3	13	no	—
78	#3752	—	3	13	no	—
79	#3752	—	3	26	no	—
80	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
81	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
82	#3750	—	3	26	no	—
83	#3750	—	3	26	no	—
84	#3750	—	3	13	no	—
85	#3750	—	3	26	no	—
86	#3750	—	3	26	no	—
87	C₄×C₃⋊Dic₃	—	3	13	no	—
88	C₄×C₃⋊Dic₃	—	3	26	no	—
89	C₂²×C₃³	—	2	1	no	—
90	C₂²×C₃³	—	2	1	no	—
91	C₂²×C₃³	C₂×Q₈	1	2	yes	—
92	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
93	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
94	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
95	C₂×C₃⋊(C₃⋊D₃)	C₂×Q₈	1	2	yes	—
96	C₃³×C₄	—	2	1	no	—
97	C₃³×C₄	—	2	2	no	—
98	C₃³×C₄	—	2	2	no	—
99	C₃³×C₄	#41	1	2	yes	—
100	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
101	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
102	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
103	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
104	C₃⋊(C₃⋊Dic₃)	C₂×Q₈	1	2	yes	—
105	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	13	yes	complemented
106	C₄×#67	—	9	13	no	—
107	C₄×#67	—	9	26	no	—
108	#924	—	9	26	no	—
109	#924	—	9	13	no	—
110	#897	—	9	13	no	—
111	#896	—	9	26	no	—
112	D₃×C₄⋊C₄	—	9	13	no	—
113	#889	—	9	26	no	—
114	#888	—	9	13	no	—
115	#887	—	9	13	no	—
116	C₂³×C₃²	C₂²×D₃	1	13	yes	—
117	C₂²×C₃⋊D₃	—	3	13	no	—
118	C₂×C₃²×C₄	—	2	26	no	—
119	C₂×C₃²×C₄	—	4	26	no	—
120	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
121	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
123	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
124	#638	—	3	52	no	—
125	C₂×C₃⋊Dic₃	—	3	13	no	—
126	C₂×C₃⋊Dic₃	—	3	26	no	—
127	C₂×C₃⋊Dic₃	—	3	13	no	—
128	C₂×C₃⋊Dic₃	—	3	13	no	—
129	C₂×C₃⋊Dic₃	—	3	26	no	—
130	C₄×C₃⋊D₃	—	6	26	no	—
131	C₄×C₃⋊D₃	—	12	26	no	—
132	#634	—	6	26	no	—
133	#634	—	12	26	no	—
134	#554	—	27	1	no	Sylow
135	C₂×C₃³	—	2	2	no	—
136	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₃³	C₂×#41	1	1	yes	characteristic
138	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
139	C₃⋊(C₃⋊D₃)	—	2	2	no	—
140	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
141	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
142	C₂×#67	—	9	13	no	—
143	C₂×C₄×D₃	—	9	13	no	—
144	C₂×C₄×D₃	—	9	26	no	—
145	C₂×Dic₆	—	9	13	no	—
146	C₂×Dic₆	—	9	26	no	—
147	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
148	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
149	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
150	#217	—	9	26	no	—
151	#217	—	9	13	no	—
152	(C₂×D₃)⋊C₄	—	9	13	no	—
153	(C₂×D₃)⋊C₄	—	9	26	no	—
154	C₄⋊Dic₃	—	9	13	no	—
155	C₄⋊Dic₃	—	9	26	no	—
156	Dic₃⋊C₄	—	9	26	no	—
157	Dic₃⋊C₄	—	9	26	no	—
158	Dic₃⋊C₄	—	9	13	no	—
159	Dic₃⋊C₄	—	9	26	no	—
160	Dic₃⋊C₄	—	9	26	no	—
161	C₄×Dic₃	—	9	13	no	—
162	C₄×Dic₃	—	9	26	no	—
163	C₂²×C₃²	—	2	13	no	—
164	C₂²×C₃²	—	2	13	no	—
165	C₂²×C₃²	D₃×Q₈	1	26	yes	—
166	C₂²×C₃²	C₂³×D₃	1	13	yes	—
167	C₂×C₃⋊D₃	—	3	26	no	—
168	C₂×C₃⋊D₃	—	6	13	no	—
169	C₂×C₃⋊D₃	—	6	13	no	—
170	C₃²×C₄	—	2	13	no	—
171	C₃²×C₄	—	2	26	no	—
172	C₃²×C₄	—	2	26	no	—
173	C₃²×C₄	#235	1	26	yes	—
174	C₃⋊Dic₃	—	3	26	no	—
175	C₃⋊Dic₃	—	6	26	no	—
176	C₃⋊Dic₃	—	6	13	no	—
177	C₃⋊Dic₃	—	6	13	no	—
178	C₃⋊Dic₃	—	6	26	no	—
179	C₄⋊Q₈	—	27	1	no	—
180	#125	—	27	2	no	—
181	C₂²⋊Q₈	—	27	2	no	—
182	C₂²⋊Q₈	—	27	1	no	—
183	C₂²⋊Q₈	—	27	2	no	—
184	C₂²⋊Q₈	—	27	1	no	—
185	C₄×Q₈	—	27	1	no	—
186	C₄×D₄	—	27	2	no	—
187	C₄×D₄	—	27	1	no	—
188	C₂×C₄⋊C₄	—	27	1	no	—
189	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
190	C₃³	#554	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
192	C₂²×D₃	—	9	13	no	—
193	C₂×C₃×C₄	—	2	26	no	—
194	C₂×C₃×C₄	—	4	26	no	—
195	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
