Finite Group Databasebeta

Subgroups — #85680

Order 384.

1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 509 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	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
2	#5791	C₂	1	6	yes	maximal, complemented
3	#5479	C₂	1	12	yes	maximal, complemented
4	#5477	C₂	1	3	yes	maximal, complemented
5	C₄×#235	C₂	1	6	yes	maximal, complemented
6	#5470	C₂	1	3	yes	maximal, complemented
7	#3444	—	3	1	no	maximal, Sylow
8	C₂³×C₃×C₄	C₂²	1	3	yes	—
9	C₂×#235	C₂²	1	6	yes	—
10	C₂×C₃×C₄²	C₂²	1	3	yes	—
11	C₂×C₃×C₄²	C₂²	1	3	yes	—
12	C₂×C₃×C₄²	C₂²	1	6	yes	—
13	#953	C₂²	1	4	yes	—
14	#930	C₂²	1	12	yes	—
15	#929	C₂²	1	12	yes	—
16	C₄×#67	C₂²	1	24	yes	—
17	#926	C₂²	1	6	yes	—
18	#924	C₂²	1	12	yes	—
19	#876	C₂²	1	8	yes	complemented
20	#875	C₂²	1	6	yes	complemented
21	#874	C₂²	1	1	yes	characteristic, complemented
22	C₄×D₁₂	C₂²	1	12	yes	complemented
23	#872	C₂²	1	12	yes	complemented
24	C₄²×D₃	C₂²	1	4	yes	complemented
25	#870	C₂²	1	6	yes	complemented
26	#869	C₂²	1	3	yes	complemented
27	C₄×Dic₆	C₂²	1	12	yes	complemented
28	#533	—	3	3	no	—
29	#532	—	3	3	no	—
30	#529	—	3	12	no	—
31	#525	—	3	6	no	—
32	C₄×#41	—	3	6	no	—
33	C₂²×C₄²	D₃	1	1	yes	characteristic, complemented
34	C₂⁴×C₃	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄	—	2	12	no	—
36	C₂²×C₃×C₄	C₂³	1	6	yes	—
37	C₂²×C₃×C₄	C₂³	1	6	yes	—
38	C₂²×C₃×C₄	C₂³	1	3	yes	—
39	C₂²×C₃×C₄	C₂³	1	3	yes	—
40	C₂×#67	C₂³	1	12	yes	—
41	#235	—	2	24	no	—
42	C₂×D₁₂	C₂³	1	6	yes	—
43	C₂×C₄×D₃	C₂³	1	12	yes	—
44	C₂×Dic₆	C₂³	1	6	yes	—
45	C₃×C₄²	C₂³	1	6	yes	—
46	C₃×C₄²	C₂³	1	1	yes	characteristic
47	C₃×C₄²	C₂³	1	3	yes	—
48	C₃×C₄²	C₂³	1	3	yes	—
49	C₃×C₄²	C₂³	1	2	yes	—
50	C₃×C₄²	C₂³	1	1	yes	characteristic
51	#217	C₂³	1	12	yes	—
52	(C₂×D₃)⋊C₄	C₂³	1	24	yes	—
53	C₄⋊Dic₃	C₂³	1	12	yes	—
54	Dic₃⋊C₄	C₂³	1	24	yes	—
55	C₄×Dic₃	C₂³	1	12	yes	—
56	C₂×#41	—	3	6	no	—
57	C₂³×C₄	C₂×D₃	1	3	yes	—
58	C₄⋊Q₈	—	3	3	no	—
59	#127	—	3	1	no	—
60	#126	—	3	8	no	—
61	#125	—	3	6	no	—
62	#124	—	3	6	no	—
63	#123	—	3	12	no	—
64	C₂²⋊Q₈	—	3	12	no	—
65	#121	—	3	12	no	—
66	#120	—	3	4	no	—
67	C₄×Q₈	—	3	12	no	—
68	C₄×D₄	—	3	24	no	—
69	C₄×D₄	—	3	12	no	—
70	#117	—	3	6	no	—
71	#117	—	3	12	no	—
72	C₂×C₄²	—	3	4	no	—
73	C₂×C₄²	C₂×D₃	1	6	yes	—
74	C₂×C₄²	C₂×D₃	1	3	yes	—
75	C₂×C₄²	C₂×D₃	1	3	yes	—
76	C₂³×C₃	—	2	6	no	—
77	C₂³×C₃	C₂⁴	1	3	yes	—
78	C₂²×D₃	C₂⁴	1	4	yes	—
79	C₂×C₃×C₄	—	2	6	no	—
80	C₂×C₃×C₄	—	2	12	no	—
81	C₂×C₃×C₄	—	2	12	no	—
82	C₂×C₃×C₄	#41	1	12	yes	complemented
