Subgroups — C₂×#11720269 #13139339

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6226	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720269	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63567	—	3	1	no	maximal
6	C₂×#63368	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	—	2	1	no	—
9	C₂×#6226	—	2	1	no	—
10	C₂×#6226	—	2	2	no	—
11	C₂×#6226	C₄	1	4	yes	complemented
12	C₂×#6226	C₂²	1	2	yes	—
13	C₂²×C₉×Dic₃	—	2	1	no	—
14	C₉×C₄⋊Dic₃	—	2	1	no	—
15	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₃×C₉⋊C₈	—	2	2	no	—
17	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
18	C₂²×#634	—	3	1	no	—
19	C₂×C₃×C₄⋊Dic₃	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
21	C₂²×Dic₁₈	—	3	1	no	—
22	C₂²×Dic₁₈	—	6	1	no	—
23	#63567	—	3	4	no	—
24	C₂×C₉×C₄⋊C₄	—	3	1	no	—
25	C₂²×C₉⋊C₈	D₃	1	1	yes	characteristic
26	#63368	—	3	4	no	—
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₃⋊Dic₉	—	2	4	no	—
33	C₂×C₃⋊Dic₉	—	2	2	no	—
34	#6226	—	2	4	no	—
35	#6226	—	2	2	no	—
36	#6226	C₂×C₄	1	4	yes	complemented
37	C₂×C₉×Dic₃	—	2	2	no	—
38	C₂×C₉×Dic₃	—	4	2	no	—
39	C₃×C₉⋊C₈	—	2	2	no	—
40	C₂×#816	—	9	1	no	—
41	C₂×#810	—	9	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃⋊Dic₃	—	6	1	no	—
44	C₂×#634	—	3	4	no	—
45	C₂×#634	—	3	2	no	—
46	C₂×#634	—	6	2	no	—
47	C₂×#634	—	6	1	no	—
48	C₂²×C₃×Dic₃	—	2	1	no	—
49	C₃×C₄⋊Dic₃	—	2	1	no	—
50	C₃×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
51	C₂×C₃×C₃⋊C₈	—	3	2	no	—
52	C₂×C₃×C₃⋊C₈	—	6	2	no	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	—	6	1	no	—
55	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂²×Dic₉	—	6	2	no	—
57	C₂²×Dic₉	—	6	1	no	—
58	C₂×Dic₁₈	—	3	4	no	—
59	C₂×Dic₁₈	—	3	2	no	—
60	C₂×Dic₁₈	—	6	4	no	—
61	C₂×Dic₁₈	—	6	2	no	—
62	C₂×Dic₁₈	—	6	2	no	—
63	C₂×Dic₁₈	—	6	4	no	—
64	C₂×Dic₁₈	—	6	1	no	—
65	C₂×Dic₁₈	—	6	2	no	—
66	C₉×C₄⋊C₄	—	3	2	no	—
67	C₉×C₄⋊C₄	—	6	1	no	—
68	C₂×C₉⋊C₈	—	2	2	no	—
69	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
70	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
72	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
73	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
74	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
75	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₃⋊Dic₉	—	2	4	no	—
78	C₉×Dic₃	—	4	2	no	—
79	C₂²×Dic₆	—	9	1	no	—
80	C₂²×Dic₆	—	9	1	no	—
81	C₂²×Dic₆	—	18	1	no	—
82	C₂²×C₃×C₈	—	9	1	no	—
83	C₂×C₃×C₄⋊C₄	—	3	1	no	—
84	C₂×C₄⋊Dic₃	D₉	1	1	yes	characteristic
85	C₂²×C₃⋊C₈	—	3	1	no	—
86	#816	—	9	4	no	—
87	#810	—	9	4	no	—
88	C₂³×C₃²	#67	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
90	C₂×C₃²×C₄	D₁₂	1	2	yes	—
91	C₂×C₃²×C₄	#67	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₃⋊Dic₃	—	6	4	no	—
94	C₂×C₃⋊Dic₃	—	6	2	no	—
95	#634	—	3	4	no	—
96	#634	—	6	4	no	—
97	#634	—	6	2	no	—
98	C₂×C₃×Dic₃	—	2	2	no	—
99	C₂×C₃×Dic₃	—	4	2	no	—
100	C₂³×C₉	—	2	1	no	—
101	C₂³×C₉	D₁₂	1	1	yes	characteristic
102	C₃×C₃⋊C₈	—	6	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	1	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	6	2	no	—
108	C₂×C₄×C₉	—	12	2	no	—
109	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
110	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
111	C₂×C₄×C₉	#67	1	2	yes	—
112	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
113	C₂×Dic₉	—	6	8	no	—
114	C₂×Dic₉	—	6	4	no	—
