Finite Group Databasebeta

← #10801033

Subgroups — #10801033

Order 768.

1342 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#901C211yesmaximal, characteristic, complemented
2C3×#2929C211yesmaximal, characteristic, complemented
3C2×#5145C212yesmaximal, complemented
4#82926C211yesmaximal, characteristic
5#71967C211yesmaximal, characteristic, complemented
6#71713C211yesmaximal, characteristic, complemented
7#70545C212yesmaximal, complemented
8#70011C212yesmaximal, complemented
9#69991C211yesmaximal, characteristic, complemented
10#69984C211yesmaximal, characteristic, complemented
11#69888C212yesmaximal, complemented
12#21467—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#131C2211yescharacteristic, complemented
15C23×C3×C8C2211yescharacteristic
16C3×#525C2211yescharacteristic
17C2×#950C2212yes—
18C22×#208C2211yescharacteristic
19C2×#901—21no—
20C2×#901—21no—
21C2×#901—28no—
22C2×#901C2212yes—
23C2×#901C2212yes—
24C22×ΓL(1,𝔽25)—21no—
25C22×D3×C8—21no—
26C3×#435—21no—
27C3×#435—21no—
28C3×C8×D4—21no—
29C3×C8×D4—21no—
30C3×#424C2211yescharacteristic, complemented
31C3×#408C2211yescharacteristic, complemented
32C3×#407C2211yescharacteristic
33C3×#405C2211yescharacteristic, complemented
34C2×#837C2212yescomplemented
35C4×#217C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5145—28no—
38C2×#825C2212yes—
39C2×#825C2212yes—
40C2×#822C2212yescomplemented
41C2×#815C2212yescomplemented
42#5050C411yescharacteristic, complemented
43#5047C412yescomplemented
44#5046C411yescharacteristic, complemented
45#5040C411yescharacteristic, complemented
46#5037C412yescomplemented
47#5036C411yescharacteristic, complemented
48#5002C2212yes—
49#4558C2212yescomplemented
50#4550C2212yescomplemented
51#4532C2212yescomplemented
52C22×#131—31no—
53#3002—31no—
54C2×#444—32no—
55#2929D311yescharacteristic, complemented
56#1876—32no—
57#1871—31no—
58#1866—32no—
59#1855—31no—
60#1789—31no—
61#1765—32no—
62#1752—31no—
63C2×C3×#41C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×#67—21no—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—24no—
69C2×#235—24no—
70C2×#235C2×C412yes—
71C2×#235C2312yes—
72C22×D12C2×C411yescharacteristic
73C22×C4×D3—21no—
74C22×Dic6C2×C411yescharacteristic
75C3×#131—24no—
76C2×C3×M16—21no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C22×C3×C8—24no—
80C22×C3×C8—21no—
81C22×C3×C8D411yescharacteristic
82C22×C3×C8D411yescharacteristic
83C22×C3×C8D411yescharacteristic
84C22×C3×C8D411yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#123—21no—
88C3×C22⋊Q8C2×C411yescharacteristic
89C3×C22⋊Q8C2×C411yescharacteristic
90C3×#121C2×C411yescharacteristic
91C3×#121C2×C411yescharacteristic
92C3×#120—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×#117C2311yescharacteristic
96C2×#217—21no—
97#950—24no—
98#950—24no—
99C2×#216C2×C412yes—
100C2×#215C2×C412yes—
101C2×#214C2×C412yes—
102C2×#213C2×C412yes—
103C2×C4×Dic3C2312yes—
104C2×#208—21no—
105C2×#208—24no—
106C2×#208C2312yes—
107C22×C3⋊C8—21no—
108C22×C3⋊C8C2312yes—
109#901—28no—
110#901—28no—
111#901—24no—
112#901—24no—
113#901—24no—
114#901D418yescomplemented
115C2×ΓL(1,𝔽25)—28no—
116C2×ΓL(1,𝔽25)—22no—
117C2×ΓL(1,𝔽25)—22no—
118C2×D3×C8—28no—
119C2×D3×C8—22no—
