Finite Group Databasebeta

← #10830948

Subgroups — #10830948

Order 768.

1384 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×Q16C211yesmaximal, characteristic, complemented
2C2×#4836C211yesmaximal, characteristic
3C2×#4816C212yesmaximal, complemented
4#76619C212yesmaximal
5C3×#2056C211yesmaximal, characteristic, complemented
6#72071C211yesmaximal, characteristic, complemented
7#70487C211yesmaximal, characteristic, complemented
8#70266C212yesmaximal, complemented
9#69678C212yesmaximal, complemented
10#68888C211yesmaximal, characteristic, complemented
11#68868C211yesmaximal, characteristic, complemented
12#24273—31nomaximal, Sylow
13C22×D3×Q8C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C2×#946C2212yes—
16C22×#216C2212yes—
17C2×D3×Q16—28no—
18C2×D3×Q16—24no—
19C22×Dic12C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C2×#897C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2212yes—
26C3×#391C2212yes—
27C2×#835C2212yes—
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#816C2212yes—
31#4987C2212yes—
32C2×#810C2212yes—
33C2×#808C2212yes—
34#4929C2212yes—
35C2×#803C2211yescharacteristic
36#4836—24no—
37#4816—28no—
38#4673—22no—
39#4668—22no—
40#4665—22no—
41#4656—22no—
42C3×#340C2211yescharacteristic
43#4551C2211yescharacteristic
44C23×Q16—31no—
45C2×#470—32no—
46C2×#435—31no—
47#2409—32no—
48#2056—31no—
49#2056D311yescharacteristic, complemented
50#2001—31no—
51#2001—31no—
52#1971—32no—
53#1807—31no—
54#1787—32no—
55C22×C3×Q8C2312yes—
56C2×D3×Q8—24no—
57C2×D3×Q8—24no—
58C2×D3×Q8—24no—
59C2×D3×Q8—24no—
60C2×D3×Q8—42no—
61C2×D3×Q8—42no—
62C22×D12D411yescharacteristic
63C22×C4×D3—22no—
64C22×C4×D3C2311yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6—22no—
67C22×Dic6C2312yes—
68C2×C3×Q16—24no—
69C2×C3×Q16—22no—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74#946—24no—
75#946—42no—
76C2×#216—24no—
77C2×#216—24no—
78C2×#216D418yescomplemented
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4D411yescharacteristic
83C2×(C2×D3)⋊C4D411yescharacteristic
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3D411yescharacteristic
86C2×C4⋊Dic3C2312yes—
87C2×Dic3⋊C4—22no—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90D3×Q16—232no—
91C2×Dic12—24no—
92C2×Dic12—22no—
93C2×ΓL(1,𝔽25)—24no—
94C2×ΓL(1,𝔽25)—22no—
95C2×D3×C8—24no—
96C2×D3×C8D414yes—
97#897—24no—
98#897—42no—
99C4×D12—22no—
100C4×D12—22no—
101C3×C4⋊C8—22no—
102C3×#103—24no—
103C3×#95—22no—
104C3×#95—22no—
105#835—42no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109(C2×D3)⋊C8—22no—
110(C2×D3)⋊C8—22no—
111#816—24no—
112#810—42no—
113#808—24no—
114#803—22no—
115C23×Q8—32no—
116C22×Q16—32no—
117C22×Q16—31no—
118C22×Q16—64no—
119C22×Q16—68no—
120C22×Q16C2×D311yescharacteristic
121C22×M16—31no—
122C23×C8—31no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8—32no—
125C2×C4×D4—31no—
126#470—68no—
127#435—64no—
128C2×#106—32no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103C2×D312yes—
134C2×C22⋊C8—31no—
135C2×C22⋊C8—31no—
136C2×#97—31no—
137#395—62no—
138#393—62no—
139#391—32no—
140#391C2×D312yes—
141#386—62no—
142#385—62no—
143#380—32no—
144#340—31no—
145#340C2×D311yescharacteristic
146C23×D3C2×D411yescharacteristic
147C23×D3C2×D411yescharacteristic
