Finite Group Databasebeta

Subgroups — #82939

Order 384.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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×#124	C₂	1	1	yes	maximal, characteristic
2	#5470	C₂	1	1	yes	maximal, characteristic, complemented
3	#5225	C₂	1	2	yes	maximal, complemented
4	#5215	C₂	1	2	yes	maximal
5	#5056	C₂	1	4	yes	maximal, complemented
6	#5055	C₂	1	4	yes	maximal, complemented
7	C₄×#208	C₂	1	1	yes	maximal, characteristic, complemented
8	#3140	—	3	1	no	maximal, Sylow
9	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₃×#124	—	2	2	no	—
12	C₃×#124	C₂²	1	2	yes	—
13	C₃×#124	C₂²	1	2	yes	—
14	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
15	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
16	#926	C₂²	1	2	yes	—
17	#924	—	2	2	no	—
18	C₂×#208	C₂²	1	2	yes	—
19	#870	C₂²	1	2	yes	complemented
20	#869	C₂²	1	2	yes	complemented
21	C₄×Dic₆	—	2	2	no	—
22	#835	C₂²	1	8	yes	complemented
23	#832	C₂²	1	8	yes	—
24	#803	C₂²	1	2	yes	complemented
25	C₄×C₃⋊C₈	C₂²	1	2	yes	complemented
26	#533	—	3	1	no	—
27	C₂×#124	D₃	1	1	yes	characteristic
28	#489	—	3	4	no	—
29	#488	—	3	4	no	—
30	#419	—	3	2	no	—
31	#418	—	3	2	no	—
32	C₄×M₁₆	—	3	1	no	—
33	C₂⁴×C₃	—	2	1	no	—
34	C₂×C₃×Q₈	—	2	1	no	—
35	C₂×C₃×Q₈	—	2	4	no	—
36	C₂×C₃×Q₈	C₂³	1	2	yes	—
37	C₂×C₃×D₄	—	2	1	no	—
38	C₂×C₃×D₄	—	2	4	no	—
39	C₂×C₃×D₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄	—	2	1	no	—
41	C₂²×C₃×C₄	D₄	1	2	yes	—
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂×Dic₆	—	2	2	no	—
44	C₃×C₂²⋊C₄	—	2	4	no	—
45	C₃×C₂²⋊C₄	—	2	4	no	—
46	C₃×C₄²	D₄	1	2	yes	—
47	C₃×C₄²	C₂³	1	1	yes	characteristic
48	C₃×C₄²	C₂³	1	1	yes	characteristic
49	#217	—	2	2	no	—
50	C₄⋊Dic₃	—	2	2	no	—
51	C₄⋊Dic₃	C₂³	1	4	yes	—
52	Dic₃⋊C₄	—	2	4	no	—
53	C₄×Dic₃	—	2	2	no	—
54	#208	—	2	4	no	—
55	C₂×C₃⋊C₈	C₂³	1	4	yes	—
56	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
57	C₂²×D₄	C₂×D₃	1	1	yes	characteristic
58	C₂×M₁₆	—	3	2	no	—
59	C₄⋊Q₈	—	3	2	no	—
60	#125	—	3	2	no	—
61	#124	—	2	2	no	—
62	#124	C₂×D₃	1	2	yes	—
63	#124	C₂×D₃	1	2	yes	—
64	C₂²⋊Q₈	—	6	2	no	—
65	C₄×Q₈	—	6	2	no	—
66	#117	—	3	2	no	—
67	C₂×C₂²⋊C₄	—	2	2	no	—
68	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
69	#103	—	3	8	no	—
70	#102	—	3	8	no	—
71	#97	—	3	2	no	—
72	C₄×C₈	—	3	2	no	—
73	C₂³×C₃	—	2	2	no	—
74	C₂³×C₃	—	4	4	no	—
75	C₂³×C₃	—	4	2	no	—
76	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
77	C₃×Q₈	—	2	4	no	—
78	C₃×Q₈	—	4	2	no	—
79	C₃×D₄	—	2	4	no	—
80	C₃×D₄	—	4	2	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	4	2	no	—
83	C₂×C₃×C₄	—	4	2	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	#41	1	4	yes	—
88	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₃	—	2	4	no	—
90	Dic₆	—	4	4	no	—
91	C₃⋊C₈	—	2	4	no	—
92	C₂⁴	—	2	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	2	4	no	—
95	C₂×Q₈	—	6	2	no	—
96	C₂×Q₈	C₂²×D₃	1	2	yes	—
97	C₂×D₄	—	2	4	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	C₂²×D₃	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	#67	1	2	yes	—
102	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
103	M₁₆	—	6	4	no	—
104	C₂×C₈	—	3	4	no	—
105	C₄⋊C₄	—	3	4	no	—
106	C₄⋊C₄	—	6	4	no	—
107	C₄⋊C₄	—	6	2	no	—
108	C₂²⋊C₄	—	2	4	no	—
109	C₂²⋊C₄	—	2	4	no	—
110	C₂²⋊C₄	—	6	2	no	—
111	C₄²	—	6	2	no	—
112	C₄²	#67	1	2	yes	—
113	C₄²	C₂²×D₃	1	1	yes	characteristic
114	C₄²	C₂²×D₃	1	1	yes	characteristic
115	C₂²×C₃	—	2	2	no	—
116	C₂²×C₃	—	4	2	no	—
117	C₂²×C₃	—	4	4	no	—
118	C₂²×C₃	—	8	4	no	—
119	C₂²×C₃	#124	1	2	yes	—
120	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	2	2	no	—
123	C₃×C₄	—	4	2	no	—
124	C₃×C₄	#124	1	2	yes	—
125	C₃×C₄	C₂×#41	1	2	yes	—
126	Dic₃	—	4	4	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	4	4	no	—
129	C₂³	—	4	2	no	—
130	C₂³	C₂×#67	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	12	4	no	—
134	D₄	—	2	4	no	—
135	D₄	—	4	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	6	4	no	—
140	C₂×C₄	#237	1	4	yes	—
141	C₂×C₄	C₂×#67	1	2	yes	—
142	C₂×C₄	C₂×#67	1	2	yes	—
143	C₂×C₄	C₂×#67	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
145	C₈	—	6	4	no	—
146	C₂×C₃	—	2	2	no	—
147	C₂×C₃	—	8	2	no	—
148	C₂×C₃	C₂×#124	1	1	yes	characteristic
149	C₂×C₃	#575	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	4	no	—
153	C₂²	—	8	4	no	—
154	C₂²	#936	1	2	yes	—
155	C₂²	C₂²×#67	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	12	4	no	—
160	C₄	#936	1	2	yes	—
161	C₄	C₂×#237	1	2	yes	—
162	C₃	#3140	1	1	yes	characteristic, complemented, Sylow
163	C₂	—	2	2	no	—
164	C₂	—	8	2	no	—
165	C₂	C₂×#936	1	1	yes	characteristic
166	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×#124	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×C₃×D₄	C₂²
hypercenter	C₂×#124	D₃
hypocenter	C₃	#3140
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
2. #5470	C₂	semidirect
3. #5225	C₂	semidirect
5. #5056	C₂	semidirect
6. #5055	C₂	semidirect
7. C₄×#208	C₂	semidirect
19. #870	C₂²	semidirect
20. #869	C₂²	semidirect
22. #835	C₂²	semidirect
24. #803	C₂²	semidirect
25. C₄×C₃⋊C₈	C₂²	semidirect
162. C₃	#3140	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#124	2
#5470	2
#5225	2
#5215	2
#5056	2
#5055	2
C₄×#208	2
#3140	3