120C2×D3×C8—22no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C22⋊C8—21no—
125C3×C22⋊C8—21no—
126C3×C22⋊C8—21no—
127C3×#97—21no—
128C3×C4×C8—21no—
129#837—24no—
130C3⋊#95—21no—
131#830C2×C411yescharacteristic
132#830C2×C412yes—
133#830C2×C411yescharacteristic
134#825—24no—
135#825—24no—
136#822—24no—
137#815—24no—
138#810C2×C412yes—
139#810C2×C412yes—
140#809C2×C412yes—
141#809C2×C412yes—
142C22×#41—31no—
143C2×#135—32no—
144C2×#131—32no—
145C2×#131—32no—
146C2×#131—68no—
147C2×#131—61no—
148C2×#131—61no—
149C2×#131C2×D311yescharacteristic, complemented
150C22×M16—31no—
151C22×M16—61no—
152C23×C8—61no—
153C23×C8C2×D311yescharacteristic
154#525C2×D311yescharacteristic
155#452—32no—
156#451—31no—
157#450—31no—
158#449—32no—
159#448—31no—
160#447—31no—
161#444—68no—
162#435—21no—
163#435—21no—
164C8×D4—21no—
165C8×D4—21no—
166C2×#108—32no—
167#424C2×D311yescharacteristic, complemented
168C2×#104—32no—
169C2×#104—32no—
170C2×#104—32no—
171#416—32no—
172#408C2×D311yescharacteristic, complemented
173#407C2×D311yescharacteristic
174C2×C22⋊C8—31no—
175#405C2×D311yescharacteristic, complemented
176C2×#97—32no—
177C4×C22⋊C4—31no—
178#341—32no—
179#339—32no—
180#336—32no—
181C24×C3#4111yescharacteristic
182C23×D3—21no—
183C3×#41—22no—
184C3×#41—22no—
185C2×C3×Q8C22×C411yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4C22×C411yescharacteristic
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4C22×C411yescharacteristic
194C22×C3×C4C22×C411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4#4111yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C2×#67—28no—
200C2×#67—22no—
201C2×#67—22no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204#235—24no—
205#235—28no—
206#235—24no—
207#235—22no—
208#235—22no—
209#235—24no—
210#235—24no—
211#235C2×D414yes—
212#235#4114yes—
213C2×D12—21no—
214C2×D12—24no—
215C2×D12C22×C412yes—
216C2×C4×D3—24no—
217C2×C4×D3—24no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×Dic6—21no—
221C2×Dic6—24no—
222C2×Dic6C22×C412yes—
223C3×M16—22no—
224C3×M16—22no—
225C3×M16—42no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—42no—
234C2×C3×C8C2×D414yes—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4C22×C411yescharacteristic
242C3×C4⋊C4C22×C411yescharacteristic
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C42—21no—
249#217—42no—
250#216—24no—
251#215—24no—
252#214—24no—
253#213—24no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256#208—24no—
257#208—22no—
258#208C2×D414yes—
259C2×C3⋊C8—24no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8C22×C412yes—
264C2×C3⋊C8C22×C412yes—
265ΓL(1,𝔽25)—28no—
266ΓL(1,𝔽25)—28no—
267D3×C8—28no—
268D3×C8—28no—
269C2×#41—32no—
270C2×#41—32no—
271C2×#41—64no—
272C2×#41—64no—
273C2×#41—61no—
274C2×#41—61no—
275C2×#41C22×D311yescharacteristic
276C22×Q8—31no—
277C22×D4—31no—
278C22×D4—61no—
279C23×C4—61no—
280C23×C4C22×D311yescharacteristic
281#135—64no—
282#135—64no—
283C2×Q16—32no—
284C2×SD16—32no—
285C2×SD16—32no—
286C2×D8—32no—
287#131—24no—
288#131—38no—
289#131—64no—
290#131—64no—
291#131—68no—
292#131—68no—
293#131—64no—
294C2×M16—21no—
295C2×M16—32no—
296C2×M16—68no—
297C2×M16—64no—
298C2×M16—61no—
299C2×M16—62no—
300C2×M16—62no—
