Subgroups — #11790208

Order 1152.

545 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63645	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#815	C₂	1	1	yes	maximal, characteristic
4	#10591352	C₂	1	2	yes	maximal
5	#10589164	C₂	1	2	yes	maximal
6	#70562	—	3	1	no	maximal
7	#69820	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₃⋊Dic₃	—	2	2	no	—
14	#63645	—	2	2	no	—
15	C₃×#817	C₄	1	4	yes	complemented
16	C₃×#815	—	2	2	no	—
17	#63579	—	2	2	no	—
18	C₂×C₃×#106	—	3	1	no	—
19	C₂×C₃×#97	—	3	1	no	—
20	C₂×#818	—	3	1	no	—
21	C₂×#818	—	3	1	no	—
22	C₂×#818	—	6	1	no	—
23	C₂×#817	D₃	1	1	yes	characteristic
24	C₂×#815	D₃	1	1	yes	characteristic
25	#4984	—	3	2	no	—
26	#4928	—	3	2	no	—
27	#4551	—	3	2	no	—
28	#4529	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	2	yes	—
36	C₂×C₃²×C₈	Q₈	1	2	yes	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
39	C₃×C₄⋊Dic₃	—	2	2	no	—
40	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
41	C₃×C₄×Dic₃	—	2	1	no	—
42	C₃×C₄×Dic₃	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
46	Dic₃⋊Dic₃	—	4	4	no	—
47	#1944	—	9	1	no	Sylow
48	C₂²×C₃×C₈	—	2	1	no	—
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄⋊C₄	—	3	2	no	—
53	C₂×C₃×C₄²	—	3	1	no	—
54	C₂×C₄⋊Dic₃	—	3	2	no	—
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₂×C₄⋊Dic₃	—	6	2	no	—
57	C₂×C₄⋊Dic₃	—	6	2	no	—
58	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
59	C₂×Dic₃⋊C₄	—	6	2	no	—
60	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×#106	—	3	4	no	—
63	C₃×#97	—	6	2	no	—
64	C₃⋊#95	—	6	2	no	—
65	C₃⋊#95	—	6	2	no	—
66	#818	—	6	4	no	—
67	#818	—	6	2	no	—
68	#818	—	6	2	no	—
69	#817	Dic₃	1	4	yes	complemented
70	#815	—	2	2	no	—
71	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	#41	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	4	2	no	—
79	C₂×C₃⋊Dic₃	—	4	2	no	—
80	C₂×C₃⋊Dic₃	—	4	2	no	—
81	C₂×C₃×Dic₃	—	2	4	no	—
82	C₂×C₃×Dic₃	—	2	1	no	—
83	C₂×C₃×Dic₃	—	2	1	no	—
84	C₂×C₃×Dic₃	—	2	2	no	—
85	C₂×C₃×Dic₃	—	2	2	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₂×C₃×Dic₃	—	4	2	no	—
88	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
89	C₃×C₃⋊C₈	—	2	2	no	—
90	C₂×#107	—	9	1	no	—
91	C₂×#106	—	3	1	no	—
92	C₂×#97	—	3	1	no	—
93	#389	—	9	2	no	—
94	#340	—	9	2	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	3	1	no	—
97	C₂²×C₃×C₄	—	3	1	no	—
98	C₂²×C₃×C₄	—	6	2	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	2	no	—
102	C₂²×Dic₃	—	6	2	no	—
103	C₂²×Dic₃	—	6	2	no	—
104	C₂²×Dic₃	—	12	2	no	—
105	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
106	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	3	2	no	—
111	C₂×C₃×C₈	—	6	1	no	—
112	C₂×C₃×C₈	—	6	1	no	—
113	C₂×C₃×C₈	Dic₆	1	2	yes	—
114	C₂×C₃×C₈	Dic₆	1	2	yes	—
115	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
116	C₂×C₃×C₈	D₁₂	1	2	yes	—
117	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
118	C₂×C₃×C₈	#67	1	2	yes	—
119	C₃×C₄⋊C₄	—	3	4	no	—
120	C₃×C₄⋊C₄	—	6	2	no	—
121	C₃×C₄²	—	6	1	no	—
122	C₃×C₄²	—	6	1	no	—
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄⋊Dic₃	—	6	4	no	—
