Finite Group Databasebeta

← #10802015

Subgroups — #10802015

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2930C211yesmaximal, characteristic, complemented
2C2×#5144C211yesmaximal, characteristic, complemented
3#83756C211yesmaximal, characteristic, complemented
4C2×Dic3×M16C211yesmaximal, characteristic, complemented
5#82923C211yesmaximal, characteristic
6#72009C211yesmaximal, characteristic, complemented
7#71711C211yesmaximal, characteristic, complemented
8#70538C211yesmaximal, characteristic, complemented
9#70535C211yesmaximal, characteristic, complemented
10#70000C211yesmaximal, characteristic, complemented
11#69997C211yesmaximal, characteristic, complemented
12#69983C211yesmaximal, characteristic, complemented
13#69965C211yesmaximal, characteristic, complemented
14#69889C211yesmaximal, characteristic, complemented
15#69888C211yesmaximal, characteristic, complemented
16#21549—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18C22×C3×M16C2211yescharacteristic
19C3×#525C2211yescharacteristic
20C2×#950C2211yescharacteristic
21#5791C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C2×#901C2211yescharacteristic
25C3×#436—22no—
26C3×#435—22no—
27C3×#424C2211yescharacteristic
28C3×#408C2211yescharacteristic
29C3×#407C2211yescharacteristic
30C3×#405C2211yescharacteristic
31C2×#837C2211yescharacteristic
32C2×#837C2211yescharacteristic
33#5233C2211yescharacteristic
34#5207C2211yescharacteristic
35#5144—24no—
36C2×#825C2211yescharacteristic
37C2×#825C2211yescharacteristic
38#5128—22no—
39#5127—22no—
40C2×#822C2211yescharacteristic
41C2×#822C2211yescharacteristic
42Dic3×M16—24no—
43#5110C2211yescharacteristic
44#5102C2211yescharacteristic
45#5101C2211yescharacteristic
46C2×#815C2211yescharacteristic
47C2×#815C2211yescharacteristic
48C2×C8×Dic3C2211yescharacteristic
49C2×C8×Dic3C2211yescharacteristic
50#5052C412yescomplemented
51#5049C412yescomplemented
52#5042C412yescomplemented
53#5039C412yescomplemented
54#5002C2211yescharacteristic
55#4998C2211yescharacteristic
56#4558C2211yescharacteristic
57#4558C2211yescharacteristic
58#4555C2211yescharacteristic
59#4555C2211yescharacteristic
60#4532C2211yescharacteristic
61#4532C2211yescharacteristic
62#3054—31no—
63C2×#443—31no—
64#2930—31no—
65#2930D311yescharacteristic, complemented
66C2×C4×M16—31no—
67#1975—31no—
68#1946—31no—
69#1937—31no—
70#1937—31no—
71#1896—31no—
72#1896—31no—
73#1811—31no—
74#1772—31no—
75#1765—31no—
76#1765—31no—
77C2×C3×#41C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×Dic3—21no—
80C2×#235C2311yescharacteristic
81C3×#131—22no—
82C3×#131—22no—
83C2×C3×M16—22no—
84C2×C3×M16—21no—
85C2×C3×M16—22no—
86C2×C3×M16D412yes—
87C2×C3×M16D412yes—
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8—21no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×#123—21no—
94C3×C22⋊Q8C2×C412yes—
95C3×#121C2×C412yes—
96C3×#120—21no—
97C3×C4×D4—22no—
98C3×#117C2311yescharacteristic
99#953—21no—
100#950—22no—
101#950—22no—
102C2×#216C2×C411yescharacteristic, complemented
103C2×#215C2×C411yescharacteristic, complemented
104C2×#214C2×C411yescharacteristic, complemented
105C2×#213C2×C411yescharacteristic, complemented
106#930C2×C412yes—
107#929—21no—
108C4×#67—22no—
109#926C2311yescharacteristic
110#924C2×C412yes—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—22no—
113C2×C4×Dic3—22no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C2×#208—22no—
117C2×#208—22no—
118C2×#208C2×C412yes—
119C2×#208C2×C412yes—
120C2×#208C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#901—22no—
125#901—22no—
126C2×ΓL(1,𝔽25)—21no—
127C2×D3×C8—21no—
128C3×C4⋊C8—22no—
129C3×C22⋊C8—22no—
130C3×C22⋊C8—22no—
131C3×#97—21no—
132C3×C4×C8—21no—
133#837—22no—
134#837—22no—
135#835C2×C412yescomplemented
136#832C2×C412yescomplemented
137#830—22no—
138#825—22no—
139#825—22no—
140#822—22no—
141#822—22no—
142(C2×D3)⋊C8—22no—
143#815—21no—
144#815—22no—
145#815—22no—
146Dic3⋊C8—22no—
147C8×Dic3—21no—
148C8×Dic3—22no—
149C8×Dic3D414yescomplemented
150#810C2×C412yescomplemented
