Subgroups — #81211

Order 384.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5698	C₂	1	1	yes	maximal, characteristic
2	#5572	C₂	1	1	yes	maximal, characteristic
3	C₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
4	#5063	C₂	1	1	yes	maximal, characteristic, complemented
5	#5031	C₂	1	1	yes	maximal, characteristic, complemented
6	#4888	C₂	1	1	yes	maximal, characteristic
7	#4873	C₂	1	1	yes	maximal, characteristic, complemented
8	#4868	C₂	1	1	yes	maximal, characteristic
9	#4853	C₂	1	1	yes	maximal, characteristic, complemented
10	#4849	C₂	1	1	yes	maximal, characteristic, complemented
11	#4840	C₂	1	1	yes	maximal, characteristic
12	#4809	C₂	1	1	yes	maximal, characteristic
13	#4809	C₂	1	1	yes	maximal, characteristic
14	#4790	C₂	1	1	yes	maximal, characteristic, complemented
15	#4789	C₂	1	1	yes	maximal, characteristic, complemented
16	#3127	—	3	1	no	maximal, Sylow
17	C₂×D₃×Q₈	C₂²	1	1	yes	characteristic
18	C₃×#125	C₂²	1	1	yes	characteristic
19	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×#216	C₂²	1	1	yes	characteristic
22	C₂×#216	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₂	C₂²	1	1	yes	characteristic
25	#898	—	2	1	no	—
26	#898	—	2	1	no	—
27	#897	C₂²	1	1	yes	characteristic
28	#897	C₂²	1	1	yes	characteristic
29	#892	—	2	1	no	—
30	#892	C₂²	1	1	yes	characteristic
31	#890	—	2	1	no	—
32	#890	—	2	1	no	—
33	#887	—	2	1	no	—
34	#887	C₂²	1	1	yes	characteristic
35	#887	C₂²	1	1	yes	characteristic
36	#887	C₂²	1	1	yes	characteristic
37	#872	—	2	1	no	—
38	#872	C₂²	1	1	yes	characteristic
39	#869	C₂²	1	1	yes	characteristic
40	C₄×Dic₆	—	2	1	no	—
41	C₄×Dic₆	C₂²	1	1	yes	characteristic
42	C₃×#107	C₂²	1	1	yes	characteristic
43	C₃×#106	C₂²	1	1	yes	characteristic
44	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₃×#103	C₂²	1	1	yes	characteristic
46	C₃×#103	C₂²	1	1	yes	characteristic
47	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×D₃)⋊C₈	C₂²	1	1	yes	characteristic
49	#816	C₂²	1	1	yes	characteristic
50	#816	C₂²	1	1	yes	characteristic
51	#815	C₂²	1	1	yes	characteristic
52	C₈×Dic₃	C₂²	1	1	yes	characteristic
53	#810	C₂²	1	1	yes	characteristic
54	#810	C₂²	1	1	yes	characteristic
55	#810	C₂²	1	1	yes	characteristic
56	#810	C₂²	1	1	yes	characteristic
57	#808	C₂²	1	1	yes	characteristic
58	#807	C₂²	1	1	yes	characteristic
59	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
60	#541	—	3	1	no	—
61	#519	—	3	1	no	—
62	#485	—	3	1	no	—
63	#485	—	3	1	no	—
64	#479	—	3	1	no	—
65	#479	D₃	1	1	yes	characteristic, complemented
66	#462	—	3	1	no	—
67	#462	—	3	1	no	—
68	#451	—	3	1	no	—
69	#451	—	3	1	no	—
70	#441	—	3	1	no	—
71	#441	—	3	1	no	—
72	C₄×Q₁₆	—	3	1	no	—
73	C₄×Q₁₆	—	3	1	no	—
74	#433	—	3	1	no	—
75	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
76	D₃×Q₈	—	2	2	no	—
77	D₃×Q₈	—	2	2	no	—
78	C₂×C₄×D₃	—	2	1	no	—
79	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
80	C₂×Dic₆	—	2	1	no	—
81	C₂×Dic₆	C₂³	1	1	yes	characteristic
82	C₂×Dic₆	C₂³	1	1	yes	characteristic
83	C₂×Dic₆	C₂³	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₄⋊C₄	—	2	1	no	—
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
