Subgroups — #78669

Order 384.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5832	C₂	1	1	yes	maximal, characteristic
2	C₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
3	#5234	C₂	1	1	yes	maximal, characteristic, complemented
4	#5128	C₂	1	1	yes	maximal, characteristic, complemented
5	#5127	C₂	1	1	yes	maximal, characteristic, complemented
6	#5111	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#67	C₂	1	1	yes	maximal, characteristic, complemented
8	#4730	C₂	1	1	yes	maximal, characteristic, complemented
9	#4729	C₂	1	1	yes	maximal, characteristic, complemented
10	#4728	C₂	1	1	yes	maximal, characteristic, complemented
11	#4727	C₂	1	1	yes	maximal, characteristic, complemented
12	#4727	C₂	1	1	yes	maximal, characteristic, complemented
13	#67⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#4719	C₂	1	1	yes	maximal, characteristic, complemented
15	#4718	C₂	1	1	yes	maximal, characteristic, complemented
16	#2981	—	3	1	no	maximal, Sylow
17	C₂×C₃×#41	C₂²	1	1	yes	characteristic
18	C₂×#235	C₂²	1	1	yes	characteristic
19	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
21	#947	C₄	1	1	yes	characteristic
22	#944	C₄	1	1	yes	characteristic
23	#940	C₄	1	1	yes	characteristic
24	#939	C₄	1	1	yes	characteristic
25	#939	C₄	1	1	yes	characteristic
26	#939	C₄	1	1	yes	characteristic
27	#939	C₄	1	1	yes	characteristic
28	#936	C₄	1	1	yes	characteristic
29	C₄×#67	C₂²	1	1	yes	characteristic
30	C₄×#67	C₂²	1	1	yes	characteristic
31	C₄×#67	C₂²	1	1	yes	characteristic
32	C₄×#67	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
34	C₂×#208	C₂²	1	1	yes	characteristic
35	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
37	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
38	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
39	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
40	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#830	C₂²	1	1	yes	characteristic
45	#830	C₂²	1	1	yes	characteristic
46	#830	C₂²	1	1	yes	characteristic
47	#830	C₂²	1	1	yes	characteristic
48	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
52	#815	C₂²	1	1	yes	characteristic
53	#815	C₂²	1	1	yes	characteristic
54	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
55	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
56	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
57	Dic₃⋊C₈	C₂²	1	1	yes	characteristic
58	C₈×Dic₃	C₂²	1	1	yes	characteristic
59	C₈×Dic₃	C₂²	1	1	yes	characteristic
60	#532	—	3	1	no	—
61	#436	—	3	1	no	—
62	#436	—	3	1	no	—
63	#435	—	3	1	no	—
64	#435	—	3	1	no	—
65	#435	—	3	1	no	—
66	#435	—	3	1	no	—
67	C₈×D₄	—	3	1	no	—
68	C₈×D₄	—	3	1	no	—
69	#432	—	3	1	no	—
70	#431	—	3	1	no	—
71	#408	—	3	1	no	—
72	#408	—	3	1	no	—
73	#408	—	3	1	no	—
74	#408	D₃	1	1	yes	characteristic, complemented
75	C₃×#41	—	4	1	no	—
76	C₃×#41	—	4	1	no	—
77	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
79	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
80	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
84	C₂×#67	C₂×C₄	1	1	yes	characteristic
85	C₂×#67	C₂×C₄	1	1	yes	characteristic
86	C₂×#67	C₂×C₄	1	1	yes	characteristic
87	C₂×#67	C₂×C₄	1	1	yes	characteristic
88	C₂×#67	C₂×C₄	1	1	yes	characteristic
89	C₂×#67	C₂×C₄	1	1	yes	characteristic
90	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
91	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
92	#235	—	4	1	no	—
93	#235	—	4	1	no	—
94	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
96	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
97	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
98	C₃×M₁₆	—	4	1	no	—
99	C₂×C₃×C₈	—	4	1	no	—
100	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
104	#217	C₂×C₄	1	1	yes	characteristic
105	#217	C₂×C₄	1	1	yes	characteristic
106	#217	C₂×C₄	1	1	yes	characteristic
107	#217	C₂×C₄	1	1	yes	characteristic
