Finite Group Databasebeta

Subgroups — #83017

Order 384.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#235	C₂	1	1	yes	maximal, characteristic
3	#5066	C₂	1	4	yes	maximal, complemented
4	#5063	C₂	1	4	yes	maximal, complemented
5	#4966	C₂	1	2	yes	maximal
6	#4964	C₂	1	2	yes	maximal, complemented
7	#4924	C₂	1	1	yes	maximal, characteristic, complemented
8	#3080	—	3	1	no	maximal, Sylow
9	C₂×#235	C₂²	1	1	yes	characteristic
10	C₃×#125	—	2	1	no	—
11	C₃×#125	—	2	1	no	—
12	C₃×#125	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	C₄×#67	—	2	2	no	—
17	#926	C₂²	1	1	yes	characteristic
18	C₂×#208	C₂²	1	2	yes	—
19	C₄×D₁₂	C₂²	1	2	yes	—
20	#872	—	2	1	no	—
21	C₄²×D₃	—	2	1	no	—
22	C₄×Dic₆	C₂²	1	2	yes	—
23	#810	C₂²	1	4	yes	—
24	#809	C₂²	1	4	yes	—
25	#808	C₂²	1	4	yes	—
26	#807	C₂²	1	4	yes	—
27	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
28	C₂×#125	D₃	1	1	yes	characteristic, complemented
29	C₄×#41	—	3	1	no	—
30	#479	—	3	4	no	—
31	#478	—	3	4	no	—
32	#428	—	3	2	no	—
33	#423	—	3	1	no	—
34	#417	—	3	2	no	—
35	C₂²×C₃×C₄	—	2	2	no	—
36	C₂²×C₃×C₄	D₄	1	2	yes	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂×#67	—	2	1	no	—
39	#235	—	2	2	no	—
40	#235	Q₈	1	4	yes	—
41	C₂×D₁₂	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₃	—	2	1	no	—
43	C₂×C₄×D₃	—	2	2	no	—
44	C₂×Dic₆	C₂³	1	1	yes	characteristic
45	C₃×C₄⋊C₄	—	2	4	no	—
46	C₃×C₄⋊C₄	—	2	4	no	—
47	C₃×C₄⋊C₄	—	2	2	no	—
48	C₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₃×C₄²	D₄	1	1	yes	characteristic
50	C₃×C₄²	D₄	1	1	yes	characteristic
51	C₃×C₄²	C₂³	1	2	yes	—
52	#217	—	2	1	no	—
53	(C₂×D₃)⋊C₄	—	2	2	no	—
54	C₄⋊Dic₃	C₂³	1	2	yes	—
55	Dic₃⋊C₄	—	2	2	no	—
56	C₄×Dic₃	—	2	2	no	—
57	C₄×Dic₃	—	2	1	no	—
58	#208	—	2	4	no	—
59	C₂×C₃⋊C₈	C₂³	1	4	yes	—
60	C₂×#41	—	3	1	no	—
61	C₂×M₁₆	—	3	2	no	—
62	#125	—	2	1	no	—
63	#125	—	2	1	no	—
64	#125	C₂×D₃	1	4	yes	—
65	C₄×Q₈	—	3	2	no	—
66	C₄×D₄	—	3	2	no	—
67	C₄×D₄	—	6	2	no	—
68	#117	—	3	1	no	—
69	#117	—	6	1	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
72	C₂×C₄²	—	6	1	no	—
73	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
74	#107	—	3	4	no	—
75	#106	—	3	4	no	—
76	C₄⋊C₈	—	3	4	no	—
77	#103	—	3	4	no	—
78	#102	—	3	4	no	—
79	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×D₃	—	2	1	no	—
81	C₂×C₃×C₄	—	2	4	no	—
82	C₂×C₃×C₄	—	4	2	no	—
83	C₂×C₃×C₄	—	4	4	no	—
84	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
88	C₂×C₃×C₄	#41	1	2	yes	—
89	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
90	#67	—	2	4	no	—
91	C₂×Dic₃	—	2	1	no	—
92	C₂×Dic₃	—	2	2	no	—
93	D₁₂	—	2	1	no	—
94	D₁₂	C₂×Q₈	1	2	yes	—
95	C₄×D₃	—	2	4	no	—
96	C₄×D₃	—	4	4	no	—
97	Dic₆	—	2	1	no	—
98	Dic₆	C₂×Q₈	1	2	yes	—
