Subgroups — #11791880

Order 1152.

665 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64078	C₂	1	1	yes	maximal, characteristic
2	C₃×#4924	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#208	C₂	1	1	yes	maximal, characteristic
4	#10589168	C₂	1	4	yes	maximal
5	#70524	—	3	1	no	maximal
6	#69879	—	3	1	no	maximal
7	C₂²×#634	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	#64078	—	2	1	no	—
12	#64078	—	2	1	no	—
13	#64078	—	2	1	no	—
14	#64078	—	2	1	no	—
15	C₂×C₃×#208	C₂²	1	2	yes	—
16	C₂×C₃×#208	C₂²	1	2	yes	—
17	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₃×#803	—	2	1	no	—
20	C₃×C₄×C₃⋊C₈	—	2	1	no	—
21	C₂×#869	—	3	1	no	—
22	C₂×#869	—	3	1	no	—
23	C₂×#869	—	6	1	no	—
24	C₃×#423	—	3	1	no	—
25	C₃×C₄×M₁₆	—	3	1	no	—
26	#4924	D₃	1	1	yes	characteristic
27	C₄×#208	D₃	1	1	yes	characteristic
28	#4554	—	3	4	no	—
29	#4554	—	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×#634	—	2	2	no	—
35	C₂×#634	—	2	2	no	—
36	C₂×#634	—	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	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
42	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
43	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
44	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
45	C₃×#208	—	2	2	no	—
46	C₃×#208	—	2	4	no	—
47	C₃×#208	Q₈	1	4	yes	complemented
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	C₂×C₃×C₃⋊C₈	—	2	2	no	—
50	#1997	—	9	1	no	Sylow
51	C₂²×Dic₆	—	3	2	no	—
52	C₂²×Dic₆	—	3	2	no	—
53	C₂²×Dic₆	—	6	2	no	—
54	C₂×C₃×M₁₆	—	3	2	no	—
55	C₂×C₃×M₁₆	—	3	2	no	—
56	C₂×C₃×C₄²	—	2	1	no	—
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₂×C₄⋊Dic₃	—	3	2	no	—
60	C₂×C₄⋊Dic₃	—	3	2	no	—
61	C₂×C₄⋊Dic₃	—	3	2	no	—
62	C₂×C₄⋊Dic₃	—	3	2	no	—
63	C₂×C₄⋊Dic₃	—	6	2	no	—
64	C₂×C₄⋊Dic₃	—	6	2	no	—
65	C₂×#208	C₂×D₃	1	2	yes	—
66	C₂×#208	C₂×D₃	1	2	yes	—
67	#869	—	6	2	no	—
68	#869	—	6	2	no	—
69	#869	—	6	2	no	—
70	#869	—	6	2	no	—
71	#869	—	6	1	no	—
72	#869	—	6	1	no	—
73	#869	—	6	1	no	—
74	#869	—	6	1	no	—
75	#869	—	6	1	no	—
76	#869	—	6	1	no	—
77	#869	—	6	1	no	—
78	#869	—	6	1	no	—
79	C₃×C₄⋊C₈	—	6	1	no	—
80	C₃×C₄⋊C₈	—	6	1	no	—
81	C₃×#97	—	6	1	no	—
82	C₃×C₄×C₈	—	6	1	no	—
83	C₄⋊(C₃⋊C₈)	—	2	1	no	—
84	C₄⋊(C₃⋊C₈)	—	2	1	no	—
85	#803	—	2	1	no	—
86	C₄×C₃⋊C₈	—	2	1	no	—
87	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	2	2	no	—
90	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
97	C₂×C₃²×C₄	#41	1	2	yes	—
98	C₂×C₃⋊Dic₃	—	2	4	no	—
99	C₂×C₃⋊Dic₃	—	4	4	no	—
100	#634	—	4	4	no	—
101	#634	—	4	4	no	—
102	C₃×C₃⋊C₈	—	2	4	no	—
103	C₃×C₃⋊C₈	—	4	2	no	—
104	C₂×C₄⋊Q₈	—	9	1	no	—
105	#423	—	3	1	no	—
106	C₄×M₁₆	—	3	1	no	—
107	#343	—	9	4	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	2	1	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	C₄×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₄×D₃	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	—	3	4	no	—
