Subgroups — C₂²×#88445 #13151969

Order 1728.

2770 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88445	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#3722	—	3	1	no	maximal
6	C₂²×(C₂×D₉)⋊C₄	—	3	1	no	maximal
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂³×C₃⋊D₉	—	2	4	no	—
9	C₂³×C₃⋊D₉	—	2	4	no	—
10	C₂³×C₃⋊D₉	C₄	1	8	yes	complemented
11	C₂³×C₃⋊D₉	C₂²	1	6	yes	—
12	C₂²×C₉×Dic₃	—	2	4	no	—
13	C₂²×C₉×Dic₃	C₂²	1	6	yes	complemented
14	C₂²×C₃×Dic₉	—	2	4	no	—
15	C₂²×C₃×Dic₉	C₂²	1	6	yes	complemented
16	#88445	C₂²	1	16	yes	direct product factor
17	C₂⁴×C₃⋊D₃	—	3	1	no	—
18	C₂³×C₃×Dic₃	—	3	1	no	—
19	C₂³×C₃×Dic₃	D₃	1	1	yes	characteristic
20	C₂⁴×D₉	—	3	1	no	—
21	C₂⁴×D₉	—	6	1	no	—
22	C₂×#3722	—	3	12	no	—
23	C₂³×C₄×C₉	—	3	1	no	—
24	C₂³×Dic₉	D₃	1	1	yes	characteristic, complemented
25	C₂×(C₂×D₉)⋊C₄	—	3	12	no	—
26	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
27	C₂³×C₃×C₉	D₄	1	4	yes	—
28	C₂³×C₃×C₉	D₄	1	4	yes	—
29	C₂³×C₃×C₉	C₂³	1	3	yes	—
30	C₂²×C₃⋊D₉	—	2	32	no	—
31	C₂²×C₃⋊D₉	—	2	12	no	—
32	C₂²×C₃⋊D₉	—	2	12	no	—
33	C₂²×C₃⋊D₉	C₂×C₄	1	24	yes	complemented
34	C₂²×C₃⋊D₉	C₂³	1	4	yes	—
35	C₂×C₉×Dic₃	—	2	12	no	—
36	C₂×C₉×Dic₃	C₂³	1	4	yes	complemented
37	C₂×C₃×Dic₉	—	2	12	no	—
38	C₂×C₃×Dic₉	C₂³	1	4	yes	complemented
39	C₂²×(C₂×D₃)⋊C₄	—	9	1	no	—
40	C₂²×(C₂×D₃)⋊C₄	—	9	1	no	—
41	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
42	C₂³×C₃⋊D₃	—	3	8	no	—
43	C₂³×C₃⋊D₃	—	3	6	no	—
44	C₂³×C₃⋊D₃	—	6	4	no	—
45	C₂³×C₃⋊D₃	—	6	4	no	—
46	C₂²×C₃×Dic₃	—	2	4	no	—
47	C₂²×C₃×Dic₃	—	3	6	no	—
48	C₂²×C₃×Dic₃	—	6	4	no	—
49	C₂²×C₃×Dic₃	C₂×D₃	1	6	yes	—
50	C₂⁴×C₉	—	2	1	no	—
51	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
52	C₂³×D₉	—	3	8	no	—
53	C₂³×D₉	—	3	6	no	—
54	C₂³×D₉	—	6	8	no	—
55	C₂³×D₉	—	6	6	no	—
56	C₂³×D₉	—	6	8	no	—
57	C₂³×D₉	—	6	8	no	—
58	C₂³×D₉	—	6	4	no	—
59	C₂³×D₉	—	6	4	no	—
60	#3722	—	3	16	no	—
61	C₂²×C₄×C₉	—	3	6	no	—
62	C₂²×C₄×C₉	—	6	4	no	—
63	C₂²×Dic₉	—	2	4	no	—
64	C₂²×Dic₉	C₂×D₃	1	6	yes	complemented
65	(C₂×D₉)⋊C₄	—	3	16	no	—
66	C₂²×C₃×C₉	C₂²⋊C₄	1	16	yes	complemented
67	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
68	C₂²×C₃×C₉	C₂×D₄	1	6	yes	—
69	C₂²×C₃×C₉	C₂×D₄	1	6	yes	—
70	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
