Finite Group Databasebeta

← #11713938

Subgroups — #11713938

Order 768.

1185 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1185 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C23×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#5427C212yesmaximal, complemented
3C2×#5421C211yesmaximal, characteristic, complemented
4#73151C212yesmaximal, complemented
5#73148C211yesmaximal, characteristic, complemented
6#32880—31nomaximal, Sylow
7C22×#41⋊C3—24no—
8C22×#41⋊C3—22no—
9C22×#41⋊C3C2212yes—
10C23×SL(2,𝔽3)C2211yescharacteristic
11C22×GL(2,𝔽3)C2211yescharacteristic, complemented
12#5427—24no—
13#5421—22no—
14C2×#859C2211yescharacteristic, complemented
15C2×#859C2212yescomplemented
16#5213—41nomaximal
17C23×#41D311yescharacteristic, complemented
18C2×#449—32no—
19C2×#449—32no—
20C2×#448—31no—
21C2×#447—31no—
22#2847—31no—
23#1788—32no—
24#1788—32no—
25#1787—31no—
26#1786—31no—
27#1779—31no—
28C23×C3×C4—41no—
29C22×#67—41no—
30C2×#41⋊C3—28no—
31C2×#41⋊C3—22no—
32C2×#41⋊C3—22no—
33C2×#41⋊C3D414yescomplemented
34C22×SL(2,𝔽3)—22no—
35C22×SL(2,𝔽3)D412yescomplemented
36C22×SL(2,𝔽3)C2311yescharacteristic
37C2×GL(2,𝔽3)—21no—
38C2×GL(2,𝔽3)—22no—
39C2×#217—41no—
40C2×(C2×D3)⋊C4—42no—
41#859—21no—
42#859—22no—
43C3⋊#95—42no—
44C22×#41—24no—
45C22×#41—22no—
46C22×#41—32no—
47C22×#41—62no—
48C22×#41—64no—
49C22×#41C2×D312yes—
50C23×Q8C2×D311yescharacteristic
51C23×D4—31no—
52C24×C4—31no—
53C22×SD16—31no—
54C22×SD16—31no—
55C22×D8—32no—
56C2×#121—31no—
57C2×#121—32no—
58C2×#120—61no—
59#449—64no—
60#449—68no—
61#449—64no—
62#448—62no—
63#448—62no—
64#447—64no—
65C2×#103—31no—
66C2×#103—32no—
67C2×#103—31no—
68C2×#103—32no—
69C2×#102—32no—
70C2×#102—32no—
71C2×#102—32no—
72C2×C22⋊C8—31no—
73C2×C22⋊C8—31no—
74C2×C22⋊C8—32no—
75C2×C22⋊C8—32no—
76#386—62no—
77#386—61no—
78#380—31no—
79#380—32no—
80C24×C3—41no—
81C23×D3—41no—
82C22×C3×C4—42no—
83C22×C3×C4—84no—
84C22×C3×C4—82no—
85C2×#67—44no—
86C2×#67—82no—
87C2×#67—82no—
88C22×Dic3—41no—
89C22×Dic3—42no—
90#41⋊C3—24no—
91C2×SL(2,𝔽3)—22no—
92C2×SL(2,𝔽3)C2×D412yes—
93C2×SL(2,𝔽3)C2×D411yescharacteristic
94GL(2,𝔽3)—41no—
95#217—82no—
96(C2×D3)⋊C4—84no—
97C25—31no—
98C2×#41—28no—
99C2×#41—22no—
100C2×#41—22no—
101C2×#41—34no—
102C2×#41—38no—
103C2×#41—64no—
104C2×#41—64no—
105C2×#41—64no—
106C2×#41—616no—
107C2×#41—62no—
108C2×#41—62no—
109C2×#41—64no—
110C2×#41—64no—
111C2×#41—616no—
112C2×#41—68no—
113C2×#41#6714yescomplemented
114C22×Q8—22no—
115C22×Q8—31no—
116C22×Q8—32no—
117C22×Q8—62no—
118C22×Q8#6712yes—
119C22×Q8C22×D311yescharacteristic
120C22×D4—32no—
121C22×D4—34no—
122C22×D4—31no—
123C22×D4—61no—
124C22×D4—62no—
125C22×D4—64no—
126C22×D4—64no—
127C22×D4—62no—
128C22×D4—61no—
129C23×C4—31no—
130C23×C4—32no—
131C23×C4—32no—
132C23×C4—62no—
133C23×C4—64no—
134C23×C4—62no—
135C23×C4—64no—
136C23×C4S411yescharacteristic
137C2×SD16—62no—
138C2×SD16—61no—
139C2×SD16—62no—
140C2×SD16—64no—
141C2×SD16—61no—
142C2×SD16—62no—
143C2×D8—64no—
144C2×D8—64no—
145C2×D8—64no—
146C22×C8—32no—
147C22×C8—31no—
148C22×C8—31no—
149#121—62no—
150#121—62no—
151#121—64no—
152#121—64no—
153#120—124no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—32no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—62no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—62no—
160#103—62no—
161#103—61no—
162#103—62no—
163#103—64no—
164#103—61no—
165#103—62no—
166#102—64no—
167#102—64no—
168#102—64no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—64no—
172C22⋊C8—64no—
173#95—61no—
174#95—62no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—41no—
179C23×C3—82no—
180C23×C3—82no—
181C22×D3—42no—
182C22×D3—82no—
183C22×D3—81no—
184C22×D3—81no—
185C22×D3—82no—
186C2×C3×C4—44no—
187C2×C3×C4—82no—
188C2×C3×C4—82no—
189C2×C3×C4—88no—
190#67—88no—
191C2×Dic3—42no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194C2×Dic3—82no—
195C2×Dic3—82no—
196C2×Dic3—82no—
197SL(2,𝔽3)#12011yescharacteristic, complemented
198C24—32no—
199C24—64no—
200C24—62no—
201C24—62no—
202C24—62no—
203C24—61no—
204C24—122no—
