Subgroups — #79492

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	#5548	C₂	1	1	yes	maximal, characteristic
2	C₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
3	#5129	C₂	1	1	yes	maximal, characteristic, complemented
4	#5128	C₂	1	1	yes	maximal, characteristic, complemented
5	#5110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#67	C₂	1	1	yes	maximal, characteristic, complemented
7	#5011	C₂	1	1	yes	maximal, characteristic, complemented
8	#4836	C₂	1	1	yes	maximal, characteristic, complemented
9	#4833	C₂	1	1	yes	maximal, characteristic, complemented
10	#4729	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	#4706	C₂	1	1	yes	maximal, characteristic, complemented
15	#4705	C₂	1	1	yes	maximal, characteristic, complemented
16	#2981	—	3	1	no	maximal, Sylow
17	C₂×#235	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	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
22	#940	C₄	1	1	yes	characteristic
23	#938	C₄	1	1	yes	characteristic
24	C₄×#67	C₂²	1	1	yes	characteristic
25	C₄×#67	C₂²	1	1	yes	characteristic
26	C₂×#208	C₂²	1	1	yes	characteristic
27	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
29	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
30	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
31	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
32	#895	C₄	1	1	yes	characteristic
33	#888	C₄	1	1	yes	characteristic
34	#885	C₄	1	1	yes	characteristic
35	#885	C₄	1	1	yes	characteristic
36	#884	C₄	1	1	yes	characteristic
37	#884	C₄	1	1	yes	characteristic
38	C₄×D₁₂	C₂²	1	1	yes	characteristic
39	C₄²×D₃	C₂²	1	1	yes	characteristic
40	C₃×C₄⋊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×#97	C₂²	1	1	yes	characteristic
44	#830	C₂²	1	1	yes	characteristic
45	#830	C₂²	1	1	yes	characteristic
46	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
47	(C₂×D₃)⋊C₈	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	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
53	(C₂×D₃)⋊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	#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	D₃	1	1	yes	characteristic, complemented
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	—	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	C₂×#67	C₂×C₄	1	1	yes	characteristic
81	C₂×#67	C₂×C₄	1	1	yes	characteristic
82	C₂×#67	C₂×C₄	1	1	yes	characteristic
83	C₂×#67	C₂×C₄	1	1	yes	characteristic
84	#235	—	4	1	no	—
85	#235	—	4	1	no	—
86	#235	—	4	1	no	—
87	#235	—	4	1	no	—
88	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
89	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
90	C₂×D₁₂	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
94	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
95	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
96	C₂×Dic₆	C₂×C₄	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	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
105	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
106	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
107	C₃×C₄²	C₂³	1	1	yes	characteristic
108	#217	C₂×C₄	1	1	yes	characteristic
109	#217	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	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
113	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
114	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
115	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
116	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
117	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
118	C₄×Dic₃	C₂×C₄	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	—	3	1	no	—
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	C₂×D₃	1	1	yes	characteristic
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	C₂×D₃	1	1	yes	characteristic
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	—	3	1	no	—
166	C₂²⋊C₈	—	3	1	no	—
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	C₂×D₃	1	1	yes	characteristic
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	C₂²×D₃	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×D₄	—	4	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
188	#67	—	2	2	no	—
189	#67	—	2	2	no	—
190	#67	—	4	1	no	—
191	#67	—	4	1	no	—
192	#67	—	4	1	no	—
193	#67	—	4	1	no	—
194	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
195	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
196	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
197	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
198	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
199	D₁₂	—	4	1	no	—
200	D₁₂	—	4	1	no	—
201	D₁₂	—	4	1	no	—
202	C₄×D₃	—	2	1	no	—
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	2	1	no	—
206	C₄×D₃	—	2	1	no	—
207	C₄×D₃	—	2	1	no	—
208	C₄×D₃	—	2	2	no	—
209	C₄×D₃	—	4	1	no	—
210	Dic₆	—	4	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	—	12	1	no	—
221	#41	—	12	1	no	—
222	#41	—	12	1	no	—
223	#41	—	12	1	no	—
224	C₂×Q₈	—	3	1	no	—
225	C₂×Q₈	—	3	1	no	—
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	—	3	1	no	—
234	C₂×D₄	—	3	1	no	—
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	—	3	1	no	—
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	C₄×D₃	1	1	yes	characteristic
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	C₄×D₃	1	1	yes	characteristic
270	C₂²⋊C₄	C₄×D₃	1	1	yes	characteristic
271	C₄²	—	3	1	no	—
272	C₄²	—	3	1	no	—
273	C₄²	—	3	1	no	—
274	C₄²	C₂²×D₃	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
282	C₂×D₃	—	2	1	no	—
283	C₂×D₃	—	2	1	no	—
284	C₂×D₃	—	2	1	no	—
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	—	2	1	no	—
295	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
296	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
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	—	3	1	no	—
305	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
306	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
307	Q₈	—	12	1	no	—
308	Q₈	—	12	1	no	—
309	D₄	—	4	1	no	—
310	D₄	—	6	2	no	—
311	D₄	—	6	2	no	—
312	D₄	—	12	1	no	—
313	D₄	—	12	1	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	—	3	1	no	—
324	C₂×C₄	—	3	1	no	—
325	C₂×C₄	—	3	1	no	—
326	C₂×C₄	—	3	1	no	—
327	C₂×C₄	—	3	1	no	—
328	C₂×C₄	—	6	2	no	—
329	C₂×C₄	—	6	1	no	—
330	C₂×C₄	—	6	1	no	—
331	C₂×C₄	—	6	1	no	—
332	C₂×C₄	—	6	1	no	—
333	C₂×C₄	—	6	1	no	—
334	C₂×C₄	—	6	1	no	—
335	C₂×C₄	—	12	1	no	—
336	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
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	#518	1	1	yes	characteristic
352	C₂×C₃	C₂×#131	1	1	yes	characteristic
353	C₂×C₃	#568	1	1	yes	characteristic
354	D₃	—	4	1	no	—
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	—	6	1	no	—
364	C₂²	—	6	1	no	—
365	C₂²	—	6	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	—	2	1	no	—
377	C₄	—	3	2	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	#1009	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	—	12	1	no	—
388	C₂	—	12	1	no	—
389	C₂	—	12	1	no	—
390	C₂	#5545	1	1	yes	characteristic
391	C₂	#5745	1	1	yes	characteristic
392	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	#5548	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#435	D₃
hypocenter	C₃	#2981
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. #5129	C₂	semidirect
4. #5128	C₂	semidirect
5. #5110	C₂	semidirect
6. C₈×#67	C₂	semidirect
7. #5011	C₂	semidirect
8. #4836	C₂	semidirect
9. #4833	C₂	semidirect
10. #4729	C₂	semidirect
11. #4727	C₂	semidirect
12. #4727	C₂	semidirect
13. #67⋊C₈	C₂	semidirect
14. #4706	C₂	semidirect
15. #4705	C₂	semidirect
66. #435	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
#5548	2
C₃×#435	2
#5129	2
#5128	2
#5110	2
C₈×#67	2
#5011	2
#4836	2
#4833	2
#4729	2
#4727	2
#67⋊C₈	2
#4706	2
#4705	2
#2981	3