Finite Group Databasebeta

← #11001610

Subgroups — #11001610

Order 768.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2840C211yesmaximal, characteristic
2#76861C2112yesmaximal, complemented
3#76778C2112yesmaximal, complemented
4#73845C213yesmaximal
5#73817C213yesmaximal
6#36674—31nomaximal, Sylow
7C24×Dic3C2211yescharacteristic
8C2×Q8×Dic3C2214yescomplemented
9C2×#877C2216yes—
10C3×#393C22112yes—
11C3×#379C2213yes—
12#5247C2213yes—
13#5230C22112yescomplemented
14#4949C22112yes—
15#4948C22124yes—
16#4943C2216yes—
17#4942C22112yes—
18Dic3×C22⋊C4C22112yes—
19#4659C22112yescomplemented
20#4657C22124yescomplemented
21#4649C22112yescomplemented
22#2842—33no—
23#2841—312no—
24#2840D311yescharacteristic
25#2396—312no—
26#2280—33no—
27C25×C3C2311yescharacteristic
28C22×C3×Q8C2314yes—
29C23×Dic3—44no—
30C23×Dic3Q818yescomplemented
31C23×Dic3C2313yes—
32C23×Dic3C2313yes—
33C2×C3×C22⋊C4C23112yes—
34C2×C3×C22⋊C4C2316yes—
35C2×#217C23112yes—
36C2×#217C2316yes—
37Q8×Dic3—48no—
38C2×C4⋊Dic3C23112yes—
39C2×Dic3⋊C4C23124yes—
40C2×Dic3⋊C4C23112yes—
41C2×C4×Dic3C23112yes—
42#877—412no—
43C3×#95C23112yes—
44C3⋊#95C23112yes—
45C3⋊#95C23124yes—
46C24×C4—31no—
47C2×C4×Q8—34no—
48C2×#117—36no—
49#400—312no—
50#399—312no—
51#398—324no—
52#396—324no—
53#395—312no—
54#393C2×D3112yes—
55#381—36no—
56#380—312no—
57#379—33no—
58#379C2×D313yes—
59C4×C4⋊C4—312no—
60C4×C22⋊C4—312no—
61C24×C3—44no—
62C24×C3—42no—
63C24×C3C2×Q814yes—
64C24×C3C2413yes—
65C2×C3×Q8—44no—
66C2×C3×Q8—48no—
67C22×C3×C4C24112yes—
68C22×Dic3—212no—
69C22×Dic3—416no—
70C22×Dic3—46no—
71C22×Dic3C2×Q8112yes—
72C22×Dic3C2×Q8112yes—
73C22×Dic3C24112yes—
74C22×Dic3C2413yes—
75C22×Dic3C2411yescharacteristic
76C3×C22⋊C4—46no—
77C3×C22⋊C4—412no—
78#217—46no—
79#217—412no—
80C4⋊Dic3—412no—
81Dic3⋊C4—412no—
82Dic3⋊C4—424no—
83C4×Dic3—224no—
84C25C22×D311yescharacteristic
85C22×Q8C22×D314yes—
86C23×C4—33no—
87C23×C4—38no—
88C23×C4—33no—
89C23×C4—124no—
90C4×Q8—128no—
91#117—1212no—
92C2×C4⋊C4—312no—
93C2×C4⋊C4—312no—
94C2×C4⋊C4—324no—
95C2×C22⋊C4—36no—
96C2×C22⋊C4—312no—
97C2×C22⋊C4C22×D316yes—
98C2×C22⋊C4C22×D3112yes—
99C2×C42—312no—
100#95—324no—
101#95—312no—
102#95C22×D3112yes—
103C23×C3—26no—
104C23×C3—212no—
105C23×C3—44no—
106C23×C3—48no—
107C23×C3—416no—
108C23×C3C22×Q816yes—
109C23×C3C2511yescharacteristic
110C3×Q8—416no—
111C2×C3×C4—212no—
112C2×C3×C4—224no—
113C2×Dic3—26no—
114C2×Dic3—224no—
115C2×Dic3—212no—
116C2×Dic3—224no—
117C2×Dic3—412no—
118C2×Dic3C22×Q816yes—
119C2×Dic3C22×Q812yes—
120C2×Dic3H(5,𝔽2)14yes—
121C24—42no—
122C24—44no—
123C24D3×Q814yes—
124C24C23×D313yes—
125C2×Q8—44no—
126C2×Q8—48no—
127C22×C4—31no—
128C22×C4—312no—
129C22×C4—33no—
130C22×C4—312no—
131C22×C4—312no—
132C22×C4—612no—
133C22×C4—1216no—
134C22×C4—126no—
135C22×C4C23×D3112yes—
136C4⋊C4—1212no—
137C4⋊C4—1212no—
138C4⋊C4—1224no—
139C22⋊C4—46no—
140C22⋊C4—412no—
141C22⋊C4—126no—
142C22⋊C4—1212no—
143C42—624no—
144C22×C3—26no—
145C22×C3—212no—
146C22×C3—416no—
147C22×C3—412no—
148C22×C3C23×Q811yescharacteristic
149C22×C3C2×H(5,𝔽2)12yes—
150C22×C3#58514yes—
151C3×C4—412no—
152Dic3—26no—
153Dic3—412no—
154Dic3#54314yes—
155C23—26no—
156C23—212no—
157C23—44no—
158C23—48no—
159C23—416no—
160C23C2×D3×Q816yes—
161C23C24×D311yescharacteristic
162Q8—416no—
163C2×C4—212no—
164C2×C4—224no—
165C2×C4—32no—
166C2×C4—34no—
167C2×C4—36no—
168C2×C4—624no—
169C2×C4—66no—
170C2×C4—612no—
171C2×C4—624no—
172C2×C4—1212no—
173C2×C3—46no—
174C2×C3C2×#54311yescharacteristic
175C2×C3#345812yes—
176C2×C3#352914yes—
177C22—212no—
178C22—26no—
179C22—416no—
180C22—412no—
181C22C22×D3×Q811yescharacteristic
182C22D3×H(5,𝔽2)12yes—
183C22#596614yes—
184C4—34no—
185C4—412no—
186C4—66no—
187C4—1212no—
188C3#3667411yescharacteristic, complemented, Sylow
189C2—46no—
190C2#8657112yes—
191C2D3×#54311yescharacteristic
192C2#8708514yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2840C2
prime radicalC25×C3C23
hypercenter#2840D3
hypocenterC3#36674
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #76861C2semidirect
3. #76778C2semidirect
8. C2×Q8×Dic3C22semidirect
13. #5230C22semidirect
19. #4659C22semidirect
20. #4657C22semidirect
21. #4649C22semidirect
30. C23×Dic3Q8semidirect
188. C3#36674semidirect

Maximal subgroups

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

subgroupindex
C3×#28402
#768612
#767782
#738452
#738172
#366743