Subgroups — #11789796

Order 1152.

438 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63380	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588668	C₂	1	1	yes	maximal, characteristic
4	#10588664	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#63369	C₂	1	1	yes	maximal, characteristic
6	C₉×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588206	C₂	1	1	yes	maximal, characteristic, complemented
8	#68853	—	3	1	no	maximal
9	C₂²×D₃₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₉)⋊C₄	—	2	1	no	—
14	C₂×(C₂×D₉)⋊C₄	—	2	1	no	—
15	C₂×C₄⋊Dic₉	—	2	1	no	—
16	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
18	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₉⋊#95	—	2	1	no	—
20	#63380	—	2	2	no	—
21	C₈×Dic₉	—	2	2	no	—
22	#63369	C₄	1	4	yes	complemented
23	C₂×#821	—	3	1	no	—
24	C₂×C₈×Dic₃	—	3	1	no	—
25	#4988	—	3	1	no	—
26	#4984	—	3	1	no	—
27	C₂×#809	—	3	1	no	—
28	C₃×#340	D₃	1	1	yes	characteristic
29	#4529	—	3	1	no	—
30	C₂³×D₉	—	2	1	no	—
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	—	2	1	no	—
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₉	—	2	1	no	—
35	C₂²×Dic₉	D₄	1	1	yes	characteristic
36	C₂²×Dic₉	D₄	1	1	yes	characteristic
37	C₂×D₃₆	—	2	1	no	—
38	C₂×D₃₆	—	2	1	no	—
39	C₂×D₃₆	—	2	1	no	—
40	C₂×D₃₆	—	2	1	no	—
41	C₂×D₃₆	—	2	1	no	—
42	C₂×D₃₆	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₉	—	2	1	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	D₄	1	2	yes	—
46	C₉×C₄⋊C₄	—	2	1	no	—
47	C₉×C₄⋊C₄	—	2	1	no	—
48	C₉×C₄⋊C₄	—	2	1	no	—
49	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	(C₂×D₉)⋊C₄	—	4	2	no	—
51	(C₂×D₉)⋊C₄	—	4	2	no	—
52	C₄⋊Dic₉	—	2	1	no	—
53	C₄⋊Dic₉	—	2	1	no	—
54	C₄⋊Dic₉	—	4	2	no	—
55	C₄×Dic₉	—	2	1	no	—
56	C₄×Dic₉	—	2	1	no	—
57	C₂×C₉⋊C₈	—	2	1	no	—
58	C₂×C₉⋊C₈	—	2	1	no	—
59	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
60	#1934	—	9	1	no	Sylow
61	C₂²×D₁₂	—	3	1	no	—
62	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
65	C₂×(C₂×D₃)⋊C₄	—	6	1	no	—
66	C₂×(C₂×D₃)⋊C₄	—	6	1	no	—
67	C₂×C₄⋊Dic₃	—	3	1	no	—
68	C₂×C₄⋊Dic₃	—	6	1	no	—
69	C₂×C₄×Dic₃	—	3	1	no	—
70	C₂²×C₃⋊C₈	—	3	1	no	—
71	C₃⋊#95	—	6	1	no	—
72	#821	—	6	2	no	—
73	C₈×Dic₃	—	6	2	no	—
74	#809	—	3	4	no	—
75	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×D₉	—	2	2	no	—
77	C₂²×D₉	—	4	1	no	—
78	C₂²×D₉	—	4	1	no	—
79	C₂²×D₉	—	4	1	no	—
80	C₂²×D₉	—	4	1	no	—
81	C₂²×D₉	—	4	1	no	—
82	C₂²×D₉	—	4	1	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	4	1	no	—
85	C₂×C₄×C₉	—	4	1	no	—
86	C₂×C₄×C₉	—	4	1	no	—
87	C₂×C₄×C₉	—	4	1	no	—
88	C₂×C₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	#41	1	1	yes	characteristic
