Finite Group Databasebeta

Subgroups — #69899

Order 384.

186 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₃×#405	C₂	1	1	yes	maximal, characteristic
2	#5102	C₂	1	1	yes	maximal, characteristic
3	#4998	C₂	1	1	yes	maximal, characteristic
4	#4558	C₂	1	1	yes	maximal, characteristic
5	#4555	C₂	1	1	yes	maximal, characteristic
6	#4550	C₂	1	1	yes	maximal, characteristic
7	#4532	C₂	1	1	yes	maximal, characteristic
8	#1922	—	3	1	no	maximal, Sylow
9	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₃×#117	C₂²	1	1	yes	characteristic
12	#926	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×#208	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#97	—	2	1	no	—
17	C₃×C₄×C₈	—	2	1	no	—
18	Dic₃⋊C₈	—	2	1	no	—
19	Dic₃⋊C₈	—	2	1	no	—
20	#808	C₄	1	1	yes	characteristic, complemented
21	#808	C₄	1	1	yes	characteristic, complemented
22	#807	C₄	1	1	yes	characteristic, complemented
23	#807	C₄	1	1	yes	characteristic, complemented
24	#427	—	3	1	no	—
25	#424	—	3	1	no	—
26	#405	D₃	1	1	yes	characteristic
27	#341	—	3	1	no	—
28	#339	—	3	1	no	—
29	#339	—	3	1	no	—
30	#336	—	3	1	no	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃	—	2	1	no	—
33	C₃×M₁₆	—	2	1	no	—
34	C₃×M₁₆	Q₈	1	2	yes	—
35	C₂×C₃×C₈	—	2	1	no	—
36	C₂×C₃×C₈	—	2	1	no	—
37	C₂×C₃×C₈	—	2	1	no	—
38	C₂×C₃×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₈	D₄	1	1	yes	characteristic
40	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₃×C₂²⋊C₄	—	2	1	no	—
43	C₃×C₄²	—	2	1	no	—
44	#217	—	2	1	no	—
45	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₃	—	2	1	no	—
48	C₄×Dic₃	—	2	1	no	—
49	C₄×Dic₃	—	2	1	no	—
50	#208	—	2	1	no	—
51	#208	—	2	1	no	—
52	C₂×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆	—	3	1	no	—
58	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₈	—	3	1	no	—
60	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
61	#117	—	3	1	no	—
62	#117	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄²	—	3	1	no	—
64	#107	—	3	1	no	—
65	#107	—	3	1	no	—
66	#106	—	3	1	no	—
67	#106	—	3	1	no	—
68	C₄⋊C₈	—	6	1	no	—
69	C₄⋊C₈	—	6	1	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂³×C₃	#41	1	1	yes	characteristic
73	C₂×C₃×C₄	—	2	1	no	—
74	C₂×C₃×C₄	—	2	1	no	—
75	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃×C₄	#41	1	1	yes	characteristic
79	C₂×C₃×C₄	#41	1	1	yes	characteristic
80	C₂×C₃×C₄	#41	1	1	yes	characteristic
81	C₂×Dic₃	—	2	2	no	—
82	C₂×Dic₃	—	2	1	no	—
83	C₂×Dic₃	—	2	1	no	—
84	C₂×Dic₃	—	4	1	no	—
85	C₂×Dic₃	—	4	1	no	—
86	C₃×C₈	—	2	2	no	—
87	C₃×C₈	—	2	1	no	—
88	C₃×C₈	—	2	1	no	—
89	C₃⋊C₈	—	2	1	no	—
90	C₃⋊C₈	—	2	1	no	—
91	C₃⋊C₈	—	4	1	no	—
92	C₂²×C₄	—	6	1	no	—
93	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
94	M₁₆	—	2	1	no	—
95	M₁₆	—	6	1	no	—
96	M₁₆	—	6	1	no	—
97	M₁₆	Dic₆	1	2	yes	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	3	1	no	—
102	C₂×C₈	—	3	1	no	—
103	C₂×C₈	—	6	1	no	—
104	C₂×C₈	—	6	1	no	—
105	C₂×C₈	—	6	1	no	—
106	C₂×C₈	#67	1	1	yes	characteristic
107	C₂×C₈	#67	1	1	yes	characteristic
108	C₄⋊C₄	—	3	1	no	—
109	C₄⋊C₄	—	3	1	no	—
110	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
111	C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
112	C₂²⋊C₄	—	2	1	no	—
113	C₂²⋊C₄	—	6	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	6	1	no	—
116	C₄²	—	6	1	no	—
117	C₄²	—	6	1	no	—
118	C₂²×C₃	—	2	1	no	—
119	C₂²×C₃	—	2	1	no	—
120	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
121	C₂²×C₃	#125	1	1	yes	characteristic
122	C₂²×C₃	#126	1	1	yes	characteristic
123	C₃×C₄	—	4	1	no	—
124	C₃×C₄	—	4	1	no	—
125	C₃×C₄	#117	1	1	yes	characteristic
126	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
127	C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
128	C₃×C₄	#123	1	1	yes	characteristic
129	Dic₃	—	4	2	no	—
130	Dic₃	—	4	1	no	—
131	Dic₃	—	4	1	no	—
132	C₂³	#235	1	1	yes	characteristic
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	6	2	no	—
136	C₂×C₄	—	6	1	no	—
137	C₂×C₄	—	6	1	no	—
138	C₂×C₄	—	12	1	no	—
139	C₂×C₄	—	12	1	no	—
140	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₄	#235	1	1	yes	characteristic
143	C₂×C₄	#235	1	1	yes	characteristic
144	C₂×C₄	#235	1	1	yes	characteristic
145	C₂×C₄	C₂×#67	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	6	1	no	—
150	C₈	—	6	1	no	—
151	C₈	—	12	1	no	—
152	C₂×C₃	—	2	1	no	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	#387	1	1	yes	characteristic
155	C₂×C₃	#443	1	1	yes	characteristic
156	C₂×C₃	#444	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#870	1	1	yes	characteristic
160	C₂²	#876	1	1	yes	characteristic
161	C₂²	C₄×#67	1	1	yes	characteristic
162	C₄	—	4	1	no	—
163	C₄	—	4	1	no	—
164	C₄	—	12	2	no	—
165	C₄	—	12	1	no	—
166	C₄	—	12	1	no	—
167	C₄	C₄×Dic₆	1	1	yes	characteristic
168	C₄	#872	1	1	yes	characteristic
169	C₄	#924	1	1	yes	characteristic
170	C₄	#929	1	1	yes	characteristic
171	C₃	#1922	1	1	yes	characteristic, complemented, Sylow
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#4933	1	1	yes	characteristic
175	C₂	#5144	1	1	yes	characteristic
176	C₂	#5145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#405	C₂
socle	C₂²×C₃	C₄×D₄
prime radical	C₂³×C₃	#41
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#405	D₃
hypocenter	C₃	#1922
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
20. #808	C₄	semidirect
21. #808	C₄	semidirect
22. #807	C₄	semidirect
23. #807	C₄	semidirect
171. C₃	#1922	semidirect

Maximal subgroups

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

subgroup	index
C₃×#405	2
#5102	2
#4998	2
#4558	2
#4555	2
#4550	2
#4532	2
#1922	3