Finite Group Databasebeta

← #17218⋊C3 #11709331

Subgroups — #17218⋊C3 #11709331

Order 768.

261 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 28 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
1C2×#395⋊C3C212yesmaximal
2#1446⋊C3C211yesmaximal, characteristic
3#17218C311yesmaximal, characteristic, complemented, Sylow
4C2×C22⋊SL(2,𝔽3)C2211yescharacteristic
5A4×C4⋊C4—41nomaximal
6#395⋊C3—22no—
7C2×#395C2×C312yes—
8#1446C2×C311yescharacteristic
9#1445—31no—
10#1444—31no—
11#1441—32no—
12C22⋊SL(2,𝔽3)D411yescharacteristic
13C2×C4×A4—41no—
14C2×C4×A4—42no—
15C23×Q8C22×C311yescharacteristic
16C22×C4⋊C4—31no—
17C22×C4⋊C4—32no—
18C22×C4⋊C4A411yescharacteristic
19#395—22no—
20#395—62no—
21C2×#95—32no—
22C2×#95—31no—
23C2×#95—31no—
24C2×#95—31no—
25C2×#95—32no—
26C2×#95—31no—
27C22×A4—41no—
28C2×SL(2,𝔽3)—161no—
29C4×A4—42no—
30C4×A4—82no—
31C3×C4⋊C4—161no—
32C22×Q8—31no—
33C22×Q8—121no—
34C22×Q8—121no—
35C22×Q8C3×D411yescharacteristic
36C23×C4—31no—
37C23×C4—32no—
38C23×C4—31no—
39C23×C4C2×A412yes—
40C23×C4C2×A411yescharacteristic
41C2×C4⋊C4—34no—
42C2×C4⋊C4—62no—
43C2×C4⋊C4—62no—
44C2×C4⋊C4—121no—
45C2×C4⋊C4—121no—
46C2×C4⋊C4—121no—
47C2×C4⋊C4—122no—
48C2×C4⋊C4—122no—
49#95—62no—
50#95—62no—
51#95—62no—
52#95—121no—
53#95—121no—
54#95—122no—
55#95—121no—
56C2×A4—41no—
57C2×A4—41no—
58C2×A4—41no—
59C2×C3×C4—161no—
60C2×C3×C4—162no—
61SL(2,𝔽3)—161no—
62C24C22×A411yescharacteristic
63C2×Q8—41no—
64C2×Q8—121no—
65C2×Q8—122no—
66C2×Q8—121no—
67C2×Q8—121no—
68C2×Q8—123no—
69C2×Q8—121no—
70C22×C4—22no—
71C22×C4—34no—
72C22×C4—32no—
73C22×C4—31no—
74C22×C4—31no—
75C22×C4—61no—
76C22×C4—62no—
77C22×C4—62no—
78C22×C4—61no—
79C22×C4—62no—
80C22×C4—61no—
81C22×C4—61no—
82C22×C4—62no—
83C22×C4—62no—
84C22×C4—62no—
85C22×C4—62no—
86C22×C4—62no—
87C22×C4—62no—
88C22×C4—62no—
89C22×C4—61no—
90C22×C4—62no—
91C22×C4—61no—
92C22×C4—62no—
93C22×C4#41⋊C312yes—
94C4⋊C4—41no—
95C4⋊C4—121no—
96C4⋊C4—124no—
97C4⋊C4—124no—
98C4⋊C4—124no—
99C22×C3—161no—
100A4—41no—
101C3×C4—162no—
102C3×C4—322no—
103C23—31no—
104C23—31no—
105C23—31no—
106C23—31no—
107C23D4×A411yescharacteristic
108C23C2×#41⋊C311yescharacteristic
109C23#99411yescharacteristic
110Q8—41no—
111Q8—121no—
112Q8—121no—
113Q8—123no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—62no—
117C2×C4—62no—
118C2×C4—64no—
119C2×C4—62no—
120C2×C4—62no—
121C2×C4—64no—
122C2×C4—64no—
123C2×C4—64no—
124C2×C4—61no—
125C2×C4—61no—
126C2×C4—121no—
127C2×C4—122no—
128C2×C4—122no—
129C2×C4—122no—
130C2×C4—122no—
131C2×C4—122no—
132C2×C4—121no—
133C2×C4—122no—
134C2×C4—121no—
135C2×C4—122no—
136C2×C4—121no—
137C2×C4—122no—
138C2×C4—121no—
139C2×C3—161no—
140C2×C3—161no—
141C2×C3—161no—
142C22—31no—
143C22—31no—
144C22—31no—
145C22—32no—
146C22—31no—
147C22—31no—
148C22—31no—
149C22—31no—
150C22—31no—
151C22—31no—
152C22C4⋊(#41⋊C3)11yescharacteristic
153C22#545011yescharacteristic
154C4—42no—
155C4—82no—
156C4—124no—
157C4—122no—
158C4—121no—
159C4—121no—
160C3—161noSylow
161C2—31no—
162C2—31no—
163C2—31no—
164C2—31no—
165C2#7320311yescharacteristic
166C2#2812⋊C311yescharacteristic
167C2#2835⋊C311yescharacteristic
subgroupG/H
center, hypercenterC22#5450
commutator, cosocleC23×Q8C22×C3
Frattini, socleC24C22×A4
Fitting#17218C3
prime radicalC2×C22⋊SL(2,𝔽3)C22
hypocenterC22×Q8C3×D4
LCSG ▷ C23×Q8 ▷ C22×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. #17218C3semidirect

Maximal subgroups

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

subgroupindex
C2×#395⋊C32
#1446⋊C32
#172183
A4×C4⋊C44