Subgroups — #78603

Order 384.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#67	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#5207	C₂	1	1	yes	maximal, characteristic, complemented
4	#5111	C₂	1	2	yes	maximal, complemented
5	C₈×#67	C₂	1	2	yes	maximal, complemented
6	#4729	C₂	1	2	yes	maximal, complemented
7	#4728	C₂	1	2	yes	maximal, complemented
8	#4726	C₂	1	1	yes	maximal, characteristic, complemented
9	D₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#4719	C₂	1	1	yes	maximal, characteristic, complemented
11	#4718	C₂	1	1	yes	maximal, characteristic, complemented
12	#2974	—	3	1	no	maximal, Sylow
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂²×#67	C₄	1	1	yes	characteristic
15	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
17	C₂×#217	C₄	1	1	yes	characteristic
18	C₄×#67	C₄	1	1	yes	characteristic
19	C₄×#67	C₄	1	1	yes	characteristic
20	C₄×#67	C₄	1	1	yes	characteristic
21	C₄×#67	C₄	1	1	yes	characteristic
22	C₄×#67	C₂²	1	1	yes	characteristic
23	C₄×#67	C₂²	1	1	yes	characteristic
24	C₄×#67	C₂²	1	1	yes	characteristic
25	C₄×#67	C₂²	1	1	yes	characteristic
26	C₂×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
27	C₂×Dic₃⋊C₄	C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
29	C₂×#208	C₂²	1	2	yes	complemented
30	C₂×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
31	C₂×D₃×C₈	C₂²	1	2	yes	complemented
32	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	#830	C₂²	1	1	yes	characteristic
36	#830	C₂²	1	1	yes	characteristic
37	#830	C₂²	1	2	yes	complemented
38	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
39	(C₂×D₃)⋊C₈	C₂²	1	2	yes	complemented
40	#815	C₂²	1	2	yes	complemented
41	Dic₃⋊C₈	C₂²	1	2	yes	complemented
42	Dic₃⋊C₈	C₂²	1	2	yes	complemented
43	C₈×Dic₃	C₂²	1	2	yes	complemented
44	C₂×C₄×D₄	—	3	1	no	—
45	#436	—	3	2	no	—
46	#435	—	3	2	no	—
47	#435	—	3	2	no	—
48	C₈×D₄	—	3	2	no	—
49	#432	—	3	1	no	—
50	#431	—	3	1	no	—
51	#407	—	3	1	no	—
52	#407	—	3	1	no	—
53	C₂×C₂²⋊C₈	—	3	1	no	—
54	C₂×C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
55	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
56	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	—	4	2	no	—
58	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₂×#67	—	2	4	no	—
65	C₂×#67	C₂×C₄	1	1	yes	characteristic
66	C₂×#67	C₂×C₄	1	1	yes	characteristic
67	C₂×#67	C₂×C₄	1	1	yes	characteristic
68	C₂×#67	C₂×C₄	1	1	yes	characteristic
69	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
70	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
71	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×D₃	—	4	2	no	—
73	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₈	—	4	2	no	—
78	C₂×C₃×C₈	C₂³	1	2	yes	—
79	C₂×C₃×C₈	C₂³	1	2	yes	—
80	#217	C₂×C₄	1	1	yes	characteristic
81	#217	C₂×C₄	1	1	yes	characteristic
82	#217	C₂×C₄	1	1	yes	characteristic
83	#217	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
89	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
90	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₃	C₂³	1	1	yes	characteristic
96	#208	—	4	2	no	—
97	C₂×C₃⋊C₈	C₂³	1	2	yes	—
98	C₂×C₃⋊C₈	C₂³	1	2	yes	—
99	ΓL(1,𝔽₂₅)	—	4	2	no	—
100	D₃×C₈	—	4	2	no	—
101	C₂²×D₄	—	3	1	no	—
