Subgroups — #11791879

Order 1152.

591 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₄×#3748	C₂	1	1	yes	maximal, characteristic
3	C₃×#4924	C₂	1	1	yes	maximal, characteristic
4	#10589167	C₂	1	4	yes	maximal
5	#71790	—	3	1	no	maximal
6	#70524	—	3	1	no	maximal
7	C₂²×C₃×Dic₆	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×#3748	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	C₃×#869	—	2	1	no	—
14	C₃×#869	—	2	1	no	—
15	C₃×#869	—	2	1	no	—
16	C₃×#869	—	2	1	no	—
17	#63630	—	2	1	no	—
18	C₄×C₃⋊(C₃⋊C₈)	—	2	1	no	—
19	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
21	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
22	C₂×#869	D₃	1	1	yes	characteristic
23	C₃×#423	—	3	1	no	—
24	#4924	D₃	1	1	yes	characteristic
25	C₄×#208	—	3	1	no	—
26	C₄×#208	—	3	1	no	—
27	C₄×#208	—	6	1	no	—
28	#4554	—	3	4	no	—
29	#4530	—	3	4	no	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
34	C₂×C₃×Dic₆	—	2	2	no	—
35	C₂×C₃×Dic₆	—	2	2	no	—
36	C₂×C₃×Dic₆	—	4	4	no	—
37	C₃²×C₄²	D₄	1	1	yes	characteristic
38	C₃²×C₄²	D₄	1	1	yes	characteristic
39	C₃²×C₄²	D₄	1	1	yes	characteristic
40	C₃²×C₄²	D₄	1	1	yes	characteristic
41	#3748	—	2	2	no	—
42	#3748	Q₈	1	4	yes	complemented
43	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×C₄⋊Dic₃	—	2	2	no	—
47	C₃×C₄⋊Dic₃	—	2	2	no	—
48	C₃×#208	—	2	4	no	—
49	C₂×C₃×C₃⋊C₈	—	2	2	no	—
50	#1997	—	9	1	no	Sylow
51	C₂²×C₃×Q₈	—	3	2	no	—
52	C₂²×Dic₆	Dic₃	1	2	yes	—
53	C₂×C₃×M₁₆	—	3	2	no	—
54	C₃×C₄⋊Q₈	—	6	1	no	—
55	C₃×C₄⋊Q₈	—	6	1	no	—
56	C₃×C₄⋊Q₈	—	6	1	no	—
57	C₃×C₄⋊Q₈	—	6	1	no	—
58	C₂×C₃×C₄⋊C₄	—	3	2	no	—
59	C₂×C₃×C₄⋊C₄	—	3	2	no	—
60	C₂×C₃×C₄²	—	2	1	no	—
61	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄⋊Dic₃	Dic₃	1	2	yes	—
64	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
65	C₂×#208	—	3	2	no	—
66	C₂×#208	—	3	2	no	—
67	C₂×#208	—	6	2	no	—
68	C₂×#208	C₂×D₃	1	2	yes	—
69	#869	—	2	1	no	—
70	#869	—	2	1	no	—
71	#869	—	2	1	no	—
72	#869	—	2	1	no	—
73	C₃×C₄⋊C₈	—	6	1	no	—
74	C₃×C₄⋊C₈	—	6	1	no	—
75	C₄⋊(C₃⋊C₈)	—	2	1	no	—
76	C₄⋊(C₃⋊C₈)	—	2	1	no	—
77	#803	—	6	2	no	—
78	#803	—	6	1	no	—
79	#803	—	6	1	no	—
80	C₄×C₃⋊C₈	—	6	2	no	—
81	C₄×C₃⋊C₈	—	6	1	no	—
82	C₄×C₃⋊C₈	—	6	1	no	—
83	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
93	C₂×C₃²×C₄	#41	1	2	yes	—
94	C₂×C₃×Dic₃	—	2	4	no	—
95	C₂×C₃×Dic₃	—	4	4	no	—
96	C₃×Dic₆	—	4	4	no	—
97	C₃×Dic₆	—	4	4	no	—
98	C₃⋊(C₃⋊C₈)	—	2	4	no	—
99	C₃×C₃⋊C₈	—	4	2	no	—
100	C₂×C₄⋊Q₈	—	3	1	no	—
101	#423	—	3	1	no	—
102	C₄×M₁₆	—	9	1	no	—
103	#343	—	9	4	no	—
104	C₂×C₃×Q₈	—	6	2	no	—
105	C₂×C₃×Q₈	—	6	2	no	—
106	C₂×C₃×Q₈	—	12	4	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	3	4	no	—
111	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₂²×Dic₃	C₂×Dic₃	1	4	yes	—
118	C₂×Dic₆	—	2	2	no	—
119	C₂×Dic₆	—	2	2	no	—
120	C₂×Dic₆	—	4	4	no	—
121	C₃×M₁₆	—	6	4	no	—
122	C₂×C₃×C₈	—	6	2	no	—
123	C₃×C₄⋊C₄	—	6	2	no	—
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₄⋊C₄	—	6	2	no	—