196	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
197	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
198	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
199	#67	—	9	52	no	—
200	C₂×Dic₃	—	9	13	no	—
201	C₂×Dic₃	—	9	26	no	—
202	C₂×Dic₃	—	9	13	no	—
203	C₂×Dic₃	—	9	13	no	—
204	C₂×Dic₃	—	9	26	no	—
205	C₄×D₃	—	18	26	no	—
206	C₄×D₃	—	36	26	no	—
207	Dic₆	—	18	26	no	—
208	Dic₆	—	36	26	no	—
209	C₂×C₃²	—	2	26	no	—
210	C₂×C₃²	C₂×#235	1	13	yes	—
211	C₂×C₃²	#1004	1	13	yes	—
212	C₂×C₃²	C₂×D₃×Q₈	1	13	yes	—
213	C₃⋊D₃	—	6	26	no	—
214	C₂×Q₈	—	27	1	no	—
215	C₂×Q₈	—	27	2	no	—
216	C₂×D₄	—	27	1	no	—
217	C₂²×C₄	—	27	1	no	—
218	C₂²×C₄	—	27	2	no	—
219	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
221	C₄⋊C₄	—	27	2	no	—
222	C₄⋊C₄	—	27	1	no	—
223	C₄⋊C₄	—	27	1	no	—
224	C₄⋊C₄	—	27	2	no	—
225	C₄⋊C₄	—	27	2	no	—
226	C₄⋊C₄	—	27	2	no	—
227	C₄⋊C₄	—	27	2	no	—
228	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
229	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
230	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
231	C₂²⋊C₄	—	27	2	no	—
232	C₂²⋊C₄	—	27	1	no	—
233	C₂²⋊C₄	—	27	2	no	—
234	C₂²⋊C₄	—	27	1	no	—
235	C₄²	—	27	2	no	—
236	C₄²	—	27	1	no	—
237	C₂²×C₃	—	2	13	no	—
238	C₂²×C₃	—	2	13	no	—
239	C₂²×C₃	Q₈×C₃⋊D₃	1	26	yes	—
240	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
241	C₂×D₃	—	9	26	no	—
242	C₂×D₃	—	18	13	no	—
243	C₂×D₃	—	18	13	no	—
244	C₃×C₄	—	2	13	no	—
245	C₃×C₄	—	2	26	no	—
246	C₃×C₄	—	2	26	no	—
247	C₃×C₄	#3828	1	26	yes	—
248	Dic₃	—	9	26	no	—
249	Dic₃	—	18	26	no	—
250	Dic₃	—	18	13	no	—
251	Dic₃	—	18	13	no	—
252	Dic₃	—	18	26	no	—
253	C₃²	#5511	1	13	yes	complemented
254	C₂³	—	27	1	no	—
255	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
256	Q₈	—	54	2	no	—
257	Q₈	—	108	2	no	—
258	D₄	—	27	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	27	1	no	—
262	C₂×C₄	—	27	2	no	—
263	C₂×C₄	—	27	1	no	—
264	C₂×C₄	—	27	1	no	—
265	C₂×C₄	—	27	2	no	—
266	C₂×C₄	—	54	2	no	—
267	C₂×C₄	—	108	2	no	—
268	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
269	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
271	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
272	C₂×C₃	—	2	26	no	—
273	C₂×C₃	C₂×#3828	1	13	yes	—
274	C₂×C₃	#64361	1	13	yes	—
275	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	13	yes	—
276	D₃	—	18	26	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	27	2	no	—
280	C₂²	—	54	1	no	—
281	C₂²	—	54	1	no	—
282	C₂²	Q₈×C₃⋊(C₃⋊D₃)	1	2	yes	—
283	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	27	2	no	—
288	C₄	—	54	1	no	—
289	C₄	—	54	2	no	—
290	C₄	—	54	2	no	—
291	C₄	—	54	1	no	—
292	C₄	#89076	1	2	yes	—
293	C₃	#10595982	1	13	yes	complemented
294	C₂	—	2	2	no	—
295	C₂	—	54	2	no	—
296	C₂	C₂×#89076	1	1	yes	characteristic
297	C₂	#11724679	1	1	yes	characteristic
298	C₂	C₂×Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₂×C₃³×C₄⋊C₄	C₂
prime radical	C₂×#6312	C₂²
hypercenter	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#554
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×C₄⋊C₄	C₂	semidirect
4. #11723965	C₂	semidirect
5. #11723965	C₂	semidirect
6. #11723938	C₂	semidirect
7. #11723937	C₂	semidirect
8. C₄⋊C₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
9. #11723930	C₂	semidirect
10. #11723929	C₂	semidirect
11. #11723928	C₂	semidirect
36. C₂×C₃²×C₄⋊C₄	D₃	semidirect
55. #6312	Q₈	semidirect
105. C₂×C₃×C₄⋊C₄	C₃⋊D₃	semidirect
189. C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)	semidirect
190. C₃³	#554	semidirect
253. C₃²	#5511	semidirect
293. C₃	#10595982	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×C₄⋊C₄	2
C₄×#6312	2
#11723965	2
#11723938	2
#11723937	2
C₄⋊C₄×C₃⋊(C₃⋊D₃)	2
#11723930	2
#11723929	2
#11723928	2
#10595982	3