Finite Group Databasebeta

← #10868433

Subgroups — #10868433

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76492C211yesmaximal, characteristic, complemented
2#73695C211yesmaximal, characteristic
3C3×#1928C211yesmaximal, characteristic, complemented
4#70645C211yesmaximal, characteristic, complemented
5#70623C211yesmaximal, characteristic
6#70284C211yesmaximal, characteristic, complemented
7#70282C211yesmaximal, characteristic
8#70240C211yesmaximal, characteristic
9#70222C211yesmaximal, characteristic, complemented
10#69676C211yesmaximal, characteristic
11C4×#810C211yesmaximal, characteristic, complemented
12#68863C211yesmaximal, characteristic
13#68789C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68655C211yesmaximal, characteristic, complemented
16#27069—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4988C2211yescharacteristic
32#4988C2211yescharacteristic
33#4977C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4973C2211yescharacteristic
36#4972C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41#4940—21no—
42#4940—21no—
43#4939C2211yescharacteristic
44C4×Dic3⋊C4C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4675—21no—
47#4675—21no—
48#4655—21no—
49#4652—21no—
50#4649—21no—
51#4647—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2513—31no—
61#2365—31no—
62#2097—31no—
63#2055—31no—
64#2043—31no—
65#2013—31no—
66#1968—31no—
67#1928D311yescharacteristic, complemented
68#1925—31no—
69#1920—31no—
70#1916—31no—
71#1880—31no—
72#1880—31no—
73#1805—31no—
74C4×#103—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C42—21no—
84C2×C3×C42C2311yescharacteristic
85Q8×Dic3—22no—
86Q8×Dic3—41no—
87#944—22no—
88#944—22no—
89C2×#215—22no—
90C2×#215—22no—
91C2×#215—41no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4D411yescharacteristic
106C2×Dic3⋊C4D411yescharacteristic
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×#204—22no—
115C2×#204—41no—
116C2×#204—41no—
117C4×Dic6—22no—
118C4×Dic6—41no—
119C3×#107—41no—
120C3×#103—22no—
121C3×#103—41no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125#835—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#821—22no—
135#816—41no—
136Dic3⋊C8—41no—
137#810—22no—
138#810—22no—
139#809—41no—
140#808—22no—
141C4×C3⋊C8—41no—
142C22×SD16—31no—
143C22×SD16—31no—
144C2×C4⋊Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×Q8—31no—
147C2×#107C2×D311yescharacteristic
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103C2×D311yescharacteristic
155C2×#103C2×D311yescharacteristic
156C2×#102—31no—
157C2×#102—31no—
158C2×C4×C8—31no—
159#397—61no—
160#397—61no—
161#391—31no—
162#391C2×D311yescharacteristic
163#390—31no—
164#390—31no—
165#390—31no—
166#389—31no—
167#389—61no—
168#388—61no—
169#388—61no—
170#387—61no—
171#387—61no—
172#382—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—61no—
176C4×C4⋊C4C2×D311yescharacteristic
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C23×D3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8#4112yes—
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×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6#4112yes—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—81no—
233#215—44no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3#4112yes—
245Dic3⋊C4—24no—
246Dic3⋊C4—24no—
247Dic3⋊C4—22no—
248Dic3⋊C4—22no—
249Dic3⋊C4—42no—
250Dic3⋊C4—81no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—42no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8#4112yes—
264#204—44no—
265C22×Q8—31no—
266C22×Q8C22×D311yescharacteristic
267C22×D4—31no—
268C2×SD16—62no—
269C2×SD16—62no—
270C2×SD16—62no—
271C2×SD16—121no—
272C2×SD16—121no—
273C2×SD16—121no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4⋊Q8—62no—
279C4⋊Q8—62no—
280C4×Q8—62no—
281C4×Q8—62no—
282C4×Q8—121no—
283C4×Q8—121no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C42—21no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—41no—
310#106—62no—
311C4⋊C8—121no—
312#103—22no—
313#103—41no—
314#103—62no—
315#103—62no—