108	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
109	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
110	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
111	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
112	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
113	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
114	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
115	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
116	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
117	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
118	C₄×Dic₃	C₂³	1	1	yes	characteristic
119	C₄×Dic₃	C₂³	1	1	yes	characteristic
120	#208	—	4	1	no	—
121	C₂×C₃⋊C₈	—	4	1	no	—
122	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
123	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
124	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
125	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
126	ΓL(1,𝔽₂₅)	—	4	1	no	—
127	ΓL(1,𝔽₂₅)	—	4	1	no	—
128	D₃×C₈	—	4	1	no	—
129	D₃×C₈	—	4	1	no	—
130	C₂×#41	—	3	1	no	—
131	C₂×#41	C₂×D₃	1	1	yes	characteristic
132	C₂×M₁₆	—	3	1	no	—
133	C₂×M₁₆	—	3	1	no	—
134	C₂×M₁₆	—	3	1	no	—
135	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
136	C₂²×C₈	—	3	1	no	—
137	C₂²×C₈	—	3	1	no	—
138	C₂²×C₈	—	3	1	no	—
139	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
140	C₄⋊Q₈	—	3	1	no	—
141	#127	—	3	1	no	—
142	#124	—	3	1	no	—
143	#124	—	3	1	no	—
144	#121	—	3	1	no	—
145	#121	—	3	1	no	—
146	#121	—	3	1	no	—
147	#121	—	3	1	no	—
148	C₄×D₄	—	3	1	no	—
149	C₄×D₄	—	3	1	no	—
150	C₄×D₄	—	3	1	no	—
151	C₄×D₄	—	3	1	no	—
152	C₂×C₄²	—	3	1	no	—
153	C₄⋊C₈	—	3	1	no	—
154	C₄⋊C₈	—	3	1	no	—
155	C₄⋊C₈	—	3	1	no	—
156	C₄⋊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	—	3	1	no	—
161	C₂²⋊C₈	—	3	1	no	—
162	C₂²⋊C₈	—	3	1	no	—
163	C₂²⋊C₈	—	3	1	no	—
164	C₂²⋊C₈	—	3	1	no	—
165	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
166	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
167	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
168	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
169	#97	—	3	1	no	—
170	#97	—	3	1	no	—
171	C₄×C₈	—	3	1	no	—
172	C₄×C₈	—	3	1	no	—
173	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
174	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
175	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
176	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
177	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
178	C₃×Q₈	—	4	1	no	—
179	C₃×D₄	—	4	1	no	—
180	C₃×D₄	—	4	1	no	—
181	C₃×D₄	—	4	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
192	#67	—	2	2	no	—
193	#67	—	2	2	no	—
194	#67	—	4	1	no	—
195	#67	—	4	1	no	—
196	#67	—	4	1	no	—
197	#67	—	4	1	no	—
198	C₂×Dic₃	—	2	2	no	—
199	C₂×Dic₃	—	4	1	no	—
200	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
201	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
202	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
203	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
204	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
205	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
206	D₁₂	—	4	1	no	—
207	C₄×D₃	—	2	1	no	—
208	C₄×D₃	—	2	1	no	—
209	C₄×D₃	—	2	1	no	—
210	C₄×D₃	—	2	1	no	—
211	Dic₆	—	4	1	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃⋊C₈	—	2	1	no	—
217	C₃⋊C₈	—	2	1	no	—
218	C₃⋊C₈	—	2	1	no	—
219	C₃⋊C₈	—	2	1	no	—
220	#41	—	4	1	no	—
221	#41	—	4	1	no	—
222	#41	—	12	1	no	—
223	#41	—	12	1	no	—
224	C₂×Q₈	—	3	1	no	—
225	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
226	C₂×D₄	—	3	1	no	—
227	C₂×D₄	—	3	1	no	—
228	C₂×D₄	—	3	1	no	—
229	C₂×D₄	—	3	1	no	—
230	C₂×D₄	—	3	1	no	—
231	C₂×D₄	—	3	1	no	—
232	C₂×D₄	—	3	1	no	—
233	C₂×D₄	C₄×D₃	1	1	yes	characteristic
234	C₂×D₄	C₄×D₃	1	1	yes	characteristic
235	C₂×D₄	C₄×D₃	1	1	yes	characteristic
236	C₂²×C₄	—	3	1	no	—
