Subgroups — #87239

Order 384.

1900 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1004	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#953	C₂	1	4	yes	maximal, complemented
4	#5806	C₂	1	8	yes	maximal, complemented
5	#5805	C₂	1	2	yes	maximal, complemented
6	#5804	C₂	1	4	yes	maximal, complemented
7	D₄×#67	C₂	1	8	yes	maximal, complemented
8	#5798	C₂	1	1	yes	maximal, characteristic, complemented
9	#5791	C₂	1	2	yes	maximal, complemented
10	#3484	—	3	1	no	maximal, Sylow
11	C₂⁵×C₃	C₂²	1	2	yes	complemented
12	C₂²×C₃×D₄	—	2	8	no	—
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	8	yes	complemented
15	C₂²×C₃×D₄	C₂²	1	2	yes	complemented
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
18	C₂²×#67	C₂²	1	4	yes	complemented
19	C₂²×#67	C₂²	1	8	yes	complemented
20	#1004	—	2	8	no	—
21	C₂×#237	C₂²	1	4	yes	complemented
22	C₂×D₃×D₄	C₂²	1	4	yes	complemented
23	C₂×#235	C₂²	1	2	yes	complemented
24	#953	C₂²	1	4	yes	complemented
25	#953	C₂²	1	16	yes	complemented
26	#953	C₂²	1	8	yes	complemented
27	C₂×#217	C₂²	1	4	yes	complemented
28	C₂×#217	C₂²	1	8	yes	complemented
29	#940	C₂²	1	4	yes	complemented
30	#939	C₂²	1	16	yes	complemented
31	#938	C₂²	1	8	yes	complemented
32	#937	C₂²	1	8	yes	complemented
33	#936	C₂²	1	4	yes	complemented
34	#935	C₂²	1	8	yes	complemented
35	D₄×Dic₃	C₂²	1	8	yes	complemented
36	#930	C₂²	1	4	yes	complemented
37	#929	C₂²	1	4	yes	complemented
38	C₄×#67	C₂²	1	8	yes	complemented
39	#926	C₂²	1	2	yes	complemented
40	#924	C₂²	1	4	yes	complemented
41	C₂×H(5,𝔽₂)	—	3	1	no	—
42	C₂³×D₄	D₃	1	1	yes	characteristic, complemented
43	#546	—	3	8	no	—
44	D₄²	—	3	8	no	—
45	#535	—	3	2	no	—
46	#534	—	3	4	no	—
47	#525	—	3	2	no	—
48	C₂×#120	—	3	4	no	—
49	#518	—	3	1	no	—
50	C₂⁴×C₃	—	2	8	no	—
51	C₂⁴×C₃	—	4	8	no	—
52	C₂⁴×C₃	C₂³	1	1	yes	characteristic
53	C₂⁴×C₃	C₂³	1	8	yes	—
54	C₂⁴×C₃	C₂³	1	2	yes	—
55	C₂⁴×C₃	C₂³	1	2	yes	—
56	C₂³×D₃	C₂³	1	4	yes	complemented
57	C₂×C₃×D₄	—	2	8	no	—
58	C₂×C₃×D₄	—	2	32	no	—
59	C₂×C₃×D₄	—	2	4	no	—
60	C₂×C₃×D₄	D₄	1	8	yes	complemented
61	C₂×C₃×D₄	C₂³	1	8	yes	—
62	C₂×C₃×D₄	C₂³	1	4	yes	—
63	C₂×C₃×D₄	C₂³	1	4	yes	—
64	C₂²×C₃×C₄	—	2	4	no	—
65	C₂²×C₃×C₄	C₂³	1	4	yes	—
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂×#67	—	2	16	no	—
69	C₂×#67	—	4	16	no	—
70	C₂×#67	C₂³	1	8	yes	—
71	C₂×#67	C₂³	1	4	yes	—
72	C₂×#67	C₂³	1	8	yes	complemented
73	C₂×#67	C₂³	1	16	yes	complemented
74	C₂²×Dic₃	C₂³	1	4	yes	complemented
75	C₂²×Dic₃	C₂³	1	8	yes	complemented
76	#237	—	2	16	no	—