301C2×M16C22×D311yescharacteristic
302C2×M16C22×D311yescharacteristic
303C22×C8—24no—
304C22×C8—21no—
305C22×C8—32no—
306C22×C8—68no—
307C22×C8—61no—
308C22×C8—62no—
309C22×C8—62no—
310C22×C8#6711yescharacteristic
311C22×C8#6711yescharacteristic
312C22×C8#6711yescharacteristic
313C22×C8#6711yescharacteristic
314C22×C8C22×D311yescharacteristic
315C22×C8C22×D311yescharacteristic
316#123—21no—
317C22⋊Q8C4×D311yescharacteristic
318C22⋊Q8C4×D311yescharacteristic
319#121C4×D311yescharacteristic
320#121C4×D311yescharacteristic
321#120—21no—
322C4×D4—21no—
323C4×D4—21no—
324#117C22×D311yescharacteristic
325C2×C22⋊C4—61no—
326C2×C42—32no—
327#108—64no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330#104—64no—
331#104—64no—
332#104—64no—
333#103—32no—
334#103—32no—
335#102—32no—
336#102—32no—
337C22⋊C8—21no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—31no—
342C22⋊C8—31no—
343C22⋊C8—32no—
344#97—21no—
345#97—64no—
346C4×C8—21no—
347#95—61no—
348C23×C3—22no—
349C23×C3—22no—
350C23×C3—21no—
351C23×C3—41no—
352C23×C3C4×D411yescharacteristic
353C23×C3C4×D411yescharacteristic
354C23×C3C2×#4111yescharacteristic
355C22×D3—24no—
356C22×D3—22no—
357C22×D3—42no—
358C22×D3—42no—
359C3×Q8—22no—
360C3×D4—22no—
361C3×D4—42no—
362C3×D4—42no—
363C3×D4—42no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4C4×D411yescharacteristic
377C2×C3×C4C4×D411yescharacteristic
378C2×C3×C4C23×C411yescharacteristic
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4112yes—
382#67—28no—
383#67—24no—
384#67—24no—
385C2×Dic3—24no—
386C2×Dic3—22no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390D12—24no—
391D12—22no—
392D12C4×D414yes—
393C4×D3—28no—
394C4×D3—24no—
395C4×D3—24no—
396Dic6—24no—
397Dic6—22no—
398Dic6C4×D414yes—
399C3×C8—21no—
400C3×C8—21no—
401C3×C8—41no—
402C3×C8—41no—
403C3×C8#12012yes—
404C3×C8#12012yes—
405C3⋊C8—22no—
406C3⋊C8—22no—
407C3⋊C8—24no—
408C24—61no—
409C24#23511yescharacteristic
410#41—22no—
411#41—22no—
412#41—34no—
413#41—34no—
414#41—64no—
415#41—64no—
416#41—64no—
417#41—68no—
418#41—64no—
419#41—62no—
420#41—62no—
421C2×Q8—32no—
422C2×Q8—64no—
423C2×Q8—61no—
424C2×Q8C2×C4×D311yescharacteristic
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—32no—
429C2×D4—64no—
430C2×D4—68no—
431C2×D4—62no—
432C2×D4—62no—
433C2×D4—61no—
434C2×D4C2×C4×D311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4C2×C4×D311yescharacteristic
445C22×C4C2×C4×D311yescharacteristic
446C22×C4#23511yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—64no—
451SD16—64no—
452SD16—64no—
453D8—64no—
454M16—22no—
455M16—22no—
456M16—34no—
457M16—42no—
458M16—68no—
459M16—64no—
460M16—62no—
461M16—68no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—32no—
470C2×C8—32no—
471C2×C8—42no—
472C2×C8—68no—
473C2×C8—64no—
474C2×C8—62no—
475C2×C8—62no—
476C2×C8—62no—
477C2×C8—68no—
478C2×C8D3×D414yes—
479C2×C8C2×#6711yescharacteristic
480C2×C8C2×#6711yescharacteristic
481C2×C8C2×#6711yescharacteristic
482C2×C8C2×#6711yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4C2×C4×D311yescharacteristic
486C4⋊C4C2×C4×D311yescharacteristic
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—122no—
493C42—21no—
494C42—62no—