90	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₃×C₄²	—	2	1	no	—
93	C₃×C₄²	C₂³	1	1	yes	characteristic
94	#216	—	2	2	no	—
95	#216	—	2	2	no	—
96	(C₂×D₃)⋊C₄	—	2	1	no	—
97	(C₂×D₃)⋊C₄	—	2	1	no	—
98	(C₂×D₃)⋊C₄	—	2	1	no	—
99	(C₂×D₃)⋊C₄	D₄	1	1	yes	characteristic
100	(C₂×D₃)⋊C₄	D₄	1	1	yes	characteristic
101	C₄⋊Dic₃	—	2	1	no	—
102	C₄⋊Dic₃	—	2	1	no	—
103	C₄⋊Dic₃	—	2	1	no	—
104	C₄⋊Dic₃	—	2	1	no	—
105	C₄⋊Dic₃	—	2	1	no	—
106	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
107	Dic₃⋊C₄	—	2	1	no	—
108	Dic₃⋊C₄	—	2	1	no	—
109	Dic₃⋊C₄	—	2	1	no	—
110	Dic₃⋊C₄	—	2	1	no	—
111	Dic₃⋊C₄	D₄	1	1	yes	characteristic
112	Dic₃⋊C₄	D₄	1	1	yes	characteristic
113	C₄×Dic₃	—	2	1	no	—
114	C₄×Dic₃	—	2	1	no	—
115	C₄×Dic₃	—	2	1	no	—
116	C₄×Dic₃	—	2	1	no	—
117	C₄×Dic₃	—	2	1	no	—
118	C₄×Dic₃	—	2	1	no	—
119	C₄×Dic₃	C₂³	1	1	yes	characteristic
120	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
121	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
122	Dic₁₂	—	2	2	no	—
123	Dic₁₂	—	2	2	no	—
124	C₂²×Q₈	—	3	1	no	—
125	C₂×Q₁₆	—	3	1	no	—
126	C₂×Q₁₆	—	3	1	no	—
127	C₂×Q₁₆	—	3	1	no	—
128	C₂×Q₁₆	—	3	1	no	—
129	C₄⋊Q₈	—	3	1	no	—
130	#126	—	6	1	no	—
131	#126	—	6	1	no	—
132	#125	—	6	1	no	—
133	#125	—	6	1	no	—
134	#125	C₂×D₃	1	1	yes	characteristic
135	C₂²⋊Q₈	—	3	1	no	—
136	C₂²⋊Q₈	—	3	1	no	—
137	C₄×Q₈	—	3	1	no	—
138	C₄×Q₈	—	3	1	no	—
139	C₄×Q₈	—	3	1	no	—
140	C₄×Q₈	—	3	1	no	—
141	C₄×Q₈	—	3	1	no	—
142	C₄×Q₈	—	6	1	no	—
143	C₄×Q₈	—	6	1	no	—
144	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
145	#117	—	3	1	no	—
146	#117	—	3	1	no	—
147	#117	—	6	1	no	—
148	#117	—	6	1	no	—
149	#107	—	3	1	no	—
150	#107	C₂×D₃	1	1	yes	characteristic
151	#106	—	3	1	no	—
152	#106	C₂×D₃	1	1	yes	characteristic
153	C₄⋊C₈	—	3	1	no	—
154	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
155	#103	—	3	1	no	—
156	#103	—	3	1	no	—
157	#103	—	3	1	no	—
158	#103	—	3	1	no	—
159	#103	—	3	1	no	—
160	#103	—	3	1	no	—
161	#103	C₂×D₃	1	1	yes	characteristic
162	#103	C₂×D₃	1	1	yes	characteristic
163	C₂²⋊C₈	—	3	1	no	—
164	C₂²⋊C₈	—	3	1	no	—
165	#97	—	3	1	no	—
166	C₄×C₈	—	3	1	no	—
167	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
168	C₃×Q₈	—	2	1	no	—
169	C₃×Q₈	#41	1	2	yes	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	1	no	—
181	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
182	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
183	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
184	C₄×D₃	—	2	4	no	—
185	C₄×D₃	—	2	1	no	—
186	C₄×D₃	—	2	1	no	—
187	Dic₆	—	2	4	no	—
188	Dic₆	—	2	1	no	—
189	Dic₆	—	2	1	no	—
190	Dic₆	—	2	1	no	—
191	Dic₆	—	2	1	no	—
192	Dic₆	—	2	1	no	—
193	Dic₆	#41	1	2	yes	—
194	C₃×C₈	—	2	1	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃⋊C₈	—	2	1	no	—
197	C₃⋊C₈	—	2	1	no	—
198	C₂×Q₈	—	3	1	no	—
199	C₂×Q₈	—	3	1	no	—
200	C₂×Q₈	—	3	1	no	—
201	C₂×Q₈	—	6	1	no	—
202	C₂×Q₈	—	6	2	no	—
203	C₂×Q₈	—	6	2	no	—
204	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	6	1	no	—
207	Q₁₆	—	6	2	no	—
208	Q₁₆	—	6	2	no	—
209	Q₁₆	—	6	2	no	—
210	Q₁₆	—	6	2	no	—
211	C₂×C₈	—	3	1	no	—
212	C₂×C₈	—	3	1	no	—
