Finite Group Databasebeta

← #10844662

Subgroups — #10844662

Order 768.

760 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#479C211yesmaximal, characteristic
2#83025C211yesmaximal, characteristic
3C2×#5069C211yesmaximal, characteristic
4#82475C211yesmaximal, characteristic
5#71631C211yesmaximal, characteristic
6#70643C211yesmaximal, characteristic
7#70624C211yesmaximal, characteristic
8#70589C211yesmaximal, characteristic
9#70570C211yesmaximal, characteristic
10#70469C211yesmaximal, characteristic
11#70456C211yesmaximal, characteristic
12#70246C211yesmaximal, characteristic
13#70246C211yesmaximal, characteristic
14#70229C211yesmaximal, characteristic
15#70229C211yesmaximal, characteristic
16#25290—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C3×#479—24no—
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#5069—24no—
37#4984—21no—
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4981—21no—
41#4981—21no—
42#4980—21no—
43#4980—21no—
44Dic3×C4⋊C4—21no—
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47#4973C2211yescharacteristic
48#4973C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#810C2211yescharacteristic
53C4×C4⋊Dic3C2211yescharacteristic
54C4×C4⋊Dic3C2211yescharacteristic
55#4928—21no—
56#4928—21no—
57C2×#803C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#491—31no—
63C2×#479D311yescharacteristic
64#2546—31no—
65#2370—31no—
66#2068—31no—
67#2066—31no—
68#2043—31no—
69#2043—31no—
70#1981—31no—
71#1981—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1927—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#125—22no—
83C3×#125D414yes—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94#944—22no—
95#944—22no—
96#944—41no—
97#944—41no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#869—22no—
118#869—22no—
119C3×#107—22no—
120C3×#106—22no—
121C3×C4⋊C8—22no—
122C3×#103—22no—
123C3×#103—22no—
124#835—22no—
125#835—22no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#816—22no—
138#816—22no—
139#816—41no—
140#816—41no—
141Dic3⋊C8—41no—
142Dic3⋊C8—41no—
143#810—22no—
144#810—22no—
145#810—22no—
146#810—22no—
147#803—22no—
148C2×C4⋊Q8—31no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8—31no—
151C2×#125C2×D311yescharacteristic
152C2×C4×Q8C2×D311yescharacteristic
153#491—64no—
154#479—24no—
155C2×#107C2×D311yescharacteristic
156C2×#106C2×D311yescharacteristic
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172C2×#97—31no—
173#398—61no—
174#398—61no—
175#391—31no—
176#391—31no—
177#389—31no—
178#389—31no—
179#389—61no—
180#389—61no—
181#389—61no—
182#387—61no—
183#387—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—61no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8#4112yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4C2×D412yes—
240C3×C4⋊C4C2×D412yes—
241C3×C42—42no—
242C3×C42C2×D412yes—
243C3×C42#4112yes—
244C4⋊Dic3—21no—
245C4⋊Dic3—22no—
246C4⋊Dic3—22no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—42no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252C4⋊Dic3—81no—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—42no—
263C4×Dic3—81no—
264C4×Dic3—81no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C22×Q8—31no—
272C22×Q8—31no—
273C22×Q8C22×D311yescharacteristic
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4⋊Q8—62no—
279C4⋊Q8—62no—
280C4⋊Q8—62no—
281C4⋊Q8—62no—
282C4⋊Q8—121no—
283C4⋊Q8—121no—
284#125—22no—
285#125#6714yes—
286C4×Q8—22no—
287C4×Q8—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C42—21no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—22no—
314#106—22no—
315C4⋊C8—22no—
316C4⋊C8—121no—
317C4⋊C8—121no—
318#103—22no—
319#103—22no—
320#103—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329#103—121no—
330#97—62no—
331#95—31no—
332#95—31no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3C22×D411yescharacteristic
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—41no—
346C3×Q8—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4#12112yes—
364C2×C3×C4#12112yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C42−1+411yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3#13712yes—
394C2×Dic3#13712yes—
395Dic6—41no—
396Dic6—41no—
397Dic6—41no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402Dic6—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8#23912yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4D3×D411yescharacteristic
438C22×C4D3×D411yescharacteristic
439C22×C4C23×D311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—241no—
473C4⋊C4C2×#6712yes—
474C4⋊C4C2×#6712yes—
475C42—42no—
476C42—61no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—122no—
481C42—241no—
482C42—241no—
483C42#23712yes—
484C42C2×#6712yes—
485C22×C3C2×#12111yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#54811yescharacteristic
488C22×C3C2×#13711yescharacteristic
489C22×C3C2×#13711yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—81no—
498C3×C4#52911yescharacteristic
499C3×C4#54211yescharacteristic
500C3×C4#54811yescharacteristic
501C3×C4#55011yescharacteristic
502Dic3—22no—
503Dic3—22no—
504Dic3—42no—
505Dic3—41no—
506Dic3—41no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23C2×D3×D411yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515Q8—121no—
516Q8—121no—
517Q8—121no—
518Q8—121no—
519Q8—121no—
520Q8—121no—
521Q8—121no—
522Q8—121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4#93912yes—
564C2×C4#93912yes—
565C2×C4C2×#23711yescharacteristic
566C2×C4C2×#23911yescharacteristic
567C2×C4#100711yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4#100911yescharacteristic
570C2×C4C22×#6711yescharacteristic
571C8—41no—
572C8—41no—
573C8—121no—
574C8—121no—
575C2×C3#242611yescharacteristic
576C2×C3#303911yescharacteristic
577C2×C3#304611yescharacteristic
578C2×C3#331211yescharacteristic
579C2×C3#332011yescharacteristic
580C2×C3#337211yescharacteristic
581C2×C3#337411yescharacteristic
582C22#563011yescharacteristic
583C22#563311yescharacteristic
584C22#577011yescharacteristic
585C22#577111yescharacteristic
586C22D3×#13711yescharacteristic
587C22D3×#13711yescharacteristic
588C22C2×#93911yescharacteristic
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—62no—
595C4—62no—
596C4—81no—
597C4—122no—
598C4—121no—
599C4—121no—
600C4—241no—
601C4—241no—
602C4—241no—
603C4#562411yescharacteristic
604C4#564311yescharacteristic
605C4#581711yescharacteristic
606C4#583411yescharacteristic
607C3#2529011yescharacteristic, complemented, Sylow
608C2#8121911yescharacteristic
609C2#8122011yescharacteristic
610C2#8124811yescharacteristic
611C2#8125011yescharacteristic
612C2#8275511yescharacteristic
613C2#8390211yescharacteristic
614C2#8396711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#479C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#479D3
hypocenterC3#25290
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
607. C3#25290semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4792
#830252
C2×#50692
#824752
#716312
#706432
#706242
#705892
#705702
#704692
#704562
#702462
#702292
#252903