Subgroups — #79462

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#237	C₂	1	1	yes	maximal, characteristic
2	C₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
3	#5127	C₂	1	1	yes	maximal, characteristic, complemented
4	#5118	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×#67	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#5011	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#4828	C₂	1	1	yes	maximal, characteristic, complemented
10	#4728	C₂	1	1	yes	maximal, characteristic, complemented
11	#67⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#4719	C₂	1	1	yes	maximal, characteristic, complemented
13	#4718	C₂	1	1	yes	maximal, characteristic, complemented
14	#4705	C₂	1	1	yes	maximal, characteristic, complemented
15	#4701	C₂	1	1	yes	maximal, characteristic, complemented
16	#2990	—	3	1	no	maximal, Sylow
17	C₂×#237	C₄	1	1	yes	characteristic
18	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
21	D₄×Dic₃	C₄	1	1	yes	characteristic
22	C₄×#67	C₂²	1	1	yes	characteristic
23	C₄×#67	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
26	C₂×#208	C₂²	1	1	yes	characteristic
27	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
29	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
30	#892	C₄	1	1	yes	characteristic
31	#887	C₄	1	1	yes	characteristic
32	#67⋊C₄	C₄	1	1	yes	characteristic
33	#67⋊C₄	C₄	1	1	yes	characteristic
34	#877	C₄	1	1	yes	characteristic
35	#877	C₄	1	1	yes	characteristic
36	C₄²×D₃	C₂²	1	1	yes	characteristic
37	C₄×Dic₆	C₂²	1	1	yes	characteristic
38	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃×#97	C₂²	1	1	yes	characteristic
42	#830	C₂²	1	1	yes	characteristic
43	#830	C₂²	1	1	yes	characteristic
44	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
46	#815	C₂²	1	1	yes	characteristic
47	#815	C₂²	1	1	yes	characteristic
48	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
55	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
56	C₈×Dic₃	C₂²	1	1	yes	characteristic
57	C₈×Dic₃	C₂²	1	1	yes	characteristic
58	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
59	#803	C₂²	1	1	yes	characteristic
60	C₄×#41	—	3	1	no	—
61	#446	—	3	1	no	—
62	#446	—	3	1	no	—
63	#435	—	3	1	no	—
64	#435	—	3	1	no	—
65	#435	—	3	1	no	—
66	#435	D₃	1	1	yes	characteristic, complemented
67	C₈×D₄	—	3	1	no	—
68	C₈×D₄	—	3	1	no	—
69	#432	—	3	1	no	—
70	#432	—	3	1	no	—
71	#431	—	3	1	no	—
72	#423	—	3	1	no	—
73	C₂×C₄⋊C₈	—	3	1	no	—
74	C₂×C₄⋊C₈	—	3	1	no	—
75	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂×#67	C₂×C₄	1	1	yes	characteristic
79	C₂×#67	C₂×C₄	1	1	yes	characteristic
80	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
81	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
82	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
84	#237	—	2	4	no	—
85	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
88	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
89	C₃×M₁₆	—	4	1	no	—
90	C₂×C₃×C₈	—	4	1	no	—
91	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
95	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
97	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
98	C₃×C₄²	C₂³	1	1	yes	characteristic
99	#217	C₂×C₄	1	1	yes	characteristic
100	#217	C₂×C₄	1	1	yes	characteristic
101	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
102	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
103	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
104	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
105	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
