Subgroups — C₂×C₃×#5734 #11903313

Order 1152.

2625 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×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#947	C₂	1	4	yes	maximal, complemented
3	C₃×#5734	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#895	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#892	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×C₄×D₁₂	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#555	—	3	1	no	maximal
8	C₂×#5734	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
10	C₂²×C₃×D₁₂	C₂²	1	2	yes	—
11	C₂²×C₃×D₁₂	C₂²	1	4	yes	—
12	C₂²×C₃×C₄×D₃	C₂²	1	4	yes	—
13	C₃²×C₄⋊Q₈	C₂²	1	8	yes	complemented
14	C₂×C₃²×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
16	C₃×#947	C₂²	1	32	yes	complemented
17	C₂×C₃×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
18	C₂×C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
19	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
20	C₂×C₃×C₄×Dic₃	C₂²	1	4	yes	—
21	C₃×#895	C₂²	1	32	yes	complemented
22	C₃×#892	C₂²	1	32	yes	complemented
23	C₃×C₄×D₁₂	C₂²	1	16	yes	complemented
24	C₃×#555	—	3	16	no	—
25	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
26	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
27	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×#124	—	3	4	no	—
29	C₂×C₃×#121	—	3	4	no	—
30	C₂×C₃×C₄×D₄	—	3	2	no	—
31	C₂×C₃×#117	—	3	4	no	—
32	C₂×#947	C₂×C₃	1	4	yes	complemented
33	#5734	C₂×C₃	1	16	yes	direct product factor
34	C₂×#895	C₂×C₃	1	4	yes	complemented
35	C₂×#892	C₂×C₃	1	4	yes	complemented
36	C₂×C₄×D₁₂	C₂×C₃	1	2	yes	complemented
37	C₂³×C₃×D₃	C₂³	1	4	yes	—
38	C₂×C₃²×Q₈	—	4	4	no	—
39	C₂×C₃²×Q₈	C₂³	1	8	yes	—
40	C₂²×C₃²×C₄	C₂³	1	4	yes	—
41	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×Dic₃	C₂³	1	4	yes	—
44	C₂×C₃×D₁₂	—	2	8	no	—
45	C₂×C₃×D₁₂	—	4	8	no	—
46	C₂×C₃×D₁₂	C₂³	1	8	yes	—
47	C₂×C₃×D₁₂	C₂³	1	16	yes	—
48	C₂×C₃×C₄×D₃	—	2	16	no	—
49	C₂×C₃×C₄×D₃	C₂³	1	16	yes	—
50	C₃²×C₄⋊C₄	C₂³	1	16	yes	—
51	C₃²×C₄²	C₂³	1	4	yes	—
52	C₃×(C₂×D₃)⋊C₄	C₂³	1	32	yes	—
53	C₃×(C₂×D₃)⋊C₄	C₂³	1	16	yes	—
54	C₃×C₄⋊Dic₃	C₂³	1	8	yes	—
55	C₃×C₄×Dic₃	C₂³	1	16	yes	—
56	C₂×#555	—	3	1	no	Sylow
57	C₂²×C₃×Q₈	—	2	2	no	—
58	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
59	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
60	C₂²×C₃×D₄	—	3	2	no	—
61	C₂²×C₃×D₄	—	3	4	no	—
62	C₂³×C₃×C₄	—	3	4	no	—
63	C₂²×D₁₂	C₂²×C₃	1	2	yes	—
64	C₂²×D₁₂	C₂²×C₃	1	4	yes	—
65	C₂²×C₄×D₃	C₂²×C₃	1	4	yes	—
66	C₃×C₄⋊Q₈	—	2	8	no	—
67	C₃×C₄⋊Q₈	C₂×D₃	1	8	yes	complemented
68	C₃×C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
69	C₃×#124	—	3	32	no	—
70	C₃×#121	—	3	32	no	—
71	C₃×C₄×D₄	—	3	16	no	—
72	C₃×#117	—	3	32	no	—
73	C₂×C₃×C₄⋊C₄	—	2	4	no	—
74	C₂×C₃×C₄⋊C₄	—	3	2	no	—
75	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
77	C₂×C₃×C₂²⋊C₄	—	3	8	no	—
78	C₂×C₃×C₂²⋊C₄	—	3	4	no	—
79	C₂×C₃×C₄²	—	2	1	no	—
80	C₂×C₃×C₄²	—	3	4	no	—
81	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