71	C₂×C₃⋊D₉	—	2	48	no	—
72	C₂×C₃⋊D₉	—	2	4	no	—
73	C₂×C₃⋊D₉	—	2	4	no	—
74	C₂×C₃⋊D₉	C₂²×C₄	1	8	yes	complemented
75	C₉×Dic₃	—	2	4	no	—
76	C₃×Dic₉	—	2	4	no	—
77	C₂⁴×D₃	—	9	1	no	—
78	C₂⁴×D₃	—	9	1	no	—
79	C₂⁴×D₃	—	18	1	no	—
80	C₂³×C₃×C₄	—	3	1	no	—
81	C₂³×C₃×C₄	—	9	1	no	—
82	C₂³×Dic₃	—	3	1	no	—
83	C₂³×Dic₃	D₉	1	1	yes	characteristic, complemented
84	C₂×(C₂×D₃)⋊C₄	—	9	12	no	—
85	C₂×(C₂×D₃)⋊C₄	—	9	12	no	—
86	C₂³×C₃²	C₄×D₃	1	4	yes	—
87	C₂³×C₃²	D₁₂	1	4	yes	—
88	C₂³×C₃²	#67	1	4	yes	—
89	C₂³×C₃²	C₂²×D₃	1	3	yes	—
90	C₂²×C₃⋊D₃	—	3	24	no	—
91	C₂²×C₃⋊D₃	—	3	4	no	—
92	C₂²×C₃⋊D₃	—	6	32	no	—
93	C₂²×C₃⋊D₃	—	6	12	no	—
94	C₂²×C₃⋊D₃	—	6	12	no	—
95	C₂×C₃×Dic₃	—	2	12	no	—
96	C₂×C₃×Dic₃	—	3	4	no	—
97	C₂×C₃×Dic₃	—	6	12	no	—
98	C₂×C₃×Dic₃	C₂²×D₃	1	4	yes	—
99	C₂³×C₉	—	2	4	no	—
100	C₂³×C₉	—	2	4	no	—
101	C₂³×C₉	—	2	4	no	—
102	C₂³×C₉	—	2	3	no	—
103	C₂³×C₉	C₄×D₃	1	4	yes	—
104	C₂³×C₉	D₁₂	1	4	yes	—
105	C₂³×C₉	#67	1	4	yes	—
106	C₂³×C₉	C₂²×D₃	1	3	yes	—
107	C₂²×D₉	—	3	24	no	—
108	C₂²×D₉	—	3	4	no	—
109	C₂²×D₉	—	6	24	no	—
110	C₂²×D₉	—	6	4	no	—
111	C₂²×D₉	—	6	64	no	—
112	C₂²×D₉	—	6	24	no	—
113	C₂²×D₉	—	6	24	no	—
114	C₂²×D₉	—	6	32	no	—
115	C₂²×D₉	—	6	12	no	—
116	C₂²×D₉	—	6	12	no	—
117	C₂×C₄×C₉	—	3	4	no	—
118	C₂×C₄×C₉	—	6	12	no	—
119	C₂×Dic₉	—	2	12	no	—
120	C₂×Dic₉	C₂²×D₃	1	4	yes	complemented
121	C₂²×C₂²⋊C₄	—	27	1	no	Sylow
122	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	12	yes	complemented
123	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
126	C₃⋊D₉	—	2	8	no	—
127	C₂⁴×C₃	—	2	1	no	—
128	C₂⁴×C₃	C₂×D₉	1	1	yes	characteristic
129	C₂⁴×C₃	D₃²	1	1	yes	characteristic
130	C₂³×D₃	—	9	8	no	—
131	C₂³×D₃	—	9	6	no	—
132	C₂³×D₃	—	9	8	no	—
133	C₂³×D₃	—	9	6	no	—
134	C₂³×D₃	—	18	8	no	—
135	C₂³×D₃	—	18	6	no	—
136	C₂³×D₃	—	18	8	no	—
137	C₂³×D₃	—	18	8	no	—
138	C₂³×D₃	—	18	4	no	—
139	C₂³×D₃	—	18	4	no	—
140	C₂³×D₃	—	18	4	no	—
141	C₂³×D₃	—	18	4	no	—
142	C₂²×C₃×C₄	—	3	6	no	—
143	C₂²×C₃×C₄	—	6	4	no	—
144	C₂²×C₃×C₄	—	9	6	no	—
145	C₂²×C₃×C₄	—	18	4	no	—
146	C₂²×Dic₃	—	2	4	no	—
147	C₂²×Dic₃	—	3	6	no	—
148	C₂²×Dic₃	—	6	4	no	—
149	C₂²×Dic₃	C₂×D₉	1	6	yes	complemented
150	(C₂×D₃)⋊C₄	—	9	16	no	—
151	(C₂×D₃)⋊C₄	—	9	16	no	—
152	C₂²×C₃²	(C₂×D₃)⋊C₄	1	16	yes	—
