Finite Group Databasebeta

← #10819944

Subgroups — #10819944

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2949C211yesmaximal, characteristic, complemented
2C2×#5176C211yesmaximal, characteristic
3#82548C211yesmaximal, characteristic, complemented
4#82538C211yesmaximal, characteristic, complemented
5#79167C211yesmaximal, characteristic, complemented
6#72052C211yesmaximal, characteristic, complemented
7#71661C211yesmaximal, characteristic, complemented
8#71650C211yesmaximal, characteristic, complemented
9#70660C211yesmaximal, characteristic, complemented
10#70343C211yesmaximal, characteristic, complemented
11#70335C211yesmaximal, characteristic, complemented
12#70323C211yesmaximal, characteristic, complemented
13#70316C211yesmaximal, characteristic, complemented
14#69903C211yesmaximal, characteristic, complemented
15#69890C211yesmaximal, characteristic, complemented
16#23202—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C3×#519C2211yescharacteristic
19C3×#518C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#940C2211yescharacteristic
23C22×#213C2211yescharacteristic
24C22×D24C2211yescharacteristic
25#5521C2211yescharacteristic
26#5521C2211yescharacteristic
27C3×#440—22no—
28C3×C4×SD16—22no—
29C3×#427C2211yescharacteristic
30C3×#419C2211yescharacteristic
31C3×#418C2211yescharacteristic
32C3×#405C2211yescharacteristic
33C2×#836C2211yescharacteristic
34#5225C2211yescharacteristic
35C2×#833C2211yescharacteristic
36#5215C2211yescharacteristic
37#5176C418yescomplemented
38C2×#823C2211yescharacteristic
39C2×#823C2211yescharacteristic
40C2×#822C2211yescharacteristic
41C2×#819C2211yescharacteristic
42#5101C2211yescharacteristic
43C2×#815C2211yescharacteristic
44#5026—22no—
45C4×#215—22no—
46#5014—22no—
47C4×#213—22no—
48#5001C2211yescharacteristic
49#5001C2211yescharacteristic
50#5001C2211yescharacteristic
51#5001C2211yescharacteristic
52#4998C2211yescharacteristic
53#4997C2211yescharacteristic
54#4997C2211yescharacteristic
55#4872—22no—
56#4869—22no—
57#4555C2211yescharacteristic
58#4532C2211yescharacteristic
59#4532C2211yescharacteristic
60#4532C2211yescharacteristic
61C2×#496—31no—
62#2949—31no—
63#2949D311yescharacteristic, complemented
64#2948—31no—
65#2948—31no—
66#1978—31no—
67#1978—31no—
68#1977—31no—
69#1977—31no—
70#1888—31no—
71#1887—31no—
72#1887—31no—
73#1885—31no—
74#1773—31no—
75#1773—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C22×#67—41no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C2×C3×SD16—21no—
82C2×C3×SD16—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16C2×C414yescomplemented
85C2×C3×M16C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×#117—41no—
92C3×#117C2311yescharacteristic
93C3×#117C2311yescharacteristic
94C3×#117C2311yescharacteristic
95C2×C3×C22⋊C4—41no—
96#947—41no—
97#947C2×C414yescomplemented
98C2×#215—21no—
99C2×#215—21no—
100C2×#215—22no—
101C2×#215C2×C414yescomplemented
102#940—41no—
103#940C2×C414yescomplemented
104C2×#213—21no—
105C2×#213—21no—
106C2×#213—22no—
107C2×#213C2×C414yescomplemented
108C2×(C2×D3)⋊C4—41no—
109#926C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×D24—21no—
116C2×D24—21no—
117C2×D24—22no—
118C2×D24C2×C414yescomplemented
119#893—22no—
120#893—22no—
121D3×C22⋊C4—41no—
122D3×C22⋊C4—41no—
123C4×D12—22no—
124C4×D12—22no—
125C3×#107D412yescomplemented
126C3×#106D412yescomplemented
127C3×#103—22no—
128C3×#102—22no—
129C3×#97—21no—
130C3×C4×C8—21no—
131#836—41no—
132#835—22no—
133#833—41no—
134#832—22no—
135#823—41no—
136#823—41no—
137#822—22no—
138#819—22no—
139#815—21no—
140#815C2×C414yescomplemented
141C8×Dic3—21no—
142#809—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#808D412yescomplemented
147#807D412yescomplemented
148#803—21no—
149#803—21no—
150C4×C3⋊C8—21no—
151C4×C3⋊C8—21no—
152C22×SD16—31no—
153C22×SD16C2×D311yescharacteristic
154C22×D8—31no—
155C22×D8—31no—
156C2×#127—31no—
157C2×#124—31no—
158#519C2×D311yescharacteristic
159#518—31no—
160#518—31no—
161#518C2×D311yescharacteristic
162#496—38no—
163#442—62no—
164#442—62no—
165#440—22no—
166#440—62no—
167C4×SD16—22no—
168C4×SD16—62no—
169C4×D8—62no—
170C4×D8—62no—
171C2×#108—31no—
172C2×#108—31no—
173#427—31no—
174#427C2×D311yescharacteristic
175#419—31no—
176#419C2×D311yescharacteristic
177#418—31no—
178#418—31no—
179#418—31no—
180#418—31no—
181#418—31no—
182#418C2×D311yescharacteristic
183C2×#101—31no—
184C2×#100—31no—
185C2×#100—31no—
186C2×#100—31no—
187#405—31no—
188#405—31no—
189#405—31no—
190#405C2×D311yescharacteristic
191C2×#97—31no—
192#339—31no—
193#339—31no—
194#339—31no—
195#339—31no—
196C24×C3—21no—
197C23×D3—21no—
198C23×D3—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C22×C412yes—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4C22×C412yes—
211C22×C3×C4—21no—
212C22×C3×C4—42no—
213C22×C3×C4C2411yescharacteristic
214C2×#67—44no—
215C2×#67—84no—
216C22×Dic3—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12C22×C412yes—
228C2×D12C22×C412yes—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C3×SD16—24no—
232C3×SD16—24no—
233C3×M16—21no—
234C3×M16—21no—
235C2×C3×C8—21no—
236C2×C3×C8C22×C412yes—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8#4112yes—
240C3×C4⋊C4—42no—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4C2×D412yes—
243C3×C4⋊C4C2×D412yes—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—42no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—42no—
252#217—21no—
253#215—24no—
254#215—24no—
255#213—24no—
256#213—24no—
257(C2×D3)⋊C4—44no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260C4⋊Dic3C2×D412yes—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3C22×C412yes—
264#208—21no—
265#208—21no—
