Subgroups — #79416

Order 384.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5565	C₂	1	2	yes	maximal, complemented
2	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#5115	C₂	1	1	yes	maximal, characteristic, complemented
4	#5115	C₂	1	1	yes	maximal, characteristic, complemented
5	#5106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#818	C₂	1	1	yes	maximal, characteristic, complemented
7	#4847	C₂	1	2	yes	maximal, complemented
8	#4737	C₂	1	2	yes	maximal, complemented
9	#4735	C₂	1	2	yes	maximal, complemented
10	C₄×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#4683	C₂	1	1	yes	maximal, characteristic, complemented
12	#3339	—	3	1	no	maximal, Sylow
13	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
16	#939	—	2	2	no	—
17	#930	C₂²	1	2	yes	—
18	#930	C₂²	1	2	yes	—
19	#926	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
21	C₂×D₂₄	C₂²	1	1	yes	characteristic
22	#898	—	2	2	no	—
23	#883	—	2	2	no	—
24	#67⋊C₄	—	2	2	no	—
25	C₄×D₁₂	C₂²	1	2	yes	—
26	#870	C₂²	1	2	yes	—
27	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
31	#821	C₂²	1	2	yes	—
32	#821	C₂²	1	2	yes	—
33	#821	C₂²	1	2	yes	—
34	#818	C₂²	1	1	yes	characteristic
35	#818	C₂²	1	1	yes	characteristic
36	#818	C₂²	1	1	yes	characteristic
37	#818	C₂²	1	2	yes	—
38	#817	C₂²	1	2	yes	—
39	#817	C₂²	1	2	yes	—
40	#553	—	3	2	no	—
41	#499	—	3	1	no	—
42	#482	—	3	2	no	—
43	#481	—	3	2	no	—
44	#478	—	3	2	no	—
45	#466	—	3	1	no	—
46	#466	—	3	1	no	—
47	C₄×D₈	—	3	1	no	—
48	C₈×D₄	D₃	1	1	yes	characteristic, complemented
49	#427	—	3	1	no	—
50	C₂×#107	—	3	1	no	—
51	C₂×C₃×D₄	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂×#67	—	2	2	no	—
55	C₂×#67	—	2	2	no	—
56	C₂²×Dic₃	—	2	2	no	—
57	C₂×D₁₂	C₂³	1	2	yes	—
58	C₂×C₄×D₃	—	2	2	no	—
59	C₂×C₃×C₈	—	2	2	no	—
60	C₂×C₃×C₈	—	2	2	no	—
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
67	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	#217	—	2	2	no	—
70	(C₂×D₃)⋊C₄	—	2	2	no	—
71	(C₂×D₃)⋊C₄	—	2	2	no	—
72	(C₂×D₃)⋊C₄	—	2	2	no	—
73	C₄⋊Dic₃	—	2	2	no	—
74	C₄⋊Dic₃	C₂³	1	2	yes	—
75	C₄⋊Dic₃	C₂³	1	2	yes	—
76	C₄⋊Dic₃	C₂³	1	2	yes	—
77	Dic₃⋊C₄	—	2	2	no	—
78	Dic₃⋊C₄	—	2	2	no	—
79	C₄×Dic₃	—	2	2	no	—
80	D₂₄	—	2	2	no	—
81	C₂×D₈	—	3	1	no	—
82	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
84	#126	—	6	2	no	—
85	#125	—	3	2	no	—
86	#123	—	6	2	no	—
87	#121	—	3	2	no	—
88	#121	—	3	2	no	—
89	#121	—	6	2	no	—
90	C₄×D₄	—	3	2	no	—
91	C₄×D₄	—	6	2	no	—
92	C₄×D₄	C₂×D₃	1	1	yes	characteristic
93	#117	—	3	2	no	—
94	C₂×C₄⋊C₄	—	3	2	no	—
95	#107	—	3	1	no	—
96	#107	—	3	1	no	—
97	#107	—	3	2	no	—
98	#107	—	3	1	no	—
99	#106	—	3	2	no	—
100	#106	—	3	2	no	—
101	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
102	#102	—	3	2	no	—
103	#102	—	3	2	no	—
104	#102	—	3	2	no	—
105	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
106	C₂²⋊C₈	C₂×D₃	1	1	yes	characteristic
107	C₄×C₈	C₂×D₃	1	1	yes	characteristic
108	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₃	—	2	2	no	—
111	C₃×D₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
121	#67	—	4	4	no	—
122	#67	—	8	2	no	—
123	C₂×Dic₃	—	2	2	no	—
124	C₂×Dic₃	—	2	2	no	—
125	C₂×Dic₃	—	2	2	no	—
126	C₂×Dic₃	—	8	2	no	—
127	D₁₂	—	2	2	no	—
128	D₁₂	—	2	2	no	—
129	C₄×D₃	—	8	2	no	—
