Finite Group Databasebeta

← #10867908

Subgroups — #10867908

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76205C211yesmaximal, characteristic
2#73792C211yesmaximal, characteristic
3C3×#1920C211yesmaximal, characteristic, complemented
4#70582C211yesmaximal, characteristic
5#70563C211yesmaximal, characteristic, complemented
6#69618C211yesmaximal, characteristic, complemented
7C4×#810C211yesmaximal, characteristic, complemented
8#68914C211yesmaximal, characteristic, complemented
9#68912C211yesmaximal, characteristic, complemented
10#68875C211yesmaximal, characteristic
11#68872C211yesmaximal, characteristic
12#68824C211yesmaximal, characteristic
13#68816C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68657C211yesmaximal, characteristic
16#27034—31nomaximal, Sylow
17C2×#897C2211yescharacteristic
18C2×D3×C4⋊C4C2211yescharacteristic
19C2×#876—21no—
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C8C2211yescharacteristic
25C2×(C2×D3)⋊C8C2211yescharacteristic
26C2×#817C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30#4989—21no—
31#4987C2211yescharacteristic
32#4987C2211yescharacteristic
33#4975C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4973C2211yescharacteristic
36#4972—21no—
37#4972C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C4×(C2×D3)⋊C4C2211yescharacteristic
43#4933—21no—
44#4928C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4675—21no—
47#4666—21no—
48#4658—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2496—31no—
61#2360—31no—
62#2072—31no—
63#2066—31no—
64#2035—31no—
65#2035—31no—
66#1948—31no—
67#1941—31no—
68#1923—31no—
69#1920D311yescharacteristic, complemented
70#1919—31no—
71#1865—31no—
72#1853—31no—
73#1824—31no—
74C4×#103—31no—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#897—22no—
108#897—22no—
109D3×C4⋊C4—22no—
110D3×C4⋊C4—41no—
111#876—44no—
112C4×Dic6—22no—
113C4×Dic6—41no—
114C3×#106—22no—
115C3×#95—21no—
116C3×#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95D411yescharacteristic
125C3⋊#95D411yescharacteristic
126(C2×D3)⋊C8—22no—
127(C2×D3)⋊C8—41no—
128#817—22no—
129#816—22no—
130#816—41no—
131Dic3⋊C8—22no—
132#810—22no—
133#810—22no—
134#808—22no—
135#808—41no—
136C4×C3⋊C8—41no—
137C2×#126—61no—
138C2×C22⋊Q8—31no—
139C2×C4×Q8—31no—
140C22×C4⋊C4—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×#106—31no—
144C2×#106C2×D311yescharacteristic
145C2×C4⋊C8—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×#103—31no—
149C2×#103—31no—
150C2×C22⋊C8—31no—
151C2×C22⋊C8—31no—
152C2×C4×C8—31no—
153#400—61no—
154#397—61no—
155#391—31no—
156#389—31no—
157#389—31no—
158#389—61no—
159#389C2×D311yescharacteristic
160#387—61no—
161#387—61no—
162#385—61no—
163#384—31no—
164#380—31no—
165#380—31no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4C2×D311yescharacteristic
168C4×C22⋊C4—31no—
169C4×C22⋊C4—61no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#340C2×D311yescharacteristic
178#336—31no—
179#336C2×D311yescharacteristic
180C23×D3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×C4×D3—81no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6#4112yes—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4#4112yes—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—42no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—41no—
230(C2×D3)⋊C4—42no—
231(C2×D3)⋊C4—42no—
232(C2×D3)⋊C4SD1614yes—
233C4⋊Dic3—24no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3#4112yes—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—42no—
249C4×Dic3—81no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8#4112yes—
256C22×Q8—31no—
257C23×C4—31no—
258C23×C4—61no—
259C22×C8—31no—
260C22×C8—31no—
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263#126—124no—
264C22⋊Q8—62no—
265C22⋊Q8—62no—
266C4×Q8—62no—
267C4×Q8—121no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—121no—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—21no—
291C2×C42—31no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#106—22no—
296#106—62no—
297#106—62no—
298#106—121no—
299C4⋊C8—62no—
300#103—62no—
301#103—62no—
302#103—62no—
303#103—121no—
304C22⋊C8—62no—
305C22⋊C8—121no—
306C4×C8—121no—
307#95—21no—
308#95—21no—
309#95—31no—
310#95—31no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318C23×C3C22×D411yescharacteristic
