Finite Group Databasebeta

← #11714038

Subgroups — #11714038

Order 768.

538 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 49 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×#520⋊C3C211yesmaximal, characteristic, complemented
2#85523C211yesmaximal, characteristic
3#73149C211yesmaximal, characteristic, complemented
4#73078C212yesmaximal, complemented
5#73076C212yesmaximal, complemented
6#32996—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C2×#986C2211yescharacteristic
9#520⋊C3—21no—
10#520⋊C3C2212yes—
11C2×C4×SL(2,𝔽3)—21no—
12#5427—21no—
13#5426C2211yescharacteristic
14C2×GU(2,𝔽3)C2212yes—
15#5421—21no—
16#4968—41nomaximal
17C2×#520D311yescharacteristic, complemented
18#3094—31no—
19#3006—31no—
20#2501—31no—
21#1883—32no—
22#1883—32no—
23#1859—32no—
24#1859—32no—
25#1826—31no—
26#1821—31no—
27#1789—31no—
28C2×#41⋊C3—22no—
29C2×#41⋊C3D411yescharacteristic
30C2×#41⋊C3C2311yescharacteristic
31C22×SL(2,𝔽3)D411yescharacteristic
32#986—22no—
33C2×GL(2,𝔽3)—21no—
34C2×C3×C4⋊C4—41no—
35#930—41no—
36C4×SL(2,𝔽3)—22no—
37GU(2,𝔽3)—22no—
38#859—21no—
39#830—41no—
40#809—42no—
41#808—42no—
42C22×#41C2×D311yescharacteristic
43C2×AGL(1,ℤ/8)—31no—
44C2×AGL(1,ℤ/8)—31no—
45C2×#123—62no—
46C2×#121—31no—
47#520—21no—
48#520—32no—
49#520—61no—
50#520C2×D312yes—
51C2×C4×Q8—21no—
52C2×C4×D4—61no—
53C2×#117—31no—
54C22×C4⋊C4—31no—
55#482—62no—
56#481—61no—
57#480—61no—
58#449—61no—
59#449—61no—
60#448—61no—
61#447—61no—
62#428—32no—
63C2×#104—32no—
64C2×#104—32no—
65#418—32no—
66#417—31no—
67#417—31no—
68#407—31no—
69#407—31no—
70#399—62no—
71#390—32no—
72#384—32no—
73C4×C22⋊C4—31no—
74#343—32no—
75#343—32no—
76#339—32no—
77#339—32no—
78C22×C3×C4—41no—
79C22×C3×C4—81no—
80C2×#67—81no—
81C2×D12—41no—
82#41⋊C3—21no—
83#41⋊C3#4112yes—
84C2×SL(2,𝔽3)—21no—
85C2×SL(2,𝔽3)C2×D411yescharacteristic
86GL(2,𝔽3)—41no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—81no—
89(C2×D3)⋊C4—81no—
90C4⋊Dic3—41no—
91C2×C3⋊C8—42no—
92C2×#41—22no—
93C2×#41—31no—
94C2×#41—31no—
95C2×#41—62no—
96C2×#41#6711yescharacteristic
97C2×#41C22×D311yescharacteristic
98C22×Q8#6711yescharacteristic
99C22×D4—31no—
100C22×D4—31no—
101C22×D4—61no—
102C23×C4—31no—
103C23×C4—32no—
104AGL(1,ℤ/8)—62no—
105AGL(1,ℤ/8)—64no—
106AGL(1,ℤ/8)—62no—
107C2×SD16—61no—
108C2×SD16—61no—
109C2×D8—61no—
110C2×D8—61no—
111C2×M16—31no—
112C2×M16—31no—
113C2×M16—32no—
114#123—128no—
115#121—61no—
116#121—61no—
117#121—61no—
118#121—61no—
119C4×Q8—22no—
120C4×Q8—62no—
121C4×D4—64no—
122C4×D4—62no—
123C4×D4—62no—
124#117—32no—
125#117—32no—
126#117—61no—
127#117—61no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4—32no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—64no—
136C2×C4⋊C4S411yescharacteristic
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—64no—
141C2×C42—32no—
142C2×C42—61no—
143#107—62no—
144#106—62no—
145#104—62no—
146#104—64no—
147#104—62no—
148#103—61no—
149#103—61no—
150#102—62no—
151#102—61no—
152#102—61no—
153#102—62no—
154C22⋊C8—61no—
155C22⋊C8—61no—
156C22⋊C8—61no—
157C22⋊C8—61no—
158#95—61no—
159C23×C3—41no—
160C22×D3—81no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—82no—
164C2×C3×C4—82no—
