Finite Group Databasebeta

← #10847374

Subgroups — #10847374

Order 768.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×#4864C212yesmaximal, complemented
3#76287C211yesmaximal, characteristic, complemented
4C3×#2058C211yesmaximal, characteristic, complemented
5#71794C211yesmaximal, characteristic, complemented
6#70584C212yesmaximal, complemented
7#69627C211yesmaximal, characteristic, complemented
8#68896C212yesmaximal, complemented
9#68883C212yesmaximal, complemented
10#68676C211yesmaximal, characteristic, complemented
11D3×#336C211yesmaximal, characteristic, complemented
12#25495—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#947—21no—
16C22×#204C2212yes—
17C22×D3×C8C2212yes—
18#5725—28no—
19C2×#897C2211yescharacteristic
20C2×#895—21no—
21C2×#895C2212yes—
22C2×#893—21no—
23C2×#893C2212yes—
24C2×#874C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×C3×#106C2212yes—
27C3×#389C2211yescharacteristic
28C2×#821C2212yes—
29#4989—21no—
30#4987C2211yescharacteristic
31#4980—21no—
32#4973C2211yescharacteristic
33#4972C2211yescharacteristic
34C2×#810C2212yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37#4929C2211yescharacteristic
38#4864—28no—
39#4664—21no—
40#4655—21no—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43#4550C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#465—32no—
48#2855—31no—
49#2058D311yescharacteristic, complemented
50#2041—31no—
51#2024—31no—
52#2014—31no—
53#1893—32no—
54#1877—32no—
55#1864—32no—
56C2×#336—31no—
57C22×C3×Q8—21no—
58C2×#239—28no—
59C2×#239—44no—
60C2×#239—44no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×C4×D3—22no—
66C22×C4×D3D411yescharacteristic
67C22×C4×D3D411yescharacteristic
68C22×C4×D3C2311yescharacteristic
69C22×Dic6C2311yescharacteristic
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C42C2311yescharacteristic
78#947—28no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90C2×#204—24no—
91C2×#204—24no—
92C2×#204—42no—
93C2×D3×C8—24no—
94C2×D3×C8—24no—
95C2×D3×C8—42no—
96#897—24no—
97#895—28no—
98#895—24no—
99#895D418yes—
100#893—28no—
101#893—24no—
102#893—24no—
103#874—24no—
104C42×D3—24no—
105C3×#106—24no—
106C3×#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#821—24no—
114#810—24no—
115#809—24no—
116#809—42no—
117C22×#41—61no—
118C22×SD16—32no—
119C23×C8—32no—
120#532—68no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#127—31no—
123C2×#124—61no—
124C2×C22⋊Q8—31no—
125C2×#121—32no—
126C2×#121—61no—
127C2×C4×D4—32no—
128C2×C4×D4—61no—
129C22×C42—31no—
130#465—68no—
131C2×#106C2×D312yes—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136#391—31no—
137#391—31no—
138#389—31no—
139#389C2×D311yescharacteristic
140#387—61no—
141#387—61no—
142#385—61no—
143#381—61no—
144#380—31no—
145#340—32no—
146#340C2×D312yes—
147#336—32no—
148#336—31no—
149#336C2×D311yescharacteristic
150C23×D3—22no—
151C23×D3C2×D411yescharacteristic
152C2×C3×Q8—24no—
153C2×C3×Q8—42no—
154C2×C3×Q8—42no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166#239—432no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—24no—
172C2×D12—24no—
173C2×D12—28no—
174C2×D12—22no—
175C2×D12—42no—
176C2×D12—42no—
177C2×D12—44no—
178C2×D12—44no—
179C2×D12—42no—
180C2×D12—42no—
181C2×D12C2×D414yes—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—28no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—41no—
191C2×C4×D3—44no—
192C2×C4×D3—44no—
193C2×C4×D3SD1614yescomplemented
194C2×C4×D3C2×D414yes—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8#4114yescomplemented
202C3×C4⋊C4—24no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4C2×D414yes—
209C3×C42—22no—
210(C2×D3)⋊C4—28no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—42no—
213(C2×D3)⋊C4—42no—
214C4⋊Dic3—22no—
215C4⋊Dic3—42no—
216Dic3⋊C4—42no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223#204—48no—
224D3×C8—48no—
225C2×#41—68no—
226C2×#41—124no—
227C2×#41—124no—
228C22×Q8—21no—
229C22×Q8—31no—
230C22×D4—32no—
231C22×D4—62no—
232C22×D4—61no—
233C22×D4—61no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—31no—
237C23×C4—62no—
238C2×SD16—64no—
239C2×SD16—64no—
240C2×SD16—122no—
241C22×C8—32no—
242C22×C8—64no—
243C22×C8—64no—
244C22×C8—122no—
245C22×C8C22×D312yes—
246C4⋊Q8—24no—
247#127—64no—
248#124—68no—
249C22⋊Q8—64no—
250#121—38no—
251#121—64no—
252#121—68no—
253C4×D4—64no—
254C4×D4—68no—
255C4×D4—64no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C4⋊C4C22×D311yescharacteristic
263C2×C4⋊C4C22×D312yes—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—61no—
268C2×C42—31no—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—64no—
272C2×C42C22×D311yescharacteristic
273#106—24no—
274#103—64no—
275#102—64no—
276#102—64no—
277#102—122no—
278#95—21no—
279#95—61no—
280#95—61no—
281#95—61no—
282#95—61no—
283#95—61no—
284#95—61no—
285C23×C3C22×D411yescharacteristic
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—21no—
290C22×D3—24no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3#12012yes—
