Finite Group Databasebeta

← #11714071

Subgroups — #11714071

Order 768.

425 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#85346C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic, complemented
3C4×#131⋊C3C211yesmaximal, characteristic, complemented
4#73128C214yesmaximal, complemented
5#33060—31nomaximal, Sylow
6C2×#986C2212yes—
7C2×#131⋊C3C2212yes—
8C4×#41⋊C3C2211yescharacteristic
9#5426C2212yes—
10C4⋊GL(2,𝔽3)—21no—
11C4⋊GL(2,𝔽3)—21no—
12#5391—21no—
13#5386—21no—
14#4694—41nomaximal
15C2×#486—31no—
16#3079—31no—
17#3038—31no—
18C4×#131D311yescharacteristic, complemented
19#1997—31no—
20#1965—31no—
21#1931—34no—
22#1907—34no—
23#1837—31no—
24C2×#41⋊C3C2311yescharacteristic
25#986—24no—
26C2×GL(2,𝔽3)—22no—
27#874—41no—
28#869—41no—
29#131⋊C3—22no—
30C4×SL(2,𝔽3)D411yescharacteristic
31C4×SL(2,𝔽3)D411yescharacteristic
32#859—22no—
33C3×C4×C8—41no—
34#821—44no—
35C2×AGL(1,ℤ/8)—32no—
36C2×#131C2×D312yes—
37C2×C4⋊Q8—31no—
38C2×#127—31no—
39C4×#41C2×D311yescharacteristic
40#486—62no—
41#486—61no—
42#486—61no—
43#477—61no—
44#476—61no—
45#475—61no—
46#474—61no—
47#460—61no—
48#460—61no—
49#459—61no—
50#459—61no—
51#428—32no—
52#423—31no—
53#423—31no—
54#418—32no—
55#417—32no—
56C2×#102—34no—
57C2×#100—34no—
58#405—32no—
59C4×M16—31no—
60C2×C4×C8—31no—
61#343—34no—
62#341—34no—
63C2×D12—42no—
64C2×D12—81no—
65C2×D12—81no—
66C2×Dic6—81no—
67#41⋊C3#4112yes—
68C2×SL(2,𝔽3)C2×D411yescharacteristic
69GL(2,𝔽3)—42no—
70C2×C3×C8—42no—
71C3×C42—41no—
72C4⋊Dic3—42no—
73C4⋊Dic3—81no—
74C2×#41C22×D311yescharacteristic
75C22×Q8—61no—
76C22×D4—32no—
77C22×D4—61no—
78C22×D4—61no—
79AGL(1,ℤ/8)—64no—
80AGL(1,ℤ/8)—64no—
81C2×SD16—62no—
82C2×D8—62no—
83#131—22no—
84#131—62no—
85C2×M16—32no—
86C2×M16—32no—
87C2×M16—32no—
88C22×C8—32no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—61no—
92C4⋊Q8—61no—
93#127—61no—
94#127—61no—
95#127—61no—
96#127—61no—
97C4×Q8D1211yescharacteristic
98C4×Q8D1211yescharacteristic
99C4×D4—31no—
100C4×D4—31no—
101#117—31no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—61no—
104C2×C42—31no—
105#107—62no—
106#106—62no—
107C4⋊C8—61no—
108C4⋊C8—61no—
109C4⋊C8—61no—
110C4⋊C8—61no—
111#103—62no—
112#102—62no—
113#102—62no—
114#102—64no—
115#102—62no—
116#102—64no—
117#100—68no—
118#97—61no—
119C4×C8—32no—
120C4×C8—31no—
121C4×C8S411yescharacteristic
122C22×D3—82no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
125C2×C3×C4—41no—
126C2×Dic3—82no—
127D12—84no—
128D12—162no—
129D12—162no—
130Dic6—162no—
131SL(2,𝔽3)#12411yescharacteristic
132C3×C8—82no—
133C24—62no—
134#41—32no—
135#41#23512yes—
136C2×Q8—121no—
137C2×Q8—121no—
138C2×Q8—242no—
139C2×Q8C2×D1211yescharacteristic
140C2×D4—31no—
141C2×D4—64no—
142C2×D4—62no—
143C2×D4—64no—
144C2×D4—62no—
145C2×D4—121no—
146C2×D4—121no—
147C2×D4—121no—
148C2×D4—121no—
149C2×D4—242no—
150C2×D4—242no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—62no—
155SD16—122no—
156SD16—122no—
157D8—124no—
158M16—34no—
159M16—62no—
160M16—64no—
161M16—64no—
162C2×C8—32no—
163C2×C8—62no—
164C2×C8—62no—
165C2×C8—62no—
166C2×C8—62no—
167C2×C8C2×S412yes—
168C4⋊C4—31no—
169C4⋊C4—31no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—121no—
173C4⋊C4—121no—
174C22⋊C4—61no—
175C42—31no—
176C42—31no—
177C42—31no—
178C42C2×S411yescharacteristic
179C22×C3—41no—
180C2×D3—162no—
181C2×D3—162no—
182C2×D3—162no—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4—42no—
186Dic3—162no—
187C23—31no—
188C23—124no—
189C23—122no—
190C23—242no—
191C23—242no—
192Q8—31no—
193Q8—242no—
194Q8—242no—
195Q8#87511yescharacteristic
196D4—32no—
197D4—61no—
198D4—122no—
199D4—122no—
200D4—124no—
201D4—242no—
202D4—242no—
203D4—242no—
204D4—242no—
205C2×C4—31no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—122no—
213C2×C4—242no—
214C2×C4C4⋊S411yescharacteristic
215C2×C4C4⋊S411yescharacteristic
216C2×C4C22×S411yescharacteristic
217C8—22no—
218C8—62no—
219C8—122no—
220C8—122no—
221C2×C3—41no—
222C2×C3—41no—
223C2×C3—41no—
224D3—322no—
225C22—32no—
226C22—61no—
227C22—61no—
228C22—242no—
229C22—242no—
230C22—242no—
231C22—484no—
232C22C2×C4⋊S411yescharacteristic
233C4—31no—
234C4—31no—
235C4—61no—
236C4—61no—
237C4—242no—
238C4#540612yes—
239C4#540712yes—
240C4#591212yes—
241C3—41noSylow
242C2—62no—
243C2—482no—
244C2#8529011yescharacteristic
245C2C2×#540611yescharacteristic
246C2C2×#540711yescharacteristic
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×#131D3
prime radical#85346C2
hypercenterC4×C8S4
hypocenterSL(2,𝔽3)#124
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
2. #85337C2semidirect
3. C4×#131⋊C3C2semidirect
4. #73128C2semidirect
18. C4×#131D3semidirect

Maximal subgroups

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

subgroupindex
#853462
#853372
C4×#131⋊C32
#731282
#330603
#46944