Finite Group Databasebeta

← #10858709

Subgroups — #10858709

Order 768.

714 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#79210C211yesmaximal, characteristic
2#79091C211yesmaximal, characteristic
3#78950C211yesmaximal, characteristic, complemented
4#78774C211yesmaximal, characteristic
5C3×#2064C211yesmaximal, characteristic, complemented
6#70722C211yesmaximal, characteristic
7#70701C211yesmaximal, characteristic, complemented
8#70666C211yesmaximal, characteristic, complemented
9#70534C211yesmaximal, characteristic, complemented
10#70348C211yesmaximal, characteristic
11#70249C211yesmaximal, characteristic, complemented
12#70123C211yesmaximal, characteristic
13#70102C211yesmaximal, characteristic, complemented
14#69002C211yesmaximal, characteristic
15#68980C211yesmaximal, characteristic, complemented
16#26406—31nomaximal, Sylow
17#5812C2211yescharacteristic
18C2×#942C2211yescharacteristic
19C2×#907C2211yescharacteristic
20#5523C2211yescharacteristic
21#5519C2211yescharacteristic
22#5504C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C3×#424C2211yescharacteristic
26C3×#419C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#101C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#825C2211yescharacteristic
32#5133C2211yescharacteristic
33#5121C2211yescharacteristic
34#5121C2211yescharacteristic
35#5121C2211yescharacteristic
36#5119C2211yescharacteristic
37#5113C2211yescharacteristic
38C2×#818C2211yescharacteristic
39#5101C2211yescharacteristic
40#4998C2211yescharacteristic
41#4975C2211yescharacteristic
42#4964C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4881—21no—
45D3×#107—21no—
46#4861—21no—
47#4860—21no—
48#4850—21no—
49#4847—21no—
50#4846—21no—
51#4829—21no—
52#4817—21no—
53#4799—21no—
54#4796—21no—
55#4793—21no—
56#4791—21no—
57#4790—21no—
58#4789—21no—
59#4788—21no—
60C3×#343C2211yescharacteristic
61#4558C2211yescharacteristic
62#4555C2211yescharacteristic
63#4555C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4550C2211yescharacteristic
67#4532C2211yescharacteristic
68#3083—31no—
69#3082—31no—
70#2947—31no—
71#2913—31no—
72#2095—31no—
73#2089—31no—
74#2064D311yescharacteristic, complemented
75#2060—31no—
76#1952—31no—
77#1922—31no—
78#1908—31no—
79#1882—31no—
80#1866—31no—
81#1859—31no—
82#1857—31no—
83C22×C3×Q8C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#117C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91Q8×Dic3—21no—
92Q8×Dic3—21no—
93Q8×Dic3—21no—
94Q8×Dic3—21no—
95C2×#216—21no—
96#942—22no—
97#942Q814yescomplemented
98C2×#215—21no—
99C4×#67—41no—
100#926—41no—
101#926C2311yescharacteristic
102#926C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—41no—
106C2×C4×Dic3C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#907—22no—
110#907—22no—
111C2×ΓL(1,𝔽25)—21no—
112C2×D3×C8—21no—
113#893—21no—
114#892—21no—
115#892—21no—
116D3×C4⋊C4—21no—
117D3×C4⋊C4—21no—
118#890—21no—
119#888—21no—
120#887—21no—
121#67⋊C4—41no—
122#878—41no—
123C4×D12—21no—
124#870—21no—
125#869—21no—
126C4×Dic6—21no—
127C3×#108—22no—
128C3×#107—21no—
129C3×#106—21no—
130C3×C4⋊C8—21no—
131C3×C4⋊C8—21no—
132C3×#103—21no—
133C3×#103—21no—
134C3×#103—21no—
135C3×#103—21no—
136C3×#101—22no—
137#835—21no—
138#835—21no—
139C3⋊#95—41no—
140#825—22no—
141#821—21no—
142#821D411yescharacteristic
143#821D411yescharacteristic
144#818—21no—
145#818—21no—
146#818—21no—
147#818—21no—
148#818—21no—
149#817—21no—
150#817—21no—
151#817—21no—
152#816—21no—
153#816D411yescharacteristic
154#816D411yescharacteristic
155#815—21no—
156Dic3⋊C8—21no—
157Dic3⋊C8—21no—
158C8×Dic3—21no—
159#808—21no—
160#808D411yescharacteristic
161#808D411yescharacteristic
162#807—21no—
163#807—21no—
164#807—21no—
165#807D411yescharacteristic
166#807D411yescharacteristic
167C2×#137—31no—