495C42—62no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—44no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3#44412yescomplemented
508C22×C3C2×C4×D411yescharacteristic
509C22×C3#52512yes—
510C2×D3—24no—
511C2×D3—48no—
512C2×D3—42no—
513C2×D3—42no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4C2×C4×D412yes—
520C3×C4C2×#12011yescharacteristic
521C3×C4#52511yescharacteristic
522Dic3—24no—
523Dic3—82no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—41no—
528C23—64no—
529C23—62no—
530C23—122no—
531C23—122no—
532C23C4×#6711yescharacteristic
533C23C4×#6711yescharacteristic
534C23C2×#23511yescharacteristic
535Q8—22no—
536Q8—34no—
537Q8—64no—
538Q8—62no—
539D4—22no—
540D4—34no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—64no—
545D4—68no—
546D4—64no—
547D4—64no—
548D4—62no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—68no—
562C2×C4—64no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4C4×#6711yescharacteristic
570C2×C4C4×#6711yescharacteristic
571C2×C4C22×C4×D311yescharacteristic
572C2×C4C2×D3×D412yes—
573C2×C4D3×#4112yes—
574C2×C4C22×#6711yescharacteristic
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—64no—
580C8—62no—
581C8—62no—
582C8#93712yes—
583C8#93712yes—
584C2×C3—22no—
585C2×C3—22no—
586C2×C3—41no—
587C2×C3—81no—
588C2×C3C4×#12011yescharacteristic
589C2×C3C2×#44411yescharacteristic
590C2×C3#295511yescharacteristic
591D3—44no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—44no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—64no—
604C22—128no—
605C22—122no—
606C22—122no—
607C22#514512yescomplemented
608C22#554712yes—
609C22C2×C4×#6711yescharacteristic
610C4—21no—
611C4—21no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—64no—
616C4—242no—
617C4C4×D3×D412yes—
618C4C2×#93711yescharacteristic
619C4#582811yescharacteristic
620C3#2146711yescharacteristic, complemented, Sylow
621C2—22no—
622C2—22no—
623C2—41no—
624C2—81no—
625C2—124no—
626C2C4×#93711yescharacteristic
627C2C2×#514511yescharacteristic
628C2#8379211yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2929C2
socleC22×C3C2×C4×D4
prime radical#82926C2
cosocleC22×C3×C4C24
hypercenter#2929D3
hypocenterC3#21467
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. C22×#901C2semidirect
2. C3×#2929C2semidirect
3. C2×#5145C2semidirect
5. #71967C2semidirect
6. #71713C2semidirect
7. #70545C2semidirect
8. #70011C2semidirect
9. #69991C2semidirect
10. #69984C2semidirect
11. #69888C2semidirect
14. C2×C3×#131C22semidirect
30. C3×#424C22semidirect
31. C3×#408C22semidirect
33. C3×#405C22semidirect
34. C2×#837C22semidirect
40. C2×#822C22semidirect
41. C2×#815C22semidirect
42. #5050C4semidirect
43. #5047C4semidirect
44. #5046C4semidirect
45. #5040C4semidirect
46. #5037C4semidirect
47. #5036C4semidirect
49. #4558C22semidirect
50. #4550C22semidirect
51. #4532C22semidirect
55. #2929D3semidirect
114. #901D4semidirect
149. C2×#131C2×D3semidirect
167. #424C2×D3semidirect
172. #408C2×D3semidirect
175. #405C2×D3semidirect
507. C22×C3#444semidirect
607. C22#5145semidirect
620. C3#21467semidirect

Maximal subgroups

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

subgroupindex
C22×#9012
C3×#29292
C2×#51452
#829262
#719672
#717132
#705452
#700112
#699912
#699842
#698882
#214673