Subgroups — #11797026

Order 1152.

345 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×C₃×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#10590692	C₂	1	1	yes	maximal, characteristic
3	C₂×#63543	C₂	1	1	yes	maximal, characteristic
4	#10589367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589133	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	#71401	—	3	1	no	maximal
9	#69285	—	3	1	no	maximal
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	#63855	C₄	1	1	yes	characteristic
12	C₂×#3711	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₃×D₃×C₁₆	—	2	2	no	—
20	#63555	C₄	1	1	yes	characteristic
21	#63554	C₄	1	1	yes	characteristic
22	#63543	—	2	2	no	—
23	C₂²×C₃×C₁₆	—	3	1	no	—
24	#5101	—	3	1	no	—
25	C₂×#812	—	3	1	no	—
26	C₂×#798	—	3	1	no	—
27	C₂×D₃×C₁₆	D₃	1	1	yes	characteristic
28	#4689	—	3	1	no	—
29	C₃×C₄⋊C₁₆	—	3	1	no	—
30	Dic₃⋊C₁₆	—	3	1	no	—
31	Dic₃⋊C₁₆	—	3	1	no	—
32	Dic₃⋊C₁₆	—	3	1	no	—
33	Dic₃⋊C₁₆	—	3	1	no	—
34	Dic₃⋊C₁₆	—	3	1	no	—
35	Dic₃⋊C₁₆	—	6	1	no	—
36	Dic₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
37	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×D₃×C₈	D₄	1	2	yes	—
46	C₃×D₃×C₈	Q₈	1	2	yes	—
47	Dic₃⋊Dic₃	C₈	1	1	yes	characteristic
48	Dic₃⋊Dic₃	C₈	1	1	yes	characteristic
49	(C₂×D₃)⋊Dic₃	C₈	1	1	yes	characteristic
50	(C₂×D₃)⋊Dic₃	C₈	1	1	yes	characteristic
51	#3711	—	2	2	no	—
52	C₃²×C₁₆	—	2	1	no	—
53	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
54	C₃×C₃⋊C₁₆	—	2	1	no	—
55	C₃×C₃⋊C₁₆	—	2	1	no	—
56	#2151	—	9	1	no	Sylow
57	C₂²×C₃×C₈	—	3	1	no	—
58	#926	—	3	1	no	—
59	C₂×#208	—	3	1	no	—
60	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
61	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
62	#872	—	3	1	no	—
63	C₂×C₃×C₁₆	—	2	1	no	—
64	C₂×C₃×C₁₆	—	3	1	no	—
65	C₂×C₃×C₁₆	—	3	1	no	—
66	C₂×C₃×C₁₆	—	6	2	no	—
67	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
68	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
69	C₃×C₄×C₈	—	3	1	no	—
70	#815	—	3	1	no	—
71	#815	—	3	1	no	—
72	#815	—	3	1	no	—
73	C₈×Dic₃	—	3	1	no	—
74	C₈×Dic₃	—	3	1	no	—
75	C₈×Dic₃	—	6	1	no	—
76	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
77	#812	—	6	2	no	—
78	C₂×C₃⋊C₁₆	—	3	1	no	—
79	C₂×C₃⋊C₁₆	—	3	1	no	—
80	C₂×C₃⋊C₁₆	—	6	1	no	—
81	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
82	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
83	#803	—	3	1	no	—
84	#798	—	6	2	no	—
85	D₃×C₁₆	—	2	2	no	—
86	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
93	C₃×C₄×D₃	C₄⋊C₄	1	2	yes	—
94	C₃×C₄×D₃	M₁₆	1	2	yes	—
95	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₃⋊(C₃⋊C₈)	—	2	1	no	—
98	C₃×C₃⋊C₈	—	2	1	no	—
99	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
100	C₂×M₃₂	—	9	1	no	—
101	C₂²×C₁₆	—	3	1	no	—
102	#405	—	9	1	no	—
103	C₄⋊C₁₆	—	3	1	no	—
104	C₄⋊C₁₆	—	9	1	no	—
105	C₄⋊C₁₆	—	9	1	no	—
106	C₄⋊C₁₆	—	9	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	3	1	no	—
111	C₂×C₃×C₈	—	3	2	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	—	3	1	no	—
