Finite Group Databasebeta

← #10858888

Subgroups — #10858888

Order 768.

774 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82172C211yesmaximal, characteristic
2#82162C211yesmaximal, characteristic
3C2×C3×#340C211yesmaximal, characteristic
4#71974C211yesmaximal, characteristic
5#71972C211yesmaximal, characteristic
6#71559C211yesmaximal, characteristic
7#71555C211yesmaximal, characteristic
8#68913C211yesmaximal, characteristic, complemented
9#68911C211yesmaximal, characteristic, complemented
10#68906C211yesmaximal, characteristic, complemented
11#68905C211yesmaximal, characteristic, complemented
12#68844C211yesmaximal, characteristic, complemented
13#68807C211yesmaximal, characteristic, complemented
14#68801C211yesmaximal, characteristic, complemented
15#68787C211yesmaximal, characteristic, complemented
16#26418—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×#887C2211yescharacteristic
23#5212—21no—
24#5212C2211yescharacteristic
25#5210—21no—
26#5210—21no—
27C4×#217—21no—
28C4×#217C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4986—21no—
35#4984C2211yescharacteristic
36#4982—21no—
37#4978—21no—
38#4977C2211yescharacteristic
39#4976—21no—
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2442—31no—
61#2360—31no—
62#2101—31no—
63#2086—31no—
64#2068—31no—
65#2026—31no—
66#1989—31no—
67#1918—31no—
68#1892—31no—
69#1872—31no—
70#1863—31no—
71#1852—31no—
72#1850—31no—
73#1805—31no—
74C2×#340D311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×C3×C8—22no—
80C22×C3×C8—41no—
81C22×C3×C8—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—41no—
89C2×C3×C4⋊C4—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217D411yescharacteristic
97C2×#217D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103#924—22no—
104#924Q814yes—
105C2×Dic3⋊C4—21no—
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×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#890—22no—
117#890—41no—
118#887—22no—
119#887—41no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#830—22no—
132#816—41no—
133#816—41no—
134Dic3⋊C8—41no—
135Dic3⋊C8—41no—
136#810—22no—
137#810—41no—
138#807—22no—
139#807—41no—
140C23×C8C2×D311yescharacteristic
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C22×C4⋊C4C2×D311yescharacteristic
145C22×C4⋊C4C2×D311yescharacteristic
146C2×#107—31no—
147C2×#107—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×C22⋊C8—31no—
155#400—61no—
156#398—61no—
157#391—31no—
158#389—31no—
159#389—31no—
160#389—31no—
161#389—61no—
162#387—61no—
163#385—61no—
164#385—61no—
165#382—31no—
166#380—31no—
167#380—61no—
168C4×C4⋊C4—31no—
169C4×C22⋊C4—31no—
170C4×C22⋊C4—61no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—81no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×Q812yes—
200C22×C3×C4#4112yes—
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6C2×Q812yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—44no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4#4112yes—
232C3×C4⋊C4#4112yes—
233#217—24no—
234#217—22no—
235#217—22no—
236#217—42no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3—81no—
244C4⋊Dic3C2×Q812yes—
245Dic3⋊C4—24no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—42no—
255C4×Dic3—81no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C22×Q8—31no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4C22×D311yescharacteristic
266C22×C8—22no—
267C22×C8—31no—
268C22×C8—31no—
269C22×C8—41no—
270C22×C8—41no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273#125—62no—
274#125—121no—
275C22⋊Q8—34no—
276C22⋊Q8—62no—
277C4×Q8—62no—
278C4×Q8—121no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—41no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—31no—
300C2×C22⋊C4—31no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—61no—
306C2×C42—61no—
307#107—62no—
308#107—121no—
309C4⋊C8—121no—
310C4⋊C8—121no—
311#103—62no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—62no—
316#95—31no—