77	D₃×D₄	—	2	16	no	—
78	#235	—	2	8	no	—
79	C₂×D₁₂	C₂³	1	2	yes	complemented
80	C₂×C₄×D₃	C₂³	1	4	yes	complemented
81	C₂×Dic₆	C₂³	1	2	yes	complemented
82	#217	C₂³	1	8	yes	—
83	#217	C₂³	1	4	yes	—
84	#217	C₂³	1	8	yes	complemented
85	#217	C₂³	1	16	yes	complemented
86	(C₂×D₃)⋊C₄	C₂³	1	8	yes	complemented
87	C₄⋊Dic₃	C₂³	1	4	yes	complemented
88	Dic₃⋊C₄	C₂³	1	8	yes	complemented
89	C₄×Dic₃	C₂³	1	4	yes	complemented
90	C₂⁵	C₂×D₃	1	2	yes	complemented
91	H(5,𝔽₂)	—	6	8	no	—
92	C₂×#41	—	3	4	no	—
93	C₂×#41	—	3	2	no	—
94	C₂²×D₄	—	2	8	no	—
95	C₂²×D₄	—	3	4	no	—
96	C₂²×D₄	—	3	8	no	—
97	C₂²×D₄	—	3	4	no	—
98	C₂²×D₄	C₂×D₃	1	1	yes	characteristic
99	C₂²×D₄	C₂×D₃	1	2	yes	complemented
100	C₂²×D₄	C₂×D₃	1	1	yes	characteristic, complemented
101	C₂²×D₄	C₂×D₃	1	8	yes	complemented
102	C₂³×C₄	C₂×D₃	1	1	yes	characteristic, complemented
103	#127	—	3	4	no	—
104	#124	—	3	4	no	—
105	#123	—	3	8	no	—
106	#123	—	3	4	no	—
107	C₂²⋊Q₈	—	3	4	no	—
108	#121	—	3	16	no	—
109	#121	—	3	8	no	—
110	#121	—	3	4	no	—
111	#120	—	3	4	no	—
112	#120	—	3	8	no	—
113	#120	—	3	16	no	—
114	#120	—	3	8	no	—
115	C₄×D₄	—	3	8	no	—
116	C₄×D₄	—	3	8	no	—
117	#117	—	3	2	no	—
118	C₂×C₂²⋊C₄	—	3	4	no	—
119	C₂×C₂²⋊C₄	—	3	8	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	—	2	4	no	—
122	C₂³×C₃	—	2	16	no	—
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	—	2	16	no	—
125	C₂³×C₃	—	4	4	no	—
126	C₂³×C₃	—	4	8	no	—
127	C₂³×C₃	—	4	4	no	—
128	C₂³×C₃	—	4	32	no	—
129	C₂³×C₃	C₂×D₄	1	8	yes	complemented
130	C₂³×C₃	C₂⁴	1	4	yes	—
131	C₂³×C₃	C₂⁴	1	1	yes	characteristic
132	C₂³×C₃	C₂⁴	1	4	yes	—
133	C₂³×C₃	C₂⁴	1	2	yes	—
134	C₂²×D₃	—	2	8	no	—
135	C₂²×D₃	—	4	8	no	—
136	C₂²×D₃	C₂⁴	1	4	yes	—
137	C₃×D₄	—	2	16	no	—
138	C₃×D₄	—	2	8	no	—
139	C₃×D₄	C₂×D₄	1	16	yes	complemented
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	8	no	—
142	C₂×C₃×C₄	C₂×D₄	1	4	yes	complemented
143	C₂×C₃×C₄	C₂⁴	1	2	yes	—
144	C₂×C₃×C₄	C₂⁴	1	2	yes	—
145	#67	—	2	8	no	—
146	#67	—	2	16	no	—
147	#67	—	2	16	no	—
148	#67	—	4	16	no	—
149	C₂×Dic₃	—	2	8	no	—
150	C₂×Dic₃	—	4	8	no	—
151	C₂×Dic₃	C₂⁴	1	8	yes	—
152	C₂×Dic₃	C₂⁴	1	4	yes	—
153	D₁₂	—	2	4	no	—
154	C₄×D₃	—	2	8	no	—
155	Dic₆	—	2	4	no	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	—	3	4	no	—
158	C₂⁴	—	4	8	no	—
159	C₂⁴	C₂²×D₃	1	1	yes	characteristic
160	C₂⁴	C₂²×D₃	1	8	yes	—
161	C₂⁴	C₂²×D₃	1	2	yes	—
162	C₂⁴	C₂²×D₃	1	2	yes	—
163	#41	—	6	16	no	—
164	#41	—	6	8	no	—
165	C₂×Q₈	—	3	2	no	—
166	C₂×D₄	—	2	8	no	—
167	C₂×D₄	—	2	32	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	3	8	no	—
