Finite Group Databasebeta

← #11711850

Subgroups — #11711850

Order 768.

382 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C4×#986C211yesmaximal, characteristic
2#2988⋊C3C211yesmaximal, characteristic, complemented
3#73068C211yesmaximal, characteristic, complemented
4#73020C212yesmaximal, complemented
5#73018C212yesmaximal, complemented
6#17875—31nomaximal, Sylow
7C2×#986C411yescharacteristic
8C4×#41⋊C3C2211yescharacteristic
9#5426C411yescharacteristic
10#5394C412yes—
11C4×GL(2,𝔽3)C2212yes—
12#4838—41nomaximal
13C8×SL(2,𝔽3)C2212yescomplemented
14SL(2,𝔽3)⋊C8C2211yescharacteristic, complemented
15SL(2,𝔽3)⋊C8C2211yescharacteristic, complemented
16#2988D311yescharacteristic, complemented
17#2975—31no—
18C4×AGL(1,ℤ/8)—31no—
19#1588—32no—
20#1586—32no—
21#1583—32no—
22#1581—32no—
23#1565—31no—
24#1476—31no—
25C2×D4⋊C8—31no—
26#1451—31no—
27C2×#41⋊C3C2×C411yescharacteristic
28#986—22no—
29C2×GL(2,𝔽3)C2×C412yes—
30C2×ΓL(1,𝔽25)—42no—
31C4×D12—41no—
32C4×SL(2,𝔽3)C2×C411yescharacteristic
33C4×SL(2,𝔽3)C2311yescharacteristic
34#859C2×C412yes—
35C3×C4⋊C8—41no—
36(C2×D3)⋊C8—42no—
37C4×C3⋊C8—41no—
38C2×AGL(1,ℤ/8)—31no—
39C4×#41C2×D311yescharacteristic
40C2×C4×D4—31no—
41C8×Q8C2×D312yes—
42#442—32no—
43#440—32no—
44C4×SD16—32no—
45C4×D8—32no—
46#436—32no—
47#436—32no—
48#436—61no—
49#436—61no—
50C8×D4—32no—
51#431—31no—
52#431—31no—
53#428—31no—
54C2×C4⋊C8—31no—
55#418—31no—
56#417—31no—
57#407—62no—
58C4×M16—31no—
59C2×C4×C8—31no—
60#335—32no—
61#334—32no—
62Q8⋊C8—31no—
63Q8⋊C8—31no—
64D4⋊C8—31no—
65D4⋊C8—32no—
66D4⋊C8—31no—
67D4⋊C8—32no—
68#322—32no—
69#322—32no—
70C2×D12—41no—
71C2×C4×D3—42no—
72#41⋊C3M1612yescomplemented
73C2×SL(2,𝔽3)C22×C411yescharacteristic
74GL(2,𝔽3)—22no—
75C2×C3×C8—42no—
76C3×C42—41no—
77(C2×D3)⋊C4—42no—
78C4⋊Dic3—41no—
79C2×C3⋊C8—41no—
80C2×C3⋊C8—41no—
81ΓL(1,𝔽25)—84no—
82C2×#41C4×D311yescharacteristic
83C22×D4—31no—
84C23×C4—61no—
85AGL(1,ℤ/8)—62no—
86AGL(1,ℤ/8)—62no—
87C2×SD16—32no—
88C2×D8—32no—
89C2×M16—31no—
90C2×M16—31no—
91C2×M16—62no—
92C2×M16—62no—
93C22×C8—61no—
94C22×C8—61no—
95C4×Q8C4×D311yescharacteristic
96C4×Q8C22×D311yescharacteristic
97C4×D4—31no—
98C4×D4—31no—
99C4×D4—32no—
100C4×D4—32no—
101C4×D4—61no—
102C4×D4—61no—
103#117—31no—
104C2×C4⋊C4—31no—
105C2×C22⋊C4—61no—
106C2×C42—31no—
107#107—32no—
108#106—32no—
109C4⋊C8—32no—
110C4⋊C8—31no—
111C4⋊C8—31no—
112C4⋊C8—31no—
113C4⋊C8S411yescharacteristic
114#103—32no—
115#102—32no—
116#102—32no—
117#102—32no—
118C22⋊C8—32no—
119C22⋊C8—62no—
120C22⋊C8—61no—
121C22⋊C8—62no—
122#97—32no—
123C4×C8—32no—
124C4×C8—32no—
125C4×C8—31no—
126C4×C8—31no—
127C4×C8—61no—
128C4×C8—61no—
129C22×D3—42no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×Dic3—42no—