205C24C2×S411yescharacteristic
206#41—24no—
207#41—68no—
208#41—616no—
209#41—616no—
210#41—616no—
211#41—64no—
212#41—64no—
213#41—64no—
214#41—616no—
215C2×Q8—22no—
216C2×Q8—32no—
217C2×Q8—31no—
218C2×Q8—32no—
219C2×Q8—34no—
220C2×Q8—61no—
221C2×Q8—62no—
222C2×Q8—64no—
223C2×Q8—62no—
224C2×Q8—64no—
225C2×Q8C2×#6712yes—
226C2×Q8C2×#6711yescharacteristic
227C2×D4—34no—
228C2×D4—34no—
229C2×D4—34no—
230C2×D4—62no—
231C2×D4—62no—
232C2×D4—62no—
233C2×D4—64no—
234C2×D4—61no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—61no—
238C2×D4—616no—
239C2×D4—68no—
240C2×D4—68no—
241C2×D4—68no—
242C2×D4—124no—
243C2×D4—124no—
244C2×D4—124no—
245C2×D4—122no—
246C2×D4—122no—
247C2×D4—122no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—31no—
251C22×C4—32no—
252C22×C4—32no—
253C22×C4—31no—
254C22×C4—34no—
255C22×C4—34no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—64no—
261C22×C4—64no—
262C22×C4—68no—
263C22×C4—616no—
264C22×C4—68no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—64no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4C2×S412yes—
271SD16—122no—
272SD16—121no—
273SD16—124no—
274SD16—121no—
275D8—128no—
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—61no—
282C2×C8—61no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—61no—
287C22⋊C4—122no—
288C22⋊C4—124no—
289C22⋊C4—122no—
290C22⋊C4—124no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—41no—
296C22×C3—44no—
297C22×C3—82no—
298C22×C3—84no—
299C22×C3—82no—
300C22×C3—84no—
301C2×D3—82no—
302C2×D3—84no—
303C2×D3—161no—
304C2×D3—162no—
305C2×D3—161no—
306C3×C4—84no—
307Dic3—82no—
308Dic3—162no—
309C23—22no—
310C23—22no—
311C23—34no—
312C23—68no—
313C23—62no—
314C23—62no—
315C23—62no—
316C23—62no—
317C23—62no—
318C23—62no—
319C23—128no—
320C23—128no—
321C23—124no—
322C23—124no—
323C23—121no—
324C23—121no—
325C23—122no—
326C23—121no—
327C23—122no—
328C23#98512yes—
329C23#98612yes—
330C23GL(2,ℤ/4)12yes—
331C23C22×S411yescharacteristic
332Q8—32no—
333Q8—31no—
334Q8—34no—
335Q8—61no—
336Q8—62no—
337Q8—64no—
338Q8#95311yescharacteristic
339D4—38no—
340D4—64no—
341D4—64no—
342D4—64no—
343D4—616no—
344D4—122no—
345D4—121no—
346D4—121no—
347D4—248no—
348D4—244no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—31no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—31no—
356C2×C4—32no—
357C2×C4—64no—
358C2×C4—64no—
359C2×C4—64no—
360C2×C4—64no—
361C2×C4—616no—
362C2×C4—62no—
363C2×C4—64no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—121no—
371C2×C4—122no—
372C2×C4GL(2,ℤ/4)14yes—
373C8—122no—
374C8—121no—
375C8—121no—
376C2×C3—41no—
377C2×C3—42no—
378C2×C3—41no—
379C2×C3—42no—
380C2×C3—41no—
381C2×C3—84no—
382D3—162no—
383C22—24no—
384C22—22no—
385C22—24no—
386C22—22no—
387C22—64no—
388C22—64no—
389C22—64no—
390C22—1216no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22—122no—
395C22—241no—
396C22—242no—
397C22—241no—
398C22—242no—
399C22—241no—
400C22#542714yescomplemented
401C22C2×#98511yescharacteristic
402C22C2×#98611yescharacteristic
403C22#592312yes—
404C22C2×GL(2,ℤ/4)11yescharacteristic
405C22C2×GL(2,ℤ/4)12yes—
406C4—24no—
407C4—32no—
408C4—31no—
409C4—31no—
410C4—62no—
411C4—241no—
412C4—242no—
413C3—41noSylow
414C2—24no—
415C2—124no—
416C2—481no—
417C2C2×#542711yescharacteristic
418C2#8552112yes—
419C2#8552211yescharacteristic
420C2#8552312yes—
421C2#8554611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC23×#41D3
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)#120
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×#41⋊C3C2semidirect
2. C2×#5427C2semidirect
3. C2×#5421C2semidirect
4. #73151C2semidirect
5. #73148C2semidirect
11. C22×GL(2,𝔽3)C22semidirect
14. C2×#859C22semidirect
15. C2×#859C22semidirect
17. C23×#41D3semidirect
33. C2×#41⋊C3D4semidirect
35. C22×SL(2,𝔽3)D4semidirect
113. C2×#41#67semidirect
197. SL(2,𝔽3)#120semidirect
400. C22#5427semidirect

Maximal subgroups

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

subgroupindex
C23×#41⋊C32
C2×#54272
C2×#54212
#731512
#731482
#328803
#52134