Finite Group Databasebeta

← #10858820

Subgroups — #10858820

Order 768.

746 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#82464C211yesmaximal, characteristic
2#74162C211yesmaximal, characteristic
3C3×#1805C211yesmaximal, characteristic, complemented
4#71629C211yesmaximal, characteristic, complemented
5#70477C211yesmaximal, characteristic
6#70463C211yesmaximal, characteristic, complemented
7#70270C211yesmaximal, characteristic
8#70269C211yesmaximal, characteristic
9#70248C211yesmaximal, characteristic, complemented
10#70231C211yesmaximal, characteristic, complemented
11#69542C211yesmaximal, characteristic, complemented
12#68877C211yesmaximal, characteristic
13#68866C211yesmaximal, characteristic
14#68800C211yesmaximal, characteristic, complemented
15#68789C211yesmaximal, characteristic, complemented
16#26413—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#382C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4990—21no—
34#4987C2211yescharacteristic
35#4978—21no—
36Dic3×C4⋊C4C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4940—21no—
40C4×(C2×D3)⋊C4C2211yescharacteristic
41C4×Dic3⋊C4—21no—
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4676—21no—
45#4673—21no—
46#4652—21no—
47#4646—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2439—31no—
61#2362—31no—
62#2101—31no—
63#2087—31no—
64#2068—31no—
65#2024—31no—
66#1988—31no—
67#1920—31no—
68#1892—31no—
69#1873—31no—
70#1865—31no—
71#1854—31no—
72#1850—31no—
73#1805D311yescharacteristic, complemented
74#1735—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×Dic6—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4×Q8—22no—
81C3×C4×Q8—41no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88#946—22no—
89#946Q814yescomplemented
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×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×D3×C8—22no—
112C2×D3×C8—41no—
113C2×D3×C8—41no—
114#892—22no—
115#892—22no—
116#892—41no—
117#892—41no—
118#888—22no—
119#888—41no—
120C3×C4⋊C8—41no—
121C3×#103—41no—
122C3×#103—41no—
123C3×#95—21no—
124C3×#95—21no—
125#835—22no—
126#835—41no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136(C2×D3)⋊C8—22no—
137#818—41no—
138Dic3⋊C8—41no—
139#808—22no—
140C23×C8—31no—
141C2×C4⋊Q8—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8C2×D311yescharacteristic
144C2×#117—31no—
145C2×#117—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#103C2×D311yescharacteristic
154C2×C22⋊C8—31no—
155#396—61no—
156#393—61no—
157#391—31no—
158#391—61no—
159#389—31no—
160#389—31no—
161#389—31no—
162#388—61no—
163#388—61no—
164#387—61no—
165#387—61no—
166#382C2×D311yescharacteristic
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C23×D3C2×D411yescharacteristic
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×Dic3—21no—
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3C2×Q812yes—
209C2×C4×D3#4112yes—
210C2×Dic6—42no—
211C2×Dic6—42no—
212C2×Dic6—81no—
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—42no—
225C3×C4⋊C4#4112yes—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—42no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—41no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—81no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3C2×Q812yes—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—24no—
244Dic3⋊C4—42no—
245Dic3⋊C4—42no—
246Dic3⋊C4—42no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C4×Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259D3×C8—44no—
260C22×Q8—61no—
261C22×Q8C22×D311yescharacteristic
262C23×C4—31no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—62no—
266C22×C8—121no—
267C22×C8—121no—
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270C4⋊Q8—62no—
271C4⋊Q8—121no—
272C22⋊Q8—34no—
273C22⋊Q8—62no—
274C4×Q8—22no—
275C4×Q8—41no—
276#117—62no—
277#117—62no—
278#117—121no—
279#117—121no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—21no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#107—121no—
306#106—62no—
307C4⋊C8—41no—
308C4⋊C8—121no—
309#103—41no—
310#103—41no—
311#103—62no—
312#103—121no—
313C22⋊C8—62no—
314#95—21no—