125	C₄⋊Dic₃	—	6	2	no	—
126	C₄⋊Dic₃	—	6	2	no	—
127	C₄⋊Dic₃	—	6	2	no	—
128	C₄⋊Dic₃	—	6	2	no	—
129	C₄⋊Dic₃	—	12	2	no	—
130	C₄⋊Dic₃	—	12	4	no	—
131	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
132	Dic₃⋊C₄	—	12	4	no	—
133	C₄×Dic₃	—	2	1	no	—
134	C₄×Dic₃	—	2	1	no	—
135	C₂×C₃⋊C₈	—	2	1	no	—
136	C₂×C₃⋊C₈	—	2	1	no	—
137	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
138	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₃²	#121	1	1	yes	characteristic
140	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
142	C₂²×C₃²	#137	1	2	yes	—
143	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
145	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
146	C₃²×C₄	#125	1	1	yes	characteristic
147	C₃⋊Dic₃	—	8	2	no	—
148	C₃×Dic₃	—	4	4	no	—
149	C₃×Dic₃	—	4	1	no	—
150	C₃×Dic₃	—	4	1	no	—
151	C₂²×C₈	—	3	1	no	—
152	C₂²×C₈	D₃²	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	3	2	no	—
154	C₂×C₄⋊C₄	—	9	2	no	—
155	C₂×C₄⋊C₄	—	18	2	no	—
156	C₂×C₄²	—	3	1	no	—
157	#107	—	18	2	no	—
158	#106	—	3	4	no	—
159	#97	—	6	2	no	—
160	#95	—	18	2	no	—
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
163	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	6	4	no	—
171	C₂×C₃×C₄	—	6	1	no	—
172	C₂×C₃×C₄	—	6	1	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	—	6	2	no	—
175	C₂×C₃×C₄	—	12	2	no	—
176	C₂×C₃×C₄	—	12	2	no	—
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
182	C₂×C₃×C₄	#235	1	1	yes	characteristic
183	C₂×C₃×C₄	#235	1	1	yes	characteristic
184	C₂×C₃×C₄	#237	1	1	yes	characteristic
185	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
186	C₂×C₃×C₄	#239	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
189	C₂×Dic₃	—	2	2	no	—
190	C₂×Dic₃	—	2	2	no	—
191	C₂×Dic₃	—	2	1	no	—
192	C₂×Dic₃	—	2	1	no	—
193	C₂×Dic₃	—	2	4	no	—
194	C₂×Dic₃	—	4	2	no	—
195	C₂×Dic₃	—	4	2	no	—
196	C₂×Dic₃	—	12	4	no	—
197	C₂×Dic₃	—	12	2	no	—
198	C₂×Dic₃	—	12	2	no	—
199	C₂×Dic₃	—	12	2	no	—
200	C₂×Dic₃	—	12	2	no	—
201	C₂×Dic₃	—	12	2	no	—
202	C₂×Dic₃	—	12	2	no	—
203	C₂×Dic₃	—	24	2	no	—
204	C₂×Dic₃	—	24	2	no	—
205	C₃×C₈	—	2	4	no	—
206	C₃×C₈	—	6	2	no	—
207	C₃×C₈	Dic₃⋊C₄	1	4	yes	complemented
208	C₃×C₈	C₄⋊Dic₃	1	4	yes	complemented
209	C₃⋊C₈	—	2	2	no	—
210	C₂×C₃²	#389	1	1	yes	characteristic
211	C₂×C₃²	#440	1	2	yes	—
212	C₂×C₃²	#469	1	1	yes	characteristic
213	C₂×C₃²	#470	1	1	yes	characteristic
214	C₂×C₃²	#501	1	2	yes	—
215	C₂²×C₄	—	3	1	no	—
216	C₂²×C₄	—	3	1	no	—
217	C₂²×C₄	—	6	2	no	—
218	C₂²×C₄	—	18	2	no	—
219	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
220	C₂×C₈	—	3	2	no	—
221	C₂×C₈	—	6	1	no	—
222	C₂×C₈	—	6	1	no	—
223	C₂×C₈	D₃×Dic₃	1	2	yes	—
224	C₂×C₈	#626	1	2	yes	—
225	C₂×C₈	#627	1	2	yes	—
226	C₄⋊C₄	—	3	4	no	—
227	C₄⋊C₄	—	6	2	no	—
228	C₄⋊C₄	—	18	2	no	—
229	C₄⋊C₄	—	18	2	no	—
230	C₄⋊C₄	—	36	4	no	—
231	C₄²	—	6	1	no	—
232	C₄²	—	6	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	#869	1	1	yes	characteristic
239	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
240	C₂²×C₃	#888	1	1	yes	characteristic
241	C₂²×C₃	#895	1	1	yes	characteristic
242	C₂²×C₃	#908	1	2	yes	—
