Subgroups — #79374

Order 384.

402 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	D₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#5000	C₂	1	1	yes	maximal, characteristic, complemented
4	#4839	C₂	1	1	yes	maximal, characteristic, complemented
5	#4839	C₂	1	1	yes	maximal, characteristic, complemented
6	#4835	C₂	1	1	yes	maximal, characteristic, complemented
7	#4833	C₂	1	1	yes	maximal, characteristic, complemented
8	#4726	C₂	1	1	yes	maximal, characteristic
9	D₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
10	#4720	C₂	1	1	yes	maximal, characteristic, complemented
11	#4720	C₂	1	1	yes	maximal, characteristic, complemented
12	#4708	C₂	1	1	yes	maximal, characteristic, complemented
13	#4705	C₂	1	1	yes	maximal, characteristic, complemented
14	#4689	C₂	1	1	yes	maximal, characteristic, complemented
15	#4685	C₂	1	1	yes	maximal, characteristic, complemented
16	#2924	—	3	1	no	maximal, Sylow
17	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
18	C₃×#117	C₂²	1	1	yes	characteristic
19	#926	C₂²	1	1	yes	characteristic
20	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
22	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
23	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
24	#892	C₄	1	1	yes	characteristic
25	#892	C₄	1	1	yes	characteristic
26	D₃×C₄⋊C₄	C₄	1	1	yes	characteristic
27	D₃×C₄⋊C₄	C₄	1	1	yes	characteristic
28	D₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
29	D₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
30	#877	C₄	1	1	yes	characteristic
31	#877	C₄	1	1	yes	characteristic
32	#872	C₂²	1	1	yes	characteristic
33	#872	C₂²	1	1	yes	characteristic
34	C₄²×D₃	C₂²	1	1	yes	characteristic
35	C₄²×D₃	C₂²	1	1	yes	characteristic
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₃×#97	C₂²	1	1	yes	characteristic, complemented
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
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	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
47	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
48	#815	C₂²	1	1	yes	characteristic, complemented
49	#815	C₂²	1	1	yes	characteristic, complemented
50	Dic₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	Dic₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	Dic₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
53	Dic₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
54	C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
55	C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
57	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
58	#803	C₂²	1	1	yes	characteristic, complemented
59	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
60	C₂×#117	—	3	1	no	—
61	#433	—	3	1	no	—
62	#433	—	3	1	no	—
63	#433	—	3	1	no	—
64	#433	D₃	1	1	yes	characteristic, complemented
65	#432	—	3	1	no	—
66	#432	—	3	1	no	—
67	#431	—	3	1	no	—
68	#431	—	3	1	no	—
69	#424	—	3	1	no	—
70	#424	—	3	1	no	—
71	#407	—	3	1	no	—
72	C₂×C₂²⋊C₈	—	3	1	no	—
73	#405	—	3	1	no	—
74	#405	—	3	1	no	—
75	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
77	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×D₃	—	2	2	no	—
79	C₂×C₄×D₃	—	2	2	no	—
80	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
87	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
92	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
95	C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
96	C₃×C₄²	C₂³	1	1	yes	characteristic
97	C₃×C₄²	C₂³	1	1	yes	characteristic