213	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
214	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	—	3	1	no	—
221	C₄⋊C₄	—	6	1	no	—
222	C₄⋊C₄	—	6	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄⋊C₄	—	6	1	no	—
225	C₄⋊C₄	—	6	1	no	—
226	C₄⋊C₄	—	6	1	no	—
227	C₄⋊C₄	—	6	1	no	—
228	C₄⋊C₄	—	6	1	no	—
229	C₄⋊C₄	—	6	1	no	—
230	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
231	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
232	C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
233	C₂²⋊C₄	—	3	1	no	—
234	C₂²⋊C₄	—	3	1	no	—
235	C₂²⋊C₄	—	6	1	no	—
236	C₂²⋊C₄	—	6	1	no	—
237	C₂²⋊C₄	—	6	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	3	1	no	—
240	C₄²	—	6	1	no	—
241	C₄²	—	6	1	no	—
242	C₄²	—	6	1	no	—
243	C₄²	—	6	1	no	—
244	C₄²	—	6	1	no	—
245	C₄²	—	6	1	no	—
246	C₄²	C₂²×D₃	1	1	yes	characteristic
247	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	—	2	1	no	—
250	C₂×D₃	—	2	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	4	1	no	—
255	C₃×C₄	—	4	1	no	—
256	C₃×C₄	—	4	1	no	—
257	C₃×C₄	C₂×#41	1	1	yes	characteristic
258	C₃×C₄	C₂×#41	1	1	yes	characteristic
259	Dic₃	—	2	1	no	—
260	Dic₃	—	2	2	no	—
261	Dic₃	—	2	1	no	—
262	Dic₃	—	2	1	no	—
263	Dic₃	—	4	1	no	—
264	Dic₃	—	4	1	no	—
265	Dic₃	—	4	1	no	—
266	C₂³	—	3	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	3	2	no	—
269	Q₈	—	6	4	no	—
270	Q₈	—	6	1	no	—
271	Q₈	—	6	1	no	—
272	Q₈	—	6	1	no	—
273	Q₈	—	6	1	no	—
274	Q₈	—	6	1	no	—
275	Q₈	#239	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	3	1	no	—
283	C₂×C₄	—	6	4	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	—	6	1	no	—
286	C₂×C₄	—	6	1	no	—
287	C₂×C₄	—	6	1	no	—
288	C₂×C₄	—	6	1	no	—
289	C₂×C₄	—	6	1	no	—
290	C₂×C₄	D₃×D₄	1	1	yes	characteristic
291	C₂×C₄	D₃×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	6	1	no	—
296	C₈	—	6	1	no	—
297	C₂×C₃	#525	1	1	yes	characteristic
298	C₂×C₃	#578	1	1	yes	characteristic
299	C₂×C₃	#578	1	1	yes	characteristic
300	D₃	—	4	1	no	—
301	C₂²	—	6	1	no	—
302	C₂²	—	6	1	no	—
303	C₂²	—	6	1	no	—
304	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	6	1	no	—
312	C₄	—	6	1	no	—
313	C₄	—	6	1	no	—
314	C₄	—	6	2	no	—
315	C₄	—	12	1	no	—
316	C₄	—	12	1	no	—
317	C₄	—	12	1	no	—
318	C₄	C₂×#239	1	1	yes	characteristic
319	C₄	D₃×#41	1	1	yes	characteristic
320	C₃	#3127	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	12	1	no	—
322	C₂	#5627	1	1	yes	characteristic
323	C₂	#5771	1	1	yes	characteristic
324	C₂	#5779	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#479	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
hypercenter	#479	D₃
hypocenter	C₃	#3127
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. C₃×#479	C₂	semidirect
4. #5063	C₂	semidirect
5. #5031	C₂	semidirect
7. #4873	C₂	semidirect
9. #4853	C₂	semidirect
10. #4849	C₂	semidirect
14. #4790	C₂	semidirect
15. #4789	C₂	semidirect
65. #479	D₃	semidirect
320. C₃	#3127	semidirect

Maximal subgroups

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

subgroup	index
#5698	2
#5572	2
C₃×#479	2
#5063	2
#5031	2
#4888	2
#4873	2
#4868	2
#4853	2
#4849	2
#4840	2
#4809	2
#4790	2
#4789	2
#3127	3