114	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
116	C₃×C₄²	—	3	1	no	—
117	#217	—	3	1	no	—
118	#217	—	3	1	no	—
119	(C₂×D₃)⋊C₄	—	3	1	no	—
120	(C₂×D₃)⋊C₄	—	3	1	no	—
121	C₄⋊Dic₃	—	3	1	no	—
122	C₄⋊Dic₃	—	3	1	no	—
123	Dic₃⋊C₄	—	3	1	no	—
124	Dic₃⋊C₄	—	3	1	no	—
125	C₄×Dic₃	—	3	1	no	—
126	C₄×Dic₃	—	3	1	no	—
127	C₄×Dic₃	—	6	1	no	—
128	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
129	#208	—	6	2	no	—
130	C₂×C₃⋊C₈	—	3	1	no	—
131	C₂×C₃⋊C₈	—	3	1	no	—
132	C₂×C₃⋊C₈	—	6	1	no	—
133	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
134	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
135	ΓL(1,𝔽₂₅)	—	6	2	no	—
136	D₃×C₈	Dic₆	1	2	yes	—
137	D₃×C₈	#67	1	2	yes	—
138	C₃×C₁₆	—	2	1	no	—
139	C₃×C₁₆	—	2	1	no	—
140	C₃×C₁₆	—	4	1	no	—
141	C₃×C₁₆	—	6	1	no	—
142	C₃×C₁₆	—	6	1	no	—
143	C₃⋊C₁₆	—	2	1	no	—
144	C₃⋊C₁₆	—	2	1	no	—
145	C₃⋊C₁₆	—	6	1	no	—
146	C₃⋊C₁₆	—	6	1	no	—
147	C₃⋊C₁₆	—	12	1	no	—
148	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₃×D₃	—	2	1	no	—
150	C₂×C₃×D₃	—	2	1	no	—
151	C₂×C₃×D₃	C₄⋊C₈	1	2	yes	—
152	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
154	C₃⋊Dic₃	—	2	1	no	—
155	C₃⋊Dic₃	—	2	1	no	—
156	C₃×Dic₃	—	2	1	no	—
157	C₃×Dic₃	—	2	1	no	—
158	C₃×Dic₃	C₄⋊C₈	1	2	yes	—
159	C₂×M₁₆	—	9	1	no	—
160	C₂²×C₈	—	3	1	no	—
161	#117	—	9	1	no	—
162	M₃₂	—	18	2	no	—
163	C₂×C₁₆	—	3	1	no	—
164	C₂×C₁₆	—	3	1	no	—
165	C₂×C₁₆	—	6	2	no	—
166	C₂×C₁₆	—	9	1	no	—
167	C₂×C₁₆	D₃²	1	1	yes	characteristic
168	#97	—	9	1	no	—
169	#97	—	9	1	no	—
170	C₄×C₈	—	3	1	no	—
171	C₄×C₈	—	9	1	no	—
172	C₂³×C₃	—	3	1	no	—
173	C₂²×D₃	D₃×C₈	1	1	yes	characteristic
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	2	no	—
177	C₂×C₃×C₄	—	3	2	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	3	1	no	—
180	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×Dic₃	—	3	1	no	—
183	C₂×Dic₃	—	3	1	no	—
184	C₂×Dic₃	—	3	1	no	—
185	C₂×Dic₃	—	3	1	no	—
186	C₂×Dic₃	—	6	1	no	—
187	C₂×Dic₃	—	6	1	no	—
188	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
189	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
190	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
191	C₄×D₃	ΓL(1,𝔽₂₅)	1	2	yes	—
192	C₄×D₃	Dic₃⋊C₄	1	2	yes	—
193	C₃×C₈	—	2	1	no	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	—	3	2	no	—
196	C₃×C₈	—	6	1	no	—
197	C₃×C₈	C₂×Dic₆	1	1	yes	characteristic
198	C₃×C₈	D₃×D₄	1	1	yes	characteristic
199	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
200	C₃×C₈	C₂×#67	1	1	yes	characteristic
201	C₃⋊C₈	—	2	1	no	—
202	C₃⋊C₈	—	6	1	no	—
203	C₃⋊C₈	—	6	1	no	—
204	C₃⋊C₈	—	12	1	no	—
205	C₃⋊C₈	Dic₃⋊C₄	1	2	yes	—
206	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
207	C₂×C₃²	#504	1	1	yes	characteristic
208	C₂×C₃²	#504	1	1	yes	characteristic
209	C₃×D₃	—	2	2	no	—
210	C₂²×C₄	—	3	1	no	—
211	M₁₆	—	18	2	no	—
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	—	3	2	no	—
214	C₂×C₈	—	3	2	no	—
215	C₂×C₈	—	3	1	no	—
216	C₂×C₈	—	9	1	no	—
