Finite Group Databasebeta

← #10714576

Subgroups — #10714576

Order 768.

436 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 101 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×C3×#494C211yesmaximal, characteristic
2C2×#4565C212yesmaximal
3#72061C212yesmaximal
4#69794C211yesmaximal, characteristic
5#67658C211yesmaximal, characteristic
6#13415—31nomaximal, Sylow
7C22×C3×Q16—21no—
8C22×C3×Q16C2212yes—
9C2×C3×C4⋊Q8—21no—
10C3×#494—24no—
11C2×C3×C4×C8C2211yescharacteristic
12C2×#819C2211yescharacteristic
13#5106C2211yescharacteristic
14#5102—21no—
15C2×#812C2212yes—
16#4565—24no—
17#4564—24no—
18#4523—21no—
19C2×#494D311yescharacteristic
20C2×#362—32no—
21#2141—32no—
22#1945—31no—
23#1376—31no—
24C22×C3×Q8—22no—
25C2×C3×Q16—24no—
26C2×C3×Q16—22no—
27C2×C3×Q16—22no—
28C2×C3×Q16—22no—
29C2×C3×Q16—22no—
30C2×C3×Q16—22no—
31C2×C3×Q16—42no—
32C2×C3×Q16—42no—
33C2×C3×Q16D414yes—
34C22×C3×C8D411yescharacteristic
35C22×C3×C8C2311yescharacteristic
36C3×C4⋊Q8—24no—
37C3×C4⋊Q8—42no—
38C2×C3×C4⋊C4—42no—
39C2×C3×C42D411yescharacteristic
40#926—21no—
41C2×#208—21no—
42C3×C4×C8—22no—
43#819—22no—
44#818D412yes—
45#817—21no—
46Dic3⋊C8—42no—
47#812—22no—
48#812—22no—
49C2×C3⋊C16—22no—
50C22×Q16—21no—
51C22×Q16C2×D312yes—
52C2×C4⋊Q8—21no—
53C2×M32—32no—
54#494—24no—
55C2×#108—31no—
56#427—31no—
57#424—61no—
58C2×C4×C8C2×D311yescharacteristic
59#362—64no—
60#358—64no—
61#339—61no—
62C2×C3×Q8—24no—
63C2×C3×Q8—42no—
64C2×C3×Q8—42no—
65C2×C3×Q8—42no—
66C2×C3×Q8—42no—
67C2×C3×Q8—42no—
68C22×C3×C4—21no—
69C22×C3×C4—42no—
70C22×C3×C4C2×D411yescharacteristic
71C3×Q16—24no—
72C3×Q16—24no—
73C3×Q16—48no—
74C3×Q16—42no—
75C3×Q16—42no—
76C2×C3×C8—21no—
77C2×C3×C8—21no—
78C2×C3×C8D812yes—
79C2×C3×C8C2×D411yescharacteristic
80C2×C3×C8C2×D412yes—
81C2×C3×C8#4112yes—
82C2×C3×C8#4111yescharacteristic
83C3×C4⋊C4—48no—
84C3×C42—21no—
85C3×C42SD1612yes—
86#217—41no—
87C4⋊Dic3—22no—
88C4×Dic3—41no—
89#208—41no—
90#208—41no—
91C2×C3⋊C8—41no—
92C3⋊C16—42no—
93C22×Q8—22no—
94C2×Q16—24no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—42no—
101C2×Q16—42no—
102C2×Q16#6714yes—
103C2×M16—61no—
104C22×C8#6711yescharacteristic
105C22×C8C22×D311yescharacteristic
106C4⋊Q8—24no—
107C4⋊Q8—42no—
108#117—61no—
109C2×C4⋊C4—42no—
110C2×C42#6711yescharacteristic
111M32—62no—
112M32—62no—
113C2×C16—62no—
114#108—62no—
115#107—32no—
116#106—61no—
117C4⋊C8—122no—
118C4×C8—22no—
119C23×C3AGL(1,ℤ/8)11yescharacteristic
120C3×Q8—44no—
121C3×Q8—44no—
122C3×Q8—82no—
123C3×Q8—82no—
124C2×C3×C4—24no—
125C2×C3×C4—22no—
126C2×C3×C4—44no—
127C2×C3×C4—82no—
128C2×C3×C4—82no—
129C2×C3×C4#12011yescharacteristic
130C2×C3×C4#12111yescharacteristic
131C2×C3×C4#12411yescharacteristic
132C2×C3×C4C2×D811yescharacteristic
133C2×C3×C4C2×SD1611yescharacteristic
134C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
135C2×Dic3—42no—
136C3×C8—22no—
137C3×C8#12112yes—
138C3×C8#12312yes—
139C3⋊C8—81no—
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4C2×#6711yescharacteristic
149Q16—24no—
150Q16—24no—
151Q16—48no—
152Q16—42no—
153Q16—42no—
154M16—121no—
155M16—121no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—121no—
159C2×C8#21312yes—
160C2×C8D3×D411yescharacteristic
161C2×C8#23711yescharacteristic
162C2×C8#23712yes—
163C2×C8C2×#6712yes—
164C4⋊C4—48no—
165C4⋊C4—62no—
166C22⋊C4—121no—
167C42—21no—
168C42—121no—
169C42#21412yes—
170C16—122no—
171C22×C3—22no—
172C22×C3#45011yescharacteristic
173C22×C3#45911yescharacteristic
174C22×C3#49011yescharacteristic
175C3×C4—24no—
176C3×C4—84no—
177C3×C4#39411yescharacteristic
178C3×C4#44711yescharacteristic
179C3×C4#46011yescharacteristic
180C3×C4#48611yescharacteristic
181Dic3—82no—
182C23#93211yescharacteristic
183Q8—44no—
184Q8—44no—
185Q8—82no—
186Q8—82no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—122no—
193C2×C4C2×#21311yescharacteristic
194C2×C4#93211yescharacteristic
195C2×C4C2×#21411yescharacteristic
196C2×C4#93611yescharacteristic
197C2×C4#93811yescharacteristic
198C2×C4#95311yescharacteristic
199C8—22no—
200C8—241no—
201C8#93512yes—
202C8#93912yes—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3#204111yescharacteristic
206C2×C3#221412yes—
207C22—22no—
208C22#507111yescharacteristic
209C22#507211yescharacteristic
210C22#521811yescharacteristic
211C4—24no—
212C4—84no—
213C4—242no—
214C4#507011yescharacteristic
215C4#507511yescharacteristic
216C4#521711yescharacteristic
217C4#522211yescharacteristic
218C3#1341511yescharacteristic, complemented, Sylow
219C2—21no—
220C2—21no—
221C2#7131712yes—
222C2#7176211yescharacteristic
subgroupG/H
centerC22#5072
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC2×C3×#494C2
socleC22×C3#459
prime radicalC23×C3AGL(1,ℤ/8)
hypercenterC2×#494D3
hypocenterC3#13415
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
218. C3#13415semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4942
C2×#45652
#720612
#697942
#676582
#134153