92	C₂×C₄×C₉	#41	1	1	yes	characteristic
93	C₂×C₄×C₉	#41	1	1	yes	characteristic
94	C₂×C₄×C₉	#41	1	1	yes	characteristic
95	C₂×Dic₉	—	2	1	no	—
96	C₂×Dic₉	—	2	1	no	—
97	C₂×Dic₉	—	2	1	no	—
98	C₂×Dic₉	—	2	1	no	—
99	C₂×Dic₉	—	4	1	no	—
100	C₂×Dic₉	—	4	1	no	—
101	C₂×Dic₉	—	4	1	no	—
102	C₂×Dic₉	D₈	1	2	yes	—
103	C₂×Dic₉	SD₁₆	1	2	yes	—
104	D₃₆	—	2	2	no	—
105	D₃₆	—	2	2	no	—
106	D₃₆	—	4	1	no	—
107	D₃₆	—	4	1	no	—
108	C₈×C₉	—	2	2	no	—
109	C₉⋊C₈	—	2	2	no	—
110	C₂×#102	—	9	1	no	—
111	C₂×#102	—	9	1	no	—
112	C₂×C₄×C₈	—	9	1	no	—
113	#390	—	9	1	no	—
114	#389	—	9	1	no	—
115	#340	—	9	1	no	—
116	#340	D₉	1	1	yes	characteristic, complemented
117	C₂³×D₃	—	6	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
121	C₂²×Dic₃	—	3	1	no	—
122	C₂²×Dic₃	—	3	1	no	—
123	C₂²×Dic₃	—	6	1	no	—
124	C₂×D₁₂	—	3	2	no	—
125	C₂×D₁₂	—	6	1	no	—
126	C₂×D₁₂	—	6	1	no	—
127	C₂×D₁₂	—	6	1	no	—
128	C₂×D₁₂	—	6	1	no	—
129	C₂×D₁₂	—	6	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	D₁₂	1	2	yes	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₄⋊C₄	—	2	1	no	—
136	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
137	(C₂×D₃)⋊C₄	—	12	2	no	—
138	(C₂×D₃)⋊C₄	—	12	2	no	—
139	C₄⋊Dic₃	—	6	1	no	—
140	C₄⋊Dic₃	—	6	1	no	—
141	C₄⋊Dic₃	—	12	2	no	—
142	C₄×Dic₃	—	6	1	no	—
143	C₄×Dic₃	—	6	1	no	—
144	C₂×C₃⋊C₈	—	3	2	no	—
145	C₂×C₃⋊C₈	—	6	1	no	—
146	C₂×C₃⋊C₈	—	6	1	no	—
147	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₉	#121	1	1	yes	characteristic
149	C₂²×C₉	#124	1	1	yes	characteristic
150	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
151	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
152	C₂²×C₉	#135	1	1	yes	characteristic
153	C₂²×C₉	#135	1	1	yes	characteristic
154	C₂×D₉	—	4	2	no	—
155	C₂×D₉	—	4	2	no	—
156	C₂×D₉	—	4	2	no	—
157	C₂×D₉	—	8	1	no	—
158	C₂×D₉	—	8	1	no	—
159	C₂×D₉	—	8	1	no	—
160	C₂×D₉	—	8	1	no	—
161	C₄×C₉	—	4	2	no	—
162	C₄×C₉	—	8	1	no	—
163	C₄×C₉	#117	1	1	yes	characteristic
164	C₄×C₉	C₄×D₄	1	1	yes	characteristic
165	C₄×C₉	#123	1	1	yes	characteristic
166	C₄×C₉	#126	1	1	yes	characteristic
167	Dic₉	—	2	2	no	—
168	Dic₉	—	2	2	no	—
169	Dic₉	—	8	1	no	—
170	C₂²×D₄	—	9	1	no	—
171	C₂²×C₈	—	9	1	no	—
172	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	9	1	no	—
174	C₂×C₄⋊C₄	—	18	1	no	—
175	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	18	1	no	—
178	C₂×C₂²⋊C₄	—	18	1	no	—
179	C₂×C₄²	—	9	1	no	—
180	#102	—	9	4	no	—
181	#102	—	18	2	no	—
182	C₄×C₈	—	18	2	no	—
183	#95	—	18	1	no	—
184	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
185	C₂²×D₃	—	6	2	no	—
186	C₂²×D₃	—	12	1	no	—