217	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
218	C₄⋊C₄	—	9	1	no	—
219	C₄⋊C₄	—	9	1	no	—
220	C₂²⋊C₄	—	9	1	no	—
221	C₂²⋊C₄	—	9	1	no	—
222	C₄²	—	3	1	no	—
223	C₄²	—	9	1	no	—
224	C₁₆	—	2	1	no	—
225	C₁₆	—	6	1	no	—
226	C₁₆	—	6	1	no	—
227	C₁₆	—	18	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	3	2	no	—
230	C₂²×C₃	—	6	1	no	—
231	C₂²×C₃	—	6	1	no	—
232	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
233	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
234	C₂×D₃	—	2	1	no	—
235	C₂×D₃	—	2	1	no	—
236	C₂×D₃	Dic₃⋊C₈	1	2	yes	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	3	2	no	—
240	C₃×C₄	—	6	1	no	—
241	C₃×C₄	—	6	1	no	—
242	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
243	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
244	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
245	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
246	Dic₃	—	2	1	no	—
247	Dic₃	—	2	1	no	—
248	Dic₃	—	6	2	no	—
249	Dic₃	—	6	1	no	—
250	Dic₃	—	6	1	no	—
251	Dic₃	—	6	1	no	—
252	Dic₃	—	6	1	no	—
253	Dic₃	—	12	1	no	—
254	Dic₃	Dic₃⋊C₈	1	2	yes	—
255	C₃²	#2151	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	3	1	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	9	1	no	—
263	C₂×C₄	—	9	1	no	—
264	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
265	C₈	—	3	2	no	—
266	C₈	—	6	1	no	—
267	C₈	—	18	1	no	—
268	C₈	D₃×Dic₆	1	1	yes	characteristic
269	C₈	D₃×#67	1	1	yes	characteristic
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	6	2	no	—
274	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
275	C₂×C₃	#4901	1	1	yes	characteristic
276	C₂×C₃	#4901	1	1	yes	characteristic
277	C₂×C₃	#4901	1	1	yes	characteristic
278	C₂×C₃	#4907	1	1	yes	characteristic
279	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
280	D₃	—	2	2	no	—
281	C₂²	—	3	2	no	—
282	C₂²	—	6	1	no	—
283	C₂²	—	6	1	no	—
284	C₂²	D₃²×C₈	1	1	yes	characteristic
285	C₄	—	3	2	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	1	no	—
288	C₄	—	18	1	no	—
289	C₄	—	18	1	no	—
290	C₄	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
291	C₄	D₃×Dic₃⋊C₄	1	1	yes	characteristic
292	C₃	—	2	1	no	—
293	C₃	#69285	1	1	yes	characteristic, complemented
294	C₃	#71401	1	1	yes	characteristic, complemented
295	C₂	—	6	2	no	—
296	C₂	#10590227	1	1	yes	characteristic
297	C₂	#10590228	1	1	yes	characteristic
298	C₂	D₃×Dic₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₁₆	C₂²
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₁₆	D₃²
hypocenter	C₃²	#2151
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. #10589367	C₂	semidirect
5. C₃×Dic₃⋊C₁₆	C₂	semidirect
6. #10589133	C₂	semidirect
7. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
36. Dic₃⋊C₁₆	D₃	semidirect
255. C₃²	#2151	semidirect
293. C₃	#69285	semidirect
294. C₃	#71401	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×C₁₆	2
#10590692	2
C₂×#63543	2
#10589367	2
C₃×Dic₃⋊C₁₆	2
#10589133	2
Dic₃⋊(C₃⋊C₁₆)	2
#71401	3
#69285	3