83	C₂×C₃×C₄	C₂⁴	1	6	yes	—
84	C₂×C₃×C₄	C₂⁴	1	3	yes	—
85	C₂×C₃×C₄	C₂⁴	1	3	yes	—
86	#67	—	2	24	no	—
87	C₂×Dic₃	C₂⁴	1	12	yes	—
88	D₁₂	—	2	12	no	—
89	C₄×D₃	—	2	24	no	—
90	Dic₆	—	2	12	no	—
91	C₂⁴	C₂²×D₃	1	1	yes	characteristic
92	#41	—	6	24	no	—
93	C₂×Q₈	—	3	6	no	—
94	C₂×D₄	—	3	12	no	—
95	C₂×D₄	—	3	6	no	—
96	C₂²×C₄	—	2	12	no	—
97	C₂²×C₄	—	3	12	no	—
98	C₂²×C₄	C₂²×D₃	1	6	yes	—
99	C₂²×C₄	C₂²×D₃	1	6	yes	—
100	C₂²×C₄	C₂²×D₃	1	3	yes	—
101	C₂²×C₄	C₂²×D₃	1	3	yes	—
102	C₄⋊C₄	—	3	12	no	—
103	C₄⋊C₄	—	3	24	no	—
104	C₂²⋊C₄	—	3	12	no	—
105	C₂²⋊C₄	—	3	24	no	—
106	C₄²	—	3	12	no	—
107	C₄²	C₂²×D₃	1	3	yes	—
108	C₄²	C₂²×D₃	1	6	yes	—
109	C₄²	C₂²×D₃	1	3	yes	—
110	C₄²	C₂²×D₃	1	2	yes	—
111	C₄²	C₂²×D₃	1	1	yes	characteristic
112	C₄²	C₂²×D₃	1	1	yes	characteristic
113	C₂²×C₃	—	2	8	no	—
114	C₂²×C₃	—	2	6	no	—
115	C₂²×C₃	C₂×#41	1	6	yes	—
116	C₂²×C₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₃	—	2	12	no	—
118	C₃×C₄	—	2	6	no	—
119	C₃×C₄	C₂×#41	1	6	yes	—
120	C₃×C₄	C₂×#41	1	6	yes	—
121	Dic₃	—	2	12	no	—
122	C₂³	—	2	6	no	—
123	C₂³	—	3	4	no	—
124	C₂³	C₂³×D₃	1	3	yes	—
125	Q₈	—	6	12	no	—
126	D₄	—	6	24	no	—
127	D₄	—	6	12	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	12	no	—
130	C₂×C₄	—	2	12	no	—
131	C₂×C₄	—	3	12	no	—
132	C₂×C₄	—	6	24	no	—
133	C₂×C₄	#235	1	12	yes	complemented
134	C₂×C₄	C₂³×D₃	1	6	yes	—
135	C₂×C₄	C₂³×D₃	1	3	yes	—
136	C₂×C₄	C₂³×D₃	1	3	yes	—
137	C₂×C₃	—	2	6	no	—
138	C₂×C₃	C₂²×#41	1	3	yes	—
139	D₃	—	4	4	no	—
140	C₂²	—	2	8	no	—
141	C₂²	—	2	6	no	—
142	C₂²	—	6	12	no	—
143	C₂²	C₂×#235	1	6	yes	—
144	C₂²	C₂⁴×D₃	1	1	yes	characteristic
145	C₄	—	2	6	no	—
146	C₄	—	6	12	no	—
147	C₄	C₂×#235	1	6	yes	—
148	C₄	C₂×#235	1	6	yes	—
149	C₃	#3444	1	1	yes	characteristic, complemented, Sylow
150	C₂	—	2	6	no	—
151	C₂	—	12	4	no	—
152	C₂	C₂²×#235	1	3	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	#3444
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄²	C₂	semidirect
2. #5791	C₂	semidirect
3. #5479	C₂	semidirect
4. #5477	C₂	semidirect
5. C₄×#235	C₂	semidirect
6. #5470	C₂	semidirect
19. #876	C₂²	semidirect
20. #875	C₂²	semidirect
21. #874	C₂²	semidirect
22. C₄×D₁₂	C₂²	semidirect
23. #872	C₂²	semidirect
24. C₄²×D₃	C₂²	semidirect
25. #870	C₂²	semidirect
26. #869	C₂²	semidirect
27. C₄×Dic₆	C₂²	semidirect
33. C₂²×C₄²	D₃	semidirect
82. C₂×C₃×C₄	#41	semidirect
133. C₂×C₄	#235	semidirect
149. C₃	#3444	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄²	2
#5791	2
#5479	2
#5477	2
C₄×#235	2
#5470	2
#3444	3