187	C₂²×D₃	—	12	1	no	—
188	C₂²×D₃	—	12	1	no	—
189	C₂²×D₃	—	12	1	no	—
190	C₂²×D₃	—	12	1	no	—
191	C₂²×D₃	—	12	1	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	4	1	no	—
194	C₂×C₃×C₄	—	4	1	no	—
195	C₂×C₃×C₄	—	4	1	no	—
196	C₂×C₃×C₄	—	4	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
200	C₂×C₃×C₄	#235	1	1	yes	characteristic
201	C₂×C₃×C₄	#237	1	1	yes	characteristic
202	C₂×C₃×C₄	#237	1	1	yes	characteristic
203	C₂×C₃×C₄	#239	1	1	yes	characteristic
204	C₂×Dic₃	—	3	2	no	—
205	C₂×Dic₃	—	3	2	no	—
206	C₂×Dic₃	—	6	1	no	—
207	C₂×Dic₃	—	6	1	no	—
208	C₂×Dic₃	—	6	1	no	—
209	C₂×Dic₃	—	6	1	no	—
210	C₂×Dic₃	—	12	1	no	—
211	C₂×Dic₃	—	12	1	no	—
212	C₂×Dic₃	—	12	1	no	—
213	D₁₂	—	6	2	no	—
214	D₁₂	—	6	2	no	—
215	D₁₂	—	12	1	no	—
216	D₁₂	—	12	1	no	—
217	C₃×C₈	—	2	2	no	—
218	C₃⋊C₈	—	6	2	no	—
219	C₂×C₉	#388	1	1	yes	characteristic
220	C₂×C₉	C₄×D₈	1	1	yes	characteristic
221	C₂×C₉	C₄×SD₁₆	1	1	yes	characteristic
222	C₂×C₉	#465	1	1	yes	characteristic
223	C₂×C₉	#466	1	1	yes	characteristic
224	C₂×C₉	#486	1	1	yes	characteristic
225	C₂×C₉	#488	1	1	yes	characteristic
226	D₉	—	8	2	no	—
227	C₂⁴	—	18	1	no	—
228	C₂×D₄	—	9	2	no	—
229	C₂×D₄	—	18	1	no	—
230	C₂×D₄	—	18	1	no	—
231	C₂×D₄	—	18	1	no	—
232	C₂×D₄	—	18	1	no	—
233	C₂×D₄	—	18	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	9	1	no	—
237	C₂²×C₄	—	9	1	no	—
238	C₂²×C₄	—	18	1	no	—
239	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	9	2	no	—
243	C₂×C₈	—	18	1	no	—
244	C₂×C₈	—	18	1	no	—
245	C₂×C₈	D₃₆	1	2	yes	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	18	1	no	—
250	C₄⋊C₄	—	18	1	no	—
251	C₄⋊C₄	—	36	2	no	—
252	C₄⋊C₄	C₄×D₉	1	2	yes	—
253	C₂²⋊C₄	—	36	2	no	—
254	C₂²⋊C₄	—	36	2	no	—
255	C₄²	—	18	1	no	—
256	C₄²	—	18	1	no	—
257	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
258	C₂²×C₃	#885	1	1	yes	characteristic
259	C₂²×C₃	#895	1	1	yes	characteristic
260	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
261	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
262	C₂²×C₃	#916	1	1	yes	characteristic
263	C₂²×C₃	#919	1	1	yes	characteristic
264	C₂×D₃	—	12	2	no	—
265	C₂×D₃	—	12	2	no	—
266	C₂×D₃	—	12	2	no	—
267	C₂×D₃	—	24	1	no	—
268	C₂×D₃	—	24	1	no	—
269	C₂×D₃	—	24	1	no	—
270	C₂×D₃	—	24	1	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	8	1	no	—
273	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
274	C₃×C₄	#886	1	1	yes	characteristic
275	C₃×C₄	#892	1	1	yes	characteristic
276	C₃×C₄	#898	1	1	yes	characteristic
277	Dic₃	—	6	2	no	—
278	Dic₃	—	6	2	no	—
279	Dic₃	—	24	1	no	—
280	C₉	#1934	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	18	2	no	—