317#95—31no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3C22⋊Q812yes—
332C23×C3#13512yes—
333C23×C3#13712yes—
334C23×C3C22×D411yescharacteristic
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—84no—
354C2×C3×C4—84no—
355C2×C3×C4C22⋊Q812yes—
356C2×C3×C4C22×Q811yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386Dic6—22no—
387Dic6—22no—
388Dic6—41no—
389Dic6—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24D3×D411yescharacteristic
395C2×Q8—32no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4#23512yes—
424C22×C4D3×D411yescharacteristic
425C22×C4D3×Q812yes—
426C22×C4C23×D311yescharacteristic
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—44no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—32no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—64no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C4⋊C4—241no—
461C4⋊C4#23512yes—
462C4⋊C4#23512yes—
463C22⋊C4—64no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—122no—
467C42—61no—
468C42—61no—
469C42—61no—
470C42—61no—
471C42—122no—
472C42—241no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3#47914yes—
480C22×C3C2×C22⋊Q811yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3#54611yescharacteristic
483C22×C3C2×#13511yescharacteristic
484C22×C3C2×#13711yescharacteristic
485C22×C3#57611yescharacteristic
486C22×C3#57811yescharacteristic
487C3×C4—22no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#52911yescharacteristic
493C3×C4#53811yescharacteristic
494C3×C4#55411yescharacteristic
495C3×C4#55411yescharacteristic
496Dic3—42no—
497Dic3—42no—
498Dic3—41no—
499Dic3—41no—
500Dic3—41no—
501Dic3—41no—
502Dic3—81no—
503Dic3—81no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23#89612yes—
509C23#91512yes—
510C23#91612yes—
511C23C2×D3×D411yescharacteristic
512Q8—62no—
513Q8—62no—
514Q8—121no—
515Q8—121no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—121no—
551C2×C4—121no—
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#89612yes—
561C2×C4C2×#23511yescharacteristic
562C2×C4C2×#23511yescharacteristic
563C2×C4C2×#23511yescharacteristic
564C2×C4#100411yescharacteristic
565C2×C4#100411yescharacteristic
566C2×C4C2×D3×Q811yescharacteristic
567C8—41no—
568C8—41no—
569C8—121no—
570C8—121no—
571C2×C3—24no—
572C2×C3#245011yescharacteristic
573C2×C3C2×#47911yescharacteristic
574C2×C3#308311yescharacteristic
575C2×C3#329711yescharacteristic
576C2×C3#330011yescharacteristic
577C2×C3#331211yescharacteristic
578C2×C3#331411yescharacteristic
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22#485714yes—
586C22#549011yescharacteristic
587C22#549011yescharacteristic
588C22C2×#89611yescharacteristic
589C22#575711yescharacteristic
590C22C2×#91511yescharacteristic
591C22C2×#91611yescharacteristic
592C22#576611yescharacteristic
593C4—22no—
594C4—42no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—122no—
599C4—122no—
600C4—121no—
601C4—121no—
602C4—121no—
603C4—121no—
604C4—241no—
605C4—241no—
606C4#547911yescharacteristic
607C4#549411yescharacteristic
608C4#551111yescharacteristic
609C4#551111yescharacteristic
610C3#2641811yescharacteristic, complemented, Sylow
611C2—24no—
612C2#7335111yescharacteristic
613C2#7876011yescharacteristic
614C2#7876311yescharacteristic
615C2#7880711yescharacteristic
616C2#7881011yescharacteristic
617C2C2×#485711yescharacteristic
618C2#7916211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#340C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#340D3
hypocenterC3#26418
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
8. #68913C2semidirect
9. #68911C2semidirect
10. #68906C2semidirect
11. #68905C2semidirect
12. #68844C2semidirect
13. #68807C2semidirect
14. #68801C2semidirect
15. #68787C2semidirect
610. C3#26418semidirect

Maximal subgroups

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

subgroupindex
#821722
#821622
C2×C3×#3402
#719742
#719722
#715592
#715552
#689132
#689112
#689062
#689052
#688442
#688072
#688012
#687872
#264183