Finite Group Databasebeta

← #10824362

Subgroups — #10824362

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Q16C211yesmaximal, characteristic
2C4×#944C211yesmaximal, characteristic
3#82451C211yesmaximal, characteristic
4C2×C4×#216C211yesmaximal, characteristic
5#72073C211yesmaximal, characteristic
6#72068C211yesmaximal, characteristic
7C4×#835C211yesmaximal, characteristic
8#70643C211yesmaximal, characteristic
9#70619C211yesmaximal, characteristic
10#70285C211yesmaximal, characteristic
11#70262C211yesmaximal, characteristic
12#70246C211yesmaximal, characteristic
13#70233C211yesmaximal, characteristic
14C4×#816C211yesmaximal, characteristic
15C4×Dic3⋊C8C211yesmaximal, characteristic
16#23679—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×C4×Q16—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C4×#216—24no—
39#4984C2211yescharacteristic
40#4984C2211yescharacteristic
41#4980—21no—
42#4980—21no—
43#4976—21no—
44#4976—21no—
45#4975C2211yescharacteristic
46#4973C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C4×C4⋊Dic3—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51#4932—21no—
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×Q16—31no—
63C2×C4×Q16D311yescharacteristic
64C4×C4⋊Q8—31no—
65#2295—31no—
66#2083—31no—
67#2072—31no—
68#2068—31no—
69#2068—31no—
70#2056—31no—
71#2056—31no—
72#2043—31no—
73#2016—31no—
74C4×C4⋊C8—31no—
75C4×#103—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8D414yes—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96#944—22no—
97#944—22no—
98C2×#216—22no—
99C2×#216—22no—
100C2×#216—22no—
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×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#107—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×C4×C8—22no—
127#835—22no—
128#835—22no—
129#835—22no—
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—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#816—22no—
142#816—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145#810—22no—
146#807—22no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×Q16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×Q8C2×D311yescharacteristic
154C4×Q16—24no—
155C4×Q16—64no—
156C2×#107—31no—
157C2×#107C2×D311yescharacteristic
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×C4×C8—31no—
169C2×C4×C8C2×D311yescharacteristic
170#391—31no—
171#391—31no—
172#391—31no—
173#391—31no—
174#389—31no—
175#389—31no—
176#389—61no—
177#389—61no—
178#387—61no—
179#387—61no—
180#384—31no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4—61no—
185#376—61no—
186C43—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8C2×D412yes—
204C2×C3×Q8#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6#4112yes—
226C3×Q16—28no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8#4112yes—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4C2×D412yes—
239C3×C42—24no—
240C3×C42—42no—
241C3×C42C2×D412yes—
242C3×C42#4112yes—
243#216—28no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252Dic3⋊C4—24no—
253Dic3⋊C4—24no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257C4×Dic3—22no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—22no—
263C4×Dic3—41no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C4×Dic3Q1614yescomplemented
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8#4112yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×Q8C22×D311yescharacteristic
276C2×Q16—22no—
277C2×Q16—22no—
278C2×Q16—22no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—62no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C4⋊Q8—62no—
287C4⋊Q8—62no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8#6714yes—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—22no—
320#107—62no—
321C4⋊C8—62no—
322C4⋊C8—62no—
323#103—22no—
324#103—22no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331C4×C8—22no—
332C4×C8—62no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×Q8—41no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
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×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4#12112yes—
368C2×C3×C4#12112yes—
369C2×C3×C4C2×Q1612yes—
370C2×C3×C4#13712yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
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×Dic3C2×Q1612yes—
399C2×Dic3C2×Q1612yes—
400C2×Dic3#13512yes—
401Dic6—22no—
402Dic6—22no—
403Dic6—41no—
404Dic6—41no—
405C3×C8—22no—
406C3×C8—41no—
407C3⋊C8—22no—
408C3⋊C8—41no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8#23512yes—
426C2×Q8C2×#6712yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4D3×D411yescharacteristic
440C22×C4D3×D411yescharacteristic
441C22×C4C23×D311yescharacteristic
442Q16—28no—
443Q16—68no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—32no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8#23512yes—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4C2×#6712yes—
475C42—24no—
476C42—34no—
477C42—42no—
478C42—62no—
479C42—62no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42—61no—
484C42—121no—
485C42—122no—
486C42—122no—
487C42#23712yes—
488C42C2×#6712yes—
489C22×C3C2×#12111yescharacteristic
490C22×C3#52511yescharacteristic
491C22×C3#53211yescharacteristic
492C22×C3C22×Q1611yescharacteristic
493C22×C3C2×#13511yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57511yescharacteristic
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—81no—
501C3×C4#46214yes—
502C3×C4#52511yescharacteristic
503C3×C4#52511yescharacteristic
504C3×C4#52911yescharacteristic
505C3×C4#52911yescharacteristic
506Dic3—22no—
507Dic3—22no—
508Dic3—22no—
509Dic3—42no—
510Dic3—41no—
511Dic3—81no—
512Dic3—81no—
513Dic3—81no—
514C23C2×D3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—62no—
524Q8—62no—
525Q8—121no—
526Q8—121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4D3×Q1612yes—
566C2×C4#91812yes—
567C2×C4#93912yes—
568C2×C4#93912yes—
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23511yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4C22×#6711yescharacteristic
575C8—22no—
576C8—41no—
577C8—62no—
578C8—121no—
579C2×C3#239211yescharacteristic
580C2×C3C2×#46211yescharacteristic
581C2×C3#303511yescharacteristic
582C2×C3#310411yescharacteristic
583C2×C3#310811yescharacteristic
584C2×C3#317011yescharacteristic
585C2×C3#317211yescharacteristic
586C22#554711yescharacteristic
587C22#554811yescharacteristic
588C22C2×D3×Q1611yescharacteristic
589C22C2×#91811yescharacteristic
590C22D3×#13511yescharacteristic
591C22#576811yescharacteristic
592C22C2×#93911yescharacteristic
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—62no—
598C4—62no—
599C4—62no—
600C4—81no—
601C4—121no—
602C4—122no—
603C4—241no—
604C4—241no—
605C4—241no—
606C4#518214yes—
607C4#553811yescharacteristic
608C4#555711yescharacteristic
609C4#579111yescharacteristic
610C4#582811yescharacteristic
611C3#2367911yescharacteristic, complemented, Sylow
612C2#7972211yescharacteristic
613C2#7972311yescharacteristic
614C2#7972511yescharacteristic
615C2#7972611yescharacteristic
616C2#8234511yescharacteristic
617C2C2×#518211yescharacteristic
618C2#8390011yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×C4×Q16D3
hypocenterC3#23679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
266. C4×Dic3Q16semidirect
611. C3#23679semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q162
C4×#9442
#824512
C2×C4×#2162
#720732
#720682
C4×#8352
#706432
#706192
#702852
#702622
#702462
#702332
C4×#8162
C4×Dic3⋊C82
#236793