Subgroups — #11790175

Order 1152.

545 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63644	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	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	#70613	—	3	1	no	maximal
7	#69819	—	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	#63644	—	2	2	no	—
15	C₃×#818	C₄	1	4	yes	complemented
16	C₃×#815	—	2	2	no	—
17	#63579	—	2	2	no	—
18	C₂×C₃×#107	—	3	1	no	—
19	C₂×C₃×#97	—	3	1	no	—
20	C₂×#818	D₃	1	1	yes	characteristic
21	C₂×#817	—	3	1	no	—
22	C₂×#817	—	3	1	no	—
23	C₂×#817	—	6	1	no	—
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	2	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₂×C₃²×C₈	Q₈	1	2	yes	—
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
40	C₃×C₄×Dic₃	—	2	1	no	—
41	C₃×C₄×Dic₃	—	2	1	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
45	Dic₃⋊Dic₃	—	4	4	no	—
46	#1943	—	9	1	no	Sylow
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄²	—	3	1	no	—
53	C₂×C₄⋊Dic₃	—	3	2	no	—
54	C₂×C₄⋊Dic₃	—	3	2	no	—
55	C₂×C₄⋊Dic₃	—	6	2	no	—
56	C₂×C₄⋊Dic₃	—	6	2	no	—
57	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
58	C₂×Dic₃⋊C₄	—	6	2	no	—
59	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
61	C₃×#107	—	3	4	no	—
62	C₃×#97	—	6	2	no	—
63	C₃⋊#95	—	6	2	no	—
64	C₃⋊#95	—	6	2	no	—
65	#818	Dic₃	1	4	yes	complemented
66	#817	—	6	4	no	—
67	#817	—	6	2	no	—
68	#817	—	6	2	no	—
69	#815	—	2	2	no	—
70	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃²×C₄	#41	1	1	yes	characteristic
76	C₂×C₃²×C₄	#41	1	1	yes	characteristic
77	C₂×C₃⋊Dic₃	—	4	2	no	—
78	C₂×C₃⋊Dic₃	—	4	2	no	—
79	C₂×C₃⋊Dic₃	—	4	2	no	—
80	C₂×C₃×Dic₃	—	2	4	no	—
81	C₂×C₃×Dic₃	—	2	2	no	—
82	C₂×C₃×Dic₃	—	2	2	no	—
83	C₂×C₃×Dic₃	—	2	2	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
87	C₃×C₃⋊C₈	—	2	2	no	—
88	C₂×#107	—	3	1	no	—
89	C₂×#106	—	9	1	no	—
90	C₂×#97	—	3	1	no	—
91	#389	—	9	2	no	—
92	#340	—	9	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	3	2	no	—
95	C₂²×C₃×C₄	—	6	2	no	—
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
98	C₂²×Dic₃	—	2	2	no	—
99	C₂²×Dic₃	—	6	2	no	—
100	C₂²×Dic₃	—	6	2	no	—
101	C₂²×Dic₃	—	12	2	no	—
102	C₂²×Dic₃	D₁₂	1	2	yes	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	—	6	1	no	—
108	C₂×C₃×C₈	—	6	1	no	—
109	C₂×C₃×C₈	Dic₆	1	2	yes	—
110	C₂×C₃×C₈	Dic₆	1	2	yes	—
111	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
112	C₂×C₃×C₈	D₁₂	1	2	yes	—
113	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
114	C₂×C₃×C₈	#67	1	2	yes	—
115	C₃×C₄⋊C₄	—	3	4	no	—
116	C₃×C₄⋊C₄	—	6	2	no	—
117	C₃×C₄²	—	6	1	no	—
118	C₃×C₄²	—	6	1	no	—
119	C₄⋊Dic₃	—	2	2	no	—
120	C₄⋊Dic₃	—	6	4	no	—
121	C₄⋊Dic₃	—	6	2	no	—
122	C₄⋊Dic₃	—	6	2	no	—
123	C₄⋊Dic₃	—	6	2	no	—
124	C₄⋊Dic₃	—	6	2	no	—
125	C₄⋊Dic₃	—	12	2	no	—
126	C₄⋊Dic₃	—	12	4	no	—
127	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