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	#41	1	2	yes	—
134	C₂×D₄	—	3	2	no	—
135	C₂×D₄	—	6	2	no	—
136	C₂×D₄	—	6	2	no	—
137	C₂×D₄	D₁₂	1	1	yes	characteristic
138	C₂²×C₄	—	6	2	no	—
139	C₂²×C₄	—	6	2	no	—
140	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
142	D₈	—	6	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
147	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
148	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
149	C₄⋊C₄	—	3	2	no	—
150	C₄⋊C₄	—	3	2	no	—
151	C₄⋊C₄	—	3	2	no	—
152	C₄⋊C₄	—	6	2	no	—
153	C₄⋊C₄	—	6	2	no	—
154	C₄⋊C₄	—	6	2	no	—
155	C₄⋊C₄	D₁₂	1	1	yes	characteristic
156	C₂²⋊C₄	—	6	2	no	—
157	C₂²⋊C₄	—	6	2	no	—
158	C₂²⋊C₄	—	6	2	no	—
159	C₂²⋊C₄	—	6	2	no	—
160	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
161	C₂²⋊C₄	D₁₂	1	1	yes	characteristic
162	C₄²	—	6	2	no	—
163	C₄²	C₂²×D₃	1	1	yes	characteristic
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	#135	1	2	yes	—
170	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×D₃	—	4	2	no	—
172	C₂×D₃	—	4	2	no	—
173	C₂×D₃	—	4	2	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	C₂×#41	1	1	yes	characteristic
179	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
180	Dic₃	—	4	2	no	—
181	Dic₃	—	4	2	no	—
182	Dic₃	—	4	2	no	—
183	C₂³	—	6	2	no	—
184	C₂³	C₂×D₁₂	1	1	yes	characteristic
185	C₂³	C₂×D₁₂	1	1	yes	characteristic
186	D₄	—	2	2	no	—
187	D₄	—	6	2	no	—
188	D₄	—	6	2	no	—
189	D₄	—	12	4	no	—
190	D₄	—	24	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	6	2	no	—
196	C₂×C₄	—	6	2	no	—
197	C₂×C₄	—	6	2	no	—
198	C₂×C₄	—	24	2	no	—
199	C₂×C₄	—	24	2	no	—
200	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
201	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
202	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
204	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	#237	1	2	yes	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	4	1	no	—
211	C₂×C₃	#547	1	1	yes	characteristic
212	C₂×C₃	C₂×#135	1	1	yes	characteristic
213	C₂×C₃	#576	1	1	yes	characteristic
214	D₃	—	8	2	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	12	2	no	—
221	C₂²	—	12	2	no	—
222	C₂²	—	12	2	no	—
223	C₂²	#904	1	2	yes	—
224	C₂²	C₂²×D₁₂	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	12	2	no	—
230	C₄	—	12	2	no	—
231	C₄	—	12	2	no	—
232	C₄	C₂×#237	1	1	yes	characteristic
233	C₄	#1010	1	1	yes	characteristic
234	C₃	#3339	1	1	yes	characteristic, complemented, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	24	2	no	—
238	C₂	#5555	1	1	yes	characteristic
239	C₂	C₂×#904	1	1	yes	characteristic
240	C₂	#5752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₈×D₄	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₈×D₄	D₃
hypocenter	C₃	#3339
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #5565	C₂	semidirect
2. C₃×C₈×D₄	C₂	semidirect
3. #5115	C₂	semidirect
4. #5115	C₂	semidirect
5. #5106	C₂	semidirect
6. C₂×#818	C₂	semidirect
7. #4847	C₂	semidirect
8. #4737	C₂	semidirect
9. #4735	C₂	semidirect
10. C₄×D₂₄	C₂	semidirect
11. #4683	C₂	semidirect
48. C₈×D₄	D₃	semidirect
234. C₃	#3339	semidirect

Maximal subgroups

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

subgroup	index
#5565	2
C₃×C₈×D₄	2
#5115	2
#5106	2
C₂×#818	2
#4847	2
#4737	2
#4735	2
C₄×D₂₄	2
#4683	2
#3339	3