316#103—121no—
317#103—121no—
318#102—62no—
319#102—121no—
320C4×C8—121no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332C23×C3C22×D411yescharacteristic
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—41no—
343C3×Q8—41no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4#13712yes—
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3#13512yes—
389D12—41no—
390D12—41no—
391D12—41no—
392D12—41no—
393Dic6—22no—
394Dic6—22no—
395Dic6—41no—
396Dic6—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—22no—
400C3⋊C8—41no—
401C24—61no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—32no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8#23512yes—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C23×D311yescharacteristic
434SD16—124no—
435SD16—124no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—32no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—32no—
455C4⋊C4—62no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C22⋊C4—122no—
466C22⋊C4—122no—
467C22⋊C4—241no—
468C22⋊C4—241no—
469C22⋊C4—241no—
470C22⋊C4—241no—
471C42—21no—
472C42—21no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—81no—
478C42—122no—
479C42—122no—
480C42—122no—
481C22×C3#52511yescharacteristic
482C22×C3#54611yescharacteristic
483C22×C3#55011yescharacteristic
484C22×C3C2×#13511yescharacteristic
485C22×C3C2×#13711yescharacteristic
486C22×C3#57711yescharacteristic
487C22×C3#57711yescharacteristic
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4#52911yescharacteristic
502C3×C4#53811yescharacteristic
503C3×C4#55111yescharacteristic
504C3×C4#55311yescharacteristic
505Dic3—22no—
506Dic3—41no—
507Dic3—42no—
508Dic3—41no—
509Dic3—42no—
510Dic3—41no—
511Dic3—81no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23—121no—
518C23—121no—
519C23C2×D3×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524Q8—62no—
525Q8—62no—
526Q8—121no—
527Q8—121no—
528D4—121no—
529D4—121no—
530D4—121no—
531D4—121no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—32no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—62no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4#91812yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4#100911yescharacteristic
577C8—41no—
578C8—41no—
579C8—62no—
580C8—121no—
581C2×C3#246711yescharacteristic
582C2×C3#310811yescharacteristic
583C2×C3#312611yescharacteristic
584C2×C3#330311yescharacteristic
585C2×C3#332011yescharacteristic
586C2×C3#339411yescharacteristic
587C2×C3#340011yescharacteristic
588D3—161no—
589C22—241no—
590C22—241no—
591C22—241no—
592C22—241no—
593C22—241no—
594C22—241no—
595C22—241no—
596C22#554711yescharacteristic
597C22#559911yescharacteristic
598C22#565411yescharacteristic
599C22C2×#91811yescharacteristic
600C22D3×#13511yescharacteristic
601C22#577511yescharacteristic
602C22#577811yescharacteristic
603C4—22no—
604C4—42no—
605C4—41no—
606C4—62no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—122no—
611C4—121no—
612C4—121no—
613C4—122no—
614C4—121no—
615C4—241no—
616C4#552711yescharacteristic
617C4#554011yescharacteristic
618C4#567811yescharacteristic
619C4#570611yescharacteristic
620C3#2706911yescharacteristic, complemented, Sylow
621C2—481no—
622C2#7585911yescharacteristic
623C2#7972311yescharacteristic
624C2#7978911yescharacteristic
625C2#8047011yescharacteristic
626C2#8052411yescharacteristic
627C2#8149111yescharacteristic
628C2#8154511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1928C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1928D3
hypocenterC3#27069
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
1. #76492C2semidirect
3. C3×#1928C2semidirect
4. #70645C2semidirect
6. #70284C2semidirect
9. #70222C2semidirect
11. C4×#810C2semidirect
13. #68789C2semidirect
15. #68655C2semidirect
67. #1928D3semidirect
620. C3#27069semidirect

Maximal subgroups

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

subgroupindex
#764922
#736952
C3×#19282
#706452
#706232
#702842
#702822
#702402
#702222
#696762
C4×#8102
#688632
#687892
#686862
#686552
#270693