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520—31no—
172#519—31no—
173#479—61no—
174#479—61no—
175#479—61no—
176#478—61no—
177#477—61no—
178#477—61no—
179#475—61no—
180#474—61no—
181#441—61no—
182#440—61no—
183C4×Q16—61no—
184C4×SD16—61no—
185C2×#108C2×D311yescharacteristic
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—61no—
191#428—61no—
192#428C2×D311yescharacteristic
193#427—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—31no—
197C2×#107—61no—
198#424—31no—
199#424C2×D311yescharacteristic
200C2×#104—31no—
201#419C2×D311yescharacteristic
202#417—31no—
203#416—31no—
204C2×#103—31no—
205C2×#103C2×D311yescharacteristic
206C2×#101C2×D311yescharacteristic
207#405—31no—
208#391—31no—
209#384—31no—
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8C2×Q812yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4C2411yescharacteristic
227C2×#67—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231#235—21no—
232#235—21no—
233#235C2×Q812yes—
234#235#4112yes—
235C2×D12C2×D411yescharacteristic
236C2×C4×D3—21no—
237C2×C4×D3—41no—
238C2×C4×D3—41no—
239C2×Dic6—41no—
240C2×Dic6C2×D411yescharacteristic
241C3×M16—21no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16#4112yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8C2×D411yescharacteristic
253C2×C3×C8C2×D411yescharacteristic
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C22⋊C4—21no—
259C3×C42—21no—
260#217—21no—
261#217—21no—
262#216—24no—
263#215—24no—
264(C2×D3)⋊C4—41no—
265(C2×D3)⋊C4—41no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—41no—
271C4⋊Dic3—41no—
272C4⋊Dic3—41no—
273C4⋊Dic3C2×D411yescharacteristic
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279Dic3⋊C4—41no—
280Dic3⋊C4—41no—
281Dic3⋊C4—82no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—41no—
287C4×Dic3—41no—
288C4×Dic3—41no—
289C4×Dic3—41no—
290#208—21no—
291#208C2×Q812yes—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8C2×D411yescharacteristic
296C2×C3⋊C8C2×D411yescharacteristic
297ΓL(1,𝔽25)—24no—
298D3×C8—24no—
299C2×#41—31no—
300C22×Q8C22×D311yescharacteristic
301#137—34no—
302#137—62no—
303C2×Q16—61no—
304C2×SD16—61no—
305#131—62no—
306#131—62no—
307C2×M16—31no—
308C2×M16—61no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C22×C8—31no—
313C22×C8—61no—
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317#125—61no—
318#125—61no—
319C22⋊Q8—121no—
320C4×Q8—61no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—121no—
329C4×D4—121no—
330#117—31no—
331#117—31no—
332#117—61no—
333#117—61no—
334#117—121no—
335#117C22×D311yescharacteristic
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—121no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C42—31no—
343#108—22no—
344#107—21no—
345#107—31no—
346#107—31no—
347#107—61no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107—61no—
354#107—61no—
355#106—21no—
356#106—31no—
357#106—31no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#104—62no—
367#103—21no—
368#103—21no—
369#103—21no—
370#103—21no—
371#103—31no—
372#103—31no—
373#103—61no—
374#103—61no—
375#103—61no—
376#102—31no—
377#102—31no—
378#102—61no—
379#101—22no—
380#97—61no—
381C4×C8—61no—
382#95—121no—
383C23×C3H(5,𝔽2)11yescharacteristic
384C22×D3—21no—
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—41no—
388C3×Q8—41no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—81no—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×Q811yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402#67—22no—
403#67—22no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—81no—
415C2×Dic3—81no—
416D12—21no—
417D12C22⋊Q812yes—
418C4×D3—22no—
419C4×D3—22no—
420C4×D3—161no—
421C4×D3—161no—
422Dic6—21no—
423Dic6—161no—
424Dic6C22⋊Q812yes—
425C3×C8—22no—
426C3×C8—21no—
427C3×C8—21no—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—22no—
