Finite Group Databasebeta

← #10802017

Subgroups — #10802017

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
1#83795C211yesmaximal, characteristic, complemented
2C2×#5144C211yesmaximal, characteristic, complemented
3#78839C211yesmaximal, characteristic
4#78554C211yesmaximal, characteristic, complemented
5C2×#4707C211yesmaximal, characteristic, complemented
6C3×#1765C211yesmaximal, characteristic, complemented
7#70382C211yesmaximal, characteristic, complemented
8#70374C211yesmaximal, characteristic, complemented
9#70287C211yesmaximal, characteristic, complemented
10#69889C211yesmaximal, characteristic, complemented
11#69839C211yesmaximal, characteristic, complemented
12#68940C211yesmaximal, characteristic, complemented
13#68918C211yesmaximal, characteristic, complemented
14#68756C211yesmaximal, characteristic, complemented
15#68748C211yesmaximal, characteristic, complemented
16#21549—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18#5828C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#901C2211yescharacteristic
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23#5524C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C3×#416C2211yescharacteristic
27C3×#405C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×#837C2211yescharacteristic
30#5234C2211yescharacteristic
31#5144—24no—
32C2×#825C2211yescharacteristic
33#5129C2211yescharacteristic
34#5128—22no—
35#5126C2211yescharacteristic
36#5126C2211yescharacteristic
37C2×#822C2211yescharacteristic
38C2×#822C2211yescharacteristic
39#5119C2211yescharacteristic
40#5113C2211yescharacteristic
41#5111—22no—
42#5101C2211yescharacteristic
43C2×C8×Dic3C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45#4999C2211yescharacteristic
46#4998C2211yescharacteristic
47C2×#805C2211yescharacteristic
48C2×#803C2211yescharacteristic
49#4816C412yescomplemented
50#4812C412yescomplemented
51#4782C412yescomplemented
52#4780C412yescomplemented
53#4707—24no—
54#4706—22no—
55#4688—22no—
56C3×#341C2211yescharacteristic
57C3×#339C2211yescharacteristic
58#4555C2211yescharacteristic
59#4552C2211yescharacteristic
60#4532C2211yescharacteristic
61#4523C2211yescharacteristic
62#3054—31no—
63C2×#443—31no—
64#2930—31no—
65#2930—31no—
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#1765D311yescharacteristic, complemented
77C2×C3×#41C2311yescharacteristic
78C2×#235C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3C2311yescharacteristic
81C3×#131—22no—
82C3×#131—22no—
83C2×C3×M16—21no—
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C22×C3×C8—21no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#117C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91#950—22no—
92#950—22no—
93#946C2×C412yes—
94C2×#216C2×C411yescharacteristic, complemented
95C2×#215C2×C411yescharacteristic, complemented
96#938C2×C412yes—
97#937—21no—
98#935—21no—
99C2×#214C2×C411yescharacteristic, complemented
100C2×#213C2×C411yescharacteristic, complemented
101C4×#67—22no—
102#926C2311yescharacteristic
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#901—22no—
110#901—22no—
111C2×ΓL(1,𝔽25)—22no—
112C2×ΓL(1,𝔽25)—22no—
113C2×ΓL(1,𝔽25)—22no—
114C2×ΓL(1,𝔽25)—22no—
115C2×ΓL(1,𝔽25)—21no—
116C2×ΓL(1,𝔽25)C2×C412yes—
117C2×ΓL(1,𝔽25)C2×C412yes—
118C2×ΓL(1,𝔽25)D412yes—
119C2×ΓL(1,𝔽25)D412yes—
120C2×D3×C8—21no—
121#897C2×C412yes—
122#895C2×C412yes—
123#886—21no—
124#882—21no—
125C4×D12—22no—
126C42×D3—22no—
127C42×D3—22no—
128C3×#104—22no—
129C3×#103C2×C412yescomplemented
130C3×#102C2×C412yescomplemented
131C3×#97—21no—
132C3×#97—22no—
133C3×C4×C8—21no—
134#837—22no—
135#830—22no—
136#825—22no—
137#822—22no—
138#822—22no—
139#821C2×C412yescomplemented
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#816C2×C412yescomplemented
144#815—21no—
145Dic3⋊C8—22no—
146C8×Dic3—21no—
147C8×Dic3—22no—
148C8×Dic3D414yescomplemented
149#808C2×C412yescomplemented
150#807C2×C412yescomplemented
151#805—22no—
152C4⋊(C3⋊C8)—22no—
153#803—22no—
154C2×#135—31no—
155C2×#131—31no—
156C2×#131C2×D311yescharacteristic
157C22×M16—31no—
158C22×M16—31no—
159#525—31no—
160#525—31no—
161C22×C42—31no—
162#470—32no—
163#469—32no—
164#468—32no—