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8C2×D414yes—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D411yescharacteristic
163D3×Q8—216no—
164D3×Q8—48no—
165D3×Q8—48no—
166D3×Q8—48no—
167C2×D12—22no—
168C2×D12—24no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—21no—
172C2×C4×D3—22no—
173C2×C4×D3—21no—
174C2×C4×D3—22no—
175C2×C4×D3—44no—
176C2×C4×D3—44no—
177C2×C4×D3—44no—
178C2×C4×D3C2×D412yes—
179C2×C4×D3#4112yes—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—44no—
186C2×Dic6—44no—
187C2×Dic6—44no—
188C2×Dic6C2×D414yes—
189C3×Q16—28no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8C2×D412yes—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C42—21no—
197C3×C42—21no—
198#216—216no—
199(C2×D3)⋊C4—22no—
200(C2×D3)⋊C4—22no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—44no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—44no—
208Dic3⋊C4—44no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8C2×D412yes—
212C2×C3⋊C8C2×D414yes—
213Dic12—28no—
214ΓL(1,𝔽25)—44no—
215D3×C8—28no—
216C22×Q8—32no—
217C22×Q8—62no—
218C22×Q8—64no—
219C22×Q8—64no—
220C22×Q8—64no—
221C22×Q8—64no—
222C22×Q8—122no—
223C22×Q8—122no—
224C22×Q8C22×D312yes—
225C22×D4—31no—
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—62no—
229C2×Q16—24no—
230C2×Q16—22no—
231C2×Q16—38no—
232C2×Q16—64no—
233C2×Q16—64no—
234C2×Q16—64no—
235C2×Q16—632no—
236C2×Q16—62no—
237C2×M16—62no—
238C2×M16—64no—
239C22×C8—31no—
240C22×C8—34no—
241C22×C8—31no—
242C22×C8—64no—
243C22×C8C22×D311yescharacteristic
244C22×C8C22×D311yescharacteristic
245C22⋊Q8—64no—
246C22⋊Q8—64no—
247C22⋊Q8—122no—
248C22⋊Q8—122no—
249C4×D4—62no—
250C4×D4—62no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—31no—
257C2×C22⋊C4—31no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C42C22×D311yescharacteristic
262#106—64no—
263C4⋊C8—22no—
264#103—24no—
265#103—64no—
266#103—122no—
267#103—122no—
268C22⋊C8—62no—
269C22⋊C8—62no—
270#97—62no—
271#95—22no—
272#95—22no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3C22×D411yescharacteristic
277C22×D3—22no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—21no—
284C22×D3—21no—
285C22×D3#12112yes—
286C22×D3#13714yes—
287C3×Q8—24no—
288C3×Q8—24no—
289C3×Q8—42no—
290C3×Q8—42no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D412yes—
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4H(5,𝔽2)12yes—
306C2×Dic3—22no—
307C2×Dic3—21no—
308C2×Dic3—22no—
309C2×Dic3—24no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3#12112yes—
316D12—44no—
317C4×D3—22no—
318C4×D3—22no—
319C4×D3—24no—
320C4×D3—416no—
321C4×D3—41no—
322C4×D3—41no—
323C4×D3—42no—
324Dic6—24no—
325Dic6—24no—
326Dic6—416no—
327Dic6—42no—
328Dic6—42no—
329C3×C8—22no—
330C3×C8—41no—
331C3⋊C8—41no—
332C3⋊C8#12014yes—
333C24—31no—
334C24—31no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—34no—
341C2×Q8—62no—
342C2×Q8—616no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—124no—
348C2×Q8—124no—
349C2×Q8—124no—
350C2×Q8—128no—
351C2×Q8—128no—
352C2×Q8—128no—
353C2×Q8D3×D414yes—
354C2×D4—62no—
355C2×D4—64no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—31no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—31no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—124no—
