Finite Group Databasebeta

← #10845377

Subgroups — #10845377

Order 768.

882 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5128C211yesmaximal, characteristic, complemented
2#82666C211yesmaximal, characteristic
3C2×#4766C211yesmaximal, characteristic, complemented
4#73940C211yesmaximal, characteristic, complemented
5C3×#1895C211yesmaximal, characteristic, complemented
6#71988C211yesmaximal, characteristic, complemented
7#71679C211yesmaximal, characteristic, complemented
8#70203C211yesmaximal, characteristic, complemented
9#70176C211yesmaximal, characteristic, complemented
10Dic3×#102C211yesmaximal, characteristic, complemented
11#69507C211yesmaximal, characteristic, complemented
12#68870C211yesmaximal, characteristic, complemented
13#68830C211yesmaximal, characteristic, complemented
14#68804C211yesmaximal, characteristic, complemented
15#68797C211yesmaximal, characteristic, complemented
16#25344—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×#938C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×C4×#67C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C2×#890C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#380C2211yescharacteristic
27#5223—21no—
28#5220—21no—
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#5128—24no—
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42#4987C2211yescharacteristic
43#4977C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4959—21no—
48#4958—21no—
49#4952—21no—
50#4951—21no—
51#4766—24no—
52#4670—21no—
53#4658—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#436—31no—
64#2562—31no—
65#2535—31no—
66#2077—31no—
67#2075—31no—
68#2034—31no—
69#2033—31no—
70#1991—31no—
71#1949—31no—
72#1895—31no—
73#1895D311yescharacteristic, complemented
74#1809—31no—
75#1809—31no—
76C4×#102—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67D411yescharacteristic
82C23×Dic3—21no—
83C22×C4×D3C2311yescharacteristic
84C2×C3×M16—22no—
85C2×C3×M16—22no—
86C2×C3×M16—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#121—22no—
90C3×#121—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217D411yescharacteristic
101#938—22no—
102#938—41no—
103D4×Dic3—22no—
104D4×Dic3—22no—
105C4×#67—22no—
106C4×#67—22no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4D411yescharacteristic
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)—22no—
124C2×ΓL(1,𝔽25)—41no—
125#890—22no—
126#890—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×#95—21no—
130#832—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#830—22no—
141(C2×D3)⋊C8—22no—
142#818—22no—
143#817—41no—
144Dic3⋊C8D414yescomplemented
145C8×Dic3—22no—
146#808—22no—
147#807—41no—
148C22×M16—31no—
149C22×M16C2×D311yescharacteristic
150C2×#125—31no—
151C2×#121—31no—
152C2×#121C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155#478—64no—
156#436—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102C2×D311yescharacteristic
165C2×#102C2×D311yescharacteristic
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×C4×C8—31no—
169#400—61no—
170#397—61no—
171#394—61no—
172#392—61no—
173#388—61no—
174#388—61no—
175#386—61no—
176#386—61no—
177#382—31no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380C2×D311yescharacteristic
184C4×C4⋊C4—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C24×C3—21no—
194C24×C3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—42no—
202C2×C3×D4—42no—
203C2×C3×D4—81no—
204C2×C3×D4#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×#67—22no—
217C2×#67—41no—
218C2×#67—41no—
219C2×#67—42no—
220C2×#67—42no—
221C2×#67—81no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—42no—
227C22×Dic3—81no—
228C22×Dic3—81no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C3×M16—44no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C2×C3×C8#4112yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253#217—22no—
254#217—42no—
255#217—42no—
256#217—81no—
257#217—81no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271Dic3⋊C4—22no—
272Dic3⋊C4—42no—
273C4×Dic3—81no—
274C4×Dic3C2×D412yes—
275C4×Dic3#4112yes—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8C2×D412yes—
282ΓL(1,𝔽25)—44no—
283C22×D4—21no—
284C22×D4—31no—
285C22×D4—61no—
286C22×D4C22×D311yescharacteristic
287C23×C4—31no—
288C23×C4—61no—
289C23×C4C22×D311yescharacteristic
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—41no—
293C2×M16—62no—
294C2×M16—62no—
295C2×M16—121no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300#125—62no—
301#125—62no—
302#121—22no—
303#121—41no—
304#121—62no—
305#121—121no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C42—31no—
332C2×C42—61no—
333#107—62no—
334#107—121no—
335#106—62no—
336#106—121no—
337C4⋊C8—34no—
338#102—22no—
339#102—22no—