98	#217	C₂×C₄	1	1	yes	characteristic
99	#217	C₂×C₄	1	1	yes	characteristic
100	(C₂×D₃)⋊C₄	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	(C₂×D₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
104	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
105	C₄⋊Dic₃	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	Dic₃⋊C₄	C₂×C₄	1	1	yes	characteristic
109	Dic₃⋊C₄	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	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
117	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
118	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
119	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
120	ΓL(1,𝔽₂₅)	—	2	2	no	—
121	ΓL(1,𝔽₂₅)	—	2	2	no	—
122	D₃×C₈	—	2	2	no	—
123	D₃×C₈	—	2	2	no	—
124	C₂³×C₄	—	3	1	no	—
125	C₂×M₁₆	—	3	1	no	—
126	C₂×M₁₆	—	3	1	no	—
127	C₂²×C₈	—	3	1	no	—
128	C₂²×C₈	—	3	1	no	—
129	#117	—	3	1	no	—
130	#117	—	3	1	no	—
131	#117	—	3	1	no	—
132	#117	—	3	1	no	—
133	#117	—	3	1	no	—
134	#117	—	3	1	no	—
135	#117	—	3	1	no	—
136	#117	C₂×D₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	3	1	no	—
138	C₂×C₄⋊C₄	—	3	1	no	—
139	C₂×C₂²⋊C₄	—	3	1	no	—
140	C₂×C₂²⋊C₄	—	3	1	no	—
141	C₂×C₄²	—	3	1	no	—
142	C₂×C₄²	—	3	1	no	—
143	C₄⋊C₈	—	3	1	no	—
144	C₄⋊C₈	—	3	1	no	—
145	C₄⋊C₈	—	3	1	no	—
146	C₄⋊C₈	—	3	1	no	—
147	C₄⋊C₈	—	3	1	no	—
148	C₄⋊C₈	—	3	1	no	—
149	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic, complemented
150	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic, complemented
151	C₂²⋊C₈	—	3	1	no	—
152	C₂²⋊C₈	—	3	1	no	—
153	C₂²⋊C₈	—	3	1	no	—
154	C₂²⋊C₈	—	3	1	no	—
155	C₂²⋊C₈	—	3	1	no	—
156	C₂²⋊C₈	—	3	1	no	—
157	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
159	#97	—	3	1	no	—
160	#97	—	3	1	no	—
161	#97	—	3	1	no	—
162	#97	C₂×D₃	1	1	yes	characteristic, complemented
163	C₄×C₈	—	3	1	no	—
164	C₄×C₈	—	3	1	no	—
165	C₄×C₈	—	3	1	no	—
166	C₄×C₈	C₂×D₃	1	1	yes	characteristic, complemented
167	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
168	C₂²×D₃	—	2	2	no	—
169	C₂²×D₃	—	2	2	no	—
170	C₂²×D₃	—	4	1	no	—
171	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
172	C₂²×D₃	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
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	C₂×Dic₃	—	2	2	no	—
182	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
183	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
184	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
185	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
186	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
187	C₂×Dic₃	C₂²×C₄	1	1	yes	characteristic
188	C₄×D₃	—	2	2	no	—
189	C₄×D₃	—	2	2	no	—
190	C₄×D₃	—	2	2	no	—
191	C₄×D₃	—	2	1	no	—
192	C₄×D₃	—	2	1	no	—
193	C₄×D₃	—	2	2	no	—
194	C₄×D₃	—	4	1	no	—
195	C₄×D₃	—	4	1	no	—
196	C₄×D₃	#41	1	2	yes	—
197	C₄×D₃	#41	1	2	yes	—
198	C₃×C₈	—	2	1	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	C₂⁴	—	3	1	no	—
207	C₂²×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	C₂²×C₄	—	6	2	no	—
217	C₂²×C₄	—	6	2	no	—
218	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
219	M₁₆	—	6	2	no	—
220	M₁₆	—	6	2	no	—
221	C₂×C₈	—	3	1	no	—
222	C₂×C₈	—	3	1	no	—
223	C₂×C₈	—	3	1	no	—
224	C₂×C₈	—	3	1	no	—
225	C₂×C₈	—	6	2	no	—
226	C₂×C₈	—	6	2	no	—
227	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
231	C₄⋊C₄	—	3	1	no	—
232	C₄⋊C₄	—	3	1	no	—
233	C₄⋊C₄	—	3	1	no	—
234	C₄⋊C₄	—	3	1	no	—
235	C₄⋊C₄	—	3	1	no	—
236	C₄⋊C₄	—	3	1	no	—