170	C₂×D₄	—	3	4	no	—
171	C₂×D₄	—	3	8	no	—
172	C₂×D₄	—	3	16	no	—
173	C₂×D₄	—	3	2	no	—
174	C₂×D₄	—	6	16	no	—
175	C₂×D₄	—	6	16	no	—
176	C₂×D₄	—	12	16	no	—
177	C₂×D₄	#67	1	8	yes	complemented
178	C₂×D₄	C₂²×D₃	1	8	yes	—
179	C₂×D₄	C₂²×D₃	1	4	yes	—
180	C₂×D₄	C₂²×D₃	1	4	yes	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	3	4	no	—
183	C₂²×C₄	—	3	8	no	—
184	C₂²×C₄	—	3	4	no	—
185	C₂²×C₄	C₂²×D₃	1	4	yes	—
186	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
188	C₄⋊C₄	—	3	4	no	—
189	C₄⋊C₄	—	3	8	no	—
190	C₂²⋊C₄	—	3	4	no	—
191	C₂²⋊C₄	—	3	8	no	—
192	C₂²⋊C₄	—	3	8	no	—
193	C₂²⋊C₄	—	3	16	no	—
194	C₂²⋊C₄	—	3	8	no	—
195	C₄²	—	3	4	no	—
196	C₂²×C₃	—	2	8	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	8	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	2	8	no	—
203	C₂²×C₃	—	4	8	no	—
204	C₂²×C₃	—	4	32	no	—
205	C₂²×C₃	—	4	8	no	—
206	C₂²×C₃	C₂²×D₄	1	8	yes	—
207	C₂²×C₃	C₂²×D₄	1	2	yes	—
208	C₂²×C₃	H(5,𝔽₂)	1	4	yes	complemented
209	C₂²×C₃	C₂⁵	1	1	yes	characteristic
210	C₂×D₃	—	2	4	no	—
211	C₂×D₃	—	2	8	no	—
212	C₂×D₃	—	8	8	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	C₂²×D₄	1	4	yes	—
215	Dic₃	—	2	4	no	—
216	Dic₃	—	2	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	16	no	—
220	C₂³	—	2	16	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	3	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	8	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	4	32	no	—
227	C₂³	—	6	8	no	—
228	C₂³	—	12	8	no	—
229	C₂³	C₂×#67	1	8	yes	complemented
230	C₂³	C₂³×D₃	1	4	yes	—
231	C₂³	C₂³×D₃	1	2	yes	—
232	C₂³	C₂³×D₃	1	1	yes	characteristic
233	C₂³	C₂³×D₃	1	4	yes	—
234	Q₈	—	6	4	no	—
235	D₄	—	2	16	no	—
236	D₄	—	2	8	no	—
237	D₄	—	6	16	no	—
238	D₄	—	6	8	no	—
239	D₄	—	6	16	no	—
240	D₄	—	6	4	no	—
241	D₄	—	12	16	no	—
242	D₄	C₂×#67	1	16	yes	complemented
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	3	8	no	—
246	C₂×C₄	—	3	4	no	—
247	C₂×C₄	—	6	8	no	—
248	C₂×C₄	—	6	8	no	—
249	C₂×C₄	—	12	8	no	—
250	C₂×C₄	C₂×#67	1	4	yes	complemented
251	C₂×C₄	C₂³×D₃	1	2	yes	—
252	C₂×C₄	C₂³×D₃	1	2	yes	—
253	C₂×C₃	—	2	8	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	—	2	4	no	—
256	C₂×C₃	—	4	4	no	—
257	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
258	C₂×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
259	D₃	—	4	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	8	no	—
268	C₂²	—	4	32	no	—
269	C₂²	—	4	8	no	—
270	C₂²	—	6	8	no	—
271	C₂²	—	6	4	no	—
272	C₂²	—	24	8	no	—
273	C₂²	#1004	1	4	yes	complemented
