Finite Group Databasebeta

← #10858935

Subgroups — #10858935

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82244C211yesmaximal, characteristic
2#73900C211yesmaximal, characteristic, complemented
3C3×#1854C211yesmaximal, characteristic, complemented
4#71580C211yesmaximal, characteristic, complemented
5#70638C211yesmaximal, characteristic, complemented
6#70609C211yesmaximal, characteristic, complemented
7#70585C211yesmaximal, characteristic, complemented
8#70568C211yesmaximal, characteristic, complemented
9#69950C211yesmaximal, characteristic, complemented
10#69933C211yesmaximal, characteristic
11#69515C211yesmaximal, characteristic
12#68899C211yesmaximal, characteristic, complemented
13#68892C211yesmaximal, characteristic, complemented
14#68811C211yesmaximal, characteristic, complemented
15#68801C211yesmaximal, characteristic, complemented
16#26421—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#380C2211yescharacteristic
27C4×#217C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4988C2211yescharacteristic
33#4984C2211yescharacteristic
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40#4957—21no—
41#4956—21no—
42#4949—21no—
43#4948—21no—
44#4940—21no—
45C4×(C2×D3)⋊C4—21no—
46#4655—21no—
47#4647—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2441—31no—
61#2363—31no—
62#2100—31no—
63#2088—31no—
64#2065—31no—
65#2025—31no—
66#1989—31no—
67#1918—31no—
68#1893—31no—
69#1875—31no—
70#1862—31no—
71#1854D311yescharacteristic, complemented
72#1850—31no—
73#1802—31no—
74#1735—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#117—22no—
82C3×#117—41no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×#217—21no—
91C2×#217D411yescharacteristic
92C2×#217D411yescharacteristic
93#930—22no—
94#930Q814yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99#926—22no—
100#926—41no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8—22no—
116C22×C3⋊C8—41no—
117C22×C3⋊C8—41no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#893—22no—
121#893—41no—
122#890—22no—
123#890—41no—
124C3×#107—22no—
125C3×#106—41no—
126C3×C22⋊C8—22no—
127C3×#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#821—22no—
135#821—41no—
136Dic3⋊C8—41no—
137Dic3⋊C8—41no—
138#809—41no—
139#809—41no—
140C23×C8—31no—
141C2×#125—31no—
142C2×#121—31no—
143C2×C4×D4—31no—
144C2×#117—31no—
145C2×#117C2×D311yescharacteristic
146C2×#107C2×D311yescharacteristic
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×C22⋊C8C2×D311yescharacteristic
155#399—61no—
156#396—61no—
157#390—31no—
158#390—61no—
159#389—31no—
160#389—31no—
161#389—31no—
162#389—61no—
163#388—61no—
164#387—61no—
165#386—61no—
166#384—31no—
167#380C2×D311yescharacteristic
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#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C23×D3—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—21no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×Q812yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—81no—
199C2×#67—81no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C2×Q812yes—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
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#4112yes—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—81no—
232C3×C42—42no—
233#217—22no—
234#217—41no—
235#217—42no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3—81no—
246C4⋊Dic3C2×Q812yes—
247C4⋊Dic3#4112yes—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—42no—
258C4×Dic3—81no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—44no—
266C22×D4—31no—
267C22×D4—61no—
268C23×C4—61no—
269C23×C4C22×D311yescharacteristic
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—62no—
273C22×C8—121no—
274C22×C8—121no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277#125—62no—
278#125—121no—
279#121—34no—
280#121—62no—
281C4×D4—62no—
282C4×D4—121no—
283#117—22no—
284#117—41no—
285#117—62no—
286#117—121no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C42—21no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42—61no—
315#107—22no—
316#106—41no—
317C4⋊C8—121no—
318C4⋊C8—121no—
319#102—62no—
320#102—121no—
321#102—121no—
322#102—121no—
323C22⋊C8—22no—
324#95—21no—
325#95—31no—
326#95—31no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C22×D411yescharacteristic
339C22×D3—22no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—41no—