151#809C2×C412yescomplemented
152#808C2×C412yescomplemented
153#807C2×C412yescomplemented
154C2×#135—31no—
155C2×#131—31no—
156C2×#131C2×D311yescharacteristic
157C22×M16—31no—
158C22×M16C2×D311yescharacteristic
159#525—31no—
160#525C2×D311yescharacteristic
161C22×C42—31no—
162#470—32no—
163#469—32no—
164#468—32no—
165#468—32no—
166#443—64no—
167#436—22no—
168#436—62no—
169#435—22no—
170#435—62no—
171C2×#108—31no—
172C2×#108—31no—
173#427—31no—
174#424—31no—
175#424C2×D311yescharacteristic
176C2×#104—31no—
177C2×#104—31no—
178C2×#104—31no—
179C2×#104—31no—
180#416—31no—
181#416—31no—
182#408—31no—
183#408C2×D311yescharacteristic
184#407—31no—
185#407C2×D311yescharacteristic
186#405—31no—
187#405C2×D311yescharacteristic
188C4×M16—64no—
189C2×#97—31no—
190C2×#97—31no—
191C2×C4×C8—31no—
192C2×C4×C8—31no—
193#341—31no—
194#341—31no—
195#339—31no—
196#339—31no—
197#339—31no—
198#339—31no—
199C24×C3#4111yescharacteristic
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—21no—
203C3×#41—21no—
204C2×C3×Q8C22×C411yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4C22×C411yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C22×C412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4#4111yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C2×#67—21no—
218C2×#67—22no—
219C22×Dic3—21no—
220C22×Dic3—24no—
221C22×Dic3—21no—
222C22×Dic3—44no—
223#235—21no—
224#235—21no—
225#235—21no—
226#235—21no—
227C2×D12C22×C411yescharacteristic
228C2×C4×D3—21no—
229C2×Dic6C22×C411yescharacteristic
230C3×M16—24no—
231C3×M16—24no—
232C3×M16—21no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—42no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—42no—
242C2×C3×C8C2×D412yes—
243C2×C3×C8C2×D412yes—
244C2×C3×C8C2×D411yescharacteristic
245C2×C3×C8C2×D411yescharacteristic
246C2×C3×C8#4112yes—
247C3×C4⋊C4—22no—
248C3×C4⋊C4C22×C412yes—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—22no—
252C3×C42—21no—
253#217—22no—
254#217—21no—
255#216—22no—
256#215—22no—
257#214—22no—
258#213—22no—
259(C2×D3)⋊C4—22no—
260C4⋊Dic3C22×C412yes—
261Dic3⋊C4—22no—
262C4×Dic3—24no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3C2×D412yes—
267C4×Dic3#4112yes—
268#208—24no—
269#208—24no—
270#208—21no—
271#208—21no—
272#208—21no—
273#208—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C22×C412yes—
279C2×C3⋊C8C22×C411yescharacteristic
280C2×C3⋊C8C22×C411yescharacteristic
281C2×C3⋊C8C2×D412yes—
282C2×C3⋊C8#4112yes—
283ΓL(1,𝔽25)—42no—
284D3×C8—42no—
285C2×#41—31no—
286C2×#41C22×D311yescharacteristic
287C23×C4—61no—
288C23×C4C22×D311yescharacteristic
289#135—62no—
290#135—62no—
291C2×Q16—31no—
292C2×SD16—31no—
293C2×SD16—31no—
294C2×D8—31no—
295#131—22no—
296#131—22no—
297#131—62no—
298#131—62no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16—21no—
302C2×M16—32no—
303C2×M16—32no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16—62no—
307C2×M16—62no—
308C2×M16—61no—
309C2×M16#6712yes—
310C2×M16#6712yes—
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—21no—
314C22×C8—31no—
315C22×C8—31no—
316C22×C8—61no—
317C22×C8C22×D311yescharacteristic
318C22×C8C22×D311yescharacteristic
319#123—21no—
320#123—61no—
321C22⋊Q8—32no—
322C22⋊Q8C4×D312yes—
323#121—32no—
324#121C4×D312yes—
325#120—21no—
326#120—61no—
327C4×D4—22no—
328C4×D4—62no—
329#117—31no—
330#117C22×D311yescharacteristic
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—62no—
334C2×C42—62no—
335C2×C42—61no—
336#108—62no—
337#108—62no—
338#107—32no—
339#106—32no—
340C4⋊C8—22no—
341C4⋊C8—62no—
342#104—62no—
343#104—62no—
344#104—62no—
345#104—62no—
346#103—32no—
347#103—32no—
348#102—32no—
349#102—32no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—62no—
353C22⋊C8—62no—
354#97—21no—
355#97—62no—
356#97—62no—
357#97—61no—
358C4×C8—21no—
359C4×C8—34no—
360C4×C8—62no—
361C4×C8—61no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—42no—
366C23×C3C4×D412yes—
367C23×C3C2×#4111yescharacteristic
368C22×D3—21no—
369C3×Q8—21no—