274	C₂²	C₂²×#67	1	2	yes	—
275	C₂²	C₂²×#67	1	8	yes	—
276	C₂²	C₂⁴×D₃	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	6	8	no	—
279	C₄	—	6	4	no	—
280	C₄	C₂²×#67	1	4	yes	—
281	C₃	#3484	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	2	4	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	8	no	—
285	C₂	—	4	4	no	—
286	C₂	—	12	4	no	—
287	C₂	C₂×#1004	1	2	yes	—
288	C₂	C₂³×#67	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₂³×C₃×D₄	C₂
hypercenter	C₂³×D₄	D₃
hypocenter	C₃	#3484
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×D₄	C₂	semidirect
2. C₂×#1004	C₂	semidirect
3. C₂×#953	C₂	semidirect
4. #5806	C₂	semidirect
5. #5805	C₂	semidirect
6. #5804	C₂	semidirect
7. D₄×#67	C₂	semidirect
8. #5798	C₂	semidirect
9. #5791	C₂	semidirect
11. C₂⁵×C₃	C₂²	semidirect
14. C₂²×C₃×D₄	C₂²	semidirect
15. C₂²×C₃×D₄	C₂²	semidirect
16. C₂²×C₃×D₄	C₂²	semidirect
17. C₂³×C₃×C₄	C₂²	semidirect
18. C₂²×#67	C₂²	semidirect
19. C₂²×#67	C₂²	semidirect
21. C₂×#237	C₂²	semidirect
22. C₂×D₃×D₄	C₂²	semidirect
23. C₂×#235	C₂²	semidirect
24. #953	C₂²	semidirect
25. #953	C₂²	semidirect
26. #953	C₂²	semidirect
27. C₂×#217	C₂²	semidirect
28. C₂×#217	C₂²	semidirect
29. #940	C₂²	semidirect
30. #939	C₂²	semidirect
31. #938	C₂²	semidirect
32. #937	C₂²	semidirect
33. #936	C₂²	semidirect
34. #935	C₂²	semidirect
35. D₄×Dic₃	C₂²	semidirect
36. #930	C₂²	semidirect
37. #929	C₂²	semidirect
38. C₄×#67	C₂²	semidirect
39. #926	C₂²	semidirect
40. #924	C₂²	semidirect
42. C₂³×D₄	D₃	semidirect
56. C₂³×D₃	C₂³	semidirect
60. C₂×C₃×D₄	D₄	semidirect
72. C₂×#67	C₂³	semidirect
73. C₂×#67	C₂³	semidirect
74. C₂²×Dic₃	C₂³	semidirect
75. C₂²×Dic₃	C₂³	semidirect
79. C₂×D₁₂	C₂³	semidirect
80. C₂×C₄×D₃	C₂³	semidirect
81. C₂×Dic₆	C₂³	semidirect
84. #217	C₂³	semidirect
85. #217	C₂³	semidirect
86. (C₂×D₃)⋊C₄	C₂³	semidirect
87. C₄⋊Dic₃	C₂³	semidirect
88. Dic₃⋊C₄	C₂³	semidirect
89. C₄×Dic₃	C₂³	semidirect
90. C₂⁵	C₂×D₃	semidirect
99. C₂²×D₄	C₂×D₃	semidirect
100. C₂²×D₄	C₂×D₃	semidirect
101. C₂²×D₄	C₂×D₃	semidirect
102. C₂³×C₄	C₂×D₃	semidirect
129. C₂³×C₃	C₂×D₄	semidirect
139. C₃×D₄	C₂×D₄	semidirect
142. C₂×C₃×C₄	C₂×D₄	semidirect
177. C₂×D₄	#67	semidirect
208. C₂²×C₃	H(5,𝔽₂)	semidirect
229. C₂³	C₂×#67	semidirect
242. D₄	C₂×#67	semidirect
250. C₂×C₄	C₂×#67	semidirect
273. C₂²	#1004	semidirect
281. C₃	#3484	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₄	2
C₂×#1004	2
C₂×#953	2
#5806	2
#5805	2
#5804	2
D₄×#67	2
#5798	2
#5791	2
#3484	3