118	C₂²×Dic₃	—	3	4	no	—
119	C₂²×Dic₃	—	6	4	no	—
120	C₂×Dic₆	—	6	4	no	—
121	C₂×Dic₆	—	6	4	no	—
122	C₂×Dic₆	—	6	2	no	—
123	C₂×Dic₆	—	6	2	no	—
124	C₂×Dic₆	—	6	2	no	—
125	C₂×Dic₆	—	6	2	no	—
126	C₂×Dic₆	—	12	8	no	—
127	C₂×Dic₆	—	12	4	no	—
128	C₂×Dic₆	—	12	4	no	—
129	C₃×M₁₆	—	3	4	no	—
130	C₃×M₁₆	—	6	2	no	—
131	C₃×M₁₆	—	6	4	no	—
132	C₂×C₃×C₈	—	6	2	no	—
133	C₂×C₃×C₈	—	6	2	no	—
134	C₃×C₄²	—	2	1	no	—
135	C₃×C₄²	—	2	1	no	—
136	C₃×C₄²	—	2	1	no	—
137	C₃×C₄²	—	2	1	no	—
138	C₃×C₄²	D₁₂	1	1	yes	characteristic
139	C₃×C₄²	D₁₂	1	1	yes	characteristic
140	C₃×C₄²	D₁₂	1	1	yes	characteristic
141	C₃×C₄²	D₁₂	1	1	yes	characteristic
142	C₃×C₄²	#67	1	1	yes	characteristic
143	C₃×C₄²	#67	1	1	yes	characteristic
144	C₃×C₄²	#67	1	1	yes	characteristic
145	C₃×C₄²	#67	1	1	yes	characteristic
146	C₄⋊Dic₃	—	6	4	no	—
147	C₄⋊Dic₃	—	6	4	no	—
148	C₄⋊Dic₃	—	6	4	no	—
149	C₄⋊Dic₃	—	6	4	no	—
150	C₄⋊Dic₃	—	6	2	no	—
151	C₄⋊Dic₃	—	6	2	no	—
152	C₄⋊Dic₃	—	6	2	no	—
153	C₄⋊Dic₃	—	6	2	no	—
154	C₄⋊Dic₃	—	6	2	no	—
155	C₄⋊Dic₃	—	6	2	no	—
156	C₄⋊Dic₃	—	6	2	no	—
157	C₄⋊Dic₃	—	6	2	no	—
158	#208	—	2	2	no	—
159	#208	—	2	4	no	—
160	#208	Dic₆	1	4	yes	complemented
161	C₂×C₃⋊C₈	—	2	2	no	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂²×C₃²	—	2	1	no	—
164	C₂²×C₃²	—	2	1	no	—
165	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₃²	C₄×Q₈	1	2	yes	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	—	2	1	no	—
169	C₃²×C₄	#100	1	2	yes	—
170	C₃²×C₄	#101	1	2	yes	—
171	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
172	C₃²×C₄	#124	1	1	yes	characteristic
173	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
174	C₃⋊Dic₃	—	4	4	no	—
175	C₂²×Q₈	—	9	2	no	—
176	C₂×M₁₆	—	3	2	no	—
177	C₂×M₁₆	—	3	2	no	—
178	C₄⋊Q₈	—	18	1	no	—
179	C₄⋊Q₈	—	18	1	no	—
180	C₄⋊Q₈	—	18	1	no	—
181	C₄⋊Q₈	—	18	1	no	—
182	C₂×C₄⋊C₄	—	9	2	no	—
183	C₂×C₄⋊C₄	—	9	2	no	—
184	C₂×C₄²	D₃²	1	1	yes	characteristic
185	C₄⋊C₈	—	6	1	no	—
186	C₄⋊C₈	—	6	1	no	—
187	#97	—	6	1	no	—
188	C₄×C₈	—	6	1	no	—
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
211	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
215	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
217	C₂×C₃×C₄	#235	1	2	yes	—
218	C₂×C₃×C₄	#237	1	2	yes	—
219	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
220	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
222	C₂×Dic₃	—	6	8	no	—
223	C₂×Dic₃	—	6	4	no	—
224	C₂×Dic₃	—	6	4	no	—
225	C₂×Dic₃	—	12	8	no	—
226	C₂×Dic₃	—	12	4	no	—
227	C₂×Dic₃	—	12	4	no	—
228	Dic₆	—	12	8	no	—
229	Dic₆	—	12	4	no	—
230	Dic₆	—	12	4	no	—
231	Dic₆	—	12	8	no	—
232	Dic₆	—	12	4	no	—
233	Dic₆	—	12	4	no	—
234	C₃×C₈	—	6	4	no	—
235	C₃×C₈	—	12	2	no	—
236	C₃⋊C₈	—	2	4	no	—
237	C₃⋊C₈	—	4	2	no	—
238	C₂×C₃²	—	2	2	no	—
239	C₂×C₃²	#391	1	1	yes	characteristic
240	C₂×C₃²	C₂×#100	1	1	yes	characteristic
241	C₂×C₃²	C₂×#101	1	1	yes	characteristic
242	C₂×Q₈	—	18	2	no	—
243	C₂×Q₈	—	18	2	no	—
244	C₂×Q₈	—	36	4	no	—