165C2×C3×C4—162no—
166#67—162no—
167C2×Dic3—81no—
168D12—82no—
169SL(2,𝔽3)#12311yescharacteristic, complemented
170C3⋊C8—82no—
171C24—31no—
172C24—61no—
173#41—21no—
174#41—32no—
175#41—34no—
176#41—62no—
177#41—62no—
178#41—61no—
179#41—62no—
180#41#23512yes—
181C2×Q8—21no—
182C2×Q8—31no—
183C2×Q8—61no—
184C2×Q8C2×#6711yescharacteristic
185C2×D4—31no—
186C2×D4—31no—
187C2×D4—61no—
188C2×D4—61no—
189C2×D4—62no—
190C2×D4—62no—
191C2×D4—62no—
192C2×D4—62no—
193C2×D4—61no—
194C2×D4—121no—
195C2×D4—124no—
196C2×D4—121no—
197C22×C4—21no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—34no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—61no—
208C22×C4—62no—
209C22×C4—61no—
210C22×C4—124no—
211C22×C4C2×S411yescharacteristic
212SD16—121no—
213SD16—122no—
214SD16—121no—
215D8—122no—
216D8—122no—
217M16—62no—
218M16—62no—
219M16—62no—
220M16—62no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—62no—
224C4⋊C4—21no—
225C4⋊C4—32no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—128no—
232C4⋊C4C2×S412yes—
233C22⋊C4—62no—
234C22⋊C4—62no—
235C22⋊C4—64no—
236C22⋊C4—121no—
237C22⋊C4—128no—
238C22⋊C4—121no—
239C42—62no—
240C42—62no—
241C42—62no—
242C42—62no—
243C22×C3—41no—
244C22×C3—42no—
245C22×C3—81no—
246C22×C3—81no—
247C2×D3—161no—
248C2×D3—161no—
249C2×D3—161no—
250C3×C4—42no—
251C3×C4—81no—
252C3×C4—162no—
253Dic3—161no—
254C23—31no—
255C23—31no—
256C23—62no—
257C23—62no—
258C23—121no—
259C23—122no—
260C23—122no—
261C23—122no—
262C23—121no—
263C23GL(2,ℤ/4)11yescharacteristic
264Q8—31no—
265Q8—32no—
266Q8—61no—
267Q8#92911yescharacteristic
268D4—32no—
269D4—32no—
270D4—61no—
271D4—61no—
272D4—64no—
273D4—121no—
274D4—121no—
275D4—122no—
276D4—242no—
277D4—242no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—31no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—31no—
284C2×C4—42no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—62no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—62no—
291C2×C4—64no—
292C2×C4—62no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—124no—
297C2×C4—122no—
298C2×C4—121no—
299C2×C4—242no—
300C2×C4GL(2,ℤ/4)11yescharacteristic
301C2×C4C22×S411yescharacteristic
302C8—121no—
303C8—121no—
304C8—122no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—82no—
309D3—321no—
310C22—21no—
311C22—21no—
312C22—32no—
313C22—61no—
314C22—61no—
315C22—61no—
316C22—61no—
317C22—61no—
318C22—124no—
319C22—244no—
320C22—241no—
321C22—241no—
322C22—241no—
323C22#542912yes—
324C22C2×GL(2,ℤ/4)11yescharacteristic
325C4—21no—
326C4—31no—
327C4—31no—
328C4—42no—
329C4—61no—
330C4—124no—
331C4—122no—
332C4—241no—
333C4#591212yes—
334C3—41noSylow
335C2—22no—
336C2—62no—
337C2—121no—
338C2—481no—
339C2#8548511yescharacteristic
340C2C2×#542911yescharacteristic
341C2#8553311yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC2×#520D3
prime radical#85523C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#520⋊C3C2semidirect
3. #73149C2semidirect
4. #73078C2semidirect
5. #73076C2semidirect
17. C2×#520D3semidirect
169. SL(2,𝔽3)#123semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
#855232
#731492
#730782
#730762
#329963
#49684