165#468—32no—
166#443—64no—
167#436—62no—
168#436—62no—
169#435—62no—
170#435—62no—
171C2×#108—31no—
172C2×#108—31no—
173#427—31no—
174#424—31no—
175#424—31no—
176C2×#104—31no—
177C2×#104—31no—
178C2×#104—31no—
179C2×#104C2×D311yescharacteristic
180#416—31no—
181#416C2×D311yescharacteristic
182#408—31no—
183#408—31no—
184#407—31no—
185#407—31no—
186#405—31no—
187#405C2×D311yescharacteristic
188C4×M16—64no—
189C2×#97—31no—
190C2×#97C2×D311yescharacteristic
191C2×C4×C8—31no—
192C2×C4×C8—31no—
193#341—31no—
194#341C2×D311yescharacteristic
195#339—31no—
196#339—31no—
197#339—31no—
198#339C2×D311yescharacteristic
199C23×D3#4111yescharacteristic
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—21no—
203C3×#41—21no—
204C2×C3×Q8C22×C411yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4C22×C411yescharacteristic
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2411yescharacteristic
210C2×#67—21no—
211C2×#67—22no—
212C22×Dic3—21no—
213C22×Dic3#4111yescharacteristic
214#235—21no—
215#235—21no—
216#235—21no—
217#235—21no—
218C2×D12—22no—
219C2×D12C22×C411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—24no—
225C2×C4×D3—21no—
226C2×C4×D3—44no—
227C2×C4×D3C22×C412yes—
228C2×C4×D3C2×D412yes—
229C2×Dic6C22×C411yescharacteristic
230C3×M16—21no—
231C3×M16—21no—
232C3×M16—21no—
233C3×M16—21no—
234C3×M16—42no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—42no—
240C2×C3×C8C22×C412yes—
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D412yes—
243C2×C3×C8C2×D411yescharacteristic
244C2×C3×C8#4112yes—
245C3×C4⋊C4C22×C412yes—
246C3×C22⋊C4—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42#4112yes—
250#217—21no—
251#217—22no—
252#216—22no—
253#215—22no—
254#214—22no—
255#213—22no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259C4⋊Dic3—22no—
260C4⋊Dic3C22×C412yes—
261Dic3⋊C4—22no—
262C4×Dic3—24no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3C2×D412yes—
266#208—21no—
267#208—21no—
268#208—21no—
269#208—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8C22×C411yescharacteristic
275C2×C3⋊C8C22×C411yescharacteristic
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8#4112yes—
279ΓL(1,𝔽25)—24no—
280ΓL(1,𝔽25)—24no—
281ΓL(1,𝔽25)—24no—
282ΓL(1,𝔽25)—24no—
283ΓL(1,𝔽25)—42no—
284D3×C8—42no—
285C2×#41—31no—
286C2×#41C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—61no—
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—21no—
300C2×M16—32no—
301C2×M16—32no—
302C2×M16—32no—
303C2×M16—32no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16—62no—
307C2×M16—62no—
308C2×M16—62no—
309C2×M16—62no—
310C2×M16—61no—
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—61no—
320#123—61no—
321C22⋊Q8—32no—
322C22⋊Q8—32no—
323#121—32no—
324#121—32no—
325#120—61no—
326#120—61no—
327C4×D4—62no—
328C4×D4—62no—
329#117—31no—
330#117C22×D311yescharacteristic
331C2×C42—31no—
332C2×C42—62no—
333C2×C42—61no—
334C2×C42—62no—
335C2×C42C22×D311yescharacteristic
336#108—62no—
337#108—62no—
338#107—32no—
339#106—32no—
340C4⋊C8—62no—
341C4⋊C8—62no—
342#104—22no—
343#104—62no—
344#104—62no—
345#104—62no—
346#103—32no—
347#103C4×D312yescomplemented
348#102—32no—
349#102C4×D312yescomplemented
350C22⋊C8—62no—
351C22⋊C8—62no—
352C22⋊C8—62no—
353C22⋊C8—62no—
354#97—22no—
355#97—21no—
356#97—62no—
357#97—61no—
358C4×C8—21no—
359C4×C8—34no—
360C4×C8—62no—
361C4×C8—61no—
362C23×C3—21no—
363C23×C3C2×#4111yescharacteristic
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—42no—
368C22×D3C4×D412yes—
369C3×Q8—21no—
370C3×Q8—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—41no—
374C3×D4—41no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—42no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4C23×C411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387#67—41no—
388#67—41no—
389#67—82no—
390C2×Dic3—24no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3C4×D412yes—
397D12—21no—
398D12—21no—
399D12—82no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—48no—
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—31no—
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—31no—
432C2×D4—62no—
433C2×D4—61no—
