Finite Group Databasebeta

← #10896438

Subgroups — #10896438

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#82328C211yesmaximal, characteristic
2#75972C211yesmaximal, characteristic, complemented
3C3×#1864C211yesmaximal, characteristic
4#71607C211yesmaximal, characteristic
5#70590C211yesmaximal, characteristic, complemented
6#70563C211yesmaximal, characteristic, complemented
7#70184C211yesmaximal, characteristic
8#70164C211yesmaximal, characteristic
9#70162C211yesmaximal, characteristic
10#70162C211yesmaximal, characteristic
11#69632C211yesmaximal, characteristic, complemented
12#68911C211yesmaximal, characteristic, complemented
13#68796C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68606C211yesmaximal, characteristic, complemented
16#29269—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×#890C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#389C2211yescharacteristic
25#5221C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28#5210—21no—
29#5210—21no—
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4976—21no—
36#4975C2211yescharacteristic
37#4975C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41#4972—21no—
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4944—21no—
47#4944—21no—
48#4932C2211yescharacteristic
49#4928C2211yescharacteristic
50#4653—21no—
51#4652—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2330—31no—
61#2326—31no—
62#2096—31no—
63#2094—31no—
64#2026—31no—
65#2025—31no—
66#1983—31no—
67#1983—31no—
68#1948—31no—
69#1948—31no—
70#1865—31no—
71#1865—31no—
72#1864—31no—
73#1864D311yescharacteristic
74#1832—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4×D4—41no—
81C3×C4×D4—41no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92#936—22no—
93#936Q814yes—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
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#890—22no—
113#890—22no—
114C4×Dic6—41no—
115C4×Dic6—41no—
116C3×#106—41no—
117C3×#102—22no—
118C3×#102—41no—
119C3×#95—21no—
120#832—22no—
121#832—41no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#817—41no—
132#816—22no—
133#816—41no—
134Dic3⋊C8—22no—
135Dic3⋊C8—22no—
136#810—22no—
137#810—41no—
138#808—41no—
139#808—41no—
140C2×#125—31no—
141C2×#124—31no—
142C2×C4×Q8—31no—
143C2×C4×D4C2×D311yescharacteristic
144C2×#106—31no—
145C2×#106—31no—
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
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#390—31no—
160#389—31no—
161#389—61no—
162#389—61no—
163#389C2×D311yescharacteristic
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⋊C4—31no—
174#376—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#336—31no—
180#336—31no—
181#336—31no—
182#336C2×D311yescharacteristic
183C24×C3—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4C2×Q812yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—42no—
194C22×C3×C4—81no—
195C22×C3×C4—81no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
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—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—81no—
225C3×C4⋊C4#4112yes—
226C3×C22⋊C4—42no—
227C3×C42—21no—
228C3×C42—21no—
229#217—24no—
230#217—24no—
231#217—81no—
232#217—81no—
233C4⋊Dic3—24no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3#4112yes—
240Dic3⋊C4—24no—
241Dic3⋊C4—42no—
242Dic3⋊C4—81no—
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—31no—
261C22×D4C22×D311yescharacteristic
262C23×C4—21no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#125—62no—
268#125—62no—
269#124—34no—
270#124—62no—
271C4×Q8—121no—
272C4×Q8—121no—
273C4×D4—41no—
274C4×D4—41no—
275C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—61no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#106—41no—
302#106—121no—
303#106—121no—
304#106—121no—
305C4⋊C8—62no—
306C4⋊C8—62no—
307#103—62no—
308#103—62no—
309#103—121no—
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×D4—22no—
334C3×D4—22no—
335C3×D4—41no—
336C3×D4—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—84no—
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—22no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
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×Dic3C22⋊Q812yes—
382C2×Dic3C22⋊Q812yes—
383Dic6—22no—
384Dic6—22no—
385Dic6—41no—
386Dic6—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24—21no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4D3×Q812yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—42no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—81no—
417C22×C4—81no—
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—21no—
437C4⋊C4—21no—
438C4⋊C4—32no—
439C4⋊C4—64no—
440C4⋊C4—64no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—81no—
447C4⋊C4—122no—
448C4⋊C4—241no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23512yes—
454C22⋊C4—42no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—241no—
458C22⋊C4—241no—
459C42—21no—
460C42—21no—
461C42—32no—
462C42—32no—
463C42—61no—
464C42—61no—
465C42—122no—
466C42—241no—
467C42—241no—
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—42no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4#55411yescharacteristic
488C3×C4#55411yescharacteristic
489C3×C4#55911yescharacteristic
490C3×C4#55911yescharacteristic
491Dic3—22no—
492Dic3—22no—
493Dic3—42no—
494Dic3—41no—
495Dic3—41no—
496Dic3—42no—
497Dic3—81no—
498Dic3—81no—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D3×D411yescharacteristic
507Q8—62no—
508Q8—62no—
509Q8—121no—
510Q8—121no—
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—21no—
521C2×C4—21no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—61no—
533C2×C4—61no—
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—62no—
543C2×C4—62no—
544C2×C4—84no—
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×C4C2×#23511yescharacteristic
556C2×C4#100411yescharacteristic
557C2×C4#100411yescharacteristic
558C2×C4C2×D3×Q811yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4D3×#4111yescharacteristic
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—
580C22#554711yescharacteristic
581C22#559211yescharacteristic
582C22D3×C22⋊Q811yescharacteristic
583C22#576211yescharacteristic
584C22#577011yescharacteristic
585C22#577511yescharacteristic
586C22#577511yescharacteristic
587C4—42no—
588C4—41no—
589C4—41no—
590C4—62no—
591C4—62no—
592C4—81no—
593C4—81no—
594C4—121no—
595C4—121no—
596C4—122no—
597C4—122no—
598C4—241no—
599C4—241no—
600C4#551111yescharacteristic
601C4#555911yescharacteristic
602C4#563611yescharacteristic
603C4#566411yescharacteristic
604C3#2926911yescharacteristic, complemented, Sylow
605C2—82no—
606C2#7448711yescharacteristic
607C2#7969811yescharacteristic
608C2#7987511yescharacteristic
609C2#8012711yescharacteristic
610C2#8014211yescharacteristic
611C2#8106311yescharacteristic
612C2#8117111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1864D3
hypocenterC3#29269
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #75972C2semidirect
5. #70590C2semidirect
6. #70563C2semidirect
11. #69632C2semidirect
12. #68911C2semidirect
13. #68796C2semidirect
14. #68684C2semidirect
15. #68606C2semidirect
604. C3#29269semidirect

Maximal subgroups

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

subgroupindex
#823282
#759722
C3×#18642
#716072
#705902
#705632
#701842
#701642
#701622
#696322
#689112
#687962
#686842
#686062
#292693