115	C₂×Dic₉	—	6	4	no	—
116	C₂×Dic₉	—	6	2	no	—
117	Dic₁₈	—	3	4	no	—
118	Dic₁₈	—	6	4	no	—
119	Dic₁₈	—	6	8	no	—
120	Dic₁₈	—	6	4	no	—
121	Dic₁₈	—	6	4	no	—
122	Dic₁₈	—	6	2	no	—
123	C₉⋊C₈	—	2	2	no	—
124	C₂×#103	—	27	1	no	Sylow
125	C₂×C₃×C₉	#103	1	4	yes	complemented
126	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	—	6	1	no	—
131	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
132	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
133	C₂²×Dic₃	—	2	1	no	—
134	C₂²×Dic₃	—	18	2	no	—
135	C₂²×Dic₃	—	18	1	no	—
136	C₂²×Dic₃	—	18	1	no	—
137	C₂×Dic₆	—	9	4	no	—
138	C₂×Dic₆	—	9	2	no	—
139	C₂×Dic₆	—	9	4	no	—
140	C₂×Dic₆	—	9	2	no	—
141	C₂×Dic₆	—	18	4	no	—
142	C₂×Dic₆	—	18	2	no	—
143	C₂×Dic₆	—	18	4	no	—
144	C₂×Dic₆	—	18	2	no	—
145	C₂×Dic₆	—	18	2	no	—
146	C₂×Dic₆	—	18	1	no	—
147	C₂×Dic₆	—	18	2	no	—
148	C₂×Dic₆	—	18	1	no	—
149	C₂×C₃×C₈	—	9	2	no	—
150	C₂×C₃×C₈	—	18	2	no	—
151	C₃×C₄⋊C₄	—	3	2	no	—
152	C₃×C₄⋊C₄	—	6	1	no	—
153	C₄⋊Dic₃	—	2	1	no	—
154	C₄⋊Dic₃	C₂×D₉	1	2	yes	—
155	C₂×C₃⋊C₈	—	3	2	no	—
156	C₂×C₃⋊C₈	—	6	2	no	—
157	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
158	C₂²×C₃²	#214	1	2	yes	—
159	C₂²×C₃²	#216	1	2	yes	—
160	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
161	C₃²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
162	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
163	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
164	C₃⋊Dic₃	—	6	4	no	—
165	C₃×Dic₃	—	4	2	no	—
166	C₂²×C₉	—	2	2	no	—
167	C₂²×C₉	—	2	2	no	—
168	C₂²×C₉	—	2	2	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	#204	1	2	yes	—
171	C₂²×C₉	Dic₁₂	1	2	yes	—
172	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
173	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
174	C₄×C₉	—	2	2	no	—
175	C₄×C₉	—	2	1	no	—
176	C₄×C₉	—	2	1	no	—
177	C₄×C₉	—	12	2	no	—
178	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
179	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
180	C₄×C₉	C₂×#67	1	1	yes	characteristic
181	Dic₉	—	6	8	no	—
182	Dic₉	—	6	4	no	—
183	C₂²×Q₈	—	27	1	no	—
184	C₂²×C₈	—	9	1	no	—
185	C₂×C₄⋊C₄	—	3	1	no	—
186	#103	—	27	4	no	—
187	C₃×C₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₃	—	2	1	no	—
189	C₂³×C₃	#611	1	1	yes	characteristic
190	C₂³×C₃	#626	1	1	yes	characteristic
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	6	2	no	—
196	C₂×C₃×C₄	—	12	2	no	—
197	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
198	C₂×C₃×C₄	D₃₆	1	2	yes	—
199	C₂×C₃×C₄	#611	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
201	C₂×C₃×C₄	#624	1	2	yes	—
202	C₂×C₃×C₄	#626	1	2	yes	—
203	C₂×C₃×C₄	#626	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
205	C₂×Dic₃	—	2	2	no	—
206	C₂×Dic₃	—	4	2	no	—
207	C₂×Dic₃	—	18	4	no	—
208	C₂×Dic₃	—	18	8	no	—
209	C₂×Dic₃	—	18	4	no	—
210	C₂×Dic₃	—	18	2	no	—
211	C₂×Dic₃	—	18	4	no	—
212	C₂×Dic₃	—	18	2	no	—
213	Dic₆	—	9	4	no	—
214	Dic₆	—	9	4	no	—
215	Dic₆	—	18	4	no	—
216	Dic₆	—	18	4	no	—
217	Dic₆	—	18	8	no	—
218	Dic₆	—	18	4	no	—
219	Dic₆	—	18	2	no	—
220	Dic₆	—	18	4	no	—
221	Dic₆	—	18	2	no	—
222	C₃×C₈	—	18	2	no	—
223	C₃⋊C₈	—	6	2	no	—
224	C₂×C₃²	#810	1	4	yes	—
225	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
226	C₂×C₃²	C₂×#214	1	1	yes	characteristic
227	C₂×C₃²	C₂×#216	1	1	yes	characteristic
228	C₂×C₉	—	2	4	no	—
229	C₂×C₉	—	2	1	no	—
230	C₂×C₉	—	2	1	no	—
231	C₂×C₉	—	2	1	no	—