282	C₂³	—	36	1	no	—
283	C₂³	—	36	1	no	—
284	C₂³	—	36	1	no	—
285	C₂³	—	36	1	no	—
286	C₂³	—	36	1	no	—
287	C₂³	—	36	1	no	—
288	C₂³	D₄×D₉	1	1	yes	characteristic
289	D₄	—	18	2	no	—
290	D₄	—	18	2	no	—
291	D₄	—	36	1	no	—
292	D₄	—	36	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	9	2	no	—
300	C₂×C₄	—	9	2	no	—
301	C₂×C₄	—	18	1	no	—
302	C₂×C₄	—	18	1	no	—
303	C₂×C₄	—	18	1	no	—
304	C₂×C₄	—	18	1	no	—
305	C₂×C₄	—	36	1	no	—
306	C₂×C₄	—	36	1	no	—
307	C₂×C₄	—	36	1	no	—
308	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
310	C₂×C₄	#3697	1	1	yes	characteristic
311	C₂×C₄	#3699	1	1	yes	characteristic
312	C₂×C₄	#3699	1	1	yes	characteristic
313	C₂×C₄	#3701	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	18	2	no	—
316	C₂×C₃	#4670	1	1	yes	characteristic
317	C₂×C₃	#4756	1	1	yes	characteristic
318	C₂×C₃	#4760	1	1	yes	characteristic
319	C₂×C₃	#4788	1	1	yes	characteristic
320	C₂×C₃	#4800	1	1	yes	characteristic
321	C₂×C₃	#4864	1	1	yes	characteristic
322	C₂×C₃	#4883	1	1	yes	characteristic
323	D₃	—	24	2	no	—
324	C₂²	—	36	2	no	—
325	C₂²	—	36	2	no	—
326	C₂²	—	36	2	no	—
327	C₂²	—	72	1	no	—
328	C₂²	—	72	1	no	—
329	C₂²	—	72	1	no	—
330	C₂²	—	72	1	no	—
331	C₂²	#611⋊C₄	1	1	yes	characteristic
332	C₂²	#63447	1	1	yes	characteristic
333	C₂²	#63457	1	1	yes	characteristic
334	C₂²	D₈×D₉	1	1	yes	characteristic
335	C₂²	SD₁₆×D₉	1	1	yes	characteristic
336	C₂²	#63478	1	1	yes	characteristic
337	C₂²	#63481	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	8	1	no	—
340	C₄	—	18	2	no	—
341	C₄	—	18	2	no	—
342	C₄	—	72	1	no	—
343	C₄	C₄×D₃₆	1	1	yes	characteristic
344	C₄	#63448	1	1	yes	characteristic
345	C₄	#63454	1	1	yes	characteristic
346	C₄	#63460	1	1	yes	characteristic
347	C₃	#68853	1	1	yes	characteristic
348	C₂	—	72	2	no	—
349	C₂	#10588350	1	1	yes	characteristic
350	C₂	#10588436	1	1	yes	characteristic
351	C₂	#10588440	1	1	yes	characteristic
352	C₂	#10588468	1	1	yes	characteristic
353	C₂	#10588480	1	1	yes	characteristic
354	C₂	#10588544	1	1	yes	characteristic
355	C₂	#10588563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#340	C₂
socle	C₂³×C₃	D₃×D₄
prime radical	C₂²×D₃₆	C₂²
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#340	D₉
hypocenter	C₉	#1934
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Dic₉	C₂	semidirect
4. #10588664	C₂	semidirect
6. C₉×#340	C₂	semidirect
7. #10588206	C₂	semidirect
22. #63369	C₄	semidirect
116. #340	D₉	semidirect
280. C₉	#1934	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63380	2
C₂×C₈×Dic₉	2
#10588668	2
#10588664	2
C₂×#63369	2
C₉×#340	2
#10588206	2
#68853	3