Finite Group Databasebeta

← #10828504

Subgroups — #10828504

Order 768.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic, complemented
3C2×#5033C211yesmaximal, characteristic
4C2×#4928C211yesmaximal, characteristic, complemented
5#71739C212yesmaximal, complemented
6#71722C212yesmaximal, complemented
7#71692C211yesmaximal, characteristic, complemented
8#71677C211yesmaximal, characteristic
9#71628C212yesmaximal, complemented
10#71607C212yesmaximal, complemented
11#71531C211yesmaximal, characteristic, complemented
12#24066—31nomaximal, Sylow
13C3×#529—24no—
14C3×#529—24no—
15C2×C3×#126—21no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C22×C3×C42C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C22×Dic3⋊C4—21no—
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#832C2212yes—
32C2×#832C2212yes—
33C2×C3⋊#95—21no—
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#5043—24no—
37#5033—24no—
38C2×#808C2212yes—
39C2×#807C2212yes—
40#4928—21no—
41#4928—21no—
42#4928C2212yes—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#529D311yescharacteristic
49C2×#483—31no—
50C2×#480—31no—
51C2×#389—31no—
52#2059—32no—
53#2042—32no—
54#1874—31no—
55#1872—31no—
56#1864—32no—
57#1863—32no—
58#1817—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C23×Dic3—21no—
68C3×#126—28no—
69C3×#125—22no—
70C3×#125—22no—
71C3×#124—22no—
72C3×#124—22no—
73C3×#123—28no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8D414yes—
76C3×#121—22no—
77C3×#121D414yes—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×D4—28no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×#117—28no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—22no—
96C2×C3×C42—22no—
97C2×C3×C42—21no—
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—24no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2312yes—
109C2×Dic3⋊C4—22no—
110C2×Dic3⋊C4—22no—
111C2×Dic3⋊C4—44no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114#835—24no—
115#835—24no—
116#832—24no—
117#832—24no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#830—22no—
121#830—22no—
122#808—24no—
123#807—24no—
124#529—24no—
125#529—24no—
126C2×#126—21no—
127C2×#125C2×D311yescharacteristic
128C2×#124C2×D311yescharacteristic
129C2×#123—21no—
130C2×C22⋊Q8C2×D311yescharacteristic
131C2×#121C2×D311yescharacteristic
132C2×C4×Q8C2×D311yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4C2×D311yescharacteristic
135C2×#117—21no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4—61no—
139C22×C42C2×D311yescharacteristic
140#483—64no—
141#480—64no—
142C2×#107—32no—
143C2×#106—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150#389—32no—
151#389—32no—
152#389—61no—
153#389—61no—
154C2×#95—61no—
155#340—32no—
156#340—32no—
157#336—32no—
158#336—32no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8C2×D412yes—
165C2×C3×D4—24no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4C2×D412yes—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—41no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—24no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—42no—
198C22×Dic3—42no—
199C22×Dic3—42no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4C2×D412yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C42—24no—
216C3×C42—22no—
217C3×C42—22no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C4⋊Dic3—24no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—42no—
227C4⋊Dic3#4114yes—
228Dic3⋊C4—44no—
229Dic3⋊C4—44no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C22×Q8C22×D311yescharacteristic
237C22×D4—21no—
238C22×D4C22×D311yescharacteristic
239C23×C4—21no—
240C23×C4—61no—
241C23×C4—61no—
242C23×C4#6711yescharacteristic
243C23×C4#6711yescharacteristic
244C23×C4C22×D311yescharacteristic
245C22×C8—32no—
246C22×C8—32no—
247#126—28no—
248#125—22no—
249#125—22no—
250#124—22no—
251#124—22no—
252#123—28no—
253C22⋊Q8—22no—
254C22⋊Q8#6714yes—
255#121—22no—
256#121#6714yes—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—28no—
260C4×D4—22no—
261C4×D4—22no—
262#117—28no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—124no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42#6711yescharacteristic