370C3×Q8—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—82no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4C4×D412yes—
387C2×C3×C4C23×C411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393#67—41no—
394#67—41no—
395#67—82no—
396C2×Dic3—24no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—48no—
402C2×Dic3—42no—
403D12—21no—
404D12—21no—
405C4×D3—41no—
406C4×D3—41no—
407Dic6—21no—
408Dic6—21no—
409C3×C8—22no—
410C3×C8—41no—
411C3×C8—41no—
412C3×C8#12112yes—
413C3×C8#12112yes—
414C3⋊C8—22no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C3⋊C8C4×D412yes—
418C3⋊C8C4×D412yes—
419C24#23511yescharacteristic
420#41—21no—
421#41—21no—
422#41—21no—
423#41—21no—
424#41—61no—
425#41—61no—
426#41—61no—
427#41—61no—
428C2×Q8—31no—
429C2×Q8C2×C4×D311yescharacteristic
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—31no—
433C2×D4—61no—
434C2×D4—62no—
435C2×D4C2×C4×D311yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—61no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—124no—
446C22×C4C2×C4×D312yes—
447C22×C4#23511yescharacteristic
448C22×C4C2×#6712yes—
449C22×C4C23×D311yescharacteristic
450Q16—62no—
451SD16—62no—
452SD16—62no—
453D8—62no—
454M16—24no—
455M16—24no—
456M16—21no—
457M16—21no—
458M16—21no—
459M16—21no—
460M16—42no—
461M16—64no—
462M16—64no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16—61no—
467M16—122no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—31no—
473C2×C8—31no—
474C2×C8—32no—
475C2×C8—32no—
476C2×C8—32no—
477C2×C8—42no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—122no—
483C2×C8D3×D412yes—
484C2×C8#23712yes—
485C2×C8C2×#6712yes—
486C2×C8C2×#6711yescharacteristic
487C2×C8C2×#6711yescharacteristic
488C4⋊C4—22no—
489C4⋊C4—32no—
490C4⋊C4—62no—
491C4⋊C4C2×C4×D312yes—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—62no—
496C22⋊C4—61no—
497C22⋊C4—62no—
498C42—21no—
499C42—32no—
500C42—32no—
501C42—64no—
502C42—61no—
503C42—61no—
504C42—61no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—41no—
513C22×C3C2×C4×D411yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3#52911yescharacteristic
516C2×D3—42no—
517C2×D3—41no—
518C3×C4—22no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4C2×C4×D411yescharacteristic
522C3×C4C4×#4111yescharacteristic
523C3×C4C2×#12111yescharacteristic
524C3×C4#53211yescharacteristic
525Dic3—24no—
526Dic3—42no—
527Dic3—42no—
528Dic3—41no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—42no—
533C23—61no—
534C23C4×#6712yes—
535C23C2×#23511yescharacteristic
536Q8—21no—
537Q8—21no—
538Q8—61no—
539Q8—61no—
540D4—21no—
541D4—21no—
542D4—41no—
543D4—41no—
544D4—61no—
545D4—61no—
546D4—82no—
547D4—121no—
548D4—121no—
549D4—242no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—128no—
566C2×C4—122no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4C4×#6712yes—
570C2×C4C22×C4×D311yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×#23711yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4C22×#6711yescharacteristic
576C8—22no—
577C8—32no—
578C8—32no—
579C8—41no—
580C8—41no—
581C8—62no—
582C8—121no—
583C8—121no—
584C8#93912yes—
585C8#93912yes—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—42no—
589C2×C3—81no—
590C2×C3C4×#12111yescharacteristic
591C2×C3#295612yes—
592D3—81no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—41no—
601C22—122no—
602C22—121no—
603C22#553811yescharacteristic
604C22#554711yescharacteristic
605C22C2×C4×#6711yescharacteristic
606C4—22no—
607C4—42no—
608C4—41no—
609C4—64no—
610C4—122no—
611C4—122no—
612C4—121no—
613C4C4×#23711yescharacteristic
614C4C4×D3×D411yescharacteristic
615C4C2×#93911yescharacteristic
616C4#583211yescharacteristic
617C3#2154911yescharacteristic, complemented, Sylow
618C2—21no—
619C2—21no—
620C2—42no—
621C2—81no—
622C2—241no—
623C2C4×#93911yescharacteristic
624C2#8381112yes—
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2930C2
socleC22×C3C2×C4×D4
prime radical#82923C2
cosocleC22×C3×C4C24
hypercenter#2930D3
hypocenterC3#21549
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×M16 ◁ #2930