237	C₂²×C₄	—	3	1	no	—
238	C₂²×C₄	—	3	1	no	—
239	C₂²×C₄	—	3	1	no	—
240	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
243	M₁₆	—	4	1	no	—
244	M₁₆	—	12	1	no	—
245	M₁₆	—	12	1	no	—
246	M₁₆	—	12	1	no	—
247	C₂×C₈	—	3	1	no	—
248	C₂×C₈	—	3	1	no	—
249	C₂×C₈	—	3	1	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	12	1	no	—
253	C₂×C₈	—	12	1	no	—
254	C₂×C₈	—	12	1	no	—
255	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
259	C₄⋊C₄	—	3	1	no	—
260	C₄⋊C₄	—	3	1	no	—
261	C₄⋊C₄	—	3	1	no	—
262	C₄⋊C₄	—	3	1	no	—
263	C₂²⋊C₄	—	3	1	no	—
264	C₂²⋊C₄	—	3	1	no	—
265	C₂²⋊C₄	—	3	1	no	—
266	C₂²⋊C₄	—	3	1	no	—
267	C₂²⋊C₄	—	3	1	no	—
268	C₂²⋊C₄	—	3	1	no	—
269	C₂²⋊C₄	—	3	1	no	—
270	C₂²⋊C₄	—	3	1	no	—
271	C₄²	—	3	1	no	—
272	C₄²	—	3	1	no	—
273	C₄²	—	3	1	no	—
274	C₄²	—	3	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	—	2	1	no	—
281	C₂²×C₃	—	2	1	no	—
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
285	C₂×D₃	—	2	1	no	—
286	C₂×D₃	—	2	1	no	—
287	C₂×D₃	—	2	1	no	—
288	C₂×D₃	—	2	1	no	—
289	C₂×D₃	—	2	1	no	—
290	C₂×D₃	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
295	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
296	Dic₃	—	2	1	no	—
297	Dic₃	—	2	1	no	—
298	Dic₃	—	2	1	no	—
299	Dic₃	—	2	1	no	—
300	Dic₃	—	2	1	no	—
301	Dic₃	#131	1	2	yes	—
302	C₂³	—	3	1	no	—
303	C₂³	—	3	1	no	—
304	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
305	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
306	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
307	Q₈	—	4	1	no	—
308	Q₈	—	12	1	no	—
309	D₄	—	4	1	no	—
310	D₄	—	4	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	6	2	no	—
313	D₄	—	6	2	no	—
314	D₄	—	12	1	no	—
315	D₄	—	12	1	no	—
316	D₄	—	12	1	no	—
317	D₄	—	12	1	no	—
318	D₄	—	12	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	3	1	no	—
326	C₂×C₄	—	3	1	no	—
327	C₂×C₄	—	3	1	no	—
328	C₂×C₄	—	3	1	no	—
329	C₂×C₄	—	3	1	no	—
330	C₂×C₄	—	3	1	no	—
331	C₂×C₄	—	6	2	no	—
332	C₂×C₄	—	6	1	no	—
333	C₂×C₄	—	6	1	no	—
334	C₂×C₄	—	6	1	no	—
335	C₂×C₄	—	6	1	no	—
336	C₂×C₄	—	12	1	no	—
337	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	6	1	no	—
346	C₈	—	6	1	no	—
347	C₈	—	6	1	no	—
348	C₈	—	6	1	no	—
349	C₂×C₃	—	4	1	no	—
350	C₂×C₃	—	4	1	no	—
351	C₂×C₃	—	4	1	no	—
352	C₂×C₃	#518	1	1	yes	characteristic
353	C₂×C₃	C₂×#131	1	1	yes	characteristic
354	C₂×C₃	#568	1	1	yes	characteristic
355	D₃	—	4	1	no	—
356	D₃	—	4	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	6	1	no	—
367	C₂²	—	6	1	no	—
368	C₂²	—	6	1	no	—
369	C₂²	—	6	1	no	—
370	C₂²	—	6	1	no	—
371	C₂²	—	6	1	no	—
372	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	3	2	no	—
377	C₄	—	6	1	no	—
378	C₄	—	6	1	no	—
379	C₄	—	6	1	no	—
380	C₄	—	6	1	no	—
381	C₄	—	6	1	no	—
382	C₄	#1004	1	1	yes	characteristic
383	C₄	#1004	1	1	yes	characteristic
384	C₃	#2981	1	1	yes	characteristic, complemented, Sylow
385	C₂	—	4	1	no	—
386	C₂	—	4	1	no	—
387	C₂	—	4	1	no	—
388	C₂	—	12	1	no	—
389	C₂	—	12	1	no	—
390	C₂	#5486	1	1	yes	characteristic
391	C₂	D₃×#131	1	1	yes	characteristic
392	C₂	#5750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#408	C₂
prime radical	#5832	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#408	D₃
hypocenter	C₃	#2981
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₃×#408	C₂	semidirect
3. #5234	C₂	semidirect
4. #5128	C₂	semidirect
5. #5127	C₂	semidirect
6. #5111	C₂	semidirect
7. C₈×#67	C₂	semidirect
8. #4730	C₂	semidirect
9. #4729	C₂	semidirect
10. #4728	C₂	semidirect
11. #4727	C₂	semidirect
12. #4727	C₂	semidirect
13. #67⋊C₈	C₂	semidirect
14. #4719	C₂	semidirect
15. #4718	C₂	semidirect
74. #408	D₃	semidirect
384. C₃	#2981	semidirect

Maximal subgroups

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

subgroup	index
#5832	2
C₃×#408	2
#5234	2
#5128	2
#5127	2
#5111	2
C₈×#67	2
#4730	2
#4729	2
#4728	2
#4727	2
#67⋊C₈	2
#4719	2
#4718	2
#2981	3