243	C₂²×C₃	#909	1	2	yes	—
244	C₂²×C₃	#914	1	2	yes	—
245	C₂²×C₃	#915	1	2	yes	—
246	C₂²×C₃	#930	1	1	yes	characteristic
247	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	12	1	no	—
253	C₃×C₄	—	12	1	no	—
254	C₃×C₄	—	12	4	no	—
255	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
256	C₃×C₄	#870	1	1	yes	characteristic
257	C₃×C₄	#890	1	1	yes	characteristic
258	C₃×C₄	#897	1	1	yes	characteristic
259	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
260	C₃×C₄	#924	1	1	yes	characteristic
261	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
262	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
263	Dic₃	—	4	1	no	—
264	Dic₃	—	4	1	no	—
265	Dic₃	—	4	4	no	—
266	Dic₃	—	24	4	no	—
267	Dic₃	—	24	2	no	—
268	Dic₃	—	24	2	no	—
269	C₃²	#1944	1	1	yes	characteristic, complemented, Sylow
270	C₂³	D₃×D₁₂	1	1	yes	characteristic
271	C₂×C₄	—	6	2	no	—
272	C₂×C₄	—	6	2	no	—
273	C₂×C₄	—	6	4	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	12	2	no	—
277	C₂×C₄	—	12	2	no	—
278	C₂×C₄	—	36	2	no	—
279	C₂×C₄	—	36	2	no	—
280	C₂×C₄	—	36	2	no	—
281	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
282	C₂×C₄	#3795	1	1	yes	characteristic
283	C₂×C₄	#3799	1	1	yes	characteristic
284	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
285	C₂×C₄	C₂×#626	1	1	yes	characteristic
286	C₂×C₄	C₂×#627	1	1	yes	characteristic
287	C₈	—	6	2	no	—
288	C₈	Dic₃⋊Dic₃	1	4	yes	complemented
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	#4702	1	2	yes	—
295	C₂×C₃	#4710	1	2	yes	—
296	C₂×C₃	#4856	1	2	yes	—
297	C₂×C₃	#4865	1	1	yes	characteristic
298	C₂×C₃	#4867	1	1	yes	characteristic
299	C₂×C₃	#4928	1	1	yes	characteristic
300	C₂×C₃	#4984	1	1	yes	characteristic
301	C₂×C₃	#5135	1	1	yes	characteristic
302	C₂×C₃	#5137	1	1	yes	characteristic
303	C₂×C₃	#5164	1	2	yes	—
304	C₂²	#63794	1	2	yes	—
305	C₂²	#63800	1	2	yes	—
306	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
307	C₂²	#63909	1	1	yes	characteristic
308	C₂²	#63918	1	1	yes	characteristic
309	C₄	—	12	4	no	—
310	C₄	—	12	1	no	—
311	C₄	—	12	1	no	—
312	C₄	—	72	2	no	—
313	C₄	Dic₃×Dic₆	1	1	yes	characteristic
314	C₄	#63857	1	1	yes	characteristic
315	C₄	#63869	1	1	yes	characteristic
316	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
317	C₃	—	2	1	no	—
318	C₃	#69820	1	1	yes	characteristic, complemented
319	C₃	#70562	1	1	yes	characteristic, complemented
320	C₂	#10590766	1	2	yes	—
321	C₂	#10590835	1	1	yes	characteristic
322	C₂	#10590853	1	1	yes	characteristic
323	C₂	#10590892	1	2	yes	—
324	C₂	#10591352	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₁₂
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1944
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
15. C₃×#817	C₄	semidirect
69. #817	Dic₃	semidirect
88. C₃²×C₈	C₄⋊C₄	semidirect
207. C₃×C₈	Dic₃⋊C₄	semidirect
208. C₃×C₈	C₄⋊Dic₃	semidirect
269. C₃²	#1944	semidirect
288. C₈	Dic₃⋊Dic₃	semidirect
318. C₃	#69820	semidirect
319. C₃	#70562	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63645	2
C₂×C₃×#817	2
C₂×C₃×#815	2
#10591352	2
#10589164	2
#70562	3
#69820	3