128	Dic₃⋊C₄	—	12	4	no	—
129	C₄×Dic₃	—	2	1	no	—
130	C₄×Dic₃	—	2	1	no	—
131	C₂×C₃⋊C₈	—	2	1	no	—
132	C₂×C₃⋊C₈	—	2	1	no	—
133	C₂×C₃⋊C₈	Dic₆	1	2	yes	—
134	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
135	C₂²×C₃²	#121	1	1	yes	characteristic
136	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
138	C₂²×C₃²	#137	1	2	yes	—
139	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
141	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
142	C₃²×C₄	#125	1	1	yes	characteristic
143	C₃⋊Dic₃	—	8	2	no	—
144	C₃×Dic₃	—	4	4	no	—
145	C₃×Dic₃	—	4	2	no	—
146	C₂²×C₈	—	3	1	no	—
147	C₂²×C₈	D₃²	1	1	yes	characteristic
148	C₂×C₄⋊C₄	—	3	2	no	—
149	C₂×C₄⋊C₄	—	9	2	no	—
150	C₂×C₄⋊C₄	—	18	2	no	—
151	C₂×C₄²	—	3	1	no	—
152	#107	—	3	4	no	—
153	#106	—	18	2	no	—
154	#97	—	6	2	no	—
155	#95	—	18	2	no	—
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
158	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	6	4	no	—
166	C₂×C₃×C₄	—	6	2	no	—
167	C₂×C₃×C₄	—	6	2	no	—
168	C₂×C₃×C₄	—	6	2	no	—
169	C₂×C₃×C₄	—	12	2	no	—
170	C₂×C₃×C₄	—	12	2	no	—
171	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
176	C₂×C₃×C₄	#235	1	1	yes	characteristic
177	C₂×C₃×C₄	#235	1	1	yes	characteristic
178	C₂×C₃×C₄	#237	1	1	yes	characteristic
179	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
180	C₂×C₃×C₄	#239	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
183	C₂×Dic₃	—	2	2	no	—
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	2	2	no	—
186	C₂×Dic₃	—	2	4	no	—
187	C₂×Dic₃	—	4	2	no	—
188	C₂×Dic₃	—	4	2	no	—
189	C₂×Dic₃	—	12	4	no	—
190	C₂×Dic₃	—	12	2	no	—
191	C₂×Dic₃	—	12	2	no	—
192	C₂×Dic₃	—	12	2	no	—
193	C₂×Dic₃	—	12	2	no	—
194	C₂×Dic₃	—	12	2	no	—
195	C₂×Dic₃	—	12	2	no	—
196	C₂×Dic₃	—	24	2	no	—
197	C₂×Dic₃	—	24	2	no	—
198	C₃×C₈	—	2	4	no	—
199	C₃×C₈	—	6	2	no	—
200	C₃×C₈	Dic₃⋊C₄	1	4	yes	complemented
201	C₃×C₈	C₄⋊Dic₃	1	4	yes	complemented
202	C₃⋊C₈	—	2	2	no	—
203	C₂×C₃²	#389	1	1	yes	characteristic
204	C₂×C₃²	#441	1	1	yes	characteristic
205	C₂×C₃²	#442	1	1	yes	characteristic
206	C₂×C₃²	#468	1	2	yes	—
207	C₂×C₃²	#501	1	2	yes	—
208	C₂²×C₄	—	3	2	no	—
209	C₂²×C₄	—	6	2	no	—
210	C₂²×C₄	—	18	2	no	—
211	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
212	C₂×C₈	—	3	2	no	—
213	C₂×C₈	—	6	1	no	—
214	C₂×C₈	—	6	1	no	—
215	C₂×C₈	D₃×Dic₃	1	2	yes	—
216	C₂×C₈	#626	1	2	yes	—
217	C₂×C₈	#627	1	2	yes	—
218	C₄⋊C₄	—	3	4	no	—
219	C₄⋊C₄	—	6	2	no	—
220	C₄⋊C₄	—	18	2	no	—
221	C₄⋊C₄	—	18	2	no	—
222	C₄⋊C₄	—	36	4	no	—
223	C₄²	—	6	1	no	—
224	C₄²	—	6	1	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	#869	1	1	yes	characteristic
231	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
232	C₂²×C₃	#888	1	1	yes	characteristic
233	C₂²×C₃	#895	1	1	yes	characteristic
234	C₂²×C₃	#908	1	2	yes	—
235	C₂²×C₃	#909	1	2	yes	—
236	C₂²×C₃	#911	1	2	yes	—
237	C₂²×C₃	#918	1	2	yes	—
238	C₂²×C₃	#930	1	1	yes	characteristic