288C2×C42#6711yescharacteristic
289C2×C42C22×D311yescharacteristic
290C2×C42C22×D311yescharacteristic
291#107—64no—
292#106—64no—
293#103—64no—
294#103—64no—
295#102—64no—
296#102—64no—
297C22⋊C8—62no—
298C22⋊C8—62no—
299#95—62no—
300#95—62no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3#12012yes—
313C23×C3#13512yes—
314C23×C3AGL(1,ℤ/8)12yes—
315C23×C3C22×D411yescharacteristic
316C3×Q8—24no—
317C3×Q8—42no—
318C3×D4—24no—
319C3×D4—48no—
320C3×D4—42no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—44no—
340C2×C3×C4—48no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—44no—
348C2×C3×C4#12012yes—
349C2×C3×C4#13712yes—
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C2×#4112yes—
353C2×C3×C42−1+412yes—
354C2×Dic3—24no—
355C2×Dic3—48no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—84no—
362C3⋊C8—42no—
363C3⋊C8—42no—
364C24—21no—
365C24C2×#6711yescharacteristic
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8C2×#6712yes—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4C2×#6712yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—122no—
395C22×C4—122no—
396C22×C4—122no—
397C22×C4—122no—
398C22×C4—122no—
399C22×C4C2×#6711yescharacteristic
400C22×C4C2×#6711yescharacteristic
401C22×C4C2×#6711yescharacteristic
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6712yes—
405C22×C4C2×#6712yes—
406C22×C4C23×D311yescharacteristic
407C2×C8—62no—
408C2×C8—62no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—34no—
420C4⋊C4—64no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—124no—
425C4⋊C4—124no—
426C4⋊C4—122no—
427C4⋊C4C2×#6712yes—
428C4⋊C4C2×#6712yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3#44914yes—
455C22×C3C2×#12011yescharacteristic
456C22×C3#52511yescharacteristic
457C22×C3#53611yescharacteristic
458C22×C3C2×#13511yescharacteristic
459C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
460C22×C3C2×#13711yescharacteristic
461C22×C3#57511yescharacteristic
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—42no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—42no—
470C3×C4#54712yes—
471C3×C4#54712yes—
472Dic3—44no—
473Dic3—82no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23#94912yes—
486C23#95012yes—
487C23#95312yes—
488C23C22×#6711yescharacteristic
489Q8—24no—
490Q8—42no—
491D4—24no—
492D4—48no—
493D4—42no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—64no—
522C2×C4—128no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—244no—
529C2×C4#95112yes—
530C2×C4#95312yes—
531C2×C4D3×#4112yes—
532C2×C4#101012yes—
533C2×C4C22×#6711yescharacteristic
534C2×C4C22×#6711yescharacteristic
535C8—122no—
536C8—122no—
537C2×C3—24no—
538C2×C3—82no—
539C2×C3#240611yescharacteristic
540C2×C3C2×#44911yescharacteristic
541C2×C3#300711yescharacteristic
542C2×C3#310811yescharacteristic
543C2×C3#311411yescharacteristic
544C2×C3#323211yescharacteristic
545C2×C3#323811yescharacteristic
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#523814yes—
560C22C2×#94911yescharacteristic
561C22C2×#95011yescharacteristic
562C22#582211yescharacteristic
563C22C2×#95111yescharacteristic
564C22#582511yescharacteristic
565C22#582811yescharacteristic
566C22C2×#95311yescharacteristic
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—124no—
576C4—242no—
577C4#582412yes—
578C4#582412yes—
579C3#2406611yescharacteristic, complemented, Sylow
580C2—24no—
581C2—82no—
582C2#8312111yescharacteristic
583C2#8312311yescharacteristic
584C2#8314311yescharacteristic
585C2#8315211yescharacteristic
586C2C2×#523811yescharacteristic
587C2#8408411yescharacteristic
588C2#8409011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#529D3
hypocenterC3#24066
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#5043C2semidirect
4. C2×#4928C2semidirect
5. #71739C2semidirect
6. #71722C2semidirect
7. #71692C2semidirect
9. #71628C2semidirect
10. #71607C2semidirect
11. #71531C2semidirect
579. C3#24066semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#50432
C2×#50332
C2×#49282
#717392
#717222
#716922
#716772
#716282
#716072
#715312
#240663