126	C₃×C₄⋊C₄	—	6	2	no	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	D₁₂	1	1	yes	characteristic
132	C₃×C₄²	D₁₂	1	1	yes	characteristic
133	C₃×C₄²	#67	1	1	yes	characteristic
134	C₃×C₄²	#67	1	1	yes	characteristic
135	C₃×C₄²	#67	1	1	yes	characteristic
136	C₃×C₄²	#67	1	1	yes	characteristic
137	C₃×C₄²	#67	1	1	yes	characteristic
138	C₃×C₄²	#67	1	1	yes	characteristic
139	C₄⋊Dic₃	—	2	2	no	—
140	C₄⋊Dic₃	—	2	2	no	—
141	C₄⋊Dic₃	—	2	2	no	—
142	C₄⋊Dic₃	—	2	2	no	—
143	#208	—	2	4	no	—
144	#208	—	3	4	no	—
145	#208	—	3	4	no	—
146	#208	—	6	4	no	—
147	#208	—	6	4	no	—
148	#208	—	6	2	no	—
149	#208	—	6	2	no	—
150	C₂×C₃⋊C₈	—	2	2	no	—
151	C₂×C₃⋊C₈	—	6	4	no	—
152	C₂×C₃⋊C₈	—	6	2	no	—
153	C₂×C₃⋊C₈	—	6	2	no	—
154	C₂²×C₃²	—	2	1	no	—
155	C₂²×C₃²	—	2	1	no	—
156	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₃²	C₄×Q₈	1	2	yes	—
158	C₃²×C₄	—	2	1	no	—
159	C₃²×C₄	—	2	1	no	—
160	C₃²×C₄	#100	1	2	yes	—
161	C₃²×C₄	#101	1	2	yes	—
162	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
163	C₃²×C₄	#124	1	1	yes	characteristic
164	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
165	C₃×Dic₃	—	4	4	no	—
166	C₂²×Q₈	—	3	2	no	—
167	C₂×M₁₆	—	3	2	no	—
168	C₂×M₁₆	—	9	2	no	—
169	C₄⋊Q₈	—	6	1	no	—
170	C₄⋊Q₈	—	6	1	no	—
171	C₄⋊Q₈	—	6	1	no	—
172	C₄⋊Q₈	—	6	1	no	—
173	C₂×C₄⋊C₄	—	3	2	no	—
174	C₂×C₄⋊C₄	—	3	2	no	—
175	C₂×C₄²	D₃²	1	1	yes	characteristic
176	C₄⋊C₈	—	6	1	no	—
177	C₄⋊C₈	—	6	1	no	—
178	#97	—	18	1	no	—
179	C₄×C₈	—	18	1	no	—
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
183	C₃×Q₈	—	12	4	no	—
184	C₃×Q₈	—	12	4	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	4	2	no	—
198	C₂×C₃×C₄	—	4	2	no	—
199	C₂×C₃×C₄	—	6	4	no	—
200	C₂×C₃×C₄	—	12	4	no	—
201	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₃×C₄	#217	1	1	yes	characteristic
206	C₂×C₃×C₄	#217	1	1	yes	characteristic
207	C₂×C₃×C₄	#217	1	1	yes	characteristic
208	C₂×C₃×C₄	#217	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
210	C₂×C₃×C₄	#237	1	2	yes	—
211	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
212	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
213	C₂×C₃×C₄	#239	1	2	yes	—
214	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
217	C₂×Dic₃	—	2	4	no	—
218	C₂×Dic₃	—	4	4	no	—
219	Dic₆	—	4	4	no	—
220	Dic₆	—	4	4	no	—
221	C₃×C₈	—	12	2	no	—
222	C₃⋊C₈	—	4	2	no	—
223	C₃⋊C₈	—	6	8	no	—
224	C₃⋊C₈	—	6	4	no	—
225	C₃⋊C₈	—	6	4	no	—
226	C₂×C₃²	—	2	2	no	—
227	C₂×C₃²	#391	1	1	yes	characteristic
228	C₂×C₃²	C₂×#100	1	1	yes	characteristic
229	C₂×C₃²	C₂×#101	1	1	yes	characteristic
230	C₂×Q₈	—	6	2	no	—
231	C₂×Q₈	—	6	2	no	—
232	C₂×Q₈	—	12	4	no	—
233	C₂²×C₄	—	3	4	no	—
234	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
235	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
237	M₁₆	—	6	4	no	—
238	M₁₆	—	9	4	no	—
239	M₁₆	—	18	2	no	—
240	C₂×C₈	—	6	2	no	—
241	C₂×C₈	—	18	2	no	—
242	C₄⋊C₄	—	6	2	no	—
243	C₄⋊C₄	—	6	2	no	—
