Finite Group Databasebeta

← #10896412

Subgroups — #10896412

Order 768.

734 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
1#76484C211yesmaximal, characteristic
2#74154C211yesmaximal, characteristic, complemented
3C3×#2097C211yesmaximal, characteristic, complemented
4#70638C211yesmaximal, characteristic
5#70615C211yesmaximal, characteristic, complemented
6#70584C211yesmaximal, characteristic
7#70563C211yesmaximal, characteristic, complemented
8#70237C211yesmaximal, characteristic, complemented
9#70235C211yesmaximal, characteristic, complemented
10#70230C211yesmaximal, characteristic
11#70230C211yesmaximal, characteristic
12#69668C211yesmaximal, characteristic, complemented
13#69558C211yesmaximal, characteristic
14#68678C211yesmaximal, characteristic
15#68606C211yesmaximal, characteristic, complemented
16#29267—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#382C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4975C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4939C2211yescharacteristic
43#4932C2211yescharacteristic
44#4669—21no—
45#4669—21no—
46#4668—21no—
47#4668—21no—
48#4653—21no—
49#4653—21no—
50#4646—21no—
51#4646—21no—
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2331—31no—
61#2326—31no—
62#2097D311yescharacteristic, complemented
63#2092—31no—
64#2024—31no—
65#2023—31no—
66#1983—31no—
67#1983—31no—
68#1948—31no—
69#1947—31no—
70#1865—31no—
71#1864—31no—
72#1863—31no—
73#1862—31no—
74#1832—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85#947—22no—
86#947Q814yescomplemented
87Q8×Dic3—41no—
88Q8×Dic3—41no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#893—41no—
111#893—41no—
112#869—22no—
113#869—22no—
114C3×#107—41no—
115C3×#106—41no—
116C3×#103—41no—
117C3×#103—41no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122#835—22no—
123#835—22no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#821—22no—
133#821—22no—
134Dic3⋊C8—22no—
135Dic3⋊C8—22no—
136#809—41no—
137#809—41no—
138#808—41no—
139#807—41no—
140C2×C4⋊Q8—31no—
141C2×#124—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×#107—31no—
145C2×#107C2×D311yescharacteristic
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#103C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158#391—61no—
159#391—61no—
160#391C2×D311yescharacteristic
161#390—31no—
162#389—31no—
163#389—31no—
164#388—61no—
165#388—61no—
166#387—61no—
167#387—61no—
168#385—61no—
169#385—61no—
170#384—31no—
171#382C2×D311yescharacteristic
172C4×C4⋊C4—31no—
173C4×C4⋊C4—31no—
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#336C2×D311yescharacteristic
183C23×D3—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—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12C2×Q812yes—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×Dic6—24no—
214C2×Dic6—81no—
215C2×Dic6—81no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C42—21no—
229C3×C42—21no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—81no—
234(C2×D3)⋊C4—81no—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—42no—
241C4⋊Dic3#4112yes—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—81no—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—42no—
250C4×Dic3—81no—
251C4×Dic3—81no—
252C4×Dic3C2×Q812yes—
253C4×Dic3#4112yes—
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×Q8—61no—
261C22×Q8C22×D311yescharacteristic
262C22×D4—31no—
263C23×C4—61no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C4⋊Q8—62no—
269C4⋊Q8—62no—
270#124—34no—
271#124—62no—
272C4×Q8—121no—
273C4×Q8—121no—
274C4×D4—121no—
275C4×D4—121no—
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⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—31no—
294C2×C42—31no—
295C2×C42—61no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—41no—
300#107—121no—
301#106—41no—
302#106—121no—
303C4⋊C8—62no—
304C4⋊C8—62no—
305#103—41no—
306#103—41no—
307#103—62no—
308#103—62no—
309#102—62no—
310#102—62no—
311#102—121no—
312#102—121no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—31no—
