Subgroups — #10948442

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#443	C₂	1	1	yes	maximal, characteristic
2	#83780	C₂	1	2	yes	maximal, complemented
3	#72006	C₂	1	2	yes	maximal, complemented
4	#70924	C₂	1	4	yes	maximal, complemented
5	#70923	C₂	1	4	yes	maximal, complemented
6	C₂×#4463	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#812	C₂	1	1	yes	maximal, characteristic, complemented
8	#33443	—	3	1	no	maximal, Sylow
9	C₂×C₃×#135	C₄	1	1	yes	characteristic
10	C₂×C₃×#131	C₂²	1	2	yes	—
11	C₃×#443	C₄	1	2	yes	—
12	C₃×#443	C₄	1	2	yes	—
13	C₃×#443	C₂²	1	2	yes	—
14	C₃×#443	C₂²	1	2	yes	—
15	C₂×C₃×#108	C₄	1	1	yes	characteristic
16	C₂×C₃×#104	C₄	1	2	yes	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	#5140	—	2	4	no	—
19	#5140	C₂²	1	8	yes	complemented
20	C₂×#812	—	2	4	no	—
21	C₂×#812	C₂²	1	2	yes	—
22	C₂×#812	C₂²	1	1	yes	characteristic
23	C₂×#812	C₂²	1	1	yes	characteristic
24	#4561	C₂²	1	8	yes	complemented
25	#4463	C₂²	1	2	yes	complemented
26	#4463	C₂²	1	2	yes	complemented
27	#4461	C₂²	1	2	yes	complemented
28	C₄×C₃⋊C₁₆	C₂²	1	2	yes	complemented
29	#3407	—	3	2	no	—
30	C₂×#443	D₃	1	1	yes	characteristic
31	#2171	—	3	4	no	—
32	#2170	—	3	4	no	—
33	C₂×#364	—	3	1	no	—
34	#2117	—	3	2	no	—
35	C₄×M₃₂	—	3	1	no	—
36	C₂×C₃×#41	C₂×C₄	1	2	yes	—
37	C₃×#135	C₈	1	2	yes	—
38	C₃×#135	C₈	1	2	yes	—
39	C₃×#135	C₂×C₄	1	2	yes	—
40	C₃×#135	C₂×C₄	1	2	yes	—
41	C₂×C₃×Q₁₆	C₈	1	1	yes	characteristic
42	C₂×C₃×SD₁₆	C₈	1	2	yes	—
43	C₂×C₃×D₈	C₈	1	1	yes	characteristic
44	C₃×#131	—	2	2	no	—
45	C₃×#131	—	2	2	no	—
46	C₃×#131	C₂×C₄	1	4	yes	—
47	C₃×#131	C₂³	1	4	yes	—
48	C₂×C₃×M₁₆	—	2	2	no	—
49	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
50	C₂²×C₃×C₈	—	2	2	no	—
51	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
54	C₃×#108	C₂×C₄	1	2	yes	—
55	C₃×#108	C₂×C₄	1	2	yes	—
56	C₃×#104	C₂×C₄	1	4	yes	—
57	C₃×#104	C₂×C₄	1	4	yes	—
58	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
62	#812	—	2	2	no	—
63	#812	—	2	1	no	—
64	#812	—	2	1	no	—
65	#812	—	2	8	no	—
66	#812	D₄	1	2	yes	—
67	#812	D₄	1	2	yes	—
68	#812	C₂³	1	4	yes	—
69	C₂×C₃⋊C₁₆	—	2	2	no	—
70	C₂×C₃⋊C₁₆	—	2	8	no	—
71	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
72	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
73	C₂×#135	Dic₃	1	1	yes	characteristic
74	C₂×#131	C₂×D₃	1	2	yes	—
75	#504	—	3	8	no	—
76	#504	—	6	4	no	—
77	C₂×M₃₂	—	3	1	no	—
78	C₂×M₃₂	—	3	1	no	—
79	C₂×M₃₂	—	3	2	no	—
80	C₂×M₃₂	—	6	4	no	—
81	#443	Dic₃	1	2	yes	—
82	#443	Dic₃	1	2	yes	—
83	#443	C₂×D₃	1	2	yes	—
84	#443	C₂×D₃	1	2	yes	—
85	C₂×#108	Dic₃	1	1	yes	characteristic
86	C₂×#104	Dic₃	1	2	yes	—
87	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
88	#364	—	3	2	no	—
89	#364	—	3	2	no	—
90	#350	—	3	8	no	—
91	#346	—	3	2	no	—
92	C₄×C₁₆	—	3	2	no	—
93	C₃×#41	—	2	2	no	—
94	C₃×#41	—	2	2	no	—
95	C₃×#41	C₂×C₈	1	4	yes	—
96	C₃×#41	C₂²×C₄	1	4	yes	—
97	C₂×C₃×Q₈	C₂×C₈	1	2	yes	—
98	C₂×C₃×D₄	—	2	2	no	—
99	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