106	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
107	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
108	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
109	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
110	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
111	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
112	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
113	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
114	C₄×Dic₃	C₂³	1	1	yes	characteristic
115	C₄×Dic₃	C₂³	1	1	yes	characteristic
116	C₄×Dic₃	C₂³	1	1	yes	characteristic
117	#208	—	4	1	no	—
118	C₂×C₃⋊C₈	—	4	1	no	—
119	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
120	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
122	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
123	D₃×C₈	—	4	1	no	—
124	D₃×C₈	—	4	1	no	—
125	C₂×#41	—	3	1	no	—
126	C₂×M₁₆	—	3	1	no	—
127	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
128	C₂²×C₈	—	3	1	no	—
129	C₂²×C₈	—	3	1	no	—
130	C₂²×C₈	—	3	1	no	—
131	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
132	C₄×Q₈	—	3	1	no	—
133	C₄×Q₈	—	3	1	no	—
134	C₄×D₄	—	3	1	no	—
135	C₄×D₄	—	3	1	no	—
136	C₄×D₄	—	3	1	no	—
137	C₄×D₄	—	3	1	no	—
138	C₄×D₄	—	3	1	no	—
139	C₄×D₄	C₂×D₃	1	1	yes	characteristic
140	#117	—	3	1	no	—
141	#117	—	3	1	no	—
142	#117	—	3	1	no	—
143	C₂×C₄²	—	3	1	no	—
144	C₂×C₄²	—	3	1	no	—
145	C₂×C₄²	—	3	1	no	—
146	C₄⋊C₈	—	3	1	no	—
147	C₄⋊C₈	—	3	1	no	—
148	C₄⋊C₈	—	3	1	no	—
149	C₄⋊C₈	—	3	1	no	—
150	C₄⋊C₈	—	3	1	no	—
151	C₄⋊C₈	—	3	1	no	—
152	C₄⋊C₈	—	3	1	no	—
153	C₄⋊C₈	—	3	1	no	—
154	C₄⋊C₈	—	3	1	no	—
155	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
156	C₂²⋊C₈	—	3	1	no	—
157	C₂²⋊C₈	—	3	1	no	—
158	C₂²⋊C₈	—	3	1	no	—
159	C₂²⋊C₈	—	3	1	no	—
160	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
161	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
162	#97	—	3	1	no	—
163	#97	—	3	1	no	—
164	#97	—	3	1	no	—
165	#97	C₂×D₃	1	1	yes	characteristic
166	C₄×C₈	—	3	1	no	—
167	C₄×C₈	—	3	1	no	—
168	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
169	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
170	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
171	C₃×D₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
181	#67	—	2	2	no	—
182	#67	M₁₆	1	4	yes	complemented
183	C₂×Dic₃	—	2	2	no	—
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	2	2	no	—
186	C₂×Dic₃	—	2	2	no	—
187	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
188	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
189	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
190	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
191	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
192	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
193	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
194	C₄×D₃	—	2	2	no	—
195	C₄×D₃	—	2	1	no	—
196	C₄×D₃	—	2	1	no	—
197	C₄×D₃	—	2	2	no	—
198	Dic₆	—	2	2	no	—
199	C₃×C₈	—	2	1	no	—
200	C₃×C₈	—	2	1	no	—
201	C₃×C₈	—	2	1	no	—
202	C₃×C₈	—	2	1	no	—
203	C₃⋊C₈	—	2	1	no	—
204	C₃⋊C₈	—	2	1	no	—
205	C₃⋊C₈	—	2	1	no	—
206	C₃⋊C₈	—	2	1	no	—
207	#41	—	6	4	no	—
208	C₂×Q₈	—	3	1	no	—
209	C₂×D₄	—	3	1	no	—
210	C₂×D₄	—	3	1	no	—
211	C₂×D₄	C₄×D₃	1	1	yes	characteristic
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	C₂²×C₄	—	3	1	no	—
217	C₂²×C₄	—	3	1	no	—
218	C₂²×C₄	—	3	1	no	—
219	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
221	M₁₆	—	4	1	no	—
222	M₁₆	—	12	1	no	—
223	C₂×C₈	—	3	1	no	—
224	C₂×C₈	—	3	1	no	—
225	C₂×C₈	—	3	1	no	—
226	C₂×C₈	—	3	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	12	1	no	—