153	C₂²×C₃²	C₂×C₄×D₃	1	6	yes	—
154	C₂²×C₃²	C₂×D₁₂	1	6	yes	—
155	C₂²×C₃²	C₂×#67	1	6	yes	—
156	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
157	C₂×C₃⋊D₃	—	3	8	no	—
158	C₂×C₃⋊D₃	—	6	48	no	—
159	C₂×C₃⋊D₃	—	6	4	no	—
160	C₂×C₃⋊D₃	—	6	4	no	—
161	C₃×Dic₃	—	2	4	no	—
162	C₃×Dic₃	—	6	4	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	2	16	no	—
165	C₂²×C₉	—	2	6	no	—
166	C₂²×C₉	—	2	6	no	—
167	C₂²×C₉	—	2	6	no	—
168	C₂²×C₉	(C₂×D₃)⋊C₄	1	16	yes	complemented
169	C₂²×C₉	C₂×C₄×D₃	1	6	yes	—
170	C₂²×C₉	C₂×D₁₂	1	6	yes	—
171	C₂²×C₉	C₂×#67	1	6	yes	—
172	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
173	C₂×D₉	—	3	8	no	—
174	C₂×D₉	—	6	8	no	—
175	C₂×D₉	—	6	8	no	—
176	C₂×D₉	—	6	8	no	—
177	C₂×D₉	—	6	96	no	—
178	C₂×D₉	—	6	48	no	—
179	C₂×D₉	—	6	4	no	—
180	C₂×D₉	—	6	4	no	—
181	C₄×C₉	—	6	4	no	—
182	Dic₉	—	2	4	no	—
183	C₂⁵	—	27	1	no	—
184	C₂³×C₄	—	3	1	no	—
185	C₂³×C₄	—	9	1	no	—
186	C₂×C₂²⋊C₄	—	27	12	no	—
187	C₃×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₃	—	2	4	no	—
189	C₂³×C₃	—	2	4	no	—
190	C₂³×C₃	—	2	4	no	—
191	C₂³×C₃	—	2	3	no	—
192	C₂³×C₃	C₄×D₉	1	4	yes	—
193	C₂³×C₃	D₃₆	1	4	yes	—
194	C₂³×C₃	#611	1	4	yes	—
195	C₂³×C₃	C₂²×D₉	1	3	yes	—
196	C₂³×C₃	#624	1	4	yes	—
197	C₂³×C₃	#626	1	4	yes	—
198	C₂³×C₃	#626	1	4	yes	—
199	C₂³×C₃	C₂×D₃²	1	3	yes	—
200	C₂²×D₃	—	9	4	no	—
201	C₂²×D₃	—	9	24	no	—
202	C₂²×D₃	—	9	4	no	—
203	C₂²×D₃	—	9	24	no	—
204	C₂²×D₃	—	18	4	no	—
205	C₂²×D₃	—	18	24	no	—
206	C₂²×D₃	—	18	24	no	—
207	C₂²×D₃	—	18	24	no	—
208	C₂²×D₃	—	18	64	no	—
209	C₂²×D₃	—	18	32	no	—
210	C₂²×D₃	—	18	12	no	—
211	C₂²×D₃	—	18	12	no	—
212	C₂²×D₃	—	18	32	no	—
213	C₂²×D₃	—	18	12	no	—
214	C₂²×D₃	—	18	12	no	—
215	C₂×C₃×C₄	—	3	4	no	—
216	C₂×C₃×C₄	—	6	12	no	—
217	C₂×C₃×C₄	—	9	4	no	—
218	C₂×C₃×C₄	—	18	12	no	—
219	C₂×Dic₃	—	2	12	no	—
220	C₂×Dic₃	—	3	4	no	—
221	C₂×Dic₃	—	6	12	no	—
222	C₂×Dic₃	C₂²×D₉	1	4	yes	complemented
223	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	12	yes	—
224	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
225	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
226	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
227	C₃⋊D₃	—	6	8	no	—
228	C₂×C₉	—	2	1	no	—
229	C₂×C₉	—	2	1	no	—
