Finite Group Databasebeta

← #10824394

Subgroups — #10824394

Order 768.

880 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#947C211yesmaximal, characteristic
2C2×C4×#216C211yesmaximal, characteristic, complemented
3C2×C4×#204C211yesmaximal, characteristic
4#73618C211yesmaximal, characteristic, complemented
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70285C211yesmaximal, characteristic, complemented
7#70279C211yesmaximal, characteristic
8#70278C211yesmaximal, characteristic
9#70247C211yesmaximal, characteristic, complemented
10C4×Dic3⋊C8C211yesmaximal, characteristic, complemented
11C4×#809C211yesmaximal, characteristic
12#68915C211yesmaximal, characteristic, complemented
13#68901C211yesmaximal, characteristic
14#68863C211yesmaximal, characteristic
15#68836C211yesmaximal, characteristic, complemented
16#23681—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C4×#216—24no—
35#4990—21no—
36#4990—21no—
37#4988C2211yescharacteristic
38#4977C2211yescharacteristic
39#4973C2211yescharacteristic
40#4973C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×(C2×D3)⋊C4—21no—
48C4×(C2×D3)⋊C4—21no—
49#4932—21no—
50C4×Dic3⋊C4—21no—
51C42×Dic3C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53C4×#204—24no—
54#4652—21no—
55#4652—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×Q16—31no—
63C2×C4×SD16—31no—
64C4×#124—31no—
65#2295—31no—
66#2085—31no—
67#2074—31no—
68#2067—31no—
69#2066—31no—
70#2056—31no—
71#2055—31no—
72#2042—31no—
73#2015—31no—
74C4×C4⋊C8—31no—
75C4×#103D311yescharacteristic, complemented
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92#947—22no—
93#947—22no—
94C2×#216—22no—
95C2×#216—22no—
96C2×#216—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—22no—
119C4×D12—22no—
120C4×D12—22no—
121C4×Dic6—22no—
122C4×Dic6D414yes—
123C3×#103—22no—
124C3×#103—22no—
125C3×C4×C8—22no—
126C3×#95—21no—
127#835—22no—
128C3⋊#95—21no—
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—
138#821—22no—
139#817—22no—
140#816—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#810—22no—
144#809—22no—
145#809—22no—
146#807—22no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154C4×Q16—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8—31no—
169C2×C4×C8C2×D311yescharacteristic
170#391—31no—
171#391—31no—
172#391—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#387—61no—
179#387—61no—
180#382—31no—
181C4×C4⋊C4—61no—
182C4×C4⋊C4C2×D311yescharacteristic
183C4×C22⋊C4—61no—
184C4×C22⋊C4—61no—
185#376—61no—
186C43—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12#4112yes—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6C2×D412yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8#4112yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C42—42no—
243C3×C42—42no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#216—28no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—24no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—24no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259C4×Dic3—24no—
260C4×Dic3—22no—
261C4×Dic3—22no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—22no—
267C4×Dic3—41no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8#4112yes—
274#204—28no—
275C22×Q8—31no—
276C22×Q8C22×D311yescharacteristic
277C22×D4—31no—
278C23×C4—61no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—62no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#124—62no—
290#124—62no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—34no—
294C4×Q8—62no—
295C4×D4—62no—
296C4×D4—62no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321#107—62no—
322#106—62no—
323C4⋊C8—62no—
324C4⋊C8—62no—
325#103—22no—
326#103—22no—
327#103—62no—
328#103—62no—
329#103—62no—
330#102—62no—
331#102—62no—
332#102—62no—
333C4×C8—22no—
334C4×C8—62no—
335#95—21no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4#12112yes—
373C2×C3×C4#12112yes—
374C2×C3×C4#13512yes—
375C2×C3×C4#13712yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3#13512yes—
403D12—22no—
404D12—22no—
405D12—41no—
406D12—41no—
407C4×D3—84no—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412C3×C8—22no—
413C3×C8—41no—
414C3⋊C8—22no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—32no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8#23512yes—
429C2×D4—32no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—68no—
453SD16—68no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—32no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8#23512yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—32no—
473C4⋊C4—42no—
474C4⋊C4—64no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—42no—
486C42—42no—
487C42—62no—
488C42—64no—
489C42—62no—
490C42—62no—
491C42—61no—
492C42—61no—
493C42—61no—
494C42—61no—
495C42—121no—
496C42#23512yes—
497C42D3×D412yes—
498C22×C3C2×#12111yescharacteristic
499C22×C3#52511yescharacteristic
500C22×C3#53211yescharacteristic
501C22×C3C2×#13511yescharacteristic
502C22×C3C2×#13511yescharacteristic
503C22×C3C2×#13711yescharacteristic
504C22×C3#57511yescharacteristic
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—42no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4#46414yes—
518C3×C4#52511yescharacteristic
519C3×C4#52511yescharacteristic
520C3×C4#52911yescharacteristic
521C3×C4#52911yescharacteristic
522Dic3—22no—
523Dic3—41no—
524Dic3—41no—
525Dic3—42no—
526Dic3—41no—
527Dic3—42no—
528Dic3—81no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C2×D3×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—62no—
542Q8—62no—
543Q8—121no—
544Q8—121no—
545D4—62no—
546D4—62no—
547D4—121no—
548D4—121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—32no—
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—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—244no—
585C2×C4#88412yes—
586C2×C4#88412yes—
587C2×C4#91612yes—
588C2×C4#91812yes—
589C2×C4C2×#23511yescharacteristic
590C2×C4C2×#23511yescharacteristic
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C8—22no—
596C8—41no—
597C8—62no—
598C8—121no—
599C2×C3#239211yescharacteristic
600C2×C3C2×#46411yescharacteristic
601C2×C3#303511yescharacteristic
602C2×C3#310211yescharacteristic
603C2×C3#310711yescharacteristic
604C2×C3#316811yescharacteristic
605C2×C3#317211yescharacteristic
606D3—82no—
607C22—122no—
608C22—122no—
609C22—122no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22C2×#88411yescharacteristic
615C22#554711yescharacteristic
616C22#554811yescharacteristic
617C22C2×#91611yescharacteristic
618C22C2×#91811yescharacteristic
619C22D3×#13511yescharacteristic
620C22#576811yescharacteristic
621C4—22no—
622C4—42no—
623C4—62no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—122no—
628C4—121no—
629C4—122no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4#479814yes—
634C4#547911yescharacteristic
635C4#552811yescharacteristic
636C4#554711yescharacteristic
637C4#554711yescharacteristic
638C3#2368111yescharacteristic, complemented, Sylow
639C2—242no—
640C2#7358111yescharacteristic
641C2C2×#479811yescharacteristic
642C2#7886111yescharacteristic
643C2#7966311yescharacteristic
644C2#7969411yescharacteristic
645C2#7972611yescharacteristic
646C2#7974711yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC4×#103D3
hypocenterC3#23681
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×C4×#216C2semidirect
4. #73618C2semidirect
5. C3×C4×#103C2semidirect
6. #70285C2semidirect
9. #70247C2semidirect
10. C4×Dic3⋊C8C2semidirect
12. #68915C2semidirect
15. #68836C2semidirect
75. C4×#103D3semidirect
638. C3#23681semidirect

Maximal subgroups

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

subgroupindex
C4×#9472
C2×C4×#2162
C2×C4×#2042
#736182
C3×C4×#1032
#702852
#702792
#702782
#702472
C4×Dic3⋊C82
C4×#8092
#689152
#689012
#688632
#688362
#236813