102	C₂³×C₄	—	3	1	no	—
103	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
104	C₂×M₁₆	—	3	2	no	—
105	C₂×M₁₆	—	3	2	no	—
106	C₂²×C₈	—	3	2	no	—
107	C₂²×C₈	C₂×D₃	1	2	yes	complemented
108	C₄×D₄	—	3	1	no	—
109	C₄×D₄	—	3	1	no	—
110	C₄×D₄	—	3	1	no	—
111	C₄×D₄	—	3	1	no	—
112	C₄×D₄	—	3	1	no	—
113	C₄×D₄	—	3	1	no	—
114	C₄×D₄	—	3	1	no	—
115	C₄×D₄	—	3	1	no	—
116	C₂×C₄⋊C₄	—	3	1	no	—
117	C₂×C₂²⋊C₄	—	3	1	no	—
118	C₂×C₂²⋊C₄	—	3	1	no	—
119	C₂×C₄²	—	3	1	no	—
120	C₄⋊C₈	—	3	2	no	—
121	C₄⋊C₈	—	3	2	no	—
122	C₂²⋊C₈	—	3	1	no	—
123	C₂²⋊C₈	—	3	1	no	—
124	C₂²⋊C₈	—	3	2	no	—
125	C₂²⋊C₈	—	3	2	no	—
126	C₂²⋊C₈	—	3	2	no	—
127	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
128	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
129	C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
130	#97	—	3	2	no	—
131	C₄×C₈	—	3	2	no	—
132	C₂³×C₃	—	2	2	no	—
133	C₂³×C₃	—	2	2	no	—
134	C₂³×C₃	—	4	1	no	—
135	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
136	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
137	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×D₃	—	2	2	no	—
139	C₂²×D₃	—	2	2	no	—
140	C₂²×D₃	—	4	1	no	—
141	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	4	1	no	—
147	C₂×C₃×C₄	—	4	1	no	—
148	C₂×C₃×C₄	—	4	1	no	—
149	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
153	#67	—	2	2	no	—
154	#67	—	2	2	no	—
155	#67	—	2	2	no	—
156	#67	M₁₆	1	4	yes	complemented
157	C₂×Dic₃	—	2	2	no	—
158	C₂×Dic₃	—	2	2	no	—
159	C₂×Dic₃	—	4	1	no	—
160	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
161	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
162	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
163	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
164	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
165	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
166	C₄×D₃	—	2	2	no	—
167	C₄×D₃	—	2	2	no	—
168	C₄×D₃	—	4	1	no	—
169	C₄×D₃	—	4	1	no	—
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃⋊C₈	—	2	2	no	—
173	C₃⋊C₈	—	2	2	no	—
174	C₂⁴	—	3	1	no	—
175	C₂⁴	C₄×D₃	1	1	yes	characteristic
176	C₂×D₄	—	3	1	no	—
177	C₂×D₄	—	3	1	no	—
178	C₂×D₄	—	3	1	no	—
179	C₂×D₄	—	3	1	no	—
180	C₂×D₄	—	6	4	no	—
181	C₂²×C₄	—	3	1	no	—
182	C₂²×C₄	—	3	1	no	—
183	C₂²×C₄	—	3	1	no	—
184	C₂²×C₄	—	3	1	no	—
185	C₂²×C₄	—	3	1	no	—
186	C₂²×C₄	—	3	1	no	—
187	C₂²×C₄	—	3	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	12	2	no	—
190	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
191	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
192	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
196	M₁₆	—	12	2	no	—
197	M₁₆	—	12	2	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	3	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	12	2	no	—
202	C₂×C₈	C₂²×D₃	1	2	yes	—
203	C₂×C₈	C₂²×D₃	1	2	yes	—
204	C₄⋊C₄	—	3	1	no	—
205	C₄⋊C₄	—	3	1	no	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	3	1	no	—
208	C₂²⋊C₄	—	3	1	no	—
209	C₂²⋊C₄	—	3	1	no	—
210	C₂²⋊C₄	—	3	1	no	—
211	C₂²⋊C₄	—	3	1	no	—
212	C₂²⋊C₄	—	3	1	no	—
213	C₂²⋊C₄	—	3	1	no	—
214	C₂²⋊C₄	—	3	1	no	—
215	C₂²⋊C₄	—	3	1	no	—
216	C₄²	—	3	1	no	—
217	C₄²	—	3	1	no	—
218	C₄²	—	3	1	no	—
219	C₄²	—	3	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	4	4	no	—
