Finite Group Databasebeta

← #10859129

Subgroups — #10859129

Order 768.

748 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
1#76205C211yesmaximal, characteristic
2#76000C211yesmaximal, characteristic
3C3×#2088C211yesmaximal, characteristic, complemented
4#70589C211yesmaximal, characteristic, complemented
5#70579C211yesmaximal, characteristic
6#70485C211yesmaximal, characteristic, complemented
7#70478C211yesmaximal, characteristic
8#69632C211yesmaximal, characteristic, complemented
9#69612C211yesmaximal, characteristic, complemented
10#68911C211yesmaximal, characteristic, complemented
11#68884C211yesmaximal, characteristic
12#68879C211yesmaximal, characteristic
13#68867C211yesmaximal, characteristic
14#68819C211yesmaximal, characteristic, complemented
15#68801C211yesmaximal, characteristic, complemented
16#26434—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4994—21no—
32#4989—21no—
33#4987C2211yescharacteristic
34#4984—21no—
35#4984C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4973C2211yescharacteristic
38#4972—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C4×(C2×D3)⋊C4C2211yescharacteristic
43#4935C2211yescharacteristic
44#4677—21no—
45#4666—21no—
46#4654—21no—
47#4653—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2441—31no—
61#2360—31no—
62#2100—31no—
63#2088D311yescharacteristic, complemented
64#2066—31no—
65#2026—31no—
66#1989—31no—
67#1920—31no—
68#1894—31no—
69#1873—31no—
70#1865—31no—
71#1853—31no—
72#1851—31no—
73#1803—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⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×ΓL(1,𝔽25)—22no—
110C2×ΓL(1,𝔽25)—41no—
111C2×ΓL(1,𝔽25)—41no—
112#897—22no—
113#897Q814yes—
114#892—22no—
115#892—41no—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—41no—
118#890—22no—
119#890—41no—
120C4×Dic6—22no—
121C4×Dic6—41no—
122C3×#106—41no—
123C3×C4⋊C8—41no—
124C3×#95—21no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133(C2×D3)⋊C8—22no—
134#816—41no—
135#816—41no—
136Dic3⋊C8—41no—
137#810—22no—
138#810—41no—
139#808—22no—
140C22×M16—31no—
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117—31no—
145C22×C4⋊C4—31no—
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×C22⋊C8—31no—
155#399—61no—
156#396—61no—
157#391—31no—
158#389—31no—
159#389—61no—
160#389—61no—
161#389C2×D311yescharacteristic
162#389C2×D311yescharacteristic
163#387—61no—
164#387—61no—
165#385—61no—
166#384—31no—
167#380—31no—
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#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
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—42no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4C2×Q812yes—
228C3×C4⋊C4#4112yes—
229C3×C42—21no—
230C3×C42—21no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—41no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3—42no—
244C4⋊Dic3—81no—
245C4⋊Dic3—81no—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—42no—
253C4×Dic3—42no—
254C4×Dic3—81no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261ΓL(1,𝔽25)—44no—
262C22×Q8—31no—
263C23×C4—31no—
264C23×C4—61no—
265C2×M16—62no—
266C2×M16—121no—
267C2×M16—121no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#125—62no—
273#125—121no—
274C22⋊Q8—34no—
275C22⋊Q8—62no—
276C4×Q8—62no—
277C4×Q8—121no—
278#117—62no—
279#117—121no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
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—121no—
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×C22⋊C4—61no—
304C2×C42—31no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#106—41no—
310#106—62no—
311C4⋊C8—41no—
312C4⋊C8—121no—
313#103—62no—
314#103—121no—
315#103—121no—
316#103—121no—
317C22⋊C8—62no—
318#95—21no—
319#95—21no—
320#95—31no—
321#95—31no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3C22⋊Q812yes—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
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×C4—41no—
352C2×C3×C4#13712yes—
353C2×C3×C4C22×Q811yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—22no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