245	C₂²×C₄	—	9	4	no	—
246	C₂²×C₄	#624	1	1	yes	characteristic
247	C₂²×C₄	#624	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
249	M₁₆	—	3	4	no	—
250	M₁₆	—	6	2	no	—
251	M₁₆	—	6	4	no	—
252	C₂×C₈	—	6	2	no	—
253	C₂×C₈	—	6	2	no	—
254	C₄⋊C₄	—	18	2	no	—
255	C₄⋊C₄	—	18	2	no	—
256	C₄⋊C₄	—	18	2	no	—
257	C₄⋊C₄	—	18	2	no	—
258	C₄²	#626	1	1	yes	characteristic
259	C₄²	#626	1	1	yes	characteristic
260	C₄²	#626	1	1	yes	characteristic
261	C₄²	#626	1	1	yes	characteristic
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	C₄×Dic₆	1	2	yes	—
271	C₂²×C₃	#887	1	2	yes	—
272	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
273	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	#823	1	2	yes	—
286	C₃×C₄	#823	1	2	yes	—
287	C₃×C₄	#824	1	2	yes	—
288	C₃×C₄	#824	1	2	yes	—
289	C₃×C₄	#869	1	1	yes	characteristic
290	C₃×C₄	#875	1	1	yes	characteristic
291	C₃×C₄	#897	1	2	yes	—
292	C₃×C₄	#924	1	2	yes	—
293	C₃×C₄	#936	1	1	yes	characteristic
294	C₃×C₄	#944	1	1	yes	characteristic
295	Dic₃	—	12	8	no	—
296	Dic₃	—	12	4	no	—
297	Dic₃	—	12	4	no	—
298	C₃²	#1997	1	1	yes	characteristic, complemented, Sylow
299	C₂³	C₂×#624	1	1	yes	characteristic
300	Q₈	—	36	4	no	—
301	Q₈	—	36	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	18	4	no	—
305	C₂×C₄	—	36	4	no	—
306	C₂×C₄	#3722	1	1	yes	characteristic
307	C₂×C₄	#3722	1	1	yes	characteristic
308	C₂×C₄	#3722	1	1	yes	characteristic
309	C₂×C₄	#3722	1	1	yes	characteristic
310	C₂×C₄	D₃×Dic₆	1	2	yes	—
311	C₂×C₄	#3795	1	2	yes	—
312	C₂×C₄	C₂×#626	1	1	yes	characteristic
313	C₂×C₄	C₂×#626	1	1	yes	characteristic
314	C₈	—	6	4	no	—
315	C₈	—	12	2	no	—
316	C₂×C₃	—	2	4	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	#4929	1	1	yes	characteristic
323	C₂×C₃	#4973	1	1	yes	characteristic
324	C₂×C₃	C₂×#823	1	1	yes	characteristic
325	C₂×C₃	C₂×#823	1	1	yes	characteristic
326	C₂×C₃	C₂×#824	1	1	yes	characteristic
327	C₂×C₃	C₂×#824	1	1	yes	characteristic
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	#63844	1	2	yes	—
331	C₂²	C₂×#3722	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	36	4	no	—
335	C₄	#63559	1	2	yes	—
336	C₄	#63561	1	2	yes	—
337	C₄	#63857	1	2	yes	—
338	C₄	#63860	1	1	yes	characteristic
339	C₄	#63866	1	1	yes	characteristic
340	C₃	—	2	1	no	—
341	C₃	#69879	1	1	yes	characteristic, complemented
342	C₃	#70524	1	1	yes	characteristic, complemented
343	C₂	—	2	2	no	—
344	C₂	C₂×#63559	1	1	yes	characteristic
345	C₂	C₂×#63561	1	1	yes	characteristic
346	C₂	#10591213	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#3722
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
47. C₃×#208	Q₈	semidirect
160. #208	Dic₆	semidirect
298. C₃²	#1997	semidirect
341. C₃	#69879	semidirect
342. C₃	#70524	semidirect

Maximal subgroups

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

subgroup	index
C₂×#64078	2
C₃×#4924	2
C₃×C₄×#208	2
#10589168	2
#70524	3
#69879	3