244	C₄⋊C₄	—	6	2	no	—
245	C₄⋊C₄	—	6	2	no	—
246	C₄²	#625	1	1	yes	characteristic
247	C₄²	#625	1	1	yes	characteristic
248	C₄²	#626	1	1	yes	characteristic
249	C₄²	#626	1	1	yes	characteristic
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	4	1	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	#887	1	2	yes	—
259	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
260	C₂²×C₃	Q₈×Dic₃	1	2	yes	—
261	C₂²×C₃	C₂×#217	1	1	yes	characteristic
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	4	1	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	12	4	no	—
274	C₃×C₄	#823	1	2	yes	—
275	C₃×C₄	#824	1	2	yes	—
276	C₃×C₄	#833	1	2	yes	—
277	C₃×C₄	#836	1	2	yes	—
278	C₃×C₄	#897	1	2	yes	—
279	C₃×C₄	#936	1	1	yes	characteristic
280	C₃×C₄	#944	1	1	yes	characteristic
281	C₃×C₄	#944	1	1	yes	characteristic
282	C₃×C₄	#946	1	2	yes	—
283	C₃×C₄	#947	1	1	yes	characteristic
284	Dic₃	—	4	4	no	—
285	C₃²	#1997	1	1	yes	characteristic, complemented, Sylow
286	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
287	Q₈	—	12	4	no	—
288	Q₈	—	12	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	6	4	no	—
292	C₂×C₄	—	12	4	no	—
293	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
294	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
295	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
296	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
297	C₂×C₄	#3796	1	2	yes	—
298	C₂×C₄	#3797	1	2	yes	—
299	C₂×C₄	C₂×#625	1	1	yes	characteristic
300	C₂×C₄	C₂×#626	1	1	yes	characteristic
301	C₈	—	12	2	no	—
302	C₈	—	18	4	no	—
303	C₂×C₃	—	2	2	no	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	4	2	no	—
309	C₂×C₃	#4973	1	1	yes	characteristic
310	C₂×C₃	C₂×#823	1	1	yes	characteristic
311	C₂×C₃	C₂×#824	1	1	yes	characteristic
312	C₂×C₃	C₂×#833	1	1	yes	characteristic
313	C₂×C₃	C₂×#836	1	1	yes	characteristic
314	C₂×C₃	#5229	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	#63843	1	2	yes	—
318	C₂²	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	12	4	no	—
322	C₄	#63558	1	2	yes	—
323	C₄	#63560	1	2	yes	—
324	C₄	#63863	1	1	yes	characteristic
325	C₄	#63871	1	2	yes	—
326	C₄	#63873	1	1	yes	characteristic
327	C₃	—	2	1	no	—
328	C₃	#70524	1	1	yes	characteristic, complemented
329	C₃	#71790	1	1	yes	characteristic, complemented
330	C₂	—	2	2	no	—
331	C₂	C₂×#63558	1	1	yes	characteristic
332	C₂	C₂×#63560	1	1	yes	characteristic
333	C₂	#10591221	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃)⋊Dic₃
commutator, prime radical	C₂³×C₃²	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₂²⋊C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1997
LCS	G ▷ C₂³×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
42. #3748	Q₈	semidirect
285. C₃²	#1997	semidirect
328. C₃	#70524	semidirect
329. C₃	#71790	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₄×#3748	2
C₃×#4924	2
#10589167	2
#71790	3
#70524	3