Finite Group Databasebeta

← #10896402

Subgroups — #10896402

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#83228C211yesmaximal, characteristic, complemented
2#76297C211yesmaximal, characteristic
3C3×#2023C211yesmaximal, characteristic
4#71794C211yesmaximal, characteristic, complemented
5#70170C211yesmaximal, characteristic
6#70168C211yesmaximal, characteristic
7#70162C211yesmaximal, characteristic
8#70162C211yesmaximal, characteristic
9#69654C211yesmaximal, characteristic
10#68907C211yesmaximal, characteristic, complemented
11#68906C211yesmaximal, characteristic, complemented
12#68796C211yesmaximal, characteristic, complemented
13#68795C211yesmaximal, characteristic, complemented
14#68693C211yesmaximal, characteristic, complemented
15#68606C211yesmaximal, characteristic, complemented
16#29267—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C3×#390C2211yescharacteristic
24#5229—21no—
25#5229—21no—
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34#4984C2211yescharacteristic
35#4984C2211yescharacteristic
36#4980—21no—
37#4980—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4946—21no—
46#4946—21no—
47#4945—21no—
48#4945—21no—
49#4936C2211yescharacteristic
50#4932C2211yescharacteristic
51#4929C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2331—31no—
61#2326—31no—
62#2097—31no—
63#2092—31no—
64#2024—31no—
65#2023D311yescharacteristic
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×Q8—21no—
76C22×C3×D4C2311yescharacteristic
77C23×Dic3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#936—22no—
94#936Q814yescomplemented
95D4×Dic3—41no—
96D4×Dic3—41no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#887—41no—
113#887—41no—
114C3×#102—22no—
115C3×#102—22no—
116#832—41no—
117#832—41no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130#818—41no—
131#817—41no—
132#816—41no—
133#816—41no—
134Dic3⋊C8—22no—
135Dic3⋊C8—22no—
136#810—22no—
137#810—22no—
138#808—41no—
139#807—41no—
140C2×C4⋊Q8C2×D311yescharacteristic
141C2×#124—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158#391—31no—
159#391—61no—
160#391—61no—
161#390C2×D311yescharacteristic
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#382—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4—31no—
174#376—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#336—31no—
180#336—31no—
181#336—31no—
182#336C2×D311yescharacteristic
183C24×C3—21no—
184C2×C3×Q8—24no—
185C2×C3×Q8—81no—
186C2×C3×Q8—81no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4C2×Q812yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—42no—
204C22×Dic3—81no—
205C22×Dic3—81no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6C2×Q812yes—
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—24no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4#4112yes—
228C3×C4⋊C4#4112yes—
229C3×C22⋊C4—81no—
230C3×C22⋊C4—81no—
231C3×C42—21no—
232C3×C42—21no—
233#217—24no—
234#217—24no—
235#217—42no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—81no—
243C4⋊Dic3—81no—
244Dic3⋊C4—42no—
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—21no—
261C22×Q8—31no—
262C22×D4C22×D311yescharacteristic
263C23×C4—61no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#124—34no—
271#124—62no—
272C4×Q8—121no—
273C4×Q8—121no—
274C4×D4—121no—
275C4×D4—121no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
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—21no—
289C2×C22⋊C4—21no—
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—121no—
300#107—121no—
301#106—121no—
302#106—121no—
303C4⋊C8—62no—
304C4⋊C8—62no—
305#103—62no—
306#103—62no—
307#103—121no—
308#103—121no—
309#102—22no—
310#102—22no—
311#102—121no—
312#102—121no—
313#95—31no—
314#95—31no—
315#95—31no—
316#95—31no—
317#95—61no—
318#95—61no—
319#95—61no—
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—84no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—41no—
337C3×D4—41no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
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×C4C22×Q811yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—84no—
382C2×Dic3C22⋊Q812yes—
383C2×Dic3C22⋊Q812yes—
384Dic6—22no—
385Dic6—22no—
386Dic6—41no—
387Dic6—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24—21no—
393C2×Q8—24no—
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4D3×Q812yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—61no—
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—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—122no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23712yes—
454C4⋊C4#23712yes—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C22⋊C4—122no—
460C42—21no—
461C42—21no—
462C42—32no—
463C42—32no—
464C42—61no—
465C42—61no—
466C42—122no—
467C42—241no—
468C42—241no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3C2×C22⋊Q811yescharacteristic
477C22×C3#52511yescharacteristic
478C22×C3#53511yescharacteristic
479C22×C3#57611yescharacteristic
480C22×C3#57711yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C3×C4—41no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—81no—
488C3×C4#55411yescharacteristic
489C3×C4#55411yescharacteristic
490C3×C4#55911yescharacteristic
491C3×C4#55911yescharacteristic
492Dic3—22no—
493Dic3—22no—
494Dic3—42no—
495Dic3—41no—
496Dic3—41no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C2×D3×D411yescharacteristic
508Q8—62no—
509Q8—62no—
510Q8—84no—
511Q8—121no—
512Q8—121no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—32no—
526C2×C4—32no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
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×C4C2×#23711yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4C2×D3×Q811yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C2×C4#100911yescharacteristic
562C2×C4#100911yescharacteristic
563C8—41no—
564C8—41no—
565C8—121no—
566C8—121no—
567C2×C3—82no—
568C2×C3#258011yescharacteristic
569C2×C3#308311yescharacteristic
570C2×C3#308411yescharacteristic
571C2×C3#311811yescharacteristic
572C2×C3#312111yescharacteristic
573C2×C3#322411yescharacteristic
574C2×C3#322511yescharacteristic
575C22—42no—
576C22—42no—
577C22—42no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22#558611yescharacteristic
583C22D3×C22⋊Q811yescharacteristic
584C22#567411yescharacteristic
585C22#575711yescharacteristic
586C22#576211yescharacteristic
587C22#577511yescharacteristic
588C22#577911yescharacteristic
589C4—41no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—62no—
594C4—62no—
595C4—81no—
596C4—122no—
597C4—121no—
598C4—121no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#563511yescharacteristic
603C4#563511yescharacteristic
604C4#568111yescharacteristic
605C4#568111yescharacteristic
606C3#2926711yescharacteristic, complemented, Sylow
607C2—82no—
608C2#7680211yescharacteristic
609C2#7999211yescharacteristic
610C2#8003211yescharacteristic
611C2#8096811yescharacteristic
612C2#8106311yescharacteristic
613C2#8157011yescharacteristic
614C2#8168911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2023D3
hypocenterC3#29267
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #83228C2semidirect
4. #71794C2semidirect
10. #68907C2semidirect
11. #68906C2semidirect
12. #68796C2semidirect
13. #68795C2semidirect
14. #68693C2semidirect
15. #68606C2semidirect
94. #936Q8semidirect
606. C3#29267semidirect

Maximal subgroups

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

subgroupindex
#832282
#762972
C3×#20232
#717942
#701702
#701682
#701622
#696542
#689072
#689062
#687962
#687952
#686932
#686062
#292673