434C2×D4—62no—
435C2×D4C2×C4×D311yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—64no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—124no—
449C22×C4C23×D311yescharacteristic
450Q16—62no—
451SD16—62no—
452SD16—62no—
453D8—62no—
454M16—21no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—42no—
459M16—64no—
460M16—64no—
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×C8C2×C4×D312yes—
484C2×C8#23512yes—
485C2×C8D3×D412yes—
486C2×C8D3×D411yescharacteristic
487C2×C8D3×D411yescharacteristic
488C4⋊C4—32no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4C2×C4×D312yes—
492C22⋊C4—21no—
493C22⋊C4—62no—
494C22⋊C4—62no—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C22⋊C4—61no—
498C42—21no—
499C42—21no—
500C42—32no—
501C42—64no—
502C42—61no—
503C42—61no—
504C42#23512yes—
505C22×C3—22no—
506C22×C3—42no—
507C22×C3—41no—
508C22×C3C2×C4×D411yescharacteristic
509C22×C3#52511yescharacteristic
510C22×C3#52911yescharacteristic
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—41no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4C2×C4×D411yescharacteristic
522C3×C4C4×#4111yescharacteristic
523C3×C4C2×#12111yescharacteristic
524C3×C4#53211yescharacteristic
525Dic3—24no—
526Dic3—22no—
527Dic3—42no—
528Dic3—41no—
529C23—21no—
530C23—32no—
531C23—61no—
532C23—61no—
533C23—61no—
534C23—122no—
535C23D3×#4111yescharacteristic
536Q8—21no—
537Q8—21no—
538Q8—61no—
539Q8—61no—
540D4—21no—
541D4—21no—
542D4—41no—
543D4—41no—
544D4—61no—
545D4—61no—
546D4—121no—
547D4—121no—
548D4—242no—
549D4—242no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—32no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—64no—
558C2×C4—61no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—61no—
567C2×C4—128no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4C22×C4×D311yescharacteristic
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×#23511yescharacteristic
573C2×C4C2×D3×D411yescharacteristic
574C2×C4C2×D3×D411yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C8—22no—
577C8—32no—
578C8—32no—
579C8—41no—
580C8—41no—
581C8—62no—
582C8—121no—
583C8—121no—
584C8#88412yes—
585C8#88412yes—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—81no—
589C2×C3C4×#12111yescharacteristic
590C2×C3#295612yes—
591D3—42no—
592D3—81no—
593C22—22no—
594C22—42no—
595C22—41no—
596C22—62no—
597C22—62no—
598C22—62no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—121no—
603C22#552811yescharacteristic
604C22C4×D3×D411yescharacteristic
605C22#554711yescharacteristic
606C4—42no—
607C4—42no—
608C4—41no—
609C4—64no—
610C4—62no—
611C4—122no—
612C4—121no—
613C4C4×#23511yescharacteristic
614C4C2×#88411yescharacteristic
615C4C4×D3×D411yescharacteristic
616C4#554811yescharacteristic
617C3#2154911yescharacteristic, complemented, Sylow
618C2—21no—
619C2—21no—
620C2—81no—
621C2—122no—
622C2—241no—
623C2C4×#88411yescharacteristic
624C2#7989812yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1765C2
socleC22×C3C2×C4×D4
prime radical#78839C2
cosocleC22×C3×C4C24
hypercenter#1765D3
hypocenterC3#21549
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
1. #83795C2semidirect
2. C2×#5144C2semidirect
4. #78554C2semidirect
5. C2×#4707C2semidirect
6. C3×#1765C2semidirect
7. #70382C2semidirect
8. #70374C2semidirect
9. #70287C2semidirect
10. #69889C2semidirect
11. #69839C2semidirect
12. #68940C2semidirect
13. #68918C2semidirect
14. #68756C2semidirect
15. #68748C2semidirect
49. #4816C4semidirect
50. #4812C4semidirect
51. #4782C4semidirect
52. #4780C4semidirect
76. #1765D3semidirect
94. C2×#216C2×C4semidirect
95. C2×#215C2×C4semidirect
99. C2×#214C2×C4semidirect
100. C2×#213C2×C4semidirect
129. C3×#103C2×C4semidirect
130. C3×#102C2×C4semidirect
139. #821C2×C4semidirect
143. #816C2×C4semidirect
148. C8×Dic3D4semidirect
149. #808C2×C4semidirect
150. #807C2×C4semidirect
347. #103C4×D3semidirect
349. #102C4×D3semidirect
617. C3#21549semidirect

Maximal subgroups

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

subgroupindex
#837952
C2×#51442
#788392
#785542
C2×#47072
C3×#17652
#703822
#703742
#702872
#698892
#698392
#689402
#689182
#687562
#687482
#215493