Finite Group Databasebeta

← #10828173

Subgroups — #10828173

Order 768.

1344 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5598C211yesmaximal, characteristic
2C22×#816C211yesmaximal, characteristic, complemented
3C2×#5034C211yesmaximal, characteristic, complemented
4#74028C211yesmaximal, characteristic, complemented
5C3×#1893C211yesmaximal, characteristic, complemented
6#71975C211yesmaximal, characteristic, complemented
7#70199C212yesmaximal, complemented
8#70178C212yesmaximal, complemented
9#69525C211yesmaximal, characteristic, complemented
10#68907C212yesmaximal, complemented
11#68824C212yesmaximal, complemented
12#24036—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#121C2211yescharacteristic
17C2×#939—21no—
18C2×#936C2211yescharacteristic
19C2×D4×Dic3—21no—
20C2×C4×#67C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22#5598—24no—
23#5598—24no—
24C2×#897C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×#885—21no—
27C2×#878—21no—
28C2×C3×#102C2212yes—
29C3×#380C2211yescharacteristic
30C2×#832C2212yes—
31C2×#830C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33C2×#816—24no—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#5034—24no—
37#4987C2212yes—
38#4984C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#808C2212yes—
42#4958—21no—
43#4950—21no—
44#4677—21no—
45#4659—21no—
46C3×#340C2212yes—
47#4550C2212yes—
48C2×#548—31no—
49C2×#481—31no—
50C22×#103—31no—
51#2487—31no—
52#2042—32no—
53#2035—32no—
54#1893D311yescharacteristic, complemented
55#1869—31no—
56#1865—32no—
57#1788—32no—
58#1787—31no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67D411yescharacteristic
64C23×Dic3—21no—
65C23×Dic3—21no—
66C23×Dic3—21no—
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—24no—
70C2×#237—42no—
71C22×C4×D3C2311yescharacteristic
72C22×Dic6—24no—
73C22×Dic6—21no—
74C22×Dic6—22no—
75C22×Dic6—42no—
76C22×Dic6D414yes—
77C22×Dic6C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×C3×C8—22no—
80C22×C3×C8—42no—
81C22×C3×C8C2312yes—
82C3×#121—22no—
83C3×#121D414yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#217D411yescharacteristic
91#939—28no—
92#936—22no—
93#936—22no—
94D4×Dic3—28no—
95C4×#67—22no—
96C4×#67—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3C2312yes—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111#897—22no—
112#897—22no—
113#887—22no—
114#887—22no—
115#885—28no—
116#878—28no—
117C3×#102—42no—
118C3×#95—21no—
119#832—24no—
120C3⋊#95—22no—
121C3⋊#95—22no—
122#830—22no—
123(C2×D3)⋊C8—42no—
124#816—24no—
125#816—24no—
126#810—24no—
127#810—42no—
128#808—24no—
129C22×#41—31no—
130C23×Q8—31no—
131C23×C8C2×D311yescharacteristic
132#548—64no—
133#548—64no—
134C2×#124—31no—
135C2×#124—61no—
136C2×C22⋊Q8—31no—
137C2×C22⋊Q8—61no—
138C2×#121—61no—
139C2×#121C2×D311yescharacteristic
140C2×C4×Q8—31no—
141C2×C4×D4—31no—
142C2×C4×D4—61no—
143C22×C4⋊C4—31no—
144#481—64no—
145C2×#106—32no—
146C2×#103—32no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—64no—
151C2×#102—32no—
152C2×#102C2×D312yes—
153C2×C22⋊C8—31no—
154C2×C22⋊C8—32no—
155#400—61no—
156#399—61no—
157#389—32no—
158#386—61no—
159#385—61no—
160#380—32no—
161#380C2×D311yescharacteristic
162#340C2×D312yes—
163#336—32no—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C23×D3C2×D411yescharacteristic
167C2×C3×D4—24no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—24no—
183C2×#67—22no—
184C2×#67—22no—
185C2×#67—22no—
186C2×#67—44no—
187C22×Dic3—21no—
188C22×Dic3—24no—
189C22×Dic3—24no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—44no—
193C22×Dic3—44no—
194C22×Dic3—42no—
195C22×Dic3—82no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199#237—48no—
200#237—44no—
201#237—44no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×Dic6—21no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6—28no—
211C2×Dic6—24no—
212C2×Dic6—21no—
213C2×Dic6—44no—
214C2×Dic6—48no—
215C2×Dic6—44no—
216C2×Dic6C2×D412yes—
217C2×Dic6C2×D412yes—
218C2×Dic6#4114yes—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—44no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4C2×D412yes—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—42no—
229#217—24no—
230#217—24no—
231#217—22no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—82no—
235C4⋊Dic3—24no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3—42no—
239C4⋊Dic3—82no—
240Dic3⋊C4—24no—
241Dic3⋊C4—22no—
242C4×Dic3—24no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—64no—
251C2×#41—122no—
252C22×Q8—34no—
253C22×Q8—31no—
254C22×Q8—31no—
255C22×Q8—64no—
256C22×Q8—62no—
257C22×Q8—61no—
258C22×Q8—122no—
259C22×D4—21no—
260C22×D4—31no—
261C22×D4—61no—
262C22×D4C22×D311yescharacteristic
263C23×C4—31no—
264C23×C4—61no—
265C23×C4—61no—
266C23×C4—61no—
267C23×C4C22×D311yescharacteristic
268C22×C8—22no—
269C22×C8—32no—
270C22×C8—42no—
271C22×C8C22×D312yes—
272#124—68no—
273#124—62no—
274#124—62no—
275C22⋊Q8—68no—
276C22⋊Q8—62no—
277C22⋊Q8—62no—
278#121—22no—
279#121—68no—
280#121D1214yes—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×D4—68no—
