Finite Group Databasebeta

← #10714871

Subgroups — #10714871

Order 768.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2811C211yesmaximal, characteristic
2#69672C213yesmaximal
3#67644C213yesmaximal
4#13464—31nomaximal, Sylow
5#5812C2211yescharacteristic
6C3×#395—23no—
7C3×#391—23no—
8C3×#391C2213yes—
9C2×#836C2213yes—
10#4964—23no—
11#4523—23no—
12#2811D311yescharacteristic
13#1857—33no—
14#1394—33no—
15C22×C3×Q8—21no—
16C22×C3×Q8—21no—
17C22×C3×Q8C2311yescharacteristic
18C2×C3×C4⋊C4—23no—
19C2×C3×C4⋊C4D413yes—
20C2×C3×C42D413yes—
21Q8×Dic3—41no—
22Q8×Dic3—41no—
23#926—23no—
24C2×#208—23no—
25C3×#95—23no—
26C3×#95—26no—
27#836—23no—
28#836Q816yescomplemented
29#808—43no—
30#807—43no—
31#519—31no—
32#428—63no—
33C2×#101—33no—
34#395—23no—
35#391—23no—
36#391C2×D313yes—
37#339—63no—
38C2×C3×Q8—24no—
39C2×C3×Q8—41no—
40C2×C3×Q8—41no—
41C2×C3×Q8—81no—
42C2×C3×Q8—81no—
43C2×C3×Q8—82no—
44C2×C3×Q8—81no—
45C2×C3×Q8C2×Q813yes—
46C2×C3×Q8#4111yescharacteristic
47C22×C3×C4—23no—
48C22×C3×C4—23no—
49C22×C3×C4C2×D413yes—
50C3×C4⋊C4—43no—
51C3×C4⋊C4—83no—
52C3×C42—43no—
53#217—41no—
54C4⋊Dic3—23no—
55C4⋊Dic3—23no—
56C4×Dic3—43no—
57#208—26no—
58#208—43no—
59C2×C3⋊C8—43no—
60C22×Q8—21no—
61C22×Q8—21no—
62C22×Q8C22×D311yescharacteristic
63C2×M16—63no—
64C4×Q8—121no—
65C4×Q8—121no—
66#117—63no—
67C2×C4⋊C4—23no—
68C2×C4⋊C4#6713yes—
69C2×C42#6713yes—
70#107—123no—
71#106—123no—
72#101—36no—
73#101—63no—
74#95—23no—
75#95—26no—
76C23×C3#12011yescharacteristic
77C3×Q8—43no—
78C3×Q8—41no—
79C3×Q8—84no—
80C3×Q8—84no—
81C2×C3×C4—23no—
82C2×C3×C4—23no—
83C2×C3×C4—26no—
84C2×C3×C4—43no—
85C2×C3×C4—43no—
86C2×C3×C4—46no—
87C2×C3×C4—43no—
88C2×C3×C4C22⋊Q813yes—
89C2×C3×C4C4⋊Q813yes—
90C2×Dic3—41no—
91C2×Dic3—41no—
92C3⋊C8—46no—
93C2×Q8—24no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8—81no—
97C2×Q8—81no—
98C2×Q8—82no—
99C2×Q8—81no—
100C2×Q8D3×Q813yes—
101C2×Q8#23911yescharacteristic
102C22×C4—23no—
103C22×C4—23no—
104C22×C4C2×#6713yes—
105M16—66no—
106M16—123no—
107C2×C8—123no—
108C4⋊C4—43no—
109C4⋊C4—63no—
110C4⋊C4—63no—
111C4⋊C4—83no—
112C22⋊C4—121no—
113C42—43no—
114C42—123no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3#39511yescharacteristic
118C22×C3B(4,𝔽2)11yescharacteristic
119C22×C3#45811yescharacteristic
120C3×C4—23no—
121C3×C4—23no—
122C3×C4—46no—
123C3×C4—83no—
124Dic3—81no—
125Dic3—81no—
126C23#95311yescharacteristic
127Q8—43no—
128Q8—41no—
129Q8—84no—
130Q8—84no—
131C2×C4—23no—
132C2×C4—23no—
133C2×C4—26no—
134C2×C4—43no—
135C2×C4—43no—
136C2×C4—46no—
137C2×C4—43no—
138C2×C4—121no—
139C2×C4—121no—
140C2×C4#94413yes—
141C2×C4#94613yes—
142C8—126no—
143C2×C3—21no—
144C2×C3—21no—
145C2×C3#206211yescharacteristic
146C2×C3#220211yescharacteristic
147C2×C3#220511yescharacteristic
148C22—21no—
149C22—21no—
150C22#523011yescharacteristic
151C22#524311yescharacteristic
152C22#524411yescharacteristic
153C4—23no—
154C4—23no—
155C4—46no—
156C4—83no—
157C4—241no—
158C4—241no—
159C3#1346411yescharacteristic, complemented, Sylow
160C2—21no—
161C2—21no—
162C2#7189511yescharacteristic
163C2#7194211yescharacteristic
164C2#7195111yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC22×C3×Q8C23
FrattiniC22×Q8C22×D3
FittingC3×#2811C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2811D3
hypocenterC3#13464
LCSG ▷ C22×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2811

Complements

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

NH ≅ G/Ntype
28. #836Q8semidirect
159. C3#13464semidirect

Maximal subgroups

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

subgroupindex
C3×#28112
#696722
#676442
#134643