Finite Group Databasebeta

← #10868039

Subgroups — #10868039

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82464C211yesmaximal, characteristic
2#73792C211yesmaximal, characteristic
3C3×C4×#103C211yesmaximal, characteristic, complemented
4#71629C211yesmaximal, characteristic, complemented
5#70275C211yesmaximal, characteristic
6#70248C211yesmaximal, characteristic, complemented
7#70244C211yesmaximal, characteristic
8#70233C211yesmaximal, characteristic, complemented
9#69785C211yesmaximal, characteristic
10#69770C211yesmaximal, characteristic, complemented
11#69458C211yesmaximal, characteristic, complemented
12#68877C211yesmaximal, characteristic
13#68873C211yesmaximal, characteristic
14#68795C211yesmaximal, characteristic, complemented
15#68789C211yesmaximal, characteristic, complemented
16#27043—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#876—21no—
21C2×C3×#103C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25#5229C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4990—21no—
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4978—21no—
37#4975C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4940—21no—
43C4×(C2×D3)⋊C4C2211yescharacteristic
44#4933—21no—
45C4×Dic3⋊C4—21no—
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4675—21no—
49#4658—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2496—31no—
61#2362—31no—
62#2072—31no—
63#2068—31no—
64#2036—31no—
65#2036—31no—
66#1947—31no—
67#1942—31no—
68#1923—31no—
69#1920—31no—
70#1918—31no—
71#1865—31no—
72#1854—31no—
73#1825—31no—
74C4×#103D311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C4×Q8—22no—
80C3×C4×Q8—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88#946—22no—
89#946—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#892—22no—
112#892—41no—
113#876—44no—
114C3×#103—22no—
115C3×#103—22no—
116C3×C4×C8—41no—
117C3×#95—21no—
118#835—22no—
119#835—41no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130(C2×D3)⋊C8—22no—
131(C2×D3)⋊C8—41no—
132#818—22no—
133#818—41no—
134Dic3⋊C8—22no—
135#808—22no—
136#807—22no—
137C2×#126—61no—
138C2×C22⋊Q8—31no—
139C2×C4×Q8C2×D311yescharacteristic
140C2×#117—31no—
141C2×#107—31no—
142C2×#107—31no—
143C2×#107—31no—
144C2×#106—31no—
145C2×C4⋊C8—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×#103C2×D311yescharacteristic
149C2×#103C2×D311yescharacteristic
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×C4×C8C2×D311yescharacteristic
153#400—61no—
154#397—61no—
155#391—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#388—61no—
160#388—61no—
161#387—61no—
162#387—61no—
163#384—31no—
164#380—31no—
165#380—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—61no—
168C4×C4⋊C4C2×D311yescharacteristic
169C4×C22⋊C4—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#336—31no—
179#336—31no—
180C23×D3C2×D411yescharacteristic
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8#4112yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8#4112yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4#4112yes—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—42no—
223C3×C42—42no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—22no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—42no—
229(C2×D3)⋊C4—42no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3#4112yes—
238Dic3⋊C4—24no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—42no—
247C4×Dic3—42no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C22×Q8C22×D311yescharacteristic
255C23×C4—31no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#126—124no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C4×Q8—22no—
264C4×Q8—41no—
265#117—62no—
266#117—121no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—31no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—31no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42C22×D311yescharacteristic
293#107—62no—
294#107—62no—
295#107—121no—
296#106—62no—
297C4⋊C8—62no—
298#103—22no—
299#103—22no—
300#103—62no—
301#103—121no—
302C22⋊C8—62no—
303C22⋊C8—121no—
304C4×C8—41no—
305#95—21no—
306#95—31no—
307#95—31no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316C23×C3C22×D411yescharacteristic
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—41no—
327C3×Q8—41no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4AGL(1,ℤ/8)12yes—
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C42−1+411yescharacteristic
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—22no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3#13512yes—
373C4×D3—41no—
374C4×D3—41no—
375C4×D3—41no—
376C4×D3—41no—
377C3×C8—22no—
378C3×C8—41no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C24—31no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8#23512yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8#23512yes—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—32no—
426C4⋊C4—42no—
427C4⋊C4—64no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—122no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4#23512yes—
441C22⋊C4—62no—
442C22⋊C4—64no—
443C22⋊C4—62no—
444C22⋊C4—122no—
445C22⋊C4—122no—
446C22⋊C4—122no—
447C42—21no—
448C42—21no—
449C42—42no—
450C42—42no—
451C42—61no—
452C42—61no—
453C42—122no—
454C42—122no—
455C22×C3#52511yescharacteristic
456C22×C3#54611yescharacteristic
457C22×C3#54711yescharacteristic
458C22×C3C2×#13511yescharacteristic
459C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
460C22×C3#57811yescharacteristic
461C22×C3#57811yescharacteristic
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—41no—
468C2×D3—41no—
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—41no—
472C3×C4—42no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4#52911yescharacteristic
476C3×C4#53911yescharacteristic
477C3×C4#55311yescharacteristic
478C3×C4#55611yescharacteristic
479Dic3—22no—
480Dic3—41no—
481Dic3—42no—
482Dic3—41no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486C23—61no—
487C23—61no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23C2×D3×D411yescharacteristic
494Q8—22no—
495Q8—22no—
496Q8—41no—
497Q8—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—32no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4#91412yes—
541C2×C4C2×#23511yescharacteristic
542C2×C4C2×#23511yescharacteristic
543C2×C4C2×#23511yescharacteristic
544C2×C4#100411yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4#101011yescharacteristic
547C8—22no—
548C8—41no—
549C8—121no—
550C8—121no—
551C2×C3#246611yescharacteristic
552C2×C3#310611yescharacteristic
553C2×C3#312711yescharacteristic
554C2×C3#330211yescharacteristic
555C2×C3#330811yescharacteristic
556C2×C3#333611yescharacteristic
557C2×C3#335311yescharacteristic
558D3—81no—
559C22—121no—
560C22—121no—
561C22—121no—
562C22—121no—
563C22—121no—
564C22—121no—
565C22—121no—
566C22#549011yescharacteristic
567C22#554711yescharacteristic
568C22#555111yescharacteristic
569C22C2×#91411yescharacteristic
570C22#576611yescharacteristic
571C22D3×#13511yescharacteristic
572C22#577111yescharacteristic
573C4—22no—
574C4—41no—
575C4—42no—
576C4—42no—
577C4—62no—
578C4—81no—
579C4—81no—
580C4—121no—
581C4—122no—
582C4—121no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4#547911yescharacteristic
587C4#553311yescharacteristic
588C4#554011yescharacteristic
589C4#556411yescharacteristic
590C3#2704311yescharacteristic, complemented, Sylow
591C2—241no—
592C2#7359211yescharacteristic
593C2#7880911yescharacteristic
594C2#7884911yescharacteristic
595C2#7969211yescharacteristic
596C2#7969511yescharacteristic
597C2#7972411yescharacteristic
598C2#7974111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC4×#103D3
hypocenterC3#27043
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. C3×C4×#103C2semidirect
4. #71629C2semidirect
6. #70248C2semidirect
8. #70233C2semidirect
10. #69770C2semidirect
11. #69458C2semidirect
14. #68795C2semidirect
15. #68789C2semidirect
74. C4×#103D3semidirect
590. C3#27043semidirect

Maximal subgroups

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

subgroupindex
#824642
#737922
C3×C4×#1032
#716292
#702752
#702482
#702442
#702332
#697852
#697702
#694582
#688772
#688732
#687952
#687892
#270433