Finite Group Databasebeta

← #10824340

Subgroups — #10824340

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×SD16C211yesmaximal, characteristic
2C4×#944C211yesmaximal, characteristic, complemented
3#82319C211yesmaximal, characteristic
4C2×C4×#214C211yesmaximal, characteristic
5#72057C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic
7C4×#835C211yesmaximal, characteristic, complemented
8#70589C211yesmaximal, characteristic, complemented
9#70567C211yesmaximal, characteristic, complemented
10#70261C211yesmaximal, characteristic, complemented
11#70246C211yesmaximal, characteristic, complemented
12#70217C211yesmaximal, characteristic
13#70165C211yesmaximal, characteristic
14C4×#816C211yesmaximal, characteristic, complemented
15C4×Dic3⋊C8C211yesmaximal, characteristic, complemented
16#23677—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#926—21no—
23C2×C4×Dic6C2211yescharacteristic
24C3×C4×SD16—24no—
25C2×C3×#106C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C4×#217—21no—
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C4×#214—24no—
41#4984C2211yescharacteristic
42#4984C2211yescharacteristic
43#4976—21no—
44#4976—21no—
45#4975C2211yescharacteristic
46#4973C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4951—21no—
50#4951—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C4×Dic3⋊C4—21no—
53C4×Dic3⋊C4—21no—
54C42×Dic3C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×C4×SD16D311yescharacteristic
64C4×C4⋊Q8—31no—
65#2293—31no—
66#2084—31no—
67#2071—31no—
68#2066—31no—
69#2066—31no—
70#2055—31no—
71#2055—31no—
72#2043—31no—
73#2013—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×Dic6C2311yescharacteristic
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×D4—22no—
89C3×C4×D4D414yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100#944—22no—
101#944—22no—
102C2×#214—22no—
103C2×#214—22no—
104C2×#214—22no—
105#926—44no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3D411yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#106—22no—
129C3×#103—22no—
130C3×#102—22no—
131C3×C4×C8—22no—
132#835—22no—
133#835—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#816—22no—
143#816—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—22no—
146#810—22no—
147#808—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16C2×D311yescharacteristic
151C2×C4⋊Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4C2×D311yescharacteristic
155C2×#117—61no—
156C4×SD16—24no—
157C4×SD16—64no—
158C2×#106—31no—
159C2×#106C2×D311yescharacteristic
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171C2×C4×C8C2×D311yescharacteristic
172#391—31no—
173#391—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#389—61no—
179#389—61no—
180#388—61no—
181#388—61no—
182#384—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185C4×C4⋊C4—61no—
186C4×C22⋊C4—61no—
187C43—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C2×D412yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6#4112yes—
230C3×SD16—28no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8#4112yes—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4C2×D412yes—
242C3×C22⋊C4—24no—
243C3×C42—42no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#217—42no—
247#217—42no—
248#217—42no—
249#217—42no—
250#214—28no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—42no—
258C4⋊Dic3—42no—
259Dic3⋊C4—24no—
260Dic3⋊C4—24no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264C4×Dic3—22no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—22no—
270C4×Dic3—41no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C4×Dic3SD1614yescomplemented
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8#4112yes—
280C22×Q8—31no—
281C22×Q8C22×D311yescharacteristic
282C22×D4C22×D311yescharacteristic
283C23×C4—21no—
284C2×SD16—22no—
285C2×SD16—22no—
286C2×SD16—22no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—62no—
295C4⋊Q8—62no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—62no—
299C4×Q8—62no—
300C4×D4—22no—
301C4×D4#6714yes—
302#117—124no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—21no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#106—22no—
332#106—62no—
333C4⋊C8—62no—
334C4⋊C8—62no—
335#103—22no—
336#103—62no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—22no—
342#102—62no—
343C4×C8—22no—