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—41no—
364C2×C3×C4C22⋊Q812yes—
365C2×C3×C4C22×Q811yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371#67—84no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3#13512yes—
393D12—22no—
394D12—22no—
395D12—41no—
396D12—41no—
397C4×D3—84no—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C24—61no—
403C24D3×D411yescharacteristic
404C2×D4—32no—
405C2×D4—64no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—241no—
412C2×D4—241no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4#23512yes—
432C22×C4D3×D411yescharacteristic
433C22×C4D3×Q812yes—
434C22×C4C23×D311yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—124no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—32no—
456C4⋊C4—32no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4—241no—
468C4⋊C4#23712yes—
469C22⋊C4—42no—
470C22⋊C4—62no—
471C22⋊C4—64no—
472C22⋊C4—81no—
473C22⋊C4—122no—
474C22⋊C4—122no—
475C22⋊C4—121no—
476C22⋊C4—122no—
477C22⋊C4—241no—
478C42—42no—
479C42—61no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42—122no—
484C42—241no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3C2×C22⋊Q811yescharacteristic
493C22×C3#54611yescharacteristic
494C22×C3#54611yescharacteristic
495C22×C3C2×#13511yescharacteristic
496C22×C3#57511yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57711yescharacteristic
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—81no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C3×C4—22no—
507C3×C4—42no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4#52911yescharacteristic
512C3×C4#53811yescharacteristic
513C3×C4#55411yescharacteristic
514C3×C4#55411yescharacteristic
515Dic3—22no—
516Dic3—42no—
517Dic3—41no—
518Dic3—42no—
519Dic3—41no—
520Dic3—41no—
521Dic3—81no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—62no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23#89612yes—
536C23C2×D3×D411yescharacteristic
537D4—62no—
538D4—62no—
539D4—121no—
540D4—121no—
541D4—244no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—32no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—244no—
582C2×C4#89612yes—
583C2×C4C2×#23511yescharacteristic
584C2×C4C2×#23711yescharacteristic
585C2×C4#100411yescharacteristic
586C2×C4C2×D3×Q811yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4#100911yescharacteristic
589C8—41no—
590C8—41no—
591C8—121no—
592C8—121no—
593C2×C3—42no—
594C2×C3#245011yescharacteristic
595C2×C3#307611yescharacteristic
596C2×C3#308111yescharacteristic
597C2×C3#329711yescharacteristic
598C2×C3#330311yescharacteristic
599C2×C3#330911yescharacteristic
600C2×C3#331711yescharacteristic
601D3—82no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—122no—
610C22—122no—
611C22—122no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22C2×#89611yescharacteristic
617C22#559111yescharacteristic
618C22#565411yescharacteristic
619C22D3×#13511yescharacteristic
620C22#576811yescharacteristic
621C22#577411yescharacteristic
622C22#577811yescharacteristic
623C4—22no—
624C4—42no—
625C4—62no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—121no—
630C4—121no—
631C4—121no—
632C4—122no—
633C4—122no—
634C4—241no—
635C4#553811yescharacteristic
636C4#556011yescharacteristic
637C4#563511yescharacteristic
638C4#563611yescharacteristic
639C3#2642111yescharacteristic, complemented, Sylow
640C2—42no—
641C2—242no—
642C2#7442311yescharacteristic
643C2#7919611yescharacteristic
644C2#7924211yescharacteristic
645C2#8010211yescharacteristic
646C2#8025611yescharacteristic
647C2#8140811yescharacteristic
648C2#8154511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1854C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1854D3
hypocenterC3#26421
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. #73900C2semidirect
3. C3×#1854C2semidirect
4. #71580C2semidirect
5. #70638C2semidirect
6. #70609C2semidirect
7. #70585C2semidirect
8. #70568C2semidirect
9. #69950C2semidirect
12. #68899C2semidirect
13. #68892C2semidirect
14. #68811C2semidirect
15. #68801C2semidirect
71. #1854D3semidirect
639. C3#26421semidirect

Maximal subgroups

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

subgroupindex
#822442
#739002
C3×#18542
#715802
#706382
#706092
#705852
#705682
#699502
#699332
#695152
#688992
#688922
#688112
#688012
#264213