Finite Group Databasebeta

← #11713691

Subgroups — #11713691

Order 768.

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

normal subgroups only

This group has 489 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#3499⋊C3C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic, complemented
3#73155C214yesmaximal, complemented
4C4×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
5#25876—31nomaximal, Sylow
6C2×2−1+4⋊C3C2212yes—
7D4×SL(2,𝔽3)—21no—
8D4×SL(2,𝔽3)—21no—
9C4×#41⋊C3C2211yescharacteristic
10#5426C2212yes—
11C2×GU(2,𝔽3)C2212yes—
12#5391—21no—
13#5386—21no—
14#5070—41nomaximal
15#3499D311yescharacteristic
16#3141—31no—
17#3079—31no—
18#2862—31no—
19#2038—34no—
20#1966—31no—
21#1962—31no—
22#1793—34no—
23C4×#104—31no—
242−1+4⋊C3—24no—
25C2×#41⋊C3C2311yescharacteristic
26C22×SL(2,𝔽3)—22no—
27C3×#127—41no—
28#869—41no—
29C4×SL(2,𝔽3)D411yescharacteristic
30C4×SL(2,𝔽3)D411yescharacteristic
31GU(2,𝔽3)—22no—
32#859—22no—
33#832—44no—
34C4×C3⋊C8—41no—
35C2×2−1+4C2×D312yes—
36D4×Q8—21no—
37D4×Q8—21no—
38#547—61no—
39#547—61no—
40#536—32no—
41#532—31no—
42C2×C4⋊Q8—31no—
43C2×C4⋊Q8—31no—
44C4×#41C2×D311yescharacteristic
45#489—62no—
46#487—61no—
47#486—61no—
48#477—61no—
49#476—61no—
50#475—61no—
51#474—61no—
52#428—32no—
53#423—31no—
54C2×#104—32no—
55C2×#104—34no—
56#417—34no—
57#417—32no—
58C2×#101—34no—
59C4×M16—31no—
60#391—34no—
61#391—62no—
62#382—62no—
63C43—31no—
64#339—32no—
65C2×C3×D4—42no—
66C2×C3×D4—81no—
67C2×C3×D4—81no—
68C2×Dic6—81no—
69#41⋊C3#4112yes—
70C2×SL(2,𝔽3)—42no—
71C2×SL(2,𝔽3)C2×D411yescharacteristic
72C3×C42—41no—
73C4⋊Dic3—42no—
74C4⋊Dic3—81no—
75C2×C3⋊C8—42no—
762−1+4—24no—
772−1+4—64no—
78C2×#41—32no—
79C2×#41—62no—
80C2×#41C22×D311yescharacteristic
81C22×Q8—22no—
82C22×Q8—32no—
83C22×Q8—61no—
84C2×M16—32no—
85C2×M16—32no—
86C4⋊Q8—31no—
87C4⋊Q8—61no—
88C4⋊Q8—61no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—61no—
92C4⋊Q8—61no—
93C4⋊Q8—61no—
94#127S411yescharacteristic
95#124—32no—
96#123—64no—
97C22⋊Q8—62no—
98C22⋊Q8—62no—
99#121—61no—
100#121—61no—
101C4×Q8#6711yescharacteristic
102C4×Q8#6711yescharacteristic
103C4×D4—31no—
104C4×D4—31no—
105C4×D4—61no—
106C4×D4—61no—
107#117—31no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—61no—
112C2×C42—31no—
113C2×C42—32no—
114C2×C42—61no—
115C2×C42—61no—
116#107—62no—
117#106—62no—
118C4⋊C8—61no—
119C4⋊C8—61no—
120#104—62no—
121#104—62no—
122#104—124no—
123#103—64no—
124#103—62no—
125#102—62no—
126#102—64no—
127#101—124no—
128#97—61no—
129C4×C8—61no—
130#95—68no—
131C23×C3—82no—
132C3×D4—84no—
133C3×D4—162no—
134C3×D4—162no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
138C2×Dic3—82no—
139Dic6—162no—
140SL(2,𝔽3)#12411yescharacteristic
141C3⋊C8—82no—
142#41—32no—
143#41—64no—
144#41—64no—
145#41—124no—
146#41—124no—
147#41#23712yes—
148C2×Q8—32no—
149C2×Q8—42no—
150C2×Q8—64no—
151C2×Q8—64no—
152C2×Q8—62no—
153C2×Q8—121no—
154C2×Q8—124no—
155C2×Q8—121no—
156C2×Q8—122no—
157C2×Q8C2×#6711yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—31no—
161C2×D4—62no—
162C2×D4C2×S412yes—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—34no—
167C22×C4—62no—
168C22×C4—62no—
169C22×C4—62no—
170M16—62no—
171M16—64no—
172M16—62no—
173C2×C8—62no—
174C2×C8—62no—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—62no—
183C4⋊C4—121no—
184C4⋊C4—121no—
185C22⋊C4—61no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C42—31no—
189C42—31no—
190C42—31no—
191C42—62no—
192C42—62no—
193C42—64no—
194C42—122no—
195C42C2×S411yescharacteristic
196C22×C3—41no—
197C22×C3—162no—
198C22×C3—162no—
199C22×C3—162no—
200C3×C4—42no—
201C3×C4—42no—
202C3×C4—42no—
203Dic3—162no—
204C23—22no—
205C23—31no—
206Q8—31no—
207Q8—64no—
208Q8—122no—
209Q8—122no—
210Q8—242no—
211Q8—242no—
212Q8#93611yescharacteristic
213D4—24no—
214D4—32no—
215D4—42no—
216D4—42no—
217D4—61no—
218D4—124no—
219C2×C4—31no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—68no—
227C2×C4—64no—
228C2×C4—62no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4—122no—
232C2×C4—122no—
233C2×C4—124no—
234C2×C4—122no—
235C2×C4GL(2,ℤ/4)11yescharacteristic
236C2×C4GL(2,ℤ/4)11yescharacteristic
237C2×C4C22×S411yescharacteristic
238C8—122no—
239C8—122no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—322no—
244C22—32no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—61no—
249C22—61no—
250C22C2×GL(2,ℤ/4)11yescharacteristic
251C4—31no—
252C4—31no—
253C4—61no—
254C4—61no—
255C4—124no—
256C4—122no—
257C4—242no—
258C4#543012yes—
259C4#543112yes—
260C4#591412yes—
261C3—41noSylow
262C2—62no—
263C2—82no—
264C2#8549211yescharacteristic
265C2C2×#543011yescharacteristic
266C2C2×#543111yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3499D3
prime radical#3499⋊C3C2
hypercenter#127S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #85337C2semidirect
3. #73155C2semidirect
4. C4×GU(2,𝔽3)C2semidirect

Maximal subgroups

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

subgroupindex
#3499⋊C32
#853372
#731552
C4×GU(2,𝔽3)2
#258763
#50704