344C4×C8—62no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3C22×D411yescharacteristic
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—41no—
367C3×D4—41no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—48no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4#12112yes—
381C2×C3×C4#12112yes—
382C2×C3×C4C2×SD1612yes—
383C2×C3×C4#13712yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3C2×SD1612yes—
412C2×Dic3C2×SD1612yes—
413C2×Dic3#13512yes—
414Dic6—22no—
415Dic6—22no—
416Dic6—41no—
417Dic6—41no—
418C3×C8—22no—
419C3×C8—41no—
420C3⋊C8—22no—
421C3⋊C8—41no—
422C24—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—32no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8#23512yes—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×#6712yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458SD16—28no—
459SD16—68no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—32no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8#23512yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—64no—
477C4⋊C4—64no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4C2×#6712yes—
490C22⋊C4—24no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C42—34no—
496C42—42no—
497C42—62no—
498C42—62no—
499C42—61no—
500C42—61no—
501C42—61no—
502C42—61no—
503C42—121no—
504C42—122no—
505C42—122no—
506C42#23712yes—
507C42C2×#6712yes—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3C2×#12111yescharacteristic
516C22×C3#52511yescharacteristic
517C22×C3#53211yescharacteristic
518C22×C3C22×SD1611yescharacteristic
519C22×C3C2×#13511yescharacteristic
520C22×C3C2×#13711yescharacteristic
521C22×C3#57511yescharacteristic
522C3×C4—22no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—81no—
526C3×C4#46114yes—
527C3×C4#52511yescharacteristic
528C3×C4#52511yescharacteristic
529C3×C4#52911yescharacteristic
530C3×C4#52911yescharacteristic
531Dic3—22no—
532Dic3—22no—
533Dic3—22no—
534Dic3—42no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538Dic3—81no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C2×D3×D411yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—62no—
552Q8—62no—
553Q8—121no—
554Q8—121no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—32no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—61no—
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—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4D3×SD1612yes—
596C2×C4#91512yes—
597C2×C4#93912yes—
598C2×C4#93912yes—
599C2×C4C2×#23511yescharacteristic
600C2×C4C2×#23511yescharacteristic
601C2×C4C2×#23711yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4C22×#6711yescharacteristic
605C8—22no—
606C8—41no—
607C8—62no—
608C8—121no—
609C2×C3—82no—
610C2×C3#239211yescharacteristic
611C2×C3C2×#46111yescharacteristic
612C2×C3#303511yescharacteristic
613C2×C3#310311yescharacteristic
614C2×C3#310811yescharacteristic
615C2×C3#316711yescharacteristic
616C2×C3#317211yescharacteristic
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22#554711yescharacteristic
625C22#554811yescharacteristic
626C22C2×D3×SD1611yescharacteristic
627C22C2×#91511yescharacteristic
628C22D3×#13511yescharacteristic
629C22#576811yescharacteristic
630C22C2×#93911yescharacteristic
631C4—22no—
632C4—42no—
633C4—42no—
634C4—62no—
635C4—62no—
636C4—62no—
637C4—81no—
638C4—121no—
639C4—122no—
640C4—241no—
641C4—241no—
642C4—241no—
643C4#516214yes—
644C4#553811yescharacteristic
645C4#555711yescharacteristic
646C4#579111yescharacteristic
647C4#582811yescharacteristic
648C3#2367711yescharacteristic, complemented, Sylow
649C2—82no—
650C2#7965311yescharacteristic
651C2#7965611yescharacteristic
652C2#7966211yescharacteristic
653C2#7966511yescharacteristic
654C2#8234511yescharacteristic
655C2C2×#516211yescharacteristic
656C2#8390011yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×C4×SD16D3
hypocenterC3#23677
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C4×#944C2semidirect
7. C4×#835C2semidirect
8. #70589C2semidirect
9. #70567C2semidirect
10. #70261C2semidirect
11. #70246C2semidirect
14. C4×#816C2semidirect
15. C4×Dic3⋊C8C2semidirect
273. C4×Dic3SD16semidirect
648. C3#23677semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SD162
C4×#9442
#823192
C2×C4×#2142
#720572
#720512
C4×#8352
#705892
#705672
#702612
#702462
#702172
#701652
C4×#8162
C4×Dic3⋊C82
#236773