229	C₂×C₈	—	12	1	no	—
230	C₂×C₈	—	12	1	no	—
231	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
232	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
235	C₄⋊C₄	—	3	1	no	—
236	C₄⋊C₄	—	3	1	no	—
237	C₄⋊C₄	—	3	1	no	—
238	C₄⋊C₄	—	3	1	no	—
239	C₄⋊C₄	—	3	1	no	—
240	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
241	C₂²⋊C₄	—	3	1	no	—
242	C₂²⋊C₄	—	3	1	no	—
243	C₂²⋊C₄	—	3	1	no	—
244	C₂²⋊C₄	—	3	1	no	—
245	C₂²⋊C₄	C₄×D₃	1	1	yes	characteristic
246	C₂²⋊C₄	C₄×D₃	1	1	yes	characteristic
247	C₄²	—	3	1	no	—
248	C₄²	—	3	1	no	—
249	C₄²	—	3	1	no	—
250	C₄²	—	3	1	no	—
251	C₄²	—	3	1	no	—
252	C₄²	—	3	1	no	—
253	C₄²	—	3	1	no	—
254	C₄²	—	3	1	no	—
255	C₄²	—	3	1	no	—
256	C₄²	C₂²×D₃	1	1	yes	characteristic
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	C₂×M₁₆	1	2	yes	—
263	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
264	C₂×D₃	—	2	1	no	—
265	C₂×D₃	—	2	1	no	—
266	C₂×D₃	C₂×M₁₆	1	2	yes	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
272	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
273	Dic₃	—	2	1	no	—
274	Dic₃	—	2	1	no	—
275	Dic₃	—	2	1	no	—
276	Dic₃	—	2	1	no	—
277	Dic₃	—	2	1	no	—
278	Dic₃	C₂×M₁₆	1	2	yes	—
279	Dic₃	#131	1	2	yes	—
280	C₂³	—	3	1	no	—
281	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
282	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
283	Q₈	—	6	2	no	—
284	D₄	—	2	2	no	—
285	D₄	—	3	4	no	—
286	D₄	—	6	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	3	1	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	3	1	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	6	2	no	—
300	C₂×C₄	—	6	2	no	—
301	C₂×C₄	—	6	2	no	—
302	C₂×C₄	—	6	2	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	6	1	no	—
305	C₂×C₄	—	6	1	no	—
306	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	6	1	no	—
316	C₈	—	6	1	no	—
317	C₈	—	6	1	no	—
318	C₈	—	6	1	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	4	1	no	—
321	C₂×C₃	#520	1	1	yes	characteristic
322	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
323	C₂×C₃	C₂×#131	1	1	yes	characteristic
324	D₃	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	3	2	no	—
331	C₂²	—	6	1	no	—
332	C₂²	—	6	1	no	—
333	C₂²	D₃×M₁₆	1	2	yes	—
334	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	3	2	no	—
340	C₄	—	3	2	no	—
341	C₄	—	6	1	no	—
342	C₄	—	6	1	no	—
343	C₄	—	6	1	no	—
344	C₄	—	6	1	no	—
345	C₄	—	6	1	no	—
346	C₄	#1004	1	1	yes	characteristic
347	C₄	#1010	1	1	yes	characteristic
348	C₃	#2990	1	1	yes	characteristic, complemented, Sylow
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	6	2	no	—
352	C₂	#5546	1	1	yes	characteristic
353	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic
354	C₂	D₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#435	C₂
prime radical	C₂×#237	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#435	D₃
hypocenter	C₃	#2990
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. C₃×#435	C₂	semidirect
3. #5127	C₂	semidirect
4. #5118	C₂	semidirect
5. C₈×#67	C₂	semidirect
6. C₂×Dic₃⋊C₈	C₂	semidirect
7. #5011	C₂	semidirect
8. D₃×C₄⋊C₈	C₂	semidirect
9. #4828	C₂	semidirect
10. #4728	C₂	semidirect
11. #67⋊C₈	C₂	semidirect
12. #4719	C₂	semidirect
13. #4718	C₂	semidirect
14. #4705	C₂	semidirect
15. #4701	C₂	semidirect
66. #435	D₃	semidirect
182. #67	M₁₆	semidirect
348. C₃	#2990	semidirect

Maximal subgroups

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

subgroup	index
C₄×#237	2
C₃×#435	2
#5127	2
#5118	2
C₈×#67	2
C₂×Dic₃⋊C₈	2
#5011	2
D₃×C₄⋊C₈	2
#4828	2
#4728	2
#67⋊C₈	2
#4719	2
#4718	2
#4705	2
#4701	2
#2990	3