Finite Group Databasebeta

← #10876269

Subgroups — #10876269

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#76610C211yesmaximal, characteristic
2#73760C211yesmaximal, characteristic
3C3×#1928C211yesmaximal, characteristic, complemented
4#70644C211yesmaximal, characteristic, complemented
5#70633C211yesmaximal, characteristic
6#70266C211yesmaximal, characteristic
7#70266C211yesmaximal, characteristic
8#70250C211yesmaximal, characteristic, complemented
9#70234C211yesmaximal, characteristic, complemented
10#69667C211yesmaximal, characteristic, complemented
11#69464C211yesmaximal, characteristic, complemented
12#68915C211yesmaximal, characteristic, complemented
13#68876C211yesmaximal, characteristic
14#68665C211yesmaximal, characteristic
15#68627C211yesmaximal, characteristic, complemented
16#27596—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×#872C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25#5229C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4977C2211yescharacteristic
36#4975C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4940—21no—
43C4×C4⋊Dic3C2211yescharacteristic
44#4933—21no—
45#4929C2211yescharacteristic
46#4669—21no—
47#4664—21no—
48#4664—21no—
49#4652—21no—
50#4646—21no—
51#4646—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2314—31no—
61#2310—31no—
62#2096—31no—
63#2095—31no—
64#2074—31no—
65#2074—31no—
66#1928—31no—
67#1928D311yescharacteristic, complemented
68#1927—31no—
69#1927—31no—
70#1854—31no—
71#1854—31no—
72#1818—31no—
73#1787—31no—
74#1787—31no—
75C22×C3×Q8C2311yescharacteristic
76C2×D3×Q8—22no—
77C2×D3×Q8—22no—
78C2×D3×Q8—22no—
79C2×D3×Q8—22no—
80C2×D3×Q8—41no—
81C2×D3×Q8—41no—
82C22×C4×D3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#892—41no—
111#892—41no—
112#872—22no—
113#872—22no—
114C3×#107—41no—
115C3×#103—22no—
116C3×#103—41no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120#835—22no—
121#835—41no—
122C3⋊#95—21no—
123C3⋊#95—21no—
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—
133(C2×D3)⋊C8—22no—
134(C2×D3)⋊C8—22no—
135#817—41no—
136#816—22no—
137#816—41no—
138#810—22no—
139#810—41no—
140#808—41no—
141#807—41no—
142C23×Q8—31no—
143C2×#117—31no—
144C2×#117—31no—
145C2×#107—31no—
146C2×#107C2×D311yescharacteristic
147C2×#106—31no—
148C2×#106—31no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103C2×D311yescharacteristic
156C2×#103C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#391—31no—
160#391—31no—
161#391—31no—
162#391—61no—
163#391—61no—
164#391C2×D311yescharacteristic
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#380—31no—
175C4×C4⋊C4—31no—
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×D3C2×D411yescharacteristic
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×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206D3×Q8—44no—
207D3×Q8—44no—
208D3×Q8—44no—
209D3×Q8—44no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—24no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×C4×D3#4112yes—
218C2×C4×D3#4112yes—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—24no—
224C2×Dic6—21no—
225C2×Dic6—42no—
226C2×Dic6—42no—
227C2×Dic6#4112yes—
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×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—81no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—81no—
247(C2×D3)⋊C4—81no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256Dic3⋊C4—81no—
257Dic3⋊C4—81no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
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—31no—
269C22×Q8—61no—
270C22×Q8—62no—
271C22×Q8—62no—
272C22×Q8—62no—
273C22×Q8—62no—
274C22×Q8—121no—
275C22×Q8—121no—
276C22×Q8C22×D311yescharacteristic
277C23×C4—31no—
278C23×C4—61no—
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—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C42—21no—
303C2×C42—31no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#107—41no—
308#107—121no—
309#106—121no—
310#106—121no—
311#103—22no—
312#103—41no—
313#103—62no—
314#103—62no—
315#103—62no—
316#103—121no—
317#103—121no—
318#103—121no—
319C22⋊C8—62no—
320C22⋊C8—62no—
321#95—21no—
322#95—21no—
323#95—21no—
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×C3C22×D411yescharacteristic
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3#12312yes—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—41no—