239	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	12	2	no	—
245	C₃×C₄	—	12	4	no	—
246	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
247	C₃×C₄	#870	1	1	yes	characteristic
248	C₃×C₄	#890	1	1	yes	characteristic
249	C₃×C₄	#897	1	1	yes	characteristic
250	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
251	C₃×C₄	#924	1	1	yes	characteristic
252	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
253	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
254	Dic₃	—	4	2	no	—
255	Dic₃	—	4	4	no	—
256	Dic₃	—	24	4	no	—
257	Dic₃	—	24	2	no	—
258	Dic₃	—	24	2	no	—
259	C₃²	#1943	1	1	yes	characteristic, complemented, Sylow
260	C₂³	D₃×D₁₂	1	1	yes	characteristic
261	C₂×C₄	—	6	2	no	—
262	C₂×C₄	—	6	2	no	—
263	C₂×C₄	—	6	4	no	—
264	C₂×C₄	—	6	2	no	—
265	C₂×C₄	—	12	2	no	—
266	C₂×C₄	—	12	2	no	—
267	C₂×C₄	—	36	2	no	—
268	C₂×C₄	—	36	2	no	—
269	C₂×C₄	—	36	2	no	—
270	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
271	C₂×C₄	#3795	1	1	yes	characteristic
272	C₂×C₄	#3799	1	1	yes	characteristic
273	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#626	1	1	yes	characteristic
275	C₂×C₄	C₂×#627	1	1	yes	characteristic
276	C₈	—	6	2	no	—
277	C₈	Dic₃⋊Dic₃	1	4	yes	complemented
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	2	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	#4702	1	2	yes	—
284	C₂×C₃	#4711	1	1	yes	characteristic
285	C₂×C₃	#4716	1	1	yes	characteristic
286	C₂×C₃	#4876	1	2	yes	—
287	C₂×C₃	#4886	1	2	yes	—
288	C₂×C₃	#4928	1	1	yes	characteristic
289	C₂×C₃	#4984	1	1	yes	characteristic
290	C₂×C₃	#5134	1	2	yes	—
291	C₂×C₃	#5152	1	1	yes	characteristic
292	C₂×C₃	#5184	1	1	yes	characteristic
293	C₂²	#63795	1	2	yes	—
294	C₂²	#63801	1	2	yes	—
295	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
296	C₂²	#63909	1	1	yes	characteristic
297	C₂²	#63918	1	1	yes	characteristic
298	C₄	—	12	4	no	—
299	C₄	—	12	2	no	—
300	C₄	—	72	2	no	—
301	C₄	Dic₃×Dic₆	1	1	yes	characteristic
302	C₄	#63857	1	1	yes	characteristic
303	C₄	#63869	1	1	yes	characteristic
304	C₄	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
305	C₃	—	2	1	no	—
306	C₃	#69819	1	1	yes	characteristic, complemented
307	C₃	#70613	1	1	yes	characteristic, complemented
308	C₂	#10590771	1	1	yes	characteristic
309	C₂	#10590812	1	1	yes	characteristic
310	C₂	#10590838	1	2	yes	—
311	C₂	#10590894	1	2	yes	—
312	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	#1943
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×#818	C₄	semidirect
65. #818	Dic₃	semidirect
86. C₃²×C₈	C₄⋊C₄	semidirect
200. C₃×C₈	Dic₃⋊C₄	semidirect
201. C₃×C₈	C₄⋊Dic₃	semidirect
259. C₃²	#1943	semidirect
277. C₈	Dic₃⋊Dic₃	semidirect
306. C₃	#69819	semidirect
307. C₃	#70613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63644	2
C₂×C₃×#818	2
C₂×C₃×#815	2
#10591352	2
#10589164	2
#70613	3
#69819	3