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×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3C22⋊Q812yes—
380C4×D3—22no—
381C4×D3—22no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—84no—
385Dic6—22no—
386Dic6—22no—
387Dic6—41no—
388Dic6—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C24—31no—
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—31no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4—241no—
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4C23×D311yescharacteristic
423M16—124no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—32no—
443C4⋊C4—42no—
444C4⋊C4—64no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—81no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4—122no—
456C4⋊C4—241no—
457C4⋊C4—241no—
458C4⋊C4D3×Q812yes—
459C4⋊C4#23912yes—
460C22⋊C4—64no—
461C22⋊C4—62no—
462C22⋊C4—121no—
463C22⋊C4—122no—
464C22⋊C4—241no—
465C42—21no—
466C42—21no—
467C42—61no—
468C42—61no—
469C42—122no—
470C42—122no—
471C42—241no—
472C22×C3C2×C22⋊Q811yescharacteristic
473C22×C3#54611yescharacteristic
474C22×C3#54611yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57511yescharacteristic
477C22×C3#57711yescharacteristic
478C22×C3#57711yescharacteristic
479C2×D3—22no—
480C2×D3—22no—
481C2×D3—22no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#52911yescharacteristic
493C3×C4#53811yescharacteristic
494C3×C4#55411yescharacteristic
495C3×C4#55411yescharacteristic
496Dic3—22no—
497Dic3—41no—
498Dic3—42no—
499Dic3—41no—
500Dic3—42no—
501Dic3—81no—
502Dic3—81no—
503C23—32no—
504C23—61no—
505C23—61no—
506C23—61no—
507C23—61no—
508C23—61no—
509C23—61no—
510C23C2×D3×D411yescharacteristic
511Q8—62no—
512Q8—62no—
513Q8—121no—
514Q8—121no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—32no—
523C2×C4—41no—
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—61no—
534C2×C4—61no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—244no—
558C2×C4#91512yes—
559C2×C4#100411yescharacteristic
560C2×C4C2×D3×Q811yescharacteristic
561C2×C4C2×#23911yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C2×C4#100911yescharacteristic
565C8—41no—
566C8—41no—
567C8—121no—
568C8—121no—
569C2×C3#245011yescharacteristic
570C2×C3#307811yescharacteristic
571C2×C3#308411yescharacteristic
572C2×C3#331711yescharacteristic
573C2×C3#332011yescharacteristic
574C2×C3#332311yescharacteristic
575C2×C3#333111yescharacteristic
576D3—42no—
577C22—62no—
578C22—62no—
579C22—62no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—121no—
584C22#561011yescharacteristic
585C22D3×C22⋊Q811yescharacteristic
586C22#563011yescharacteristic
587C22C2×#91511yescharacteristic
588C22#577011yescharacteristic
589C22#577511yescharacteristic
590C22#577711yescharacteristic
591C4—22no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—62no—
596C4—81no—
597C4—81no—
598C4—121no—
599C4—122no—
600C4—121no—
601C4—122no—
602C4—241no—
603C4—241no—
604C4#565711yescharacteristic
605C4#569011yescharacteristic
606C4#569111yescharacteristic
607C4#573011yescharacteristic
608C3#2643411yescharacteristic, complemented, Sylow
609C2—122no—
610C2#7795311yescharacteristic
611C2#8069411yescharacteristic
612C2#8082711yescharacteristic
613C2#8101111yescharacteristic
614C2#8108011yescharacteristic
615C2#8117111yescharacteristic
616C2#8124911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2088C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2088D3
hypocenterC3#26434
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2088

Complements

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

NH ≅ G/Ntype
3. C3×#2088C2semidirect
4. #70589C2semidirect
6. #70485C2semidirect
8. #69632C2semidirect
9. #69612C2semidirect
10. #68911C2semidirect
14. #68819C2semidirect
15. #68801C2semidirect
63. #2088D3semidirect
608. C3#26434semidirect

Maximal subgroups

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

subgroupindex
#762052
#760002
C3×#20882
#705892
#705792
#704852
#704782
#696322
#696122
#689112
#688842
#688792
#688672
#688192
#688012
#264343