Finite Group Databasebeta

Subgroups — #69998

Order 384.

222 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 75 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×#408	C₂	1	1	yes	maximal, characteristic
2	C₂×#837	C₂	1	1	yes	maximal, characteristic
3	#5233	C₂	1	1	yes	maximal, characteristic
4	#5102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	#4558	C₂	1	1	yes	maximal, characteristic, complemented
7	#4555	C₂	1	1	yes	maximal, characteristic, complemented
8	#1938	—	3	1	no	maximal, Sylow
9	C₂×C₃×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	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×C₂²⋊C₈	—	2	2	no	—
17	#837	—	2	2	no	—
18	#835	C₄	1	2	yes	complemented
19	#832	C₄	1	2	yes	complemented
20	#815	—	2	2	no	—
21	Dic₃⋊C₈	—	2	2	no	—
22	#424	—	3	1	no	—
23	C₂×#104	—	3	1	no	—
24	#416	—	3	1	no	—
25	#408	D₃	1	1	yes	characteristic
26	C₂×#97	—	3	1	no	—
27	#341	—	3	1	no	—
28	#339	—	3	1	no	—
29	C₃×#41	—	2	1	no	—
30	C₃×#41	—	2	1	no	—
31	C₃×#41	—	2	1	no	—
32	C₃×#41	—	2	1	no	—
33	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×D₄	—	2	1	no	—
35	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	—	2	1	no	—
37	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
38	C₂²×Dic₃	—	2	1	no	—
39	C₃×M₁₆	—	2	1	no	—
40	C₃×M₁₆	—	2	1	no	—
41	C₂×C₃×C₈	—	2	1	no	—
42	C₂×C₃×C₈	—	2	1	no	—
43	C₂×C₃×C₈	—	2	1	no	—
44	C₂×C₃×C₈	D₄	1	2	yes	—
45	#217	—	2	1	no	—
46	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
47	C₄×Dic₃	—	2	1	no	—
48	C₄×Dic₃	—	2	1	no	—
49	C₄×Dic₃	D₄	1	2	yes	—
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	2	yes	—
56	C₂×#41	C₂×D₃	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	C₂×C₄²	—	3	1	no	—
63	C₄⋊C₈	—	6	2	no	—
64	#104	—	6	2	no	—
65	#103	—	3	2	no	—
66	#102	—	3	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	#97	—	6	2	no	—
69	C₂³×C₃	—	2	1	no	—
70	C₂³×C₃	#41	1	1	yes	characteristic
71	C₃×Q₈	—	2	1	no	—
72	C₃×Q₈	—	2	1	no	—
73	C₃×D₄	—	2	1	no	—
74	C₃×D₄	—	2	1	no	—
75	C₃×D₄	—	4	1	no	—
76	C₃×D₄	—	4	1	no	—
77	C₂×C₃×C₄	—	2	1	no	—
78	C₂×C₃×C₄	—	4	1	no	—
79	C₂×C₃×C₄	—	4	1	no	—
80	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄	#41	1	1	yes	characteristic
84	C₂×C₃×C₄	#41	1	1	yes	characteristic
85	C₂×C₃×C₄	#41	1	1	yes	characteristic
86	C₂×Dic₃	—	2	2	no	—
87	C₂×Dic₃	—	2	2	no	—
88	C₂×Dic₃	—	4	2	no	—
89	C₃×C₈	—	2	2	no	—
90	C₃×C₈	—	4	1	no	—
91	C₃⋊C₈	—	2	2	no	—
92	C₃⋊C₈	—	4	1	no	—
93	#41	—	2	1	no	—
94	#41	—	2	1	no	—
95	#41	—	2	1	no	—
96	#41	—	2	1	no	—
97	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	C₄×D₃	1	1	yes	characteristic
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	6	1	no	—
102	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	6	1	no	—
106	M₁₆	—	6	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	3	2	no	—
111	C₂×C₈	—	6	1	no	—
112	C₂×C₈	—	6	1	no	—
113	C₂×C₈	—	6	1	no	—
114	C₂×C₈	#67	1	2	yes	—
115	C₄⋊C₄	—	3	2	no	—
116	C₂²⋊C₄	—	6	1	no	—
117	C₄²	—	3	2	no	—
118	C₄²	—	6	1	no	—
119	C₄²	—	6	1	no	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	4	1	no	—
122	C₂²×C₃	—	4	2	no	—
123	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₃	#123	1	1	yes	characteristic
125	C₂²×C₃	#126	1	1	yes	characteristic
126	C₃×C₄	—	4	1	no	—
127	C₃×C₄	#117	1	1	yes	characteristic
128	C₃×C₄	C₄×D₄	1	1	yes	characteristic
129	C₃×C₄	#121	1	1	yes	characteristic
130	C₃×C₄	#124	1	1	yes	characteristic
131	Dic₃	—	4	2	no	—
132	Dic₃	—	4	2	no	—
133	C₂³	—	2	1	no	—
134	C₂³	#235	1	1	yes	characteristic
135	Q₈	—	2	1	no	—
136	Q₈	—	2	1	no	—
137	D₄	—	2	1	no	—
138	D₄	—	2	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	6	2	no	—
145	C₂×C₄	—	6	2	no	—
146	C₂×C₄	—	12	2	no	—
147	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄	D₃×D₄	1	1	yes	characteristic
149	C₂×C₄	#237	1	1	yes	characteristic
150	C₂×C₄	#237	1	1	yes	characteristic
151	C₂×C₄	#237	1	1	yes	characteristic
152	C₂×C₄	C₂×#67	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	4	1	no	—
155	C₈	—	6	2	no	—
156	C₈	—	12	1	no	—
157	C₂×C₃	—	2	1	no	—
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	—	8	1	no	—
160	C₂×C₃	#388	1	1	yes	characteristic
161	C₂×C₃	#444	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	#879	1	1	yes	characteristic
166	C₂²	#883	1	1	yes	characteristic
167	C₂²	C₄×#67	1	1	yes	characteristic
168	C₄	—	4	1	no	—
169	C₄	—	12	2	no	—
170	C₄	—	12	2	no	—
171	C₄	#877	1	1	yes	characteristic
172	C₄	#67⋊C₄	1	1	yes	characteristic
173	C₄	#936	1	1	yes	characteristic
174	C₄	#939	1	1	yes	characteristic
175	C₃	#1938	1	1	yes	characteristic, complemented, Sylow
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	8	1	no	—
179	C₂	#4945	1	1	yes	characteristic
180	C₂	#5145	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#408	C₂
socle	C₂²×C₃	C₄×D₄
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#408	D₃
hypocenter	C₃	#1938
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #408

Complements

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

N	H ≅ G/N	type
4. #5102	C₂	semidirect
5. C₂×#815	C₂	semidirect
6. #4558	C₂	semidirect
7. #4555	C₂	semidirect
18. #835	C₄	semidirect
19. #832	C₄	semidirect
175. C₃	#1938	semidirect

Maximal subgroups

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

subgroup	index
C₃×#408	2
C₂×#837	2
#5233	2
#5102	2
C₂×#815	2
#4558	2
#4555	2
#1938	3