Finite Group Databasebeta

← #11714102

Subgroups — #11714102

Order 768.

557 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 63 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×#5406C214yesmaximal, complemented
2#85346C211yesmaximal, characteristic
3#85336C211yesmaximal, characteristic, complemented
4C4×#131⋊C3C211yesmaximal, characteristic, complemented
5#33189—31nomaximal, Sylow
6C2×#986C2212yes—
7C2×#984C2212yes—
8C2×#131⋊C3C2212yes—
9C4×#41⋊C3C2211yescharacteristic
10#5406—28no—
11C4⋊GL(2,𝔽3)—22no—
12C4⋊#226—22no—
13#4693—41nomaximal
14C2×#492—31no—
15C2×#471—34no—
16#3039—31no—
17#3038—31no—
18C4×#131D311yescharacteristic, complemented
19#1977—34no—
20#1966—31no—
21#1965—31no—
22#1848—31no—
23C2×#41⋊C3C2311yescharacteristic
24#986—24no—
25#984—24no—
26C2×GL(2,𝔽3)—22no—
27C2×#226—22no—
28C2×#204—44no—
29#874—41no—
30#869—41no—
31#131⋊C3D414yescomplemented
32C4×SL(2,𝔽3)D412yes—
33C3×C4×C8—41no—
34C2×#137—32no—
35C2×AGL(1,ℤ/8)—32no—
36C22×SD16—34no—
37C2×#131C2×D312yes—
38C2×C4⋊Q8—31no—
39C2×#127—31no—
40C4×#41C2×D311yescharacteristic
41#492—62no—
42#492—61no—
43#492—61no—
44#471—68no—
45#471—68no—
46#462—62no—
47#461—62no—
48#460—62no—
49#459—62no—
50C2×#108—34no—
51#423—31no—
52#423—31no—
53#419—32no—
54#418—32no—
55C2×#101—34no—
56C2×#100—34no—
57#405—32no—
58C4×M16—31no—
59C2×C4×C8—31no—
60C2×D12—42no—
61C2×D12—82no—
62C2×Dic6—42no—
63#41⋊C3C2×D412yes—
64C2×SL(2,𝔽3)C2×D411yescharacteristic
65GL(2,𝔽3)—42no—
66#226—42no—
67C2×C3×C8—42no—
68C3×C42—41no—
69C4⋊Dic3—82no—
70#204—88no—
71C2×#41C22×D311yescharacteristic
72C22×Q8—32no—
73C22×D4—32no—
74C22×D4—62no—
75#137—64no—
76#137—64no—
77AGL(1,ℤ/8)—64no—
78AGL(1,ℤ/8)—64no—
79C2×Q16—62no—
80C2×SD16—68no—
81C2×SD16—68no—
82C2×SD16—64no—
83C2×SD16—62no—
84C2×SD16—62no—
85C2×SD16—64no—
86C2×D8—62no—
87#131—34no—
88#131D1214yescomplemented
89C2×M16—32no—
90C2×M16—32no—
91C2×M16—32no—
92C22×C8—32no—
93C4⋊Q8—62no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96#127—62no—
97#127—61no—
98#127—61no—
99C4×Q8D1212yes—
100C4×D4—32no—
101#117—31no—
102C2×C4⋊C4—62no—
103C2×C42—31no—
104#108—68no—
105C4⋊C8—62no—
106C4⋊C8—62no—
107#103—64no—
108#102—64no—
109#101—68no—
110#100—68no—
111#97—61no—
112C4×C8—32no—
113C4×C8—31no—
114C4×C8S411yescharacteristic
115C22×D3—82no—
116C2×C3×C4—42no—
117C2×C3×C4—41no—
118C2×Dic3—82no—
119D12—84no—
120D12—164no—
121Dic6—84no—
122SL(2,𝔽3)#12711yescharacteristic
123C3×C8—44no—
124C24—62no—
125#41—32no—
126#41C2×D1212yes—
127C2×Q8—64no—
128C2×Q8—62no—
129C2×Q8—64no—
130C2×Q8—62no—
131C2×Q8C2×D1211yescharacteristic
132C2×D4—31no—
133C2×D4—64no—
134C2×D4—62no—
135C2×D4—64no—
136C2×D4—62no—
137C2×D4—122no—
138C2×D4—122no—
139C2×D4—244no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—62no—
143Q16—122no—
144Q16—122no—
145SD16—128no—
146SD16—124no—
147SD16—122no—
148SD16—122no—
149SD16—128no—
150D8—124no—
151M16—34no—
152M16—62no—
153M16—64no—
154M16—64no—
155C2×C8—32no—
156C2×C8—34no—
157C2×C8—34no—
158C2×C8—62no—
159C2×C8—62no—
160C2×C8C2×S412yes—
161C4⋊C4—32no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C22⋊C4—61no—
165C42—32no—
166C42—31no—
167C42C2×S411yescharacteristic
168C22×C3—41no—
169C2×D3—162no—
170C2×D3—164no—
171C3×C4—42no—
172C3×C4—44no—
173Dic3—162no—
174C23—31no—
175C23—124no—
176C23—122no—
177C23—244no—
178Q8—31no—
179Q8—122no—
180Q8—122no—
181Q8—124no—
182Q8#87411yescharacteristic
183D4—32no—
184D4—61no—
185D4—122no—
186D4—122no—
187D4—124no—
188D4—244no—
189D4—244no—
190C2×C4—31no—
191C2×C4—32no—
192C2×C4—32no—
193C2×C4—32no—
194C2×C4—64no—
195C2×C4—124no—
196C2×C4—122no—
197C2×C4C4⋊S412yes—
198C2×C4C22×S411yescharacteristic
199C8—34no—
200C8—122no—
201C8—122no—
202C8C4⋊S414yes—
203C2×C3—41no—
204C2×C3—42no—
205D3—322no—
206C22—32no—
207C22—62no—
208C22—242no—
209C22—244no—
210C22—484no—
211C22C2×C4⋊S411yescharacteristic
212C4—31no—
213C4—31no—
214C4—62no—
215C4—242no—
216C4#540614yescomplemented
217C4C2×C4⋊S412yes—
218C3—41noSylow
219C2—62no—
220C2—482no—
221C2#8528911yescharacteristic
222C2C2×#540612yes—
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×#131D3
prime radical#85346C2
hypercenterC4×C8S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
1. C2×#5406C2semidirect
3. #85336C2semidirect
4. C4×#131⋊C3C2semidirect
18. C4×#131D3semidirect
31. #131⋊C3D4semidirect
88. #131D12semidirect
216. C4#5406semidirect

Maximal subgroups

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

subgroupindex
C2×#54062
#853462
#853362
C4×#131⋊C32
#331893
#46934