82	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
83	#947	C₂²×C₃	1	32	yes	complemented
84	C₂×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
85	C₂×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
86	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
87	C₂×C₄×Dic₃	C₂²×C₃	1	4	yes	—
88	#895	C₂²×C₃	1	32	yes	complemented
89	#892	C₂²×C₃	1	32	yes	complemented
90	C₄×D₁₂	C₂²×C₃	1	16	yes	complemented
91	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
92	C₂²×C₃×D₃	—	2	8	no	—
93	C₂²×C₃×D₃	—	2	8	no	—
94	C₂²×C₃×D₃	—	2	8	no	—
95	C₂²×C₃×D₃	C₂⁴	1	8	yes	—
96	C₃²×Q₈	—	4	8	no	—
97	C₂×C₃²×C₄	—	2	2	no	—
98	C₂×C₃²×C₄	—	2	8	no	—
99	C₂×C₃²×C₄	#41	1	8	yes	—
100	C₂×C₃²×C₄	C₂⁴	1	8	yes	—
101	C₂×C₃²×C₄	C₂⁴	1	2	yes	—
102	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
103	C₂×C₃×Dic₃	—	2	8	no	—
104	C₂×C₃×Dic₃	C₂⁴	1	8	yes	—
105	C₃×D₁₂	—	2	16	no	—
106	C₃×D₁₂	—	4	16	no	—
107	C₃×C₄×D₃	—	2	32	no	—
108	#555	—	3	16	no	—
109	C₂×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
110	C₂×#124	—	3	4	no	—
111	C₂×#121	—	3	4	no	—
112	C₂×C₄×D₄	—	3	2	no	—
113	C₂×#117	—	3	4	no	—
114	C₂⁴×C₃	—	3	4	no	—
115	C₂³×D₃	C₂³×C₃	1	4	yes	—
116	C₂×C₃×Q₈	—	2	8	no	—
117	C₂×C₃×Q₈	—	4	4	no	—
118	C₂×C₃×Q₈	—	4	4	no	—
119	C₂×C₃×Q₈	—	4	8	no	—
120	C₂×C₃×Q₈	C₂²×D₃	1	8	yes	—
121	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
122	C₂×C₃×D₄	—	3	8	no	—
123	C₂×C₃×D₄	—	3	16	no	—
124	C₂×C₃×D₄	—	6	8	no	—
125	C₂×C₃×D₄	—	12	8	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	2	4	no	—
129	C₂²×C₃×C₄	—	3	4	no	—
130	C₂²×C₃×C₄	—	3	16	no	—
131	C₂²×C₃×C₄	—	6	16	no	—
132	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
133	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
135	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
136	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
138	C₂²×Dic₃	C₂³×C₃	1	4	yes	—
139	C₂×D₁₂	—	2	8	no	—
140	C₂×D₁₂	—	4	8	no	—
141	C₂×D₁₂	C₂³×C₃	1	8	yes	—
142	C₂×D₁₂	C₂³×C₃	1	16	yes	—
143	C₂×C₄×D₃	—	2	16	no	—
144	C₂×C₄×D₃	C₂³×C₃	1	16	yes	—
145	C₃×C₄⋊C₄	—	2	16	no	—
146	C₃×C₄⋊C₄	—	3	8	no	—
147	C₃×C₄⋊C₄	C₂²×D₃	1	16	yes	—
148	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
149	C₃×C₂²⋊C₄	—	3	32	no	—
150	C₃×C₂²⋊C₄	—	3	16	no	—
151	C₃×C₄²	—	2	4	no	—
152	C₃×C₄²	—	3	16	no	—
153	C₃×C₄²	C₂²×D₃	1	4	yes	—
154	C₃×C₄²	C₂³×C₃	1	4	yes	—
155	(C₂×D₃)⋊C₄	C₂³×C₃	1	16	yes	—
156	(C₂×D₃)⋊C₄	C₂³×C₃	1	32	yes	—
157	C₄⋊Dic₃	C₂³×C₃	1	8	yes	—
158	C₄×Dic₃	C₂³×C₃	1	16	yes	—
159	C₂²×C₃²	C₂×#41	1	4	yes	—
160	C₂²×C₃²	H(5,𝔽₂)	1	2	yes	—
161	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
162	C₂×C₃×D₃	—	2	8	no	—
163	C₂×C₃×D₃	—	2	8	no	—
164	C₂×C₃×D₃	—	2	8	no	—
165	C₂×C₃×D₃	—	4	16	no	—
166	C₃²×C₄	—	2	2	no	—
167	C₃²×C₄	—	2	8	no	—
168	C₃²×C₄	C₂×#41	1	8	yes	—
169	C₃×Dic₃	—	2	8	no	—
170	C₂²×Q₈	C₂×C₃×D₃	1	2	yes	—
171	C₂²×D₄	—	3	2	no	—
172	C₂²×D₄	—	3	4	no	—
173	C₂³×C₄	—	3	4	no	—
174	C₄⋊Q₈	C₂×C₃×D₃	1	8	yes	complemented
175	#124	—	3	32	no	—
176	#121	—	3	32	no	—