228	C₂²×C₃	C₂×M₁₆	1	2	yes	—
229	C₂²×C₃	#131	1	2	yes	—
230	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
231	C₂×D₃	—	2	1	no	—
232	C₂×D₃	—	2	1	no	—
233	C₂×D₃	—	2	1	no	—
234	C₂×D₃	—	2	1	no	—
235	C₂×D₃	—	2	1	no	—
236	C₂×D₃	—	4	4	no	—
237	C₂×D₃	C₂×M₁₆	1	2	yes	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
242	Dic₃	—	2	1	no	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	2	1	no	—
246	Dic₃	—	2	1	no	—
247	Dic₃	C₂×M₁₆	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	3	1	no	—
251	C₂³	—	3	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	6	2	no	—
254	C₂³	—	6	2	no	—
255	C₂³	—	12	1	no	—
256	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
257	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
258	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
259	D₄	—	3	4	no	—
260	D₄	—	6	2	no	—
261	D₄	—	6	2	no	—
262	D₄	—	6	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	6	2	no	—
276	C₂×C₄	—	6	2	no	—
277	C₂×C₄	—	6	2	no	—
278	C₂×C₄	—	6	2	no	—
279	C₂×C₄	—	12	1	no	—
280	C₂×C₄	—	12	1	no	—
281	C₂×C₄	—	12	1	no	—
282	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
285	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	2	2	no	—
288	C₈	—	6	2	no	—
289	C₈	—	6	2	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	4	1	no	—
293	C₂×C₃	#518	1	1	yes	characteristic
294	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
295	C₂×C₃	C₂×#131	1	1	yes	characteristic
296	D₃	—	2	2	no	—
297	D₃	—	4	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	3	2	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	6	1	no	—
308	C₂²	—	6	1	no	—
309	C₂²	—	6	1	no	—
310	C₂²	—	6	1	no	—
311	C₂²	—	6	1	no	—
312	C₂²	—	12	4	no	—
313	C₂²	#901	1	2	yes	—
314	C₂²	D₃×M₁₆	1	2	yes	—
315	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	3	2	no	—
320	C₄	—	6	1	no	—
321	C₄	—	6	1	no	—
322	C₄	—	6	1	no	—
323	C₄	—	6	1	no	—
324	C₄	—	6	1	no	—
325	C₄	#1004	1	2	yes	—
326	C₃	#2974	1	1	yes	characteristic, complemented, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	6	2	no	—
331	C₂	—	12	1	no	—
332	C₂	#5486	1	1	yes	characteristic
333	C₂	C₂×#901	1	1	yes	characteristic
334	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₂²⋊C₈	C₂
prime radical	C₂²×#67	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₃
hypocenter	C₃	#2974
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₂²⋊C₈	C₂	semidirect
3. #5207	C₂	semidirect
4. #5111	C₂	semidirect
5. C₈×#67	C₂	semidirect
6. #4729	C₂	semidirect
7. #4728	C₂	semidirect
8. #4726	C₂	semidirect
9. D₃×C₂²⋊C₈	C₂	semidirect
10. #4719	C₂	semidirect
11. #4718	C₂	semidirect
16. C₂²×C₃×C₈	C₂²	semidirect
29. C₂×#208	C₂²	semidirect
30. C₂×ΓL(1,𝔽₂₅)	C₂²	semidirect
31. C₂×D₃×C₈	C₂²	semidirect
34. C₃×C₂²⋊C₈	C₂²	semidirect
37. #830	C₂²	semidirect
38. (C₂×D₃)⋊C₈	C₂²	semidirect
39. (C₂×D₃)⋊C₈	C₂²	semidirect
40. #815	C₂²	semidirect
41. Dic₃⋊C₈	C₂²	semidirect
42. Dic₃⋊C₈	C₂²	semidirect
43. C₈×Dic₃	C₂²	semidirect
54. C₂×C₂²⋊C₈	D₃	semidirect
107. C₂²×C₈	C₂×D₃	semidirect
129. C₂²⋊C₈	C₂×D₃	semidirect
156. #67	M₁₆	semidirect
326. C₃	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#67	2
C₂×C₃×C₂²⋊C₈	2
#5207	2
#5111	2
C₈×#67	2
#4729	2
#4728	2
#4726	2
D₃×C₂²⋊C₈	2
#4719	2
#4718	2
#2974	3