431C3⋊C8—21no—
432C3⋊C8—21no—
433#41—32no—
434#41—32no—
435#41—61no—
436#41—61no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—31no—
443C2×Q8—121no—
444C2×Q8D3×Q812yes—
445C2×D4—31no—
446C2×D4—61no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—121no—
454C22×C4—121no—
455C22×C4C23×D311yescharacteristic
456Q16—64no—
457SD16—64no—
458M16—21no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—21no—
463M16—32no—
464M16—64no—
465M16—61no—
466M16#23712yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—31no—
473C2×C8—31no—
474C2×C8—64no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8D3×D411yescharacteristic
479C2×C8D3×D411yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—121no—
491C4⋊C4—121no—
492C4⋊C4—121no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—242no—
498C4⋊C4D3×D411yescharacteristic
499C4⋊C4D3×D411yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—61no—
502C22⋊C4—61no—
503C22⋊C4—121no—
504C22⋊C4—121no—
505C42—21no—
506C42—61no—
507C42—61no—
508C42—61no—
509C42—61no—
510C42—121no—
511C42—121no—
512C42—121no—
513C42—121no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3#53911yescharacteristic
517C22×C3D4211yescharacteristic
518C22×C3#55411yescharacteristic
519C2×D3—22no—
520C2×D3—41no—
521C2×D3—41no—
522C3×C4—42no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—81no—
526C3×C4C2×C22⋊Q811yescharacteristic
527C3×C4#52511yescharacteristic
528C3×C4#54711yescharacteristic
529C3×C4D4×Q811yescharacteristic
530Dic3—22no—
531Dic3—42no—
532Dic3—41no—
533Dic3—41no—
534Dic3—41no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538C23—61no—
539C23#100911yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—32no—
543Q8—41no—
544Q8—41no—
545Q8—61no—
546Q8—481no—
547D4—32no—
548D4—62no—
549D4—62no—
550D4—61no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—81no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—241no—
571C2×C4—241no—
572C2×C4—481no—
573C2×C4—481no—
574C2×C4C2×D3×D411yescharacteristic
575C2×C4C2×D3×D411yescharacteristic
576C2×C4C2×#23711yescharacteristic
577C2×C4C2×D3×Q811yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4#101011yescharacteristic
580C8—22no—
581C8—21no—
582C8—21no—
583C8—41no—
584C8—41no—
585C8—61no—
586C8—61no—
587C8—62no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3#244911yescharacteristic
591C2×C3#328811yescharacteristic
592C2×C3#329311yescharacteristic
593D3—42no—
594C22—21no—
595C22—21no—
596C22—62no—
597C22—121no—
598C22—121no—
599C22#560911yescharacteristic
600C22#563511yescharacteristic
601C22#570711yescharacteristic
602C4—41no—
603C4—41no—
604C4—42no—
605C4—62no—
606C4—81no—
607C4—122no—
608C4—121no—
609C4—121no—
610C4—121no—
611C4—121no—
612C4—241no—
613C4—241no—
614C4#560611yescharacteristic
615C4#561111yescharacteristic
616C4D3×C22⋊Q811yescharacteristic
617C4#563211yescharacteristic
618C3#2640611yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—122no—
622C2#7741911yescharacteristic
623C2#8093211yescharacteristic
624C2#8094711yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2064C2
socleC22×C3D42
prime radicalC2×#235C23
hypercenter#2064D3
hypocenterC3#26406
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2064

Complements

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

NH ≅ G/Ntype
3. #78950C2semidirect
5. C3×#2064C2semidirect
7. #70701C2semidirect
8. #70666C2semidirect
9. #70534C2semidirect
11. #70249C2semidirect
13. #70102C2semidirect
15. #68980C2semidirect
74. #2064D3semidirect
97. #942Q8semidirect
618. C3#26406semidirect

Maximal subgroups

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

subgroupindex
#792102
#790912
#789502
#787742
C3×#20642
#707222
#707012
#706662
#705342
#703482
#702492
#701232
#701022
#690022
#689802
#264063