230	C₂×C₉	—	2	1	no	—
231	C₂×C₉	—	2	12	no	—
232	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	12	yes	complemented
233	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
234	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
235	C₂×C₉	C₂²×#67	1	1	yes	characteristic
236	D₉	—	6	16	no	—
237	D₉	—	6	8	no	—
238	C₂⁴	—	27	8	no	—
239	C₂⁴	—	27	6	no	—
240	C₂⁴	—	54	4	no	—
241	C₂⁴	—	54	4	no	—
242	C₂⁴	D₃×D₉	1	1	yes	characteristic
243	C₂²×C₄	—	3	6	no	—
244	C₂²×C₄	—	6	4	no	—
245	C₂²×C₄	—	9	6	no	—
246	C₂²×C₄	—	18	4	no	—
247	C₂²⋊C₄	—	27	16	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	6	no	—
250	C₂²×C₃	—	2	6	no	—
251	C₂²×C₃	—	2	6	no	—
252	C₂²×C₃	—	2	16	no	—
253	C₂²×C₃	(C₂×D₉)⋊C₄	1	16	yes	complemented
254	C₂²×C₃	C₂×C₄×D₉	1	6	yes	—
255	C₂²×C₃	C₂×D₃₆	1	6	yes	—
256	C₂²×C₃	C₂×#611	1	6	yes	—
257	C₂²×C₃	#3722	1	16	yes	—
258	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
259	C₂²×C₃	C₂×#624	1	6	yes	—
260	C₂²×C₃	C₂×#626	1	6	yes	—
261	C₂²×C₃	C₂×#626	1	6	yes	—
262	C₂²×C₃	C₂²×D₃²	1	1	yes	characteristic
263	C₂×D₃	—	9	8	no	—
264	C₂×D₃	—	9	8	no	—
265	C₂×D₃	—	18	8	no	—
266	C₂×D₃	—	18	8	no	—
267	C₂×D₃	—	18	8	no	—
268	C₂×D₃	—	18	96	no	—
269	C₂×D₃	—	18	48	no	—
270	C₂×D₃	—	18	4	no	—
271	C₂×D₃	—	18	4	no	—
272	C₂×D₃	—	18	4	no	—
273	C₂×D₃	—	18	48	no	—
274	C₂×D₃	—	18	4	no	—
275	C₃×C₄	—	6	4	no	—
276	C₃×C₄	—	18	4	no	—
277	Dic₃	—	2	4	no	—
278	Dic₃	—	6	4	no	—
279	C₃²	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
280	C₉	—	2	1	no	—
281	C₉	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
282	C₂³	—	27	4	no	—
283	C₂³	—	27	24	no	—
284	C₂³	—	54	12	no	—
285	C₂³	—	54	12	no	—
286	C₂³	—	54	32	no	—
287	C₂³	(C₃⋊D₉)⋊C₄	1	4	yes	—
288	C₂³	#6192	1	4	yes	—
289	C₂³	#6195	1	4	yes	—
290	C₂³	C₂×D₃×D₉	1	3	yes	—
291	C₂×C₄	—	3	4	no	—
292	C₂×C₄	—	6	12	no	—
293	C₂×C₄	—	9	4	no	—
294	C₂×C₄	—	18	12	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	2	12	no	—
299	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	12	yes	complemented
300	C₂×C₃	C₂²×C₄×D₉	1	1	yes	characteristic
301	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
302	C₂×C₃	C₂²×#611	1	1	yes	characteristic
303	C₂×C₃	C₂×#3722	1	12	yes	—
304	C₂×C₃	C₂²×#624	1	1	yes	characteristic