320#95—31no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3C22×D411yescharacteristic
326C22×D3—22no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
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×C4—41no—
355C2×C3×C4C22×Q811yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3C22⋊Q812yes—
381C2×Dic3C22⋊Q812yes—
382D12—22no—
383D12—22no—
384D12—41no—
385D12—41no—
386C4×D3—84no—
387Dic6—84no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24—61no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—64no—
399C2×Q8—241no—
400C2×Q8—241no—
401C2×Q8D3×Q812yes—
402C2×D4—32no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—122no—
420C22×C4—241no—
421C22×C4—241no—
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4C23×D311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—32no—
444C4⋊C4—32no—
445C4⋊C4—64no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—122no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—32no—
463C42—32no—
464C42—61no—
465C42—61no—
466C42—122no—
467C42—241no—
468C42—241no—
469C22×C3C2×C22⋊Q811yescharacteristic
470C22×C3#52511yescharacteristic
471C22×C3#53511yescharacteristic
472C22×C3#57611yescharacteristic
473C22×C3#57711yescharacteristic
474C22×C3#57711yescharacteristic
475C22×C3#57811yescharacteristic
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3—81no—
480C2×D3—81no—
481C2×D3—81no—
482C2×D3—81no—
483C3×C4—42no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4#55411yescharacteristic
490C3×C4#55411yescharacteristic
491C3×C4#55911yescharacteristic
492C3×C4#55911yescharacteristic
493Dic3—22no—
494Dic3—22no—
495Dic3—42no—
496Dic3—42no—
497Dic3—41no—
498Dic3—41no—
499Dic3—81no—
500C23—62no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23C2×D3×D411yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512Q8—244no—
513D4—62no—
514D4—62no—
515D4—121no—
516D4—121no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—32no—
525C2×C4—32no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—244no—
557C2×C4#100411yescharacteristic
558C2×C4#100411yescharacteristic
559C2×C4C2×D3×Q811yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C8—41no—
564C8—41no—
565C8—121no—
566C8—121no—
567C2×C3#258011yescharacteristic
568C2×C3#308311yescharacteristic
569C2×C3#308411yescharacteristic
570C2×C3#311811yescharacteristic
571C2×C3#312111yescharacteristic
572C2×C3#322411yescharacteristic
573C2×C3#322511yescharacteristic
574D3—82no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22—241no—
579C22—241no—
580C22—241no—
581C22—241no—
582C22D3×C22⋊Q811yescharacteristic
583C22#565611yescharacteristic
584C22#571511yescharacteristic
585C22#576911yescharacteristic
586C22#577011yescharacteristic
587C22#577811yescharacteristic
588C22#577911yescharacteristic
589C4—41no—
590C4—41no—
591C4—42no—
592C4—62no—
593C4—62no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—122no—
598C4—122no—
599C4—121no—
600C4—121no—
601C4—241no—
602C4#563611yescharacteristic
603C4#563611yescharacteristic
604C4#566411yescharacteristic
605C4#566411yescharacteristic
606C3#2926711yescharacteristic, complemented, Sylow
607C2—242no—
608C2#7804111yescharacteristic
609C2#8123411yescharacteristic
610C2#8123511yescharacteristic
611C2#8149611yescharacteristic
612C2#8149711yescharacteristic
613C2#8182711yescharacteristic
614C2#8188011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2097C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2097D3
hypocenterC3#29267
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2097

Complements

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

NH ≅ G/Ntype
2. #74154C2semidirect
3. C3×#2097C2semidirect
5. #70615C2semidirect
7. #70563C2semidirect
8. #70237C2semidirect
9. #70235C2semidirect
12. #69668C2semidirect
15. #68606C2semidirect
62. #2097D3semidirect
86. #947Q8semidirect
606. C3#29267semidirect

Maximal subgroups

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

subgroupindex
#764842
#741542
C3×#20972
#706382
#706152
#705842
#705632
#702372
#702352
#702302
#696682
#695582
#686782
#686062
#292673