Finite Group Databasebeta

← #10876266

Subgroups — #10876266

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84062C211yesmaximal, characteristic
2#73868C211yesmaximal, characteristic
3C3×#1787C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic
5#70250C211yesmaximal, characteristic, complemented
6#70249C211yesmaximal, characteristic, complemented
7#70242C211yesmaximal, characteristic, complemented
8#70241C211yesmaximal, characteristic, complemented
9#70234C211yesmaximal, characteristic, complemented
10#70234C211yesmaximal, characteristic, complemented
11#70224C211yesmaximal, characteristic, complemented
12#70223C211yesmaximal, characteristic, complemented
13#69929C211yesmaximal, characteristic
14#69911C211yesmaximal, characteristic
15#69506C211yesmaximal, characteristic
16#27596—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C3×#380C2211yescharacteristic
26#5229—21no—
27#5229—21no—
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31#5229C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5212C2211yescharacteristic
37#5211—21no—
38#5211—21no—
39C2×#830C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42#4977C2211yescharacteristic
43#4975C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4945—21no—
49#4944—21no—
50#4654—21no—
51#4653—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2314—31no—
61#2310—31no—
62#2096—31no—
63#2095—31no—
64#2074—31no—
65#2074—31no—
66#1928—31no—
67#1928—31no—
68#1927—31no—
69#1927—31no—
70#1854—31no—
71#1854—31no—
72#1818—31no—
73#1787—31no—
74#1787D311yescharacteristic
75C22×C3×Q8—22no—
76C22×C3×Q8—22no—
77C22×C3×Q8—21no—
78C22×C3×Q8—22no—
79C22×C3×Q8—22no—
80C22×C3×Q8—41no—
81C22×C3×Q8—41no—
82C22×C3×Q8C2311yescharacteristic
83C22×C3×Q8C2311yescharacteristic
84C23×C3×C4—21no—
85C23×C3×C4C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217D411yescharacteristic
94C2×#217D411yescharacteristic
95#926—22no—
96#926—22no—
97#926—41no—
98#926—41no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C3×#103—22no—
115C3×#103—22no—
116C3×#103—41no—
117C3×#103—41no—
118C3×C22⋊C8—22no—
119C3×#95—21no—
120#835—22no—
121#835—22no—
122#835—41no—
123#835—41no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
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—
136C3⋊#95—21no—
137#830—22no—
138#818—41no—
139#817—41no—
140#808—41no—
141#807—41no—
142C23×Q8C2×D311yescharacteristic
143C2×#117—31no—
144C2×#117—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×#103C2×D311yescharacteristic
156C2×#103C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#391—31no—
160#391—31no—
161#391—31no—
162#391—31no—
163#391—61no—
164#391—61no—
165#388—61no—
166#388—61no—
167#387—61no—
168#387—61no—
169#384—31no—
170#382—31no—
171#381—61no—
172#381—61no—
173#380—31no—
174#380C2×D311yescharacteristic
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3C2×D411yescharacteristic
186C2×C3×Q8—24no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—44no—
198C2×C3×Q8—44no—
199C2×C3×Q8—44no—
200C2×C3×Q8—44no—
201C2×C3×Q8—42no—
202C2×C3×Q8—42no—
203C2×C3×Q8#4112yes—
204C2×C3×Q8#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4112yes—
218C22×C3×C4#4112yes—
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C22⋊C4—81no—
239#217—22no—
240#217—22no—
241#217—42no—
242#217—81no—
243C4⋊Dic3—22no—
244C4⋊Dic3—22no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253Dic3⋊C4—81no—
254Dic3⋊C4—81no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—42no—
260C4×Dic3—81no—
261C4×Dic3—81no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—21no—
271C22×Q8—22no—
272C22×Q8—22no—
273C22×Q8—41no—
274C22×Q8—41no—
275C22×Q8C22×D311yescharacteristic
276C22×Q8C22×D311yescharacteristic
277C23×C4—21no—
278C23×C4C22×D311yescharacteristic
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283#117—62no—
284#117—62no—
285#117—121no—
286#117—121no—
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—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—31no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307#107—121no—
308#107—121no—
309#106—121no—
310#106—121no—
311#103—22no—
312#103—22no—
313#103—41no—
314#103—41no—
315#103—62no—
316#103—62no—
317#103—121no—
318#103—121no—
319C22⋊C8—22no—
320C22⋊C8—62no—
321#95—21no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
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—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3#12312yes—
342C23×C3C22×D411yescharacteristic
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—48no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—41no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—48no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4#12312yes—
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C24D3×D411yescharacteristic
407C2×Q8—24no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8#23712yes—
425C2×Q8#23712yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4#23512yes—
446C22×C4D3×D411yescharacteristic
447C22×C4#23712yes—
448C22×C4C23×D311yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—241no—
476C4⋊C4—241no—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—81no—
480C22⋊C4—122no—
481C22⋊C4—241no—
482C42—61no—
483C42—61no—
484C42—61no—
485C42—61no—
486C42—122no—
487C42—241no—
488C42—241no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3C2×#12311yescharacteristic
497C22×C3#52511yescharacteristic
498C22×C3#53511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57711yescharacteristic
501C22×C3#57811yescharacteristic
502C22×C3#57811yescharacteristic
503C3×C4—22no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4#55111yescharacteristic
509C3×C4#55111yescharacteristic
510C3×C4#55111yescharacteristic
511C3×C4#55111yescharacteristic
512Dic3—41no—
513Dic3—41no—
514Dic3—41no—
515Dic3—41no—
516Dic3—81no—
517Dic3—81no—
518Dic3—81no—
519Dic3—81no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#88312yes—
527C23C2×D3×D411yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—48no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4#88312yes—
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×#23711yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4C2×#23711yescharacteristic
585C2×C4#100411yescharacteristic
586C2×C4#100411yescharacteristic
587C8—41no—
588C8—41no—
589C8—121no—
590C8—121no—
591C2×C3—42no—
592C2×C3#248211yescharacteristic
593C2×C3#309711yescharacteristic
594C2×C3#309711yescharacteristic
595C2×C3#312111yescharacteristic
596C2×C3#312111yescharacteristic
597C2×C3#322111yescharacteristic
598C2×C3#322311yescharacteristic
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22C2×#88311yescharacteristic
607C22#558611yescharacteristic
608C22#559211yescharacteristic
609C22#576211yescharacteristic
610C22#576611yescharacteristic
611C22#577811yescharacteristic
612C22#577911yescharacteristic
613C4—22no—
614C4—42no—
615C4—42no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—121no—
620C4—121no—
621C4—121no—
622C4—241no—
623C4—241no—
624C4—241no—
625C4—241no—
626C4#549511yescharacteristic
627C4#549511yescharacteristic
628C4#558711yescharacteristic
629C4#558711yescharacteristic
630C3#2759611yescharacteristic, complemented, Sylow
631C2—42no—
632C2#7383211yescharacteristic
633C2#7880411yescharacteristic
634C2#7895111yescharacteristic
635C2#8003211yescharacteristic
636C2#8003611yescharacteristic
637C2#8017611yescharacteristic
638C2#8018011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1787D3
hypocenterC3#27596
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
5. #70250C2semidirect
6. #70249C2semidirect
7. #70242C2semidirect
8. #70241C2semidirect
9. #70234C2semidirect
10. #70234C2semidirect
11. #70224C2semidirect
12. #70223C2semidirect
630. C3#27596semidirect

Maximal subgroups

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

subgroupindex
#840622
#738682
C3×#17872
#721322
#702502
#702492
#702422
#702412
#702342
#702242
#702232
#699292
#699112
#695062
#275963