346C3×Q8—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
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—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
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#12312yes—
392C4×D3—22no—
393C4×D3—22no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—48no—
397Dic6—22no—
398Dic6—22no—
399Dic6—48no—
400Dic6—41no—
401Dic6—41no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—41no—
405C3⋊C8—41no—
406C24—31no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—32no—
413C2×Q8—64no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—122no—
420C2×Q8—122no—
421C2×Q8—124no—
422C2×Q8—124no—
423C2×Q8—124no—
424C2×Q8—124no—
425C2×Q8#23512yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—31no—
431C22×C4—32no—
432C22×C4—32no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—64no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4D3×D411yescharacteristic
447C22×C4D3×D411yescharacteristic
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—21no—
466C4⋊C4—21no—
467C4⋊C4—62no—
468C4⋊C4—62no—
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—122no—
480C22⋊C4—241no—
481C22⋊C4—241no—
482C42—21no—
483C42—21no—
484C42—61no—
485C42—61no—
486C42—81no—
487C42—122no—
488C42—241no—
489C22×C3C2×#12311yescharacteristic
490C22×C3#52511yescharacteristic
491C22×C3#53511yescharacteristic
492C22×C3#57711yescharacteristic
493C22×C3#57711yescharacteristic
494C22×C3#57811yescharacteristic
495C22×C3#57811yescharacteristic
496C2×D3—22no—
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—41no—
500C2×D3—41no—
501C2×D3—41no—
502C2×D3—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—42no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4—81no—
509C3×C4#55111yescharacteristic
510C3×C4#55111yescharacteristic
511C3×C4#55111yescharacteristic
512C3×C4#55111yescharacteristic
513Dic3—22no—
514Dic3—41no—
515Dic3—41no—
516Dic3—42no—
517Dic3—81no—
518Dic3—81no—
519Dic3—81no—
520C23—32no—
521C23—61no—
522C23—61no—
523C23—61no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23C2×D3×D411yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532Q8—62no—
533Q8—62no—
534Q8—128no—
535Q8—121no—
536Q8—121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—32no—
545C2×C4—41no—
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—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—128no—
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—121no—
581C2×C4C2×#23511yescharacteristic
582C2×C4#100411yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4#100911yescharacteristic
587C8—41no—
588C8—41no—
589C8—121no—
590C8—121no—
591C2×C3#248211yescharacteristic
592C2×C3#309711yescharacteristic
593C2×C3#309711yescharacteristic
594C2×C3#312111yescharacteristic
595C2×C3#312111yescharacteristic
596C2×C3#322111yescharacteristic
597C2×C3#322311yescharacteristic
598D3—42no—
599C22—62no—
600C22—62no—
601C22—62no—
602C22—121no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22#554711yescharacteristic
607C22#560711yescharacteristic
608C22D3×#12311yescharacteristic
609C22#576611yescharacteristic
610C22#577011yescharacteristic
611C22#577511yescharacteristic
612C22#577911yescharacteristic
613C4—41no—
614C4—42no—
615C4—41no—
616C4—62no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—121no—
621C4—122no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4—241no—
626C4#552711yescharacteristic
627C4#555611yescharacteristic
628C4#567911yescharacteristic
629C4#570511yescharacteristic
630C3#2759611yescharacteristic, complemented, Sylow
631C2—122no—
632C2#7586611yescharacteristic
633C2#7975011yescharacteristic
634C2#7981411yescharacteristic
635C2#8048411yescharacteristic
636C2#8054511yescharacteristic
637C2#8147011yescharacteristic
638C2#8151811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1928C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1928D3
hypocenterC3#27596
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
3. C3×#1928C2semidirect
4. #70644C2semidirect
8. #70250C2semidirect
9. #70234C2semidirect
10. #69667C2semidirect
11. #69464C2semidirect
12. #68915C2semidirect
15. #68627C2semidirect
67. #1928D3semidirect
630. C3#27596semidirect

Maximal subgroups

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

subgroupindex
#766102
#737602
C3×#19282
#706442
#706332
#702662
#702502
#702342
#696672
#694642
#689152
#688762
#686652
#686272
#275963