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⋊C8C22×C412yes—
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8#4112yes—
277D24—24no—
278D24—24no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—121no—
283C22×D4C22×D311yescharacteristic
284C2×SD16—22no—
285C2×SD16—21no—
286C2×SD16—21no—
287C2×SD16—34no—
288C2×SD16—62no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16C4×D314yescomplemented
292C2×D8—34no—
293C2×D8—34no—
294C2×D8—62no—
295C2×D8—61no—
296C2×D8—61no—
297C2×D8—62no—
298C2×D8—61no—
299C2×D8—61no—
300C2×M16—31no—
301C2×M16—31no—
302C2×M16—31no—
303C2×M16C22×D311yescharacteristic
304C22×C8—31no—
305C22×C8C22×D311yescharacteristic
306#127—34no—
307#127—121no—
308#124—34no—
309#124—121no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—62no—
318#117—31no—
319#117—41no—
320#117C22×D311yescharacteristic
321#117C22×D311yescharacteristic
322#117C22×D311yescharacteristic
323C2×C22⋊C4—41no—
324C2×C22⋊C4—121no—
325C2×C22⋊C4—121no—
326C2×C22⋊C4—121no—
327C2×C42—31no—
328#108—62no—
329#108—62no—
330#107—32no—
331#107#6712yescomplemented
332#106—32no—
333#106#6712yescomplemented
334#103—22no—
335#103—62no—
336#102—22no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342#101—121no—
343#100—121no—
344#100—121no—
345#100—121no—
346#97—21no—
347#97—34no—
348#97—61no—
349#97—61no—
350#97—61no—
351C4×C8—21no—
352C4×C8—61no—
353C4×C8—61no—
354C4×C8—61no—
355C23×C3—22no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—82no—
359C23×C3H(5,𝔽2)11yescharacteristic
360C22×D3—22no—
361C22×D3—22no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—82no—
367C22×D3—82no—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—41no—
375C3×D4—41no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—162no—
383C2×C3×C4C23×C411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389#67—88no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—42no—
393D12—22no—
394D12—22no—
395D12—22no—
396D12—22no—
397D12—41no—
398D12—41no—
399D12—41no—
400D12—41no—
401C4×D3—162no—
402C4×D3—162no—
403C3×C8—41no—
404C3×C8C4×D412yes—
405C3×C8C4×D412yes—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8C4×D412yes—
410C3⋊C8C4×D412yes—
411C24—21no—
412C24—61no—
413C24—61no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8C2×C4×D312yes—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—32no—
426C2×D4—32no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—124no—
438C2×D4—244no—
439C2×D4C2×C4×D312yes—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—61no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4C23×D311yescharacteristic
446SD16—24no—
447SD16—24no—
448SD16—64no—
449SD16—64no—
450D8—64no—
451D8—64no—
452D8—64no—
453D8—64no—
454M16—21no—
455M16—21no—
456M16—61no—
457M16—61no—
458M16—61no—
459M16—61no—
460M16—61no—
461M16—61no—
462C2×C8—21no—
463C2×C8—32no—
464C2×C8—31no—
465C2×C8—31no—
466C2×C8—32no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8C2×C4×D312yes—
471C2×C8#23512yes—
472C2×C8C2×#6711yescharacteristic
473C2×C8C2×#6711yescharacteristic
474C4⋊C4—32no—
475C4⋊C4—42no—
476C4⋊C4D3×D412yes—
477C4⋊C4C2×#6712yes—
478C4⋊C4C2×#6712yes—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—42no—
483C22⋊C4—61no—
484C22⋊C4—124no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—32no—
491C42—42no—
492C42—61no—
493C42—61no—
494C22×C3—22no—
495C22×C3—42no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—161no—
499C22×C3#51811yescharacteristic
500C22×C3D4211yescharacteristic
501C22×C3#55911yescharacteristic
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—82no—
505C2×D3—82no—
506C2×D3—82no—
507C2×D3—82no—
508C2×D3—161no—
509C2×D3—161no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4C2×C4×D411yescharacteristic
515C3×C4C2×C4×D411yescharacteristic
516C3×C4#55011yescharacteristic
517C3×C4#55011yescharacteristic
518Dic3—42no—
519Dic3—42no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—62no—
524C23—62no—
525C23—82no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—242no—
531C23—242no—
532C23#100411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537D4—22no—
538D4—22no—
539D4—41no—
540D4—41no—
541D4—62no—
542D4—62no—
543D4—62no—
544D4—62no—
545D4—121no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—248no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—122no—
559C2×C4—162no—
560C2×C4—482no—
561C2×C4—482no—
562C2×C4C22×C4×D311yescharacteristic
563C2×C4C2×#23511yescharacteristic
564C2×C4C2×D3×D411yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4#100911yescharacteristic
567C2×C4C22×#6711yescharacteristic
568C8—32no—
569C8—32no—
570C8—41no—
571C8—121no—
572C8—121no—
573C8—121no—
574C8C4×#6712yes—
575C8C4×#6712yes—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—82no—
579C2×C3#236611yescharacteristic
580C2×C3#328812yes—
581D3—82no—
582D3—82no—
583C22—22no—
584C22—42no—
585C22—82no—
586C22—82no—
587C22—122no—
588C22—122no—
589C22—161no—
590C22—242no—
591C22—242no—
592C22—242no—
593C22—242no—
594C22—481no—
595C22—481no—
596C22#554511yescharacteristic
597C22#555911yescharacteristic
598C22D4×#6711yescharacteristic
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—122no—
604C4—122no—
605C4C4×D3×D411yescharacteristic
606C4#555311yescharacteristic
607C4C2×C4×#6711yescharacteristic
608C4#583411yescharacteristic
609C3#2320211yescharacteristic, complemented, Sylow
610C2—21no—
611C2—21no—
612C2—82no—
613C2—242no—
614C2—242no—
615C2#8234311yescharacteristic
616C2#8401512yes—
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2949C2
socleC22×C3D42
prime radicalC2×#5176C2
cosocleC22×C3×C4C24
hypercenter#2949D3
hypocenterC3#23202
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2949

