Finite Group Databasebeta

← #10824754

Subgroups — #10824754

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#432C211yesmaximal, characteristic
2C4×#924C211yesmaximal, characteristic
3C4×#924C211yesmaximal, characteristic
4C2×#4716C211yesmaximal, characteristic, complemented
5#70489C211yesmaximal, characteristic, complemented
6#70488C211yesmaximal, characteristic, complemented
7#70487C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#69964C211yesmaximal, characteristic
10#69940C211yesmaximal, characteristic
11#69937C211yesmaximal, characteristic
12#69936C211yesmaximal, characteristic
13#69814C211yesmaximal, characteristic, complemented
14#69813C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#23709—31nomaximal, Sylow
17C22×C3×C42C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C22×Dic12C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#432—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5212C2211yescharacteristic
31#5212C2211yescharacteristic
32#5210—21no—
33#5210—21no—
34C4×#217—21no—
35C4×#217—21no—
36C2×#818C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4933—21no—
48#4933—21no—
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4—21no—
51#4929C2211yescharacteristic
52#4929C2211yescharacteristic
53#4928C2211yescharacteristic
54#4928C2211yescharacteristic
55#4716—24no—
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#441—31no—
63C2×#432D311yescharacteristic
64C4×C22⋊Q8—31no—
65C4×C22⋊Q8—31no—
66#2088—31no—
67#2077—31no—
68#2068—31no—
69#2066—31no—
70#2056—31no—
71#2036—31no—
72#2035—31no—
73#2001—31no—
74#1776—31no—
75#1763—31no—
76#1763—31no—
77C23×C3×C4D411yescharacteristic
78C23×C3×C4D411yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102#924—22no—
103#924—22no—
104#924—22no—
105#924—22no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×Dic12—22no—
117C2×Dic12—22no—
118C2×Dic12—22no—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×C4⋊C8—22no—
124C3×C4⋊C8—22no—
125C3×C22⋊C8—22no—
126C3×C22⋊C8—22no—
127C3×#97—22no—
128C3×#97D414yes—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#818—22no—
140#817—22no—
141#817—22no—
142#816—22no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C22×C42C2×D311yescharacteristic
154#441—64no—
155#432—24no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×C4⋊C8C2×D311yescharacteristic
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×C22⋊C8C2×D311yescharacteristic
168C2×C22⋊C8C2×D311yescharacteristic
169C2×#97C2×D311yescharacteristic
170C2×#97C2×D311yescharacteristic
171#391—31no—
172#391—31no—
173#389—31no—
174#389—31no—
175#387—61no—
176#387—61no—
177#385—61no—
178#385—61no—
179#380—31no—
180#380—31no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4—61no—
185C4×C22⋊C4—61no—
186C4×C22⋊C4—61no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3C2×D411yescharacteristic
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—41no—
205C22×C3×C4—41no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6#4112yes—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C2×D412yes—
243C2×C3×C8C2×D412yes—
244C3×C42—22no—
245C3×C42—22no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250#217—24no—
251#217—24no—
252#217—42no—
253#217—42no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3#4112yes—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276Dic12—28no—
277C22×Q8—31no—
278C22×Q8—31no—
279C23×C4D1211yescharacteristic
280C23×C4D1211yescharacteristic
281C23×C4C22×D311yescharacteristic
282C2×Q16—62no—
283C2×Q16—62no—
284C2×Q16—62no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C22⋊Q8—62no—
290C22⋊Q8—62no—
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42D1211yescharacteristic
320C2×C42D1211yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#106—62no—
325#106—62no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334C22⋊C8—22no—
335C22⋊C8—22no—
336#97—22no—
337#97#6714yes—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3#13712yes—
355C23×C3C22×D411yescharacteristic
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—44no—
369C2×C3×C4—44no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4#12112yes—
375C2×C3×C4#12112yes—
376C2×C3×C4#13712yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407Dic6—22no—
408Dic6—22no—
409Dic6—22no—
410Dic6—22no—
411Dic6—41no—
412Dic6—41no—
413Dic6—41no—
414Dic6—41no—
415C3×C8—22no—
416C3×C8—22no—
417C3×C8—41no—
418C3×C8—41no—
419C24C2×D1211yescharacteristic
420C2×Q8—32no—
421C2×Q8—32no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4C2×D1211yescharacteristic
453C22×C4C2×D1211yescharacteristic
454C22×C4C2×D1211yescharacteristic
455C22×C4C2×D1211yescharacteristic
456C22×C4C2×D1211yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—68no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8C2×#6712yes—
470C2×C8C2×#6712yes—
471C4⋊C4—32no—
472C4⋊C4—32no—
473C4⋊C4—64no—
474C4⋊C4—64no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C22⋊C4—64no—
490C22⋊C4—64no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—122no—
498C42—122no—
499C42—122no—
500C42—122no—
501C42#23512yes—
502C42C2×#6712yes—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3C2×#12111yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3#52511yescharacteristic
513C22×C3C2×#13711yescharacteristic
514C22×C3C2×#13711yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57511yescharacteristic
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—22no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4#52511yescharacteristic
523C3×C4#52911yescharacteristic
524C3×C4#52911yescharacteristic
525C3×C4#53211yescharacteristic
526Dic3—42no—
527Dic3—42no—
528Dic3—42no—
529Dic3—42no—
530Dic3—81no—
531Dic3—81no—
532Dic3—81no—
533Dic3—81no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#90912yes—
541C23C22×D1211yescharacteristic
542Q8—62no—
543Q8—62no—
544Q8—62no—
545Q8—62no—
546Q8—121no—
547Q8—121no—
548Q8—121no—
549Q8—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4#90912yes—
593C2×C4#93012yes—
594C2×C4#93012yes—
595C2×C4C2×#23511yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C2×C4D3×#4111yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4C22×#6711yescharacteristic
601C8—22no—
602C8—22no—
603C8—41no—
604C8—41no—
605C2×C3—42no—
606C2×C3#239211yescharacteristic
607C2×C3#305611yescharacteristic
608C2×C3#305611yescharacteristic
609C2×C3#311111yescharacteristic
610C2×C3#311311yescharacteristic
611C2×C3#311611yescharacteristic
612C2×C3#311711yescharacteristic
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22#552411yescharacteristic
621C22#552411yescharacteristic
622C22C2×#90911yescharacteristic
623C22C2×#90911yescharacteristic
624C22#574811yescharacteristic
625C22#574811yescharacteristic
626C22C2×#93011yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—122no—
633C4—122no—
634C4—122no—
635C4—122no—
636C4—241no—
637C4—241no—
638C4—241no—
639C4—241no—
640C4#552811yescharacteristic
641C4#552811yescharacteristic
642C4#582811yescharacteristic
643C4#583211yescharacteristic
644C3#2370911yescharacteristic, complemented, Sylow
645C2—42no—
646C2#7934511yescharacteristic
647C2#7934811yescharacteristic
648C2#7935611yescharacteristic
649C2#7935711yescharacteristic
650C2#8227311yescharacteristic
651C2#8377611yescharacteristic
652C2#8377611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#432C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#432D3
hypocenterC3#23709
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
4. C2×#4716C2semidirect
5. #70489C2semidirect
6. #70488C2semidirect
7. #70487C2semidirect
8. #70464C2semidirect
13. #69814C2semidirect
14. #69813C2semidirect
15. #69813C2semidirect
644. C3#23709semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4322
C4×#9242
C2×#47162
#704892
#704882
#704872
#704642
#699642
#699402
#699372
#699362
#698142
#698132
#237093