284C4×D4—62no—
285C4×D4—62no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4C22×D312yes—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—31no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—62no—
304C2×C42—31no—
305C2×C42—61no—
306#106—64no—
307#103—64no—
308#103—64no—
309#103—64no—
310#103—122no—
311#102—42no—
312#102—64no—
313C22⋊C8—62no—
314C22⋊C8—122no—
315#95—21no—
316#95—62no—
317#95—62no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—42no—
325C23×C3#12012yes—
326C23×C3#13512yes—
327C23×C3#13712yes—
328C23×C3C22×D411yescharacteristic
329C22×D3—21no—
330C22×D3—22no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—22no—
334C3×D4—24no—
335C3×D4—48no—
336C3×D4—41no—
337C3×D4—41no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4#12012yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4112yes—
352C2×C3×C4H(5,𝔽2)12yes—
353#67—48no—
354#67—44no—
355C2×Dic3—22no—
356C2×Dic3—24no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—48no—
366C2×Dic3—48no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—84no—
372C4×D3—41no—
373C4×D3—41no—
374C4×D3—42no—
375Dic6—22no—
376Dic6—24no—
377Dic6—22no—
378Dic6—22no—
379Dic6—48no—
380Dic6—42no—
381Dic6#12314yes—
382C3×C8—42no—
383C3⋊C8—42no—
384C24—21no—
385C24—31no—
386C24D3×D411yescharacteristic
387#41—128no—
388#41—124no—
389#41—124no—
390C2×Q8—32no—
391C2×Q8—32no—
392C2×Q8—34no—
393C2×Q8—68no—
394C2×Q8—64no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×Q8—124no—
401C2×Q8—124no—
402C2×Q8—128no—
403C2×D4—24no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—64no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—124no—
414C2×D4C2×D1212yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—61no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—124no—
432C22×C4—124no—
433C22×C4—122no—
434C22×C4—242no—
435C22×C4C2×D1212yes—
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D412yes—
438C22×C4C23×D311yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—64no—
450C4⋊C4—64no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—122no—
455C4⋊C4—242no—
456C4⋊C4C2×D1212yes—
457C22⋊C4—24no—
458C22⋊C4—42no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—62no—
463C22⋊C4—62no—
464C22⋊C4—242no—
465C42—64no—
466C42—61no—
467C42—61no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3#45214yes—
478C22×C3C2×#12011yescharacteristic
479C22×C3#54612yes—
480C22×C3C2×#13511yescharacteristic
481C22×C3C2×#13711yescharacteristic
482C22×C3#57712yes—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—42no—
487C2×D3—42no—
488C3×C4—22no—
489C3×C4—42no—
490C3×C4—82no—
491C3×C4C2×#12311yescharacteristic
492C3×C4#53411yescharacteristic
493C3×C4#54612yes—
494Dic3—24no—
495Dic3—42no—
496Dic3—41no—
497Dic3—41no—
498Dic3—41no—
499Dic3—82no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—61no—
508C23—61no—
509C23—61no—
510C23—62no—
511C23—62no—
512C23#88212yes—
513C23#91212yes—
514C23#91512yes—
515C23C2×D3×D411yescharacteristic
516Q8—34no—
517Q8—64no—
518Q8—62no—
519Q8—62no—
520Q8—62no—
521Q8—128no—
522Q8—122no—
523D4—24no—
524D4—48no—
525D4—41no—
526D4—41no—
527D4—128no—
528D4—124no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—128no—
550C2×C4—128no—
551C2×C4—122no—
552C2×C4—121no—
553C2×C4—122no—
554C2×C4—121no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—244no—
559C2×C4#88212yes—
560C2×C4C22×D1211yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4#100412yes—
563C2×C4D3×#4112yes—
564C8—42no—
565C8—122no—
566C2×C3—24no—
567C2×C3—82no—
568C2×C3#240511yescharacteristic
569C2×C3C2×#45211yescharacteristic
570C2×C3#301011yescharacteristic
571C2×C3#330112yes—
572C2×C3#331612yes—
573D3—81no—
574C22—24no—
575C22—24no—
576C22—22no—
577C22—22no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—82no—
582C22—82no—
583C22—122no—
584C22—122no—
585C22—121no—
586C22—121no—
587C22—121no—
588C22#478314yes—
589C22C2×#88211yescharacteristic
590C22#561012yes—
591C22C2×#91211yescharacteristic
592C22C2×#91511yescharacteristic
593C22#577012yes—
594C4—22no—
595C4—42no—
596C4—64no—
597C4—82no—
598C4—121no—
599C4—121no—
600C4—122no—
601C4—121no—
602C4—242no—
603C4#555412yes—
604C4#558911yescharacteristic
605C4D3×#12311yescharacteristic
606C3#2403611yescharacteristic, complemented, Sylow
607C2—24no—
608C2—82no—
609C2—241no—
610C2#7468611yescharacteristic
611C2C2×#478311yescharacteristic
612C2#7883611yescharacteristic
613C2#8069212yes—
614C2#8073612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
prime radicalC2×#5598C2
hypercenter#1893D3
hypocenterC3#24036
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#816C2semidirect
3. C2×#5034C2semidirect
4. #74028C2semidirect
5. C3×#1893C2semidirect
6. #71975C2semidirect
7. #70199C2semidirect
8. #70178C2semidirect
9. #69525C2semidirect
10. #68907C2semidirect
11. #68824C2semidirect
54. #1893D3semidirect
606. C3#24036semidirect

Maximal subgroups

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

subgroupindex
C2×#55982
C22×#8162
C2×#50342
#740282
C3×#18932
#719752
#701992
#701782
#695252
#689072
#688242
#240363