315#95—21no—
316#95—31no—
317#95—31no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3C22×D411yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3C22⋊Q812yes—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
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—21no—
367C2×Dic3—22no—
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×Dic3C22⋊Q812yes—
381C2×Dic3#13512yes—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—22no—
386Dic6—84no—
387C3×C8—41no—
388C3×C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24—31no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—122no—
398C2×Q8—122no—
399C2×Q8—241no—
400C2×Q8D3×Q812yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—31no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
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—
433C2×C8—124no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—32no—
440C4⋊C4—32no—
441C4⋊C4—42no—
442C4⋊C4—64no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—122no—
450C4⋊C4—122no—
451C4⋊C4—122no—
452C4⋊C4—241no—
453C4⋊C4#23512yes—
454C22⋊C4—62no—
455C22⋊C4—64no—
456C22⋊C4—121no—
457C22⋊C4—122no—
458C22⋊C4—122no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—42no—
463C42—61no—
464C42—61no—
465C42—122no—
466C42—122no—
467C42—122no—
468C22×C3C2×C22⋊Q811yescharacteristic
469C22×C3#54611yescharacteristic
470C22×C3#54611yescharacteristic
471C22×C3C2×#13511yescharacteristic
472C22×C3#57511yescharacteristic
473C22×C3#57711yescharacteristic
474C22×C3#57811yescharacteristic
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—41no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—42no—
485C3×C4—41no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4#52911yescharacteristic
489C3×C4#53811yescharacteristic
490C3×C4#55411yescharacteristic
491C3×C4#55411yescharacteristic
492Dic3—22no—
493Dic3—22no—
494Dic3—42no—
495Dic3—42no—
496Dic3—41no—
497Dic3—81no—
498Dic3—81no—
499C23—32no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—61no—
506C23C2×D3×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511Q8—244no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—32no—
521C2×C4—32no—
522C2×C4—41no—
523C2×C4—41no—
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—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
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×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#245011yescharacteristic
566C2×C3#307611yescharacteristic
567C2×C3#308211yescharacteristic
568C2×C3#329611yescharacteristic
569C2×C3#330211yescharacteristic
570C2×C3#331311yescharacteristic
571C2×C3#332111yescharacteristic
572D3—42no—
573C22—62no—
574C22—62no—
575C22—62no—
576C22—121no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22#549011yescharacteristic
581C22#561211yescharacteristic
582C22D3×C22⋊Q811yescharacteristic
583C22D3×#13511yescharacteristic
584C22#577111yescharacteristic
585C22#577711yescharacteristic
586C22#577811yescharacteristic
587C4—41no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—62no—
592C4—62no—
593C4—81no—
594C4—81no—
595C4—122no—
596C4—122no—
597C4—121no—
598C4—241no—
599C4—241no—
600C4#551111yescharacteristic
601C4#552811yescharacteristic
602C4#561311yescharacteristic
603C4#563611yescharacteristic
604C3#2641311yescharacteristic, complemented, Sylow
605C2—122no—
606C2#7402311yescharacteristic
607C2#7884911yescharacteristic
608C2#7896911yescharacteristic
609C2#8061511yescharacteristic
610C2#8061811yescharacteristic
611C2#8122911yescharacteristic
612C2#8123211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1805D3
hypocenterC3#26413
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
3. C3×#1805C2semidirect
4. #71629C2semidirect
6. #70463C2semidirect
9. #70248C2semidirect
10. #70231C2semidirect
11. #69542C2semidirect
14. #68800C2semidirect
15. #68789C2semidirect
73. #1805D3semidirect
89. #946Q8semidirect
604. C3#26413semidirect

Maximal subgroups

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

subgroupindex
#824642
#741622
C3×#18052
#716292
#704772
#704632
#702702
#702692
#702482
#702312
#695422
#688772
#688662
#688002
#687892
#264133