Finite Group Databasebeta

← #10896431

Subgroups — #10896431

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82953C211yesmaximal, characteristic
2#73626C211yesmaximal, characteristic, complemented
3C3×#1983C211yesmaximal, characteristic
4#71721C211yesmaximal, characteristic
5#70454C211yesmaximal, characteristic, complemented
6#70444C211yesmaximal, characteristic, complemented
7#70250C211yesmaximal, characteristic, complemented
8#70238C211yesmaximal, characteristic, complemented
9#70237C211yesmaximal, characteristic, complemented
10#70232C211yesmaximal, characteristic, complemented
11#70184C211yesmaximal, characteristic
12#70170C211yesmaximal, characteristic
13#70169C211yesmaximal, characteristic
14#70164C211yesmaximal, characteristic
15#69467C211yesmaximal, characteristic, complemented
16#29269—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×#890C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4984C2211yescharacteristic
36#4975C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4951—21no—
41#4951—21no—
42#4950—21no—
43#4950—21no—
44#4935C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4933—21no—
47#4932C2211yescharacteristic
48#4928—21no—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4654—21no—
51#4647—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2330—31no—
61#2326—31no—
62#2096—31no—
63#2094—31no—
64#2026—31no—
65#2025—31no—
66#1983—31no—
67#1983D311yescharacteristic
68#1948—31no—
69#1948—31no—
70#1865—31no—
71#1865—31no—
72#1864—31no—
73#1864—31no—
74#1832—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×Dic3—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#124—22no—
81C3×#124Q814yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91Q8×Dic3—41no—
92Q8×Dic3—41no—
93D4×Dic3—41no—
94D4×Dic3—41no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#890—22no—
115#890—22no—
116C3×C4⋊C8—22no—
117C3×#103—22no—
118C3×#103—41no—
119C3×#102—22no—
120C3×#102—41no—
121C3×#95—21no—
122#835—22no—
123#835—41no—
124#832—22no—
125#832—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#817—41no—
136#817—41no—
137#808—41no—
138#808—41no—
139C4⋊(C3⋊C8)—22no—
140C2×#125—31no—
141C2×#124C2×D311yescharacteristic
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×#106—31no—
145C2×#106—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#103C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158#391—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—61no—
163#389—61no—
164#388—61no—
165#388—61no—
166#387—61no—
167#387—61no—
168#385—61no—
169#385—61no—
170#384—31no—
171#384—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4C2×D311yescharacteristic
174#376—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183C24×C3—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C2×Q812yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—42no—
207C22×Dic3—81no—
208C22×Dic3—81no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4#4112yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C42—81no—
226C3×C42C2×Q812yes—
227C3×C42#4112yes—
228#217—42no—
229#217—81no—
230#217—81no—
231C4⋊Dic3—24no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—42no—
240C4⋊Dic3—81no—
241C4⋊Dic3—81no—
242C4⋊Dic3—81no—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—42no—
253C4×Dic3—81no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×Q8C22×D311yescharacteristic
261C22×D4C22×D311yescharacteristic
262C23×C4—61no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#125—62no—
268#125—62no—
269#124—22no—
270#124Dic614yes—
271C4×Q8—121no—
272C4×Q8—121no—
273C4×D4—121no—
274C4×D4—121no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—21no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#106—121no—
302#106—121no—
303#106—121no—
304#106—121no—
305C4⋊C8—22no—
306C4⋊C8—62no—
307#103—22no—
308#103—41no—
309#103—62no—
310#103—121no—
311#102—22no—
312#102—41no—
313#102—62no—
314#102—121no—
315#95—21no—
316#95—31no—
317#95—31no—
318#95—31no—
319#95—31no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4C22⋊Q812yes—
355C2×C3×C4C22⋊Q812yes—
356C2×C3×C4C22×Q811yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—84no—
387C3×C8—41no—
388C3×C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8C2×Dic612yes—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4C2×Dic612yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—241no—
417C22×C4—241no—
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4C23×D311yescharacteristic
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—32no—
437C4⋊C4—64no—
438C4⋊C4—64no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—122no—
447C4⋊C4—241no—
448C4⋊C4—241no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23712yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—122no—
457C22⋊C4—241no—
458C22⋊C4—241no—
459C42—61no—
460C42—61no—
461C42—61no—
462C42—61no—
463C42—81no—
464C42—122no—
465C42—241no—
466C42C2×Dic612yes—
467C42#23712yes—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3C2×C22⋊Q811yescharacteristic
476C22×C3#52511yescharacteristic
477C22×C3#53511yescharacteristic
478C22×C3#57711yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57711yescharacteristic
481C22×C3#57711yescharacteristic
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—81no—
487C3×C4#55411yescharacteristic
488C3×C4#55411yescharacteristic
489C3×C4#55911yescharacteristic
490C3×C4#55911yescharacteristic
491Dic3—41no—
492Dic3—41no—
493Dic3—42no—
494Dic3—41no—
495Dic3—41no—
496Dic3—81no—
497Dic3—81no—
498Dic3—81no—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D3×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511D4—22no—
512D4—22no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—62no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—244no—
553C2×C4#87812yes—
554C2×C4#87812yes—
555C2×C4C22×Dic611yescharacteristic
556C2×C4C2×#23711yescharacteristic
557C2×C4C2×#23711yescharacteristic
558C2×C4#100411yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4#100911yescharacteristic
561C8—41no—
562C8—41no—
563C8—121no—
564C8—121no—
565C2×C3—82no—
566C2×C3#258011yescharacteristic
567C2×C3#308411yescharacteristic
568C2×C3#308411yescharacteristic
569C2×C3#312511yescharacteristic
570C2×C3#312511yescharacteristic
571C2×C3#322111yescharacteristic
572C2×C3#322211yescharacteristic
573C22—42no—
574C22—42no—
575C22—42no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22C2×#87811yescharacteristic
581C22#560611yescharacteristic
582C22#560711yescharacteristic
583C22#577011yescharacteristic
584C22#577011yescharacteristic
585C22#577511yescharacteristic
586C22#577811yescharacteristic
587C4—22no—
588C4—22no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—121no—
593C4—121no—
594C4—121no—
595C4—122no—
596C4—121no—
597C4—241no—
598C4—241no—
599C4—241no—
600C4#553911yescharacteristic
601C4#554311yescharacteristic
602C4#568111yescharacteristic
603C4#571311yescharacteristic
604C3#2926911yescharacteristic, complemented, Sylow
605C2—82no—
606C2#7636411yescharacteristic
607C2#7882411yescharacteristic
608C2#7887311yescharacteristic
609C2#8051111yescharacteristic
610C2#8054511yescharacteristic
611C2#8057511yescharacteristic
612C2#8060811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1983C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1983D3
hypocenterC3#29269
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. #73626C2semidirect
5. #70454C2semidirect
6. #70444C2semidirect
7. #70250C2semidirect
8. #70238C2semidirect
9. #70237C2semidirect
10. #70232C2semidirect
15. #69467C2semidirect
604. C3#29269semidirect

Maximal subgroups

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

subgroupindex
#829532
#736262
C3×#19832
#717212
#704542
#704442
#702502
#702382
#702372
#702322
#701842
#701702
#701692
#701642
#694672
#292693