305	C₂×C₃	C₂²×#626	1	1	yes	characteristic
306	C₂×C₃	C₂²×#626	1	1	yes	characteristic
307	D₃	—	18	16	no	—
308	D₃	—	18	8	no	—
309	D₃	—	18	8	no	—
310	C₂²	—	27	8	no	—
311	C₂²	—	54	4	no	—
312	C₂²	—	54	4	no	—
313	C₂²	—	54	48	no	—
314	C₂²	#88445	1	16	yes	direct product factor
315	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	6	yes	—
316	C₂²	C₂×#6192	1	6	yes	—
317	C₂²	C₂×#6195	1	6	yes	—
318	C₂²	C₂²×D₃×D₉	1	1	yes	characteristic
319	C₄	—	6	4	no	—
320	C₄	—	18	4	no	—
321	C₃	—	2	1	no	—
322	C₃	C₂²×(C₂×D₉)⋊C₄	1	1	yes	characteristic, complemented
323	C₃	C₂²×#3722	1	1	yes	characteristic
324	C₂	—	54	8	no	—
325	C₂	C₂×#88445	1	12	yes	direct product factor
326	C₂	C₂²×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
327	C₂	C₂²×#6192	1	1	yes	characteristic
328	C₂	C₂²×#6195	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	D₃×D₉
commutator	C₂×C₃×C₉	C₂³×C₄
Frattini	C₂²×C₃	C₂²×D₃²
Fitting	C₂⁴×C₃×C₉	C₂²
socle	C₂⁴×C₃²	C₂×D₃
prime radical	C₂⁴×C₃⋊D₉	C₂
cosocle	C₂²×C₃×C₉	C₂⁴
hypocenter	C₃×C₉	C₂²×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂³×C₉×Dic₃	C₂	semidirect
3. C₂³×C₃×Dic₉	C₂	semidirect
4. C₂×#88445	C₂	direct (N × H)
10. C₂³×C₃⋊D₉	C₄	semidirect
13. C₂²×C₉×Dic₃	C₂²	semidirect
15. C₂²×C₃×Dic₉	C₂²	semidirect
16. #88445	C₂²	direct (N × H)
24. C₂³×Dic₉	D₃	semidirect
33. C₂²×C₃⋊D₉	C₂×C₄	semidirect
36. C₂×C₉×Dic₃	C₂³	semidirect
38. C₂×C₃×Dic₉	C₂³	semidirect
64. C₂²×Dic₉	C₂×D₃	semidirect
66. C₂²×C₃×C₉	C₂²⋊C₄	semidirect
74. C₂×C₃⋊D₉	C₂²×C₄	semidirect
83. C₂³×Dic₃	D₉	semidirect
120. C₂×Dic₉	C₂²×D₃	semidirect
122. C₂×C₃×C₉	C₂×C₂²⋊C₄	semidirect
149. C₂²×Dic₃	C₂×D₉	semidirect
168. C₂²×C₉	(C₂×D₃)⋊C₄	semidirect
187. C₃×C₉	C₂²×C₂²⋊C₄	semidirect
222. C₂×Dic₃	C₂²×D₉	semidirect
232. C₂×C₉	C₂×(C₂×D₃)⋊C₄	semidirect
253. C₂²×C₃	(C₂×D₉)⋊C₄	semidirect
281. C₉	C₂²×(C₂×D₃)⋊C₄	semidirect
299. C₂×C₃	C₂×(C₂×D₉)⋊C₄	semidirect
314. C₂²	#88445	direct (N × H)
322. C₃	C₂²×(C₂×D₉)⋊C₄	semidirect
325. C₂	C₂×#88445	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃⋊D₉	2
C₂³×C₉×Dic₃	2
C₂³×C₃×Dic₉	2
C₂×#88445	2
C₂²×#3722	3
C₂²×(C₂×D₉)⋊C₄	3

Subgroups — C22×#88445 #13151969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88445 #13151969