340#102—62no—
341#102—62no—
342C22⋊C8—62no—
343C22⋊C8—62no—
344C4×C8—62no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—21no—
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3C22×D411yescharacteristic
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—84no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4H(5,𝔽2)11yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403#67—44no—
404#67—84no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—84no—
425C2×Dic3#12112yes—
426C2×Dic3#12112yes—
427C4×D3—41no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C3×C8—22no—
432C3×C8—22no—
433C3⋊C8—22no—
434C3⋊C8—41no—
435C24—21no—
436C24—31no—
437C24D3×D411yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—62no—
446C2×D4—81no—
447C2×D4—121no—
448C2×D4—121no—
449C2×D4—122no—
450C2×D4—122no—
451C2×D4—241no—
452C2×D4#23512yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—122no—
476C22×C4—241no—
477C22×C4—241no—
478C22×C4D3×D411yescharacteristic
479C22×C4C23×D311yescharacteristic
480M16—44no—
481M16—124no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—32no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8D3×D412yes—
493C2×C8#23712yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—62no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—122no—
510C4⋊C4—122no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C42—32no—
523C42—32no—
524C42—241no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—42no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3C2×#12111yescharacteristic
540C22×C3#54611yescharacteristic
541C22×C3#54711yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57511yescharacteristic
544C22×C3#57611yescharacteristic
545C22×C3#57711yescharacteristic
546C2×D3—41no—
547C2×D3—41no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C3×C4—41no—
554C3×C4—81no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4#52911yescharacteristic
558C3×C4#54211yescharacteristic
559C3×C4#54611yescharacteristic
560C3×C4#54811yescharacteristic
561Dic3—22no—
562Dic3—22no—
563Dic3—41no—
564Dic3—81no—
565Dic3—81no—
566Dic3—81no—
567Dic3—81no—
568C23—22no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23C2×D3×D411yescharacteristic
590D4—22no—
591D4—22no—
592D4—41no—
593D4—41no—
594D4—84no—
595D4—124no—
596D4—244no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—244no—
639C2×C4C2×#23511yescharacteristic
640C2×C4C2×D3×D411yescharacteristic
641C2×C4C2×#23711yescharacteristic
642C2×C4#100411yescharacteristic
643C2×C4D3×#4111yescharacteristic
644C2×C4#101011yescharacteristic
645C8—22no—
646C8—22no—
647C8—62no—
648C8—121no—
649C2×C3—81no—
650C2×C3—82no—
651C2×C3#242711yescharacteristic
652C2×C3#305411yescharacteristic
653C2×C3#306211yescharacteristic
654C2×C3#332211yescharacteristic
655C2×C3#332311yescharacteristic
656C2×C3#335111yescharacteristic
657C2×C3#335511yescharacteristic
658D3—81no—
659C22—42no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—42no—
667C22—41no—
668C22—41no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—121no—
674C22—121no—
675C22—121no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22#549011yescharacteristic
681C22D3×#12111yescharacteristic
682C22#561111yescharacteristic
683C22#577311yescharacteristic
684C22#577311yescharacteristic
685C22#577411yescharacteristic
686C22#577811yescharacteristic
687C4—41no—
688C4—62no—
689C4—62no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—121no—
694C4—241no—
695C4—241no—
696C4—241no—
697C4—241no—
698C4#553811yescharacteristic
699C4#555211yescharacteristic
700C4#559111yescharacteristic
701C4#566211yescharacteristic
702C3#2534411yescharacteristic, complemented, Sylow
703C2—81no—
704C2—82no—
705C2—241no—
706C2#7469711yescharacteristic
707C2#7890311yescharacteristic
708C2#7893811yescharacteristic
709C2#8080911yescharacteristic
710C2#8083311yescharacteristic
711C2#8086111yescharacteristic
712C2#8086311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1895C2
socleC23×C3C22×D4
prime radical#82666C2
hypercenter#1895D3
hypocenterC3#25344
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
1. C2×#5128C2semidirect
3. C2×#4766C2semidirect
4. #73940C2semidirect
5. C3×#1895C2semidirect
6. #71988C2semidirect
7. #71679C2semidirect
8. #70203C2semidirect
9. #70176C2semidirect
10. Dic3×#102C2semidirect
11. #69507C2semidirect
12. #68870C2semidirect
13. #68830C2semidirect
14. #68804C2semidirect
15. #68797C2semidirect
73. #1895D3semidirect
144. Dic3⋊C8D4semidirect
702. C3#25344semidirect

Maximal subgroups

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

subgroupindex
C2×#51282
#826662
C2×#47662
#739402
C3×#18952
#719882
#716792
#702032
#701762
Dic3×#1022
#695072
#688702
#688302
#688042
#687972
#253443