Complements

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

NH ≅ G/Ntype
1. C3×#2949C2semidirect
3. #82548C2semidirect
4. #82538C2semidirect
5. #79167C2semidirect
6. #72052C2semidirect
7. #71661C2semidirect
8. #71650C2semidirect
9. #70660C2semidirect
10. #70343C2semidirect
11. #70335C2semidirect
12. #70323C2semidirect
13. #70316C2semidirect
14. #69903C2semidirect
15. #69890C2semidirect
37. #5176C4semidirect
63. #2949D3semidirect
84. C2×C3×SD16C2×C4semidirect
97. #947C2×C4semidirect
101. C2×#215C2×C4semidirect
103. #940C2×C4semidirect
107. C2×#213C2×C4semidirect
118. C2×D24C2×C4semidirect
125. C3×#107D4semidirect
126. C3×#106D4semidirect
140. #815C2×C4semidirect
146. #808D4semidirect
147. #807D4semidirect
291. C2×SD16C4×D3semidirect
331. #107#67semidirect
333. #106#67semidirect
609. C3#23202semidirect

Maximal subgroups

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

subgroupindex
C3×#29492
C2×#51762
#825482
#825382
#791672
#720522
#716612
#716502
#706602
#703432
#703352
#703232
#703162
#699032
#698902
#232023