100	C₂²×C₃×C₄	—	2	2	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
103	C₃×Q₁₆	C₂×C₈	1	2	yes	—
104	C₃×Q₁₆	C₂×C₈	1	2	yes	—
105	C₃×SD₁₆	C₂×C₈	1	4	yes	—
106	C₃×SD₁₆	C₂×C₈	1	4	yes	—
107	C₃×D₈	C₂×C₈	1	2	yes	—
108	C₃×D₈	C₂×C₈	1	2	yes	—
109	C₃×M₁₆	—	2	2	no	—
110	C₃×M₁₆	—	2	4	no	—
111	C₃×M₁₆	—	2	4	no	—
112	C₃×M₁₆	C₂²×C₄	1	4	yes	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	#41	1	1	yes	characteristic
126	C₂×C₃×C₈	#41	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
130	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
131	C₃⋊C₁₆	—	2	4	no	—
132	C₃⋊C₁₆	—	2	2	no	—
133	C₃⋊C₁₆	C₂×D₄	1	4	yes	—
134	C₂×#41	C₂×Dic₃	1	2	yes	—
135	#135	C₃⋊C₈	1	2	yes	—
136	#135	C₃⋊C₈	1	2	yes	—
137	#135	C₂×Dic₃	1	2	yes	—
138	#135	C₂×Dic₃	1	2	yes	—
139	C₂×Q₁₆	C₃⋊C₈	1	1	yes	characteristic
140	C₂×SD₁₆	C₃⋊C₈	1	2	yes	—
141	C₂×D₈	C₃⋊C₈	1	1	yes	characteristic
142	#131	—	2	2	no	—
143	#131	—	2	2	no	—
144	#131	C₂×Dic₃	1	4	yes	—
145	#131	C₂²×D₃	1	4	yes	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
152	M₃₂	—	3	2	no	—
153	M₃₂	—	3	2	no	—
154	M₃₂	—	3	4	no	—
155	M₃₂	—	6	8	no	—
156	M₃₂	—	6	2	no	—
157	M₃₂	—	6	1	no	—
158	M₃₂	—	6	1	no	—
159	C₂×C₁₆	—	3	2	no	—
160	C₂×C₁₆	—	3	2	no	—
161	C₂×C₁₆	—	6	8	no	—
162	C₂×C₁₆	—	6	2	no	—
163	#108	C₂×Dic₃	1	2	yes	—
164	#108	C₂×Dic₃	1	2	yes	—
165	#104	C₂×Dic₃	1	4	yes	—
166	#104	C₂×Dic₃	1	4	yes	—
167	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
168	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
169	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
171	C₂³×C₃	—	2	2	no	—
172	C₂³×C₃	#131	1	1	yes	characteristic
173	C₃×Q₈	—	2	2	no	—
174	C₃×Q₈	C₂²×C₈	1	4	yes	—
175	C₃×D₄	—	2	4	no	—
176	C₃×D₄	—	2	4	no	—
177	C₃×D₄	—	2	2	no	—
178	C₃×D₄	C₂²×C₈	1	4	yes	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
189	C₂×C₃×C₄	#131	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
191	C₃×C₈	—	2	4	no	—
192	C₃×C₈	C₄×D₄	1	1	yes	characteristic
193	C₃×C₈	C₄×D₄	1	1	yes	characteristic
194	C₃×C₈	C₂²×C₈	1	1	yes	characteristic
195	C₃×C₈	C₂²×C₈	1	1	yes	characteristic
196	C₃×C₈	C₂²×C₈	1	1	yes	characteristic
197	C₃×C₈	C₂²×C₈	1	1	yes	characteristic
198	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
199	C₃×C₈	C₂×#41	1	1	yes	characteristic
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	#41	C₂×C₃⋊C₈	1	4	yes	—
203	#41	C₂²×Dic₃	1	4	yes	—
204	C₂×Q₈	C₂×C₃⋊C₈	1	2	yes	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
210	Q₁₆	C₂×C₃⋊C₈	1	2	yes	—
211	Q₁₆	C₂×C₃⋊C₈	1	2	yes	—
212	SD₁₆	C₂×C₃⋊C₈	1	4	yes	—
213	SD₁₆	C₂×C₃⋊C₈	1	4	yes	—
214	D₈	C₂×C₃⋊C₈	1	2	yes	—
215	D₈	C₂×C₃⋊C₈	1	2	yes	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	C₂²×Dic₃	1	4	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
224	C₂×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
225	C₂×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
226	C₂×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
227	C₂×C₈	D₃×D₄	1	1	yes	characteristic