177	C₄×D₄	—	3	16	no	—
178	#117	—	3	32	no	—
179	C₂×C₄⋊C₄	—	3	2	no	—
180	C₂×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
181	C₂×C₂²⋊C₄	—	3	8	no	—
182	C₂×C₂²⋊C₄	—	3	4	no	—
183	C₂×C₄²	—	3	4	no	—
184	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
185	C₂³×C₃	—	2	1	no	—
186	C₂³×C₃	—	3	8	no	—
187	C₂³×C₃	—	6	8	no	—
188	C₂³×C₃	—	6	8	no	—
189	C₂³×C₃	—	6	8	no	—
190	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
191	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
192	C₂²×D₃	—	2	8	no	—
193	C₂²×D₃	—	2	8	no	—
194	C₂²×D₃	—	2	8	no	—
195	C₂²×D₃	C₂⁴×C₃	1	8	yes	—
196	C₃×Q₈	—	4	8	no	—
197	C₃×Q₈	—	4	8	no	—
198	C₃×Q₈	—	4	16	no	—
199	C₃×D₄	—	6	16	no	—
200	C₃×D₄	—	12	16	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	8	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	8	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	8	no	—
209	C₂×C₃×C₄	—	2	8	no	—
210	C₂×C₃×C₄	—	3	8	no	—
211	C₂×C₃×C₄	—	4	8	no	—
212	C₂×C₃×C₄	—	6	8	no	—
213	C₂×C₃×C₄	—	6	32	no	—
214	C₂×C₃×C₄	#239	1	8	yes	—
215	C₂×C₃×C₄	C₃×#41	1	8	yes	—
216	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
217	C₂×C₃×C₄	C₂³×D₃	1	2	yes	—
218	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
219	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
220	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
221	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
222	C₂×Dic₃	—	2	8	no	—
223	C₂×Dic₃	C₂⁴×C₃	1	8	yes	—
224	D₁₂	—	2	16	no	—
225	D₁₂	—	4	16	no	—
226	C₄×D₃	—	2	32	no	—
227	C₂×C₃²	#555	1	4	yes	complemented
228	C₂×C₃²	C₂²×#41	1	2	yes	—
229	C₂×C₃²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
230	C₃×D₃	—	4	8	no	—
231	C₂⁴	—	3	4	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	C₂²×C₃×D₃	1	8	yes	—
234	C₂×D₄	—	3	8	no	—
235	C₂×D₄	—	3	16	no	—
236	C₂×D₄	—	6	8	no	—
237	C₂×D₄	—	12	8	no	—
238	C₂²×C₄	—	3	4	no	—
239	C₂²×C₄	—	3	16	no	—
240	C₂²×C₄	—	6	16	no	—
241	C₂²×C₄	C₂²×C₃×D₃	1	4	yes	—
242	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₃×D₃	1	2	yes	—
244	C₄⋊C₄	—	3	8	no	—
245	C₄⋊C₄	C₂²×C₃×D₃	1	16	yes	—
246	C₂²⋊C₄	—	3	32	no	—
247	C₂²⋊C₄	—	3	16	no	—
248	C₄²	—	3	16	no	—
249	C₄²	C₂²×C₃×D₃	1	4	yes	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	4	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	6	8	no	—
254	C₂²×C₃	—	6	8	no	—
255	C₂²×C₃	—	6	8	no	—
256	C₂²×C₃	—	12	16	no	—
257	C₂²×C₃	#1004	1	2	yes	—
258	C₂²×C₃	C₂×#239	1	4	yes	—
259	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
260	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
261	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
262	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
263	C₂×D₃	—	2	8	no	—
264	C₂×D₃	—	2	8	no	—
265	C₂×D₃	—	2	8	no	—
266	C₂×D₃	—	4	16	no	—
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	8	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	8	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	2	8	no	—
273	C₃×C₄	—	4	8	no	—