Complements

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

NH ≅ G/Ntype
1. C3×#2930C2semidirect
2. C2×#5144C2semidirect
3. #83756C2semidirect
4. C2×Dic3×M16C2semidirect
6. #72009C2semidirect
7. #71711C2semidirect
8. #70538C2semidirect
9. #70535C2semidirect
10. #70000C2semidirect
11. #69997C2semidirect
12. #69983C2semidirect
13. #69965C2semidirect
14. #69889C2semidirect
15. #69888C2semidirect
50. #5052C4semidirect
51. #5049C4semidirect
52. #5042C4semidirect
53. #5039C4semidirect
65. #2930D3semidirect
102. C2×#216C2×C4semidirect
103. C2×#215C2×C4semidirect
104. C2×#214C2×C4semidirect
105. C2×#213C2×C4semidirect
135. #835C2×C4semidirect
136. #832C2×C4semidirect
149. C8×Dic3D4semidirect
150. #810C2×C4semidirect
151. #809C2×C4semidirect
152. #808C2×C4semidirect
153. #807C2×C4semidirect
617. C3#21549semidirect

Maximal subgroups

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

subgroupindex
C3×#29302
C2×#51442
#837562
C2×Dic3×M162
#829232
#720092
#717112
#705382
#705352
#700002
#699972
#699832
#699652
#698892
#698882
#215493