99	C₃⋊C₈	—	2	4	no	—
100	#41	—	3	4	no	—
101	#41	—	6	2	no	—
102	C₂×Q₈	—	3	1	no	—
103	C₂×D₄	—	3	1	no	—
104	C₂×D₄	—	6	1	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	6	1	no	—
107	C₂²×C₄	—	6	2	no	—
108	C₂²×C₄	#67	1	2	yes	—
109	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
110	M₁₆	—	6	4	no	—
111	C₂×C₈	—	3	4	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	3	2	no	—
116	C₄⋊C₄	—	6	2	no	—
117	C₄⋊C₄	C₂²×D₃	1	4	yes	—
118	C₂²⋊C₄	—	6	1	no	—
119	C₂²⋊C₄	—	6	2	no	—
120	C₄²	—	6	1	no	—
121	C₄²	—	6	2	no	—
122	C₄²	#67	1	1	yes	characteristic
123	C₄²	#67	1	1	yes	characteristic
124	C₄²	C₂²×D₃	1	2	yes	—
125	C₂²×C₃	—	2	2	no	—
126	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
127	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₃	—	2	2	no	—
129	C₂×D₃	—	4	2	no	—
130	C₃×C₄	—	2	2	no	—
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	—	4	4	no	—
133	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
134	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
135	C₃×C₄	C₂×#41	1	1	yes	characteristic
136	Dic₃	—	2	2	no	—
137	Dic₃	—	4	2	no	—
138	C₂³	—	6	1	no	—
139	C₂³	C₂×#67	1	1	yes	characteristic
140	Q₈	—	3	2	no	—
141	Q₈	—	6	1	no	—
142	D₄	—	3	2	no	—
143	D₄	—	6	4	no	—
144	D₄	—	6	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	6	1	no	—
149	C₂×C₄	—	6	4	no	—
150	C₂×C₄	—	6	2	no	—
151	C₂×C₄	—	12	4	no	—
152	C₂×C₄	D₃×Q₈	1	2	yes	—
153	C₂×C₄	#239	1	2	yes	—
154	C₂×C₄	C₂×#67	1	2	yes	—
155	C₂×C₄	C₂×#67	1	2	yes	—
156	C₂×C₄	C₂×#67	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
158	C₈	—	6	4	no	—
159	C₂×C₃	—	2	2	no	—
160	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₃	#575	1	2	yes	—
162	D₃	—	4	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	6	2	no	—
165	C₂²	—	12	2	no	—
166	C₂²	#946	1	2	yes	—
167	C₂²	C₂²×#67	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	6	2	no	—
172	C₄	—	12	2	no	—
173	C₄	#946	1	2	yes	—
174	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
175	C₄	C₂×#239	1	1	yes	characteristic
176	C₃	#3080	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	2	2	no	—
178	C₂	—	12	2	no	—
179	C₂	C₂×#946	1	1	yes	characteristic
180	C₂	#5822	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#125	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×#235	C₂²
hypercenter	C₂×#125	D₃
hypocenter	C₃	#3080
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#125	C₂	semidirect
3. #5066	C₂	semidirect
4. #5063	C₂	semidirect
6. #4964	C₂	semidirect
7. #4924	C₂	semidirect
28. C₂×#125	D₃	semidirect
176. C₃	#3080	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#125	2
C₄×#235	2
#5066	2
#5063	2
#4966	2
#4964	2
#4924	2
#3080	3