274	C₃×C₄	—	6	8	no	—
275	C₃×C₄	C₂×#239	1	8	yes	—
276	C₃×C₄	C₂×C₃×#41	1	8	yes	—
277	Dic₃	—	2	8	no	—
278	C₃²	C₂×#555	1	1	yes	characteristic, complemented, Sylow
279	C₂³	—	3	8	no	—
280	C₂³	—	6	8	no	—
281	C₂³	—	6	8	no	—
282	C₂³	—	6	8	no	—
283	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
284	Q₈	—	4	8	no	—
285	D₄	—	6	16	no	—
286	D₄	—	12	16	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	3	8	no	—
290	C₂×C₄	—	6	8	no	—
291	C₂×C₄	—	6	32	no	—
292	C₂×C₄	C₃×#239	1	8	yes	—
293	C₂×C₄	C₂³×C₃×D₃	1	8	yes	—
294	C₂×C₄	C₂³×C₃×D₃	1	2	yes	—
295	C₂×C₄	C₂³×C₃×D₃	1	4	yes	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	—	2	4	no	—
299	C₂×C₃	—	12	8	no	—
300	C₂×C₃	#5734	1	4	yes	direct product factor
301	C₂×C₃	C₃×#555	1	4	yes	complemented
302	C₂×C₃	C₂×#1004	1	1	yes	characteristic
303	C₂×C₃	C₂²×#239	1	2	yes	—
304	C₂×C₃	C₂²×C₃×#41	1	2	yes	—
305	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
306	D₃	—	4	8	no	—
307	C₂²	—	6	8	no	—
308	C₂²	—	6	8	no	—
309	C₂²	—	6	8	no	—
310	C₂²	—	12	16	no	—
311	C₂²	C₃×#1004	1	2	yes	—
312	C₂²	C₂×C₃×#239	1	4	yes	—
313	C₂²	C₂⁴×C₃×D₃	1	1	yes	characteristic
314	C₄	—	2	8	no	—
315	C₄	—	2	2	no	—
316	C₄	—	6	8	no	—
317	C₄	C₂×C₃×#239	1	8	yes	—
318	C₃	—	2	1	no	—
319	C₃	C₂×#5734	1	1	yes	characteristic, direct product factor
320	C₃	C₂×C₃×#555	1	1	yes	characteristic, complemented
321	C₂	—	12	8	no	—
322	C₂	C₃×#5734	1	4	yes	direct product factor
323	C₂	C₂×C₃×#1004	1	1	yes	characteristic
324	C₂	C₂²×C₃×#239	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₃
Fitting	C₂×C₃²×C₄⋊Q₈	C₂
socle	C₂³×C₃²	C₂⁴
hypercenter	C₂×C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×C₃×#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₄⋊Q₈	C₂	semidirect
2. C₂×C₃×#947	C₂	semidirect
3. C₃×#5734	C₂	direct (N × H)
4. C₂×C₃×#895	C₂	semidirect
5. C₂×C₃×#892	C₂	semidirect
6. C₂×C₃×C₄×D₁₂	C₂	semidirect
8. C₂×#5734	C₃	direct (N × H)
13. C₃²×C₄⋊Q₈	C₂²	semidirect
16. C₃×#947	C₂²	semidirect
21. C₃×#895	C₂²	semidirect
22. C₃×#892	C₂²	semidirect
23. C₃×C₄×D₁₂	C₂²	semidirect
26. C₂×C₃×C₄⋊Q₈	D₃	semidirect
27. C₂×C₃×C₄⋊Q₈	C₂×C₃	semidirect
32. C₂×#947	C₂×C₃	semidirect
33. #5734	C₂×C₃	direct (N × H)
34. C₂×#895	C₂×C₃	semidirect
35. C₂×#892	C₂×C₃	semidirect
36. C₂×C₄×D₁₂	C₂×C₃	semidirect
67. C₃×C₄⋊Q₈	C₂×D₃	semidirect
68. C₃×C₄⋊Q₈	C₂²×C₃	semidirect
83. #947	C₂²×C₃	semidirect
88. #895	C₂²×C₃	semidirect
89. #892	C₂²×C₃	semidirect
90. C₄×D₁₂	C₂²×C₃	semidirect
109. C₂×C₄⋊Q₈	C₃×D₃	semidirect
174. C₄⋊Q₈	C₂×C₃×D₃	semidirect
227. C₂×C₃²	#555	semidirect
278. C₃²	C₂×#555	semidirect
300. C₂×C₃	#5734	direct (N × H)
301. C₂×C₃	C₃×#555	semidirect
319. C₃	C₂×#5734	direct (N × H)
320. C₃	C₂×C₃×#555	semidirect
322. C₂	C₃×#5734	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊Q₈	2
C₂×C₃×#947	2
C₃×#5734	2
C₂×C₃×#895	2
C₂×C₃×#892	2
C₂×C₃×C₄×D₁₂	2
C₂×C₃×#555	3
C₂×#5734	3

Subgroups — C2×C3×#5734 #11903313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#5734 #11903313