134D12—82no—
135C4×D3—82no—
136C4×D3—82no—
137SL(2,𝔽3)C2×M1611yescharacteristic
138C3×C8—82no—
139C3⋊C8—42no—
140C3⋊C8—81no—
141C24—61no—
142#41—32no—
143#41ΓL(1,𝔽25)12yescomplemented
144C2×Q8C2×C4×D311yescharacteristic
145C2×D4—31no—
146C2×D4—32no—
147C2×D4—62no—
148C2×D4—62no—
149C2×D4—61no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—62no—
154C22×C4—61no—
155C22×C4—62no—
156C22×C4—241no—
157C22×C4—241no—
158SD16—62no—
159SD16—62no—
160D8—64no—
161M16—62no—
162M16—62no—
163M16—242no—
164M16—242no—
165C2×C8—32no—
166C2×C8—32no—
167C2×C8—32no—
168C2×C8—61no—
169C2×C8—61no—
170C2×C8—61no—
171C2×C8—61no—
172C2×C8—122no—
173C2×C8—241no—
174C2×C8C2×S412yes—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—32no—
178C4⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—62no—
181C22⋊C4—62no—
182C42—31no—
183C42—31no—
184C42—31no—
185C42C2×S411yescharacteristic
186C22×C3—41no—
187C2×D3—82no—
188C2×D3—82no—
189C2×D3—82no—
190C3×C4—42no—
191C3×C4—41no—
192C3×C4—41no—
193C3×C4—81no—
194Dic3—82no—
195C23—31no—
196C23—62no—
197C23—122no—
198C23—241no—
199C23—241no—
200Q8—31no—
201Q8C2×ΓL(1,𝔽25)11yescharacteristic
202D4—32no—
203D4—61no—
204D4—62no—
205D4—62no—
206D4—122no—
207C2×C4—31no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4—121no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—242no—
219C2×C4—241no—
220C2×C4C4×S411yescharacteristic
221C2×C4C4×S411yescharacteristic
222C2×C4C22×S411yescharacteristic
223C8—22no—
224C8—62no—
225C8—62no—
226C8—62no—
227C8—62no—
228C8—121no—
229C8—121no—
230C8—121no—
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234D3—162no—
235C22—32no—
236C22—61no—
237C22—61no—
238C22—122no—
239C22—122no—
240C22—122no—
241C22—244no—
242C22C2×C4×S411yescharacteristic
243C4—21no—
244C4—31no—
245C4—31no—
246C4—61no—
247C4—61no—
248C4—122no—
249C4#540012yes—
250C4#592511yescharacteristic
251C4#592611yescharacteristic
252C3—41noSylow
253C2—62no—
254C2—242no—
255C2#8536611yescharacteristic
256C2C2×#540011yescharacteristic
257C2#8545311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2988D3
socleC22C2×C4×S4
prime radicalC2×#986C4
cosocleC4×SL(2,𝔽3)C23
hypercenterC4⋊C8S4
hypocenterSL(2,𝔽3)C2×M16
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #2988⋊C3C2semidirect
3. #73068C2semidirect
4. #73020C2semidirect
5. #73018C2semidirect
13. C8×SL(2,𝔽3)C22semidirect
14. SL(2,𝔽3)⋊C8C22semidirect
15. SL(2,𝔽3)⋊C8C22semidirect
16. #2988D3semidirect
72. #41⋊C3M16semidirect
143. #41ΓL(1,𝔽25)semidirect

Maximal subgroups

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

subgroupindex
C4×#9862
#2988⋊C32
#730682
#730202
#730182
#178753
#48384