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3C2×SD1612yes—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
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×C4C2×SD1612yes—
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C42−1+411yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—41no—
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×Dic3C2×SD1612yes—
371C4×D3—41no—
372C4×D3—41no—
373C4×D3—41no—
374C4×D3—41no—
375C4×D3—84no—
376Dic6—22no—
377Dic6—22no—
378Dic6—41no—
379Dic6—41no—
380C3×C8—41no—
381C3×C8—41no—
382C3⋊C8—22no—
383C3⋊C8—41no—
384C24—31no—
385C2×Q8—32no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—122no—
409C22×C4—122no—
410C22×C4—241no—
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D411yescharacteristic
413C22×C4C23×D311yescharacteristic
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—32no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—32no—
434C4⋊C4—64no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—81no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4#23512yes—
448C22⋊C4—34no—
449C22⋊C4—64no—
450C22⋊C4—121no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—21no—
454C42—21no—
455C42—42no—
456C42—61no—
457C42—61no—
458C42—122no—
459C42—241no—
460C22×C3#52511yescharacteristic
461C22×C3#54611yescharacteristic
462C22×C3#54711yescharacteristic
463C22×C3C22×SD1611yescharacteristic
464C22×C3#57511yescharacteristic
465C22×C3#57711yescharacteristic
466C22×C3#57811yescharacteristic
467C2×D3—22no—
468C2×D3—22no—
469C2×D3—22no—
470C2×D3—41no—
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—41no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4#52911yescharacteristic
481C3×C4#53911yescharacteristic
482C3×C4#55311yescharacteristic
483C3×C4#55611yescharacteristic
484Dic3—22no—
485Dic3—41no—
486Dic3—42no—
487Dic3—41no—
488Dic3—42no—
489Dic3—81no—
490Dic3—81no—
491C23—32no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23—61no—
497C23—61no—
498C23C2×D3×D411yescharacteristic
499Q8—62no—
500Q8—62no—
501Q8—121no—
502Q8—121no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—32no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—62no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
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—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—244no—
546C2×C4D3×SD1612yes—
547C2×C4C2×#23511yescharacteristic
548C2×C4#100711yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4#100911yescharacteristic
553C8—41no—
554C8—41no—
555C8—62no—
556C8—121no—
557C2×C3#246611yescharacteristic
558C2×C3#310311yescharacteristic
559C2×C3#312011yescharacteristic
560C2×C3#329811yescharacteristic
561C2×C3#331111yescharacteristic
562C2×C3#333511yescharacteristic
563C2×C3#335211yescharacteristic
564D3—42no—
565C22—62no—
566C22—62no—
567C22—62no—
568C22—121no—
569C22—121no—
570C22—121no—
571C22—121no—
572C22#554711yescharacteristic
573C22#563111yescharacteristic
574C22#565411yescharacteristic
575C22C2×D3×SD1611yescharacteristic
576C22#576811yescharacteristic
577C22#577511yescharacteristic
578C22#577911yescharacteristic
579C4—22no—
580C4—42no—
581C4—41no—
582C4—62no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—121no—
587C4—122no—
588C4—121no—
589C4—122no—
590C4—241no—
591C4—241no—
592C4#553011yescharacteristic
593C4#556811yescharacteristic
594C4#569211yescharacteristic
595C4#570611yescharacteristic
596C3#2703411yescharacteristic, complemented, Sylow
597C2—122no—
598C2#7533011yescharacteristic
599C2#7965311yescharacteristic
600C2#7978611yescharacteristic
601C2#8108111yescharacteristic
602C2#8126011yescharacteristic
603C2#8140111yescharacteristic
604C2#8145011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1920D3
hypocenterC3#27034
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
3. C3×#1920C2semidirect
5. #70563C2semidirect
6. #69618C2semidirect
7. C4×#810C2semidirect
8. #68914C2semidirect
9. #68912C2semidirect
13. #68816C2semidirect
14. #68684C2semidirect
69. #1920D3semidirect
596. C3#27034semidirect

Maximal subgroups

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

subgroupindex
#762052
#737922
C3×#19202
#705822
#705632
#696182
C4×#8102
#689142
#689122
#688752
#688722
#688242
#688162
#686842
#686572
#270343