Finite Group Databasebeta

← #10858121

Subgroups — #10858121

Order 768.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5121C211yesmaximal, characteristic
2C2×#4829C211yesmaximal, characteristic, complemented
3#73495C211yesmaximal, characteristic
4#73468C211yesmaximal, characteristic
5C3×#1851C211yesmaximal, characteristic
6#71989C211yesmaximal, characteristic
7#70489C211yesmaximal, characteristic, complemented
8#70463C211yesmaximal, characteristic, complemented
9#70455C211yesmaximal, characteristic, complemented
10#69930C211yesmaximal, characteristic
11#69929C211yesmaximal, characteristic
12#68693C211yesmaximal, characteristic, complemented
13#68684C211yesmaximal, characteristic, complemented
14#68634C211yesmaximal, characteristic, complemented
15#68622C211yesmaximal, characteristic, complemented
16#26367—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27#5212C2211yescharacteristic
28#5121—24no—
29C2×#818C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39#4946—21no—
40#4945—21no—
41#4944—21no—
42#4942—21no—
43#4936C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45#4929C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4928C2211yescharacteristic
49#4829—24no—
50#4660—21no—
51#4657—21no—
52#4652—21no—
53#4649—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#479—31no—
63C2×#428—31no—
64#2442—31no—
65#2362—31no—
66#2097—31no—
67#2087—31no—
68#2068—31no—
69#2059—31no—
70#1949—31no—
71#1919—31no—
72#1894—31no—
73#1872—31no—
74#1865—31no—
75#1854—31no—
76#1851D311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94#926—22no—
95#926—41no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—41no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106#924—22no—
107#924Q814yes—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118#870—22no—
119#870—41no—
120C4×Dic6—22no—
121C4×Dic6—41no—
122C3×C4⋊C8—22no—
123C3×C4⋊C8—22no—
124C3×C22⋊C8—22no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#818—22no—
137#818—22no—
138#818—22no—
139#817—22no—
140#817—41no—
141#817D414yes—
142#816—22no—
143#816—22no—
144#816—22no—
145#816—41no—
146C22×M16C2×D311yescharacteristic
147C2×#125—31no—
148C2×C22⋊Q8—31no—
149C2×C4×Q8—31no—
150C2×#117—31no—
151C22×C4⋊C4—31no—
152#479—64no—
153#428—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×C4⋊C8C2×D311yescharacteristic
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8C2×D311yescharacteristic
167#400—61no—
168#398—61no—
169#391—31no—
170#389—31no—
171#389—31no—
172#389—31no—
173#388—61no—
174#388—61no—
175#387—61no—
176#385—61no—
177#382—31no—
178#380—31no—
179#380—61no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—61no—
182C4×C22⋊C4C2×D311yescharacteristic
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#336C2×D311yescharacteristic
190#336C2×D311yescharacteristic
191C24×C3C2×D411yescharacteristic
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×Q812yes—
202C22×C3×C4#4112yes—
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—42no—
213C22×Dic3—42no—
214C22×Dic3—81no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C2×Q812yes—
221C3×M16—28no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C2×D412yes—
233C2×C3×C8#4112yes—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240#217—24no—
241#217—42no—
242#217—42no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3C2×D412yes—
256C4⋊Dic3C2×D412yes—
257C4⋊Dic3C2×Q812yes—
258Dic3⋊C4—24no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C22×Q8—31no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278#125—62no—
279#125—121no—
280C22⋊Q8—34no—
281C22⋊Q8—62no—
282C4×Q8—62no—
283C4×Q8—121no—
284#117—62no—
285#117—121no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—121no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4D1211yescharacteristic
307C2×C22⋊C4D1211yescharacteristic
308C2×C42—61no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#107—62no—
315#107—62no—
316#106—34no—
317#106—62no—
318#106—121no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—121no—
325C22⋊C8—22no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95D1211yescharacteristic
336#95D1211yescharacteristic
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3C22⋊Q812yes—
344C23×C3C22×D411yescharacteristic
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×Q811yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—84no—
395Dic6—22no—
396Dic6—22no—
397Dic6—41no—
398Dic6—41no—
399C3×C8—22no—
400C3×C8—22no—
401C3×C8—41no—
402C3×C8—41no—
403C24C2×D1211yescharacteristic
404C2×Q8—32no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—241no—
425C22×C4C2×D1211yescharacteristic
426C22×C4C2×D1211yescharacteristic
427C22×C4C2×D1211yescharacteristic
428C22×C4C2×D1211yescharacteristic
429C22×C4C2×D1211yescharacteristic
430C22×C4#23712yes—
431C22×C4D3×Q812yes—
432C22×C4C23×D311yescharacteristic
433M16—28no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8D3×D412yes—
445C2×C8#23712yes—
446C4⋊C4—32no—
447C4⋊C4—32no—
448C4⋊C4—32no—
449C4⋊C4—64no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C4⋊C4—241no—
467C4⋊C4—241no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—64no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—122no—
478C42—122no—
479C42—122no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3C2×C22⋊Q811yescharacteristic
488C22×C3D4211yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3#57511yescharacteristic
491C22×C3#57511yescharacteristic
492C22×C3#57711yescharacteristic
493C22×C3#57811yescharacteristic
494C3×C4—22no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4#52511yescharacteristic
500C3×C4#53911yescharacteristic
501C3×C4D4×Q811yescharacteristic
502C3×C4#55411yescharacteristic
503Dic3—42no—
504Dic3—42no—
505Dic3—42no—
506Dic3—42no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510Dic3—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23#89712yes—
518C23C22×D1211yescharacteristic
519Q8—62no—
520Q8—62no—
521Q8—121no—
522Q8—121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—62no—
542C2×C4—62no—
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×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—244no—
566C2×C4#89712yes—
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×#23711yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4C2×D3×Q811yescharacteristic
571C2×C4#100911yescharacteristic
572C2×C4#101011yescharacteristic
573C8—22no—
574C8—22no—
575C8—41no—
576C8—41no—
577C2×C3—42no—
578C2×C3#244911yescharacteristic
579C2×C3#308011yescharacteristic
580C2×C3#308211yescharacteristic
581C2×C3#326911yescharacteristic
582C2×C3#327511yescharacteristic
583C2×C3#334911yescharacteristic
584C2×C3#335111yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22C2×#89711yescharacteristic
593C22D4×D1211yescharacteristic
594C22#555511yescharacteristic
595C22#574811yescharacteristic
596C22#574811yescharacteristic
597C22#575111yescharacteristic
598C22#575311yescharacteristic
599C4—22no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—122no—
605C4—122no—
606C4—122no—
607C4—122no—
608C4—241no—
609C4—241no—
610C4—241no—
611C4—241no—
612C4#558611yescharacteristic
613C4#563211yescharacteristic
614C4#563511yescharacteristic
615C4#566511yescharacteristic
616C3#2636711yescharacteristic, complemented, Sylow
617C2—42no—
618C2#7442211yescharacteristic
619C2#7906111yescharacteristic
620C2#7910111yescharacteristic
621C2#7947511yescharacteristic
622C2#7949811yescharacteristic
623C2#7952911yescharacteristic
624C2#7955111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1851C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1851D3
hypocenterC3#26367
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. C2×#4829C2semidirect
7. #70489C2semidirect
8. #70463C2semidirect
9. #70455C2semidirect
12. #68693C2semidirect
13. #68684C2semidirect
14. #68634C2semidirect
15. #68622C2semidirect
616. C3#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#51212
C2×#48292
#734952
#734682
C3×#18512
#719892
#704892
#704632
#704552
#699302
#699292
#686932
#686842
#686342
#686222
#263673