298C22×D3C2×SD1612yes—
299C22×D3AGL(1,ℤ/8)12yes—
300C3×Q8—48no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—24no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4C2×SD1612yes—
315C2×C3×C4#13512yes—
316C2×C3×C4C22×D412yes—
317C2×C3×C4C2×#4112yes—
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3#12012yes—
334C2×Dic3C2×SD1612yes—
335D12—24no—
336D12—24no—
337D12—48no—
338D12—48no—
339D12—416no—
340D12—42no—
341D12—42no—
342C4×D3—28no—
343C4×D3—24no—
344C4×D3—24no—
345C4×D3—416no—
346C4×D3—44no—
347Dic6—42no—
348Dic6—42no—
349C3×C8—24no—
350C3⋊C8—42no—
351C24—31no—
352C24—62no—
353#41—1232no—
354C2×Q8—24no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—61no—
360C2×Q8—61no—
361C2×D4—34no—
362C2×D4—68no—
363C2×D4—64no—
364C2×D4—64no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—122no—
371C2×D4—122no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—122no—
375C2×D4—122no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—34no—
382C22×C4—34no—
383C22×C4—31no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—68no—
388C22×C4—62no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—121no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4C23×D311yescharacteristic
400SD16—128no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—62no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—128no—
407C2×C8#23914yescomplemented
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—42no—
414C4⋊C4—62no—
415C4⋊C4—122no—
416C4⋊C4—122no—
417C4⋊C4D3×D414yes—
418C22⋊C4—68no—
419C22⋊C4—68no—
420C22⋊C4—122no—
421C22⋊C4—122no—
422C42—22no—
423C42—62no—
424C42—62no—
425C42—62no—
426C22×C3C2×#12011yescharacteristic
427C22×C3#53211yescharacteristic
428C22×C3#53711yescharacteristic
429C22×C3C22×SD1611yescharacteristic
430C22×C3C2×#13511yescharacteristic
431C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
432C22×C3#57511yescharacteristic
433C2×D3—22no—
434C2×D3—22no—
435C2×D3—22no—
436C2×D3—22no—
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—44no—
440C2×D3—44no—
441C2×D3—44no—
442C2×D3—82no—
443C2×D3—82no—
444C2×D3—82no—
445C2×D3—82no—
446C2×D3#45014yes—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—44no—
450C3×C4—81no—
451C3×C4—81no—
452C3×C4#54612yes—
453C3×C4#54712yes—
454Dic3—22no—
455Dic3—22no—
456Dic3—41no—
457Dic3—81no—
458Dic3—81no—
459C23—32no—
460C23—32no—
461C23—32no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—64no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23—122no—
471C23—122no—
472C23—122no—
473C23C2×D3×D411yescharacteristic
474Q8—48no—
475Q8—122no—
476Q8—122no—
477D4—64no—
478D4—64no—
479D4—128no—
480D4—128no—
481D4—1216no—
482D4—122no—
483D4—122no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—32no—
490C2×C4—32no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—68no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—124no—
510C2×C4—1216no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4D3×SD1612yes—
518C2×C4#91612yes—
519C2×C4C2×D3×D412yes—
520C2×C4C2×#23912yes—
521C2×C4#100711yescharacteristic
522C2×C4#100911yescharacteristic
523C8—24no—
524C8—122no—
525C2×C3#243011yescharacteristic
526C2×C3C2×#45011yescharacteristic
527C2×C3#300211yescharacteristic
528C2×C3#316711yescharacteristic
529C2×C3#318011yescharacteristic
530C2×C3#324611yescharacteristic
531C2×C3#324711yescharacteristic
532D3—24no—
533D3—84no—
534C22—34no—
535C22—62no—
536C22—62no—
537C22—62no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—124no—
542C22—124no—
543C22—124no—
544C22—242no—
545C22—242no—
546C22—242no—
547C22—242no—
548C22D3×#12011yescharacteristic
549C22#562911yescharacteristic
550C22#572511yescharacteristic
551C22C2×D3×SD1611yescharacteristic
552C22C2×#91611yescharacteristic
553C22D3×AGL(1,ℤ/8)11yescharacteristic
554C22#577711yescharacteristic
555C4—22no—
556C4—22no—
557C4—44no—
558C4—62no—
559C4—62no—
560C4—81no—
561C4—81no—
562C4—121no—
563C4—241no—
564C4—241no—
565C4#563012yes—
566C4#572612yes—
567C3#2549511yescharacteristic, complemented, Sylow
568C2—64no—
569C2—244no—
570C2#7725011yescharacteristic
571C2#8003811yescharacteristic
572C2D3×#45011yescharacteristic
573C2#8100211yescharacteristic
574C2#8107111yescharacteristic
575C2#8188611yescharacteristic
576C2#8196211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2058C2
socleC23×C3C22×D4
prime radicalC2×#5725C2
hypercenter#2058D3
hypocenterC3#25495
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
2. C2×#4864C2semidirect
3. #76287C2semidirect
4. C3×#2058C2semidirect
5. #71794C2semidirect
6. #70584C2semidirect
7. #69627C2semidirect
8. #68896C2semidirect
9. #68883C2semidirect
10. #68676C2semidirect
11. D3×#336C2semidirect
49. #2058D3semidirect
193. C2×C4×D3SD16semidirect
201. C2×C3×C8#41semidirect
407. C2×C8#239semidirect
567. C3#25495semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#48642
#762872
C3×#20582
#717942
#705842
#696272
#688962
#688832
#686762
D3×#3362
#254953