Finite Group Databasebeta

← #11713674

Subgroups — #11713674

Order 768.

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

normal subgroups only

This group has 397 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#3504⋊C3C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic
3#73076C212yesmaximal
4#73075C212yesmaximal
5#73067C211yesmaximal, characteristic
6#25833—31nomaximal, Sylow
7#557⋊C3—21no—
8#557⋊C3—21no—
9#520⋊C3C2211yescharacteristic
10#520⋊C3C2211yescharacteristic
11C4×#41⋊C3C2211yescharacteristic
12#5426C2212yes—
13C2×GU(2,𝔽3)C2212yes—
14#5391—21no—
15#5386—21no—
16#5064—41nomaximal
17#3504D311yescharacteristic
18#3165—31no—
19#3079—31no—
20#2660—31no—
21#2061—32no—
22#2060—32no—
23#1998—31no—
24#1995—31no—
25#1860—32no—
26#1859—32no—
27#1759—31no—
28C2×#41⋊C3C2311yescharacteristic
29C3×#125—41no—
30#869—41no—
31C4×SL(2,𝔽3)—21no—
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)D411yescharacteristic
34C4×SL(2,𝔽3)D411yescharacteristic
35GU(2,𝔽3)Q814yescomplemented
36#859—22no—
37#808—42no—
38#807—42no—
39#803—41no—
40#557—21no—
41#557—21no—
42#554—61no—
43#554—61no—
44#544—31no—
45#544—31no—
46#533—31no—
47C2×C4⋊Q8—31no—
48C2×#125—31no—
49#520C2×D311yescharacteristic
50#520C2×D311yescharacteristic
51C4×#41C2×D311yescharacteristic
52#501—61no—
53#501—61no—
54#500—61no—
55#498—61no—
56#477—61no—
57#476—61no—
58#475—61no—
59#474—61no—
60#428—32no—
61#428—32no—
62#428—32no—
63#423—31no—
64C2×#104—32no—
65#417—32no—
66C4×M16—31no—
67#398—62no—
68#395—62no—
69#384—32no—
70#382—32no—
71#376—31no—
72#343—32no—
73#343—32no—
74#339—32no—
75#339—32no—
76#339—32no—
77C2×Dic6—81no—
78#41⋊C3C2×Q812yes—
79C2×SL(2,𝔽3)C2×D411yescharacteristic
80C3×C4⋊C4—41no—
81C3×C4⋊C4—41no—
82C3×C4⋊C4—81no—
83C3×C4⋊C4—81no—
84C3×C42—41no—
85C4⋊Dic3—42no—
86C4⋊Dic3—81no—
87C2×C3⋊C8—42no—
88C2×#41C22×D311yescharacteristic
89C22×Q8—61no—
90C2×M16—32no—
91C2×M16—32no—
92C4⋊Q8—31no—
93C4⋊Q8—31no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96C4⋊Q8—61no—
97C4⋊Q8—61no—
98C4⋊Q8—61no—
99C4⋊Q8—61no—
100#125—31no—
101#125—61no—
102#125—61no—
103#125—61no—
104#125—61no—
105#125—61no—
106#125S411yescharacteristic
107C22⋊Q8—61no—
108C22⋊Q8—61no—
109C22⋊Q8—62no—
110C22⋊Q8—62no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8—61no—
114C4×Q8—61no—
115C4×Q8#6711yescharacteristic
116C4×Q8#6711yescharacteristic
117C4×D4—31no—
118C4×D4—31no—
119C4×D4—61no—
120C4×D4—61no—
121#117—31no—
122#117—31no—
123#117—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—32no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—64no—
131C2×C42—31no—
132C2×C42—32no—
133#107—62no—
134#107—62no—
135#107—62no—
136#106—62no—
137#106—62no—
138#106—62no—
139C4⋊C8—61no—
140C4⋊C8—61no—
141#104—34no—
142#104—34no—
143#103—62no—
144#102—62no—
145#97—61no—
146C4×C8—61no—
147#95—61no—
148#95—61no—
149#95—64no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—81no—
154C2×C3×C4—81no—
155C2×Dic3—82no—
156Dic6—162no—
157SL(2,𝔽3)C4⋊Q811yescharacteristic
158C3⋊C8—44no—
159#41—32no—
160#41D3×Q812yes—
161C2×Q8—61no—
162C2×Q8—61no—
163C2×Q8—121no—
164C2×Q8—121no—
165C2×Q8—242no—
166C2×Q8C2×#6711yescharacteristic
167C2×D4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—32no—
172C22×C4—32no—
173C22×C4—62no—
174C22×C4—62no—
175M16—34no—
176M16—62no—
177M16—64no—
178C2×C8—62no—
179C2×C8—62no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—31no—
183C4⋊C4—31no—
184C4⋊C4—31no—
185C4⋊C4—31no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4—61no—
189C4⋊C4—61no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C4⋊C4—61no—
194C4⋊C4—61no—
195C4⋊C4—62no—
196C4⋊C4—121no—
197C4⋊C4—121no—
198C4⋊C4—244no—
199C4⋊C4C2×S411yescharacteristic
200C4⋊C4C2×S411yescharacteristic
201C22⋊C4—61no—
202C22⋊C4—61no—
203C22⋊C4—61no—
204C42—31no—
205C42—31no—
206C42—31no—
207C42—34no—
208C42—61no—
209C42—61no—
210C42—62no—
211C42C2×S411yescharacteristic
212C22×C3—41no—
213C3×C4—42no—
214C3×C4—81no—
215C3×C4—81no—
216C3×C4—161no—
217C3×C4—161no—
218Dic3—162no—
219C23—31no—
220Q8—31no—
221Q8—242no—
222Q8—241no—
223Q8—241no—
224Q8—242no—
225Q8#94411yescharacteristic
226D4—32no—
227D4—61no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—64no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4—124no—
245C2×C4—122no—
246C2×C4GL(2,ℤ/4)11yescharacteristic
247C2×C4GL(2,ℤ/4)11yescharacteristic
248C2×C4C22×S411yescharacteristic
249C8—64no—
250C8—122no—
251C2×C3—41no—
252C2×C3—41no—
253C2×C3—41no—
254C22—32no—
255C22—61no—
256C22—61no—
257C22C2×GL(2,ℤ/4)11yescharacteristic
258C4—21no—
259C4—21no—
260C4—31no—
261C4—31no—
262C4—41no—
263C4—41no—
264C4—61no—
265C4—61no—
266C4—124no—
267C4—121no—
268C4—121no—
269C4—242no—
270C4Q8×S412yes—
271C3—41noSylow
272C2—62no—
273C2#8550211yescharacteristic
274C2#8554211yescharacteristic
275C2#8554311yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3504D3
hypercenter#125S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
35. GU(2,𝔽3)Q8semidirect

Maximal subgroups

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

subgroupindex
#3504⋊C32
#853372
#730762
#730752
#730672
#258333
#50644