228	C₂×C₈	D₃×D₄	1	1	yes	characteristic
229	C₂×C₈	#237	1	1	yes	characteristic
230	C₂×C₈	#237	1	1	yes	characteristic
231	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
232	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
234	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
235	C₄²	—	2	1	no	—
236	C₄²	C₂²×Dic₃	1	1	yes	characteristic
237	C₄²	C₂²×Dic₃	1	1	yes	characteristic
238	C₁₆	—	3	4	no	—
239	C₁₆	—	6	4	no	—
240	C₁₆	—	6	2	no	—
241	C₂²×C₃	—	2	4	no	—
242	C₂²×C₃	—	4	1	no	—
243	C₂²×C₃	—	8	2	no	—
244	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
245	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
246	C₂²×C₃	C₂×#131	1	1	yes	characteristic
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	4	no	—
249	C₃×C₄	—	4	1	no	—
250	C₃×C₄	C₈×D₄	1	1	yes	characteristic
251	C₃×C₄	C₈×D₄	1	1	yes	characteristic
252	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
253	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
254	C₂³	—	2	2	no	—
255	C₂³	#948	1	1	yes	characteristic
256	Q₈	—	2	2	no	—
257	Q₈	C₂²×C₃⋊C₈	1	4	yes	—
258	D₄	—	2	4	no	—
259	D₄	—	2	4	no	—
260	D₄	—	2	2	no	—
261	D₄	C₂²×C₃⋊C₈	1	4	yes	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
272	C₂×C₄	#948	1	1	yes	characteristic
273	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	C₂²×C₃⋊C₈	1	1	yes	characteristic
276	C₈	C₂²×C₃⋊C₈	1	1	yes	characteristic
277	C₈	C₂²×C₃⋊C₈	1	1	yes	characteristic
278	C₈	C₂²×C₃⋊C₈	1	1	yes	characteristic
279	C₈	D₄×Dic₃	1	1	yes	characteristic
280	C₈	D₄×Dic₃	1	1	yes	characteristic
281	C₈	C₂×D₃×D₄	1	1	yes	characteristic
282	C₈	C₂×#237	1	1	yes	characteristic
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	4	4	no	—
287	C₂×C₃	C₂×C₈×D₄	1	1	yes	characteristic
288	C₂²	—	2	4	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	8	2	no	—
291	C₂²	D₄×C₃⋊C₈	1	1	yes	characteristic
292	C₂²	D₄×C₃⋊C₈	1	1	yes	characteristic
293	C₂²	C₂×#948	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	4	1	no	—
297	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
298	C₄	D₄×C₃⋊C₈	1	1	yes	characteristic
299	C₄	C₂³×C₃⋊C₈	1	1	yes	characteristic
300	C₄	C₂×D₄×Dic₃	1	1	yes	characteristic
301	C₃	#33443	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₂×D₄×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×#443	C₂
socle	C₂×C₃	C₂×C₈×D₄
prime radical	C₂×C₃×#135	C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×#443	D₃
hypocenter	C₃	#33443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #83780	C₂	semidirect
3. #72006	C₂	semidirect
4. #70924	C₂	semidirect
5. #70923	C₂	semidirect
6. C₂×#4463	C₂	semidirect
7. C₄×#812	C₂	semidirect
19. #5140	C₂²	semidirect
24. #4561	C₂²	semidirect
25. #4463	C₂²	semidirect
26. #4463	C₂²	semidirect
27. #4461	C₂²	semidirect
28. C₄×C₃⋊C₁₆	C₂²	semidirect
301. C₃	#33443	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#443	2
#83780	2
#72006	2
#70924	2
#70923	2
C₂×#4463	2
C₄×#812	2
#33443	3