371C22×C4—124no—
372C22×C4—124no—
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D411yescharacteristic
375C22×C4C23×D311yescharacteristic
376Q16—28no—
377Q16—68no—
378Q16—616no—
379M16—124no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—32no—
384C2×C8—34no—
385C2×C8—68no—
386C2×C8—61no—
387C2×C8—62no—
388C2×C8D3×D412yes—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C4⋊C4—124no—
395C4⋊C4—124no—
396C22⋊C4—62no—
397C22⋊C4—62no—
398C22⋊C4—124no—
399C22⋊C4—124no—
400C22⋊C4—124no—
401C42—21no—
402C42—21no—
403C22×C3C2×#12111yescharacteristic
404C22×C3D4212yes—
405C22×C3C2×#13712yes—
406C22×C3#57811yescharacteristic
407C22×C3#57811yescharacteristic
408C2×D3—22no—
409C2×D3—24no—
410C2×D3—24no—
411C2×D3—22no—
412C2×D3—41no—
413C2×D3—42no—
414C2×D3—42no—
415C2×D3—41no—
416C2×D3—41no—
417C2×D3#47014yescomplemented
418C3×C4—44no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—82no—
422C3×C4C2×#12011yescharacteristic
423C3×C4#53511yescharacteristic
424C3×C4#54612yes—
425Dic3—22no—
426Dic3—44no—
427Dic3—41no—
428Dic3—82no—
429C23—34no—
430C23—32no—
431C23—61no—
432C23—61no—
433C23—62no—
434C23—61no—
435C23—61no—
436C23—62no—
437C23—61no—
438C23—62no—
439C23C2×D3×D411yescharacteristic
440Q8—24no—
441Q8—24no—
442Q8—42no—
443Q8—42no—
444Q8—64no—
445Q8—64no—
446Q8—1216no—
447Q8—122no—
448Q8—122no—
449D4—124no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—32no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—62no—
463C2×C4—64no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—61no—
467C2×C4—64no—
468C2×C4—62no—
469C2×C4—1216no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4C2×D3×D412yes—
479C2×C4C2×D3×D411yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4#100912yes—
482C8—22no—
483C8—34no—
484C8—41no—
485C8—121no—
486C2×C3#240811yescharacteristic
487C2×C3C2×#47011yescharacteristic
488C2×C3#305911yescharacteristic
489C2×C3#327012yes—
490C2×C3#327012yes—
491D3—24no—
492D3—81no—
493C22—34no—
494C22—62no—
495C22—64no—
496C22—62no—
497C22—64no—
498C22—121no—
499C22—121no—
500C22—122no—
501C22—121no—
502C22—122no—
503C22D3×#12111yescharacteristic
504C22#560912yes—
505C22#576611yescharacteristic
506C22#577111yescharacteristic
507C22D3×#13712yes—
508C4—41no—
509C4—44no—
510C4—41no—
511C4—62no—
512C4—82no—
513C4—121no—
514C4—124no—
515C4—242no—
516C4D3×#12011yescharacteristic
517C4#565412yes—
518C4#567411yescharacteristic
519C3#2427311yescharacteristic, complemented, Sylow
520C2—64no—
521C2—241no—
522C2#7689511yescharacteristic
523C2#8048912yes—
524C2#8063112yes—
525C2#8076011yescharacteristic
526C2D3×#47011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#2056D3
hypocenterC3#24273
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q16C2semidirect
3. C2×#4816C2semidirect
5. C3×#2056C2semidirect
6. #72071C2semidirect
7. #70487C2semidirect
8. #70266C2semidirect
9. #69678C2semidirect
10. #68888C2semidirect
11. #68868C2semidirect
49. #2056D3semidirect
78. C2×#216D4semidirect
417. C2×D3#470semidirect
519. C3#24273semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q162
C2×#48362
C2×#48162
#766192
C3×#20562
#720712
#704872
#702662
#696782
#688882
#688682
#242733