232	C₂×C₉	#816	1	4	yes	complemented
233	C₂×C₉	C₂×#204	1	1	yes	characteristic
234	C₂×C₉	C₂×Dic₁₂	1	1	yes	characteristic
235	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
236	C₂×Q₈	—	27	4	no	—
237	C₂×Q₈	—	27	2	no	—
238	C₂×Q₈	—	54	2	no	—
239	C₂×Q₈	—	54	1	no	—
240	C₂²×C₄	—	6	1	no	—
241	C₂²×C₄	—	54	1	no	—
242	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
243	C₂×C₈	—	9	2	no	—
244	C₂×C₈	—	18	2	no	—
245	C₄⋊C₄	—	3	2	no	—
246	C₄⋊C₄	—	6	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	(C₂×D₉)⋊C₄	1	2	yes	—
252	C₂²×C₃	#3672	1	2	yes	—
253	C₂²×C₃	#3674	1	2	yes	—
254	C₂²×C₃	C₂×#611	1	1	yes	characteristic
255	C₂²×C₃	#3716	1	2	yes	—
256	C₂²×C₃	#3719	1	2	yes	—
257	C₂²×C₃	#3722	1	2	yes	—
258	C₂²×C₃	C₂×#626	1	1	yes	characteristic
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	12	2	no	—
263	C₃×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
264	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
265	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
266	C₃×C₄	#3722	1	2	yes	—
267	C₃×C₄	C₂×#624	1	1	yes	characteristic
268	C₃×C₄	C₂×#626	1	1	yes	characteristic
269	Dic₃	—	4	2	no	—
270	Dic₃	—	18	8	no	—
271	Dic₃	—	18	4	no	—
272	Dic₃	—	18	4	no	—
273	C₃²	C₂×#810	1	1	yes	characteristic
274	C₉	—	2	1	no	—
275	C₉	C₂×#816	1	1	yes	characteristic, complemented
276	C₂³	#6195	1	1	yes	characteristic
277	Q₈	—	27	4	no	—
278	Q₈	—	54	2	no	—
279	Q₈	—	54	4	no	—
280	C₂×C₄	—	6	2	no	—
281	C₂×C₄	—	12	2	no	—
282	C₂×C₄	—	54	2	no	—
283	C₂×C₄	—	54	4	no	—
284	C₂×C₄	(C₃⋊D₉)⋊C₄	1	2	yes	—
285	C₂×C₄	#6192	1	2	yes	—
286	C₂×C₄	#6195	1	1	yes	characteristic
287	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
288	C₈	—	18	2	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	4	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	#63368	1	4	yes	complemented
294	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
295	C₂×C₃	C₂×#3672	1	1	yes	characteristic
296	C₂×C₃	C₂×#3674	1	1	yes	characteristic
297	C₂×C₃	#63567	1	4	yes	—
298	C₂×C₃	C₂×#3716	1	1	yes	characteristic
299	C₂×C₃	C₂×#3719	1	1	yes	characteristic
300	C₂×C₃	C₂×#3722	1	1	yes	characteristic
301	C₂²	#88424	1	2	yes	—
302	C₂²	#88428	1	2	yes	—
303	C₂²	#88445	1	2	yes	—
304	C₂²	C₂×#6195	1	1	yes	characteristic
305	C₄	—	12	2	no	—
306	C₄	—	54	4	no	—
307	C₄	#88445	1	2	yes	—
308	C₄	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
309	C₄	C₂×#6192	1	1	yes	characteristic
310	C₃	—	2	1	no	—
311	C₃	C₂×#63368	1	1	yes	characteristic, complemented
312	C₃	C₂×#63567	1	1	yes	characteristic
313	C₂	#11720269	1	4	yes	direct product factor
314	C₂	C₂×#88424	1	1	yes	characteristic
315	C₂	C₂×#88428	1	1	yes	characteristic
316	C₂	C₂×#88445	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6195
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#103
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720269	C₂	direct (N × H)
11. C₂×#6226	C₄	semidirect
36. #6226	C₂×C₄	semidirect
125. C₂×C₃×C₉	#103	semidirect
187. C₃×C₉	C₂×#103	semidirect
232. C₂×C₉	#816	semidirect
275. C₉	C₂×#816	semidirect
293. C₂×C₃	#63368	semidirect
311. C₃	C₂×#63368	semidirect
313. C₂	#11720269	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#6226	2
C₂×C₉×C₄⋊Dic₃	2
C₂²×C₃×C₉⋊C₈	2
#11720269	2
C₂×#63567	3
C₂×#63368	3

Subgroups — C2×#11720269 #13139339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720269 #13139339