Finite Group Databasebeta

← #10859146

Subgroups — #10859146

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75984C211yesmaximal, characteristic
2#74163C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4#69623C211yesmaximal, characteristic, complemented
5#69559C211yesmaximal, characteristic, complemented
6#68913C211yesmaximal, characteristic, complemented
7#68911C211yesmaximal, characteristic, complemented
8#68884C211yesmaximal, characteristic
9#68878C211yesmaximal, characteristic
10#68876C211yesmaximal, characteristic
11#68867C211yesmaximal, characteristic
12#68816C211yesmaximal, characteristic, complemented
13#68802C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68667C211yesmaximal, characteristic
16#26435—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×(C2×D3)⋊C8C2211yescharacteristic
26C2×#817C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30#4990—21no—
31#4987C2211yescharacteristic
32Dic3×C4⋊C4—21no—
33Dic3×C4⋊C4C2211yescharacteristic
34#4973C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C4×(C2×D3)⋊C4C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42#4669—21no—
43#4668—21no—
44#4667—21no—
45#4658—21no—
46#4657—21no—
47#4654—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2442—31no—
61#2362—31no—
62#2101D311yescharacteristic, complemented
63#2087—31no—
64#2068—31no—
65#2026—31no—
66#1989—31no—
67#1919—31no—
68#1894—31no—
69#1872—31no—
70#1865—31no—
71#1854—31no—
72#1851—31no—
73#1805—31no—
74#1736—31no—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4D411yescharacteristic
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)—41no—
109C2×ΓL(1,𝔽25)—41no—
110#897—22no—
111#897Q814yes—
112#892—22no—
113#892—41no—
114D3×C4⋊C4—22no—
115D3×C4⋊C4—41no—
116#890—22no—
117#890—41no—
118C4×Dic6—22no—
119C4×Dic6—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(C2×D3)⋊C8—22no—
132#817—22no—
133#816—22no—
134#816—41no—
135Dic3⋊C8—41no—
136#810—41no—
137#810—41no—
138#807—41no—
139C4⋊(C3⋊C8)—41no—
140C22×M16—31no—
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117—31no—
145C22×C4⋊C4—31no—
146C2×#107—31no—
147C2×#106—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#389C2×D311yescharacteristic
161#388—61no—
162#388—61no—
163#387—61no—
164#385—61no—
165#382C2×D311yescharacteristic
166#380—31no—
167#380—61no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4—31no—
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#336C2×D311yescharacteristic
183C23×D3C2×D411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×C4×D3—81no—
205C2×C4×D3C2×Q812yes—
206C2×C4×D3#4112yes—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—42no—
227C3×C4⋊C4C2×Q812yes—
228C3×C42—21no—
229C3×C42—21no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—41no—
232(C2×D3)⋊C4—41no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—42no—
241C4⋊Dic3#4112yes—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C4×Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259ΓL(1,𝔽25)—44no—
260C22×Q8—31no—
261C23×C4—31no—
262C23×C4—61no—
263C2×M16—62no—
264C2×M16—121no—
265C2×M16—121no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270#125—62no—
271#125—121no—
272C22⋊Q8—34no—
273C22⋊Q8—62no—
274C4×Q8—62no—
275C4×Q8—121no—
276#117—62no—
277#117—121no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—121no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#107—121no—
306#106—62no—
307C4⋊C8—121no—
308C4⋊C8—121no—
309#103—62no—
310#103—121no—
311#103—121no—
312#103—121no—
313C22⋊C8—62no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3C22×D411yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3C22⋊Q812yes—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
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×C4C22×Q811yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4H(5,𝔽2)11yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—22no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3C22⋊Q812yes—
378C4×D3—22no—
379C4×D3—22no—
380C4×D3—22no—
381C4×D3—22no—
382C4×D3—84no—
383Dic6—22no—
384Dic6—22no—
385Dic6—41no—
386Dic6—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24—31no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—31no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—241no—
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4C23×D311yescharacteristic
421M16—124no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—42no—
444C4⋊C4—64no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—122no—
450C4⋊C4—122no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C4⋊C4D3×Q812yes—
456C22⋊C4—64no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C42—21no—
462C42—21no—
463C42—61no—
464C42—61no—
465C42—122no—
466C42—122no—
467C42—122no—
468C22×C3C2×C22⋊Q811yescharacteristic
469C22×C3#54611yescharacteristic
470C22×C3#54611yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57511yescharacteristic
473C22×C3#57711yescharacteristic
474C22×C3#57811yescharacteristic
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4#52911yescharacteristic
489C3×C4#53811yescharacteristic
490C3×C4#55411yescharacteristic
491C3×C4#55411yescharacteristic
492Dic3—22no—
493Dic3—42no—
494Dic3—41no—
495Dic3—42no—
496Dic3—42no—
497Dic3—41no—
498Dic3—81no—
499C23—32no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—61no—
506C23C2×D3×D411yescharacteristic
507Q8—62no—
508Q8—62no—
509Q8—121no—
510Q8—121no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—32no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—62no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—244no—
555C2×C4C2×D3×Q811yescharacteristic
556C2×C4D3×#4111yescharacteristic
557C2×C4D3×#4111yescharacteristic
558C2×C4D3×#4111yescharacteristic
559C2×C4#100911yescharacteristic
560C2×C4#100911yescharacteristic
561C8—41no—
562C8—41no—
563C8—121no—
564C8—121no—
565C2×C3#245011yescharacteristic
566C2×C3#308011yescharacteristic
567C2×C3#308211yescharacteristic
568C2×C3#331311yescharacteristic
569C2×C3#331511yescharacteristic
570C2×C3#332311yescharacteristic
571C2×C3#332611yescharacteristic
572D3—42no—
573C22—62no—
574C22—62no—
575C22—62no—
576C22—121no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22D3×C22⋊Q811yescharacteristic
581C22#565411yescharacteristic
582C22#567611yescharacteristic
583C22#576811yescharacteristic
584C22#577011yescharacteristic
585C22#577311yescharacteristic
586C22#577911yescharacteristic
587C4—41no—
588C4—42no—
589C4—41no—
590C4—62no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—122no—
595C4—121no—
596C4—122no—
597C4—121no—
598C4—122no—
599C4—241no—
600C4#569011yescharacteristic
601C4#569011yescharacteristic
602C4#570411yescharacteristic
603C4#570611yescharacteristic
604C3#2643511yescharacteristic, complemented, Sylow
605C2—122no—
606C2#7834411yescharacteristic
607C2#8116711yescharacteristic
608C2#8123311yescharacteristic
609C2#8141011yescharacteristic
610C2#8154711yescharacteristic
611C2#8165711yescharacteristic
612C2#8169911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2101D3
hypocenterC3#26435
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. C3×#2101C2semidirect
4. #69623C2semidirect
5. #69559C2semidirect
6. #68913C2semidirect
7. #68911C2semidirect
12. #68816C2semidirect
13. #68802C2semidirect
14. #68684C2semidirect
62. #2101D3semidirect
604. C3#26435semidirect

Maximal subgroups

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

subgroupindex
#759842
#741632
C3×#21012
#696232
#695592
#689132
#689112
#688842
#688782
#688762
#688672
#688162
#688022
#686842
#686672
#264353