237	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
238	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
239	C₂²⋊C₄	—	3	1	no	—
240	C₂²⋊C₄	—	3	1	no	—
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	C₂²×D₃	1	1	yes	characteristic
254	C₄²	C₂²×D₃	1	1	yes	characteristic
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
259	C₂×D₃	—	2	1	no	—
260	C₂×D₃	—	2	1	no	—
261	C₂×D₃	—	2	1	no	—
262	C₂×D₃	—	2	1	no	—
263	C₂×D₃	—	4	4	no	—
264	C₂×D₃	#117	1	2	yes	—
265	C₂×D₃	#131	1	2	yes	—
266	C₃×C₄	—	2	1	no	—
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	C₂×#41	1	1	yes	characteristic
272	C₃×C₄	C₂×#41	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	#117	1	2	yes	—
279	C₂³	—	3	1	no	—
280	C₂³	—	3	1	no	—
281	C₂³	—	6	2	no	—
282	C₂³	—	6	2	no	—
283	C₂³	—	12	1	no	—
284	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	2	no	—
289	C₂×C₄	—	3	2	no	—
290	C₂×C₄	—	3	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	—	6	2	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	—	6	2	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	6	2	no	—
300	C₂×C₄	—	6	1	no	—
301	C₂×C₄	—	6	1	no	—
302	C₂×C₄	—	12	1	no	—
303	C₂×C₄	—	12	1	no	—
304	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
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	C₂³×D₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	6	1	no	—
315	C₈	—	6	1	no	—
316	C₈	—	6	1	no	—
317	C₈	—	6	1	no	—
318	C₂×C₃	—	4	1	no	—
319	C₂×C₃	C₂×#117	1	1	yes	characteristic
320	C₂×C₃	C₂×#131	1	1	yes	characteristic
321	C₂×C₃	#568	1	1	yes	characteristic
322	D₃	—	2	2	no	—
323	D₃	—	2	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	3	2	no	—
328	C₂²	—	3	2	no	—
329	C₂²	—	6	1	no	—
330	C₂²	—	6	1	no	—
331	C₂²	—	6	1	no	—
332	C₂²	—	6	1	no	—
333	C₂²	—	12	4	no	—
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	—	2	1	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	D₃×#41	1	1	yes	characteristic
347	C₄	D₃×#41	1	1	yes	characteristic
348	C₃	#2924	1	1	yes	characteristic, complemented, Sylow
349	C₂	—	4	1	no	—
350	C₂	—	6	2	no	—
351	C₂	—	6	2	no	—
352	C₂	D₃×#117	1	1	yes	characteristic
353	C₂	D₃×#131	1	1	yes	characteristic
354	C₂	#5750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#433	C₂
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#433	D₃
hypocenter	C₃	#2924
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. C₃×#433	C₂	semidirect
3. #5000	C₂	semidirect
4. #4839	C₂	semidirect
5. #4839	C₂	semidirect
6. #4835	C₂	semidirect
7. #4833	C₂	semidirect
10. #4720	C₂	semidirect
11. #4720	C₂	semidirect
12. #4708	C₂	semidirect
13. #4705	C₂	semidirect
14. #4689	C₂	semidirect
15. #4685	C₂	semidirect
36. C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×C₄⋊C₈	C₂²	semidirect
40. C₃×#97	C₂²	semidirect
41. C₃×C₄×C₈	C₂²	semidirect
48. #815	C₂²	semidirect
49. #815	C₂²	semidirect
50. Dic₃⋊C₈	C₂²	semidirect
51. Dic₃⋊C₈	C₂²	semidirect
52. Dic₃⋊C₈	C₂²	semidirect
53. Dic₃⋊C₈	C₂²	semidirect
54. C₈×Dic₃	C₂²	semidirect
55. C₈×Dic₃	C₂²	semidirect
56. C₄⋊(C₃⋊C₈)	C₂²	semidirect
57. C₄⋊(C₃⋊C₈)	C₂²	semidirect
58. #803	C₂²	semidirect
59. C₄×C₃⋊C₈	C₂²	semidirect
64. #433	D₃	semidirect
149. C₄⋊C₈	C₂×D₃	semidirect
150. C₄⋊C₈	C₂×D₃	semidirect
162. #97	C₂×D₃	semidirect
166. C₄×C₈	C₂×D₃	semidirect
348. C₃	#2924	semidirect

Maximal subgroups

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

subgroup	index
D₃×#117	2
C₃×#433	2
#5000	2
#4839	2
#4835	2
#4833	2
#4726	2
D₃×C₂²⋊C₈	2
#4720	2
#4708	2
#4705	2
#4689	2
#4685	2
#2924	3