Finite Group Databasebeta

← #10829976

Subgroups — #10829976

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5568C211yesmaximal, characteristic
2C2×#5182C211yesmaximal, characteristic, complemented
3C2×#4788C211yesmaximal, characteristic
4#76548C211yesmaximal, characteristic, complemented
5C3×#1878C211yesmaximal, characteristic, complemented
6#72072C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic, complemented
8#70281C211yesmaximal, characteristic, complemented
9#70245C211yesmaximal, characteristic
10#70225C211yesmaximal, characteristic, complemented
11#69689C211yesmaximal, characteristic
12#68907C211yesmaximal, characteristic, complemented
13#68892C211yesmaximal, characteristic
14#68630C211yesmaximal, characteristic
15#68619C211yesmaximal, characteristic, complemented
16#24189—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#944C2211yescharacteristic
22C22×#216C2211yescharacteristic
23C22×#215C2211yescharacteristic
24#5568—24no—
25#5568—24no—
26C2×#898—21no—
27C2×#896—21no—
28C2×#893C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×#876—21no—
31C2×#875C2211yescharacteristic
32C2×#872—21no—
33C2×C4×Dic6C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C3×#391C2211yescharacteristic
37#5229C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40#5182—24no—
41C2×#821C2211yescharacteristic
42C2×#821C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46C2×C8×Dic3C2211yescharacteristic
47#4984C2211yescharacteristic
48#4975C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#808C2211yescharacteristic
52#4928C2211yescharacteristic
53#4788—24no—
54#4661—21no—
55#4657—21no—
56#4647—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#462—31no—
66C2×C4×SD16—31no—
67#2582—31no—
68#2042—31no—
69#2042—31no—
70#2015—31no—
71#2015—31no—
72#1984—31no—
73#1956—31no—
74#1878—31no—
75#1878D311yescharacteristic, complemented
76#1865—31no—
77#1865—31no—
78#1802—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×Dic6D411yescharacteristic
84C22×Dic6C2311yescharacteristic
85C2×C3×Q16—22no—
86C2×C3×Q16—22no—
87C2×C3×Q16—41no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97#947—22no—
98#947—22no—
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101#944—22no—
102#944D414yes—
103C2×#216—22no—
104C2×#216—22no—
105C2×#216—41no—
106C2×#215—22no—
107C2×#215—22no—
108C2×#215—22no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3D411yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4D411yescharacteristic
126C2×Dic3⋊C4D411yescharacteristic
127C2×C4×Dic3—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133#898—28no—
134#896—28no—
135#893—22no—
136#893D414yes—
137#887—22no—
138#887—22no—
139#876—28no—
140#875—22no—
141#875—22no—
142#872—28no—
143C4×Dic6—22no—
144C4×Dic6—22no—
145C3×#103—22no—
146C3×#103—41no—
147C3×#95—21no—
148C3×#95—21no—
149#835—22no—
150#835—22no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155#821—22no—
156#821—41no—
157#816—22no—
158#816—41no—
159Dic3⋊C8—22no—
160C8×Dic3—22no—
161#810—22no—
162#809—41no—
163#808—22no—
164C22×Q16—31no—
165C22×Q16C2×D311yescharacteristic
166C22×SD16—31no—
167#556—64no—
168#556—64no—
169C2×C4⋊Q8—31no—
170C2×#126—61no—
171C2×#126—61no—
172C2×#124—31no—
173C2×#124—31no—
174C2×C22⋊Q8—61no—
175C2×C4×Q8—31no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×D4—31no—
180C2×#117—61no—
181#462—64no—
182C4×SD16—64no—
183C2×#106—31no—
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103C2×D311yescharacteristic
191C2×#103C2×D311yescharacteristic
192C2×#102—31no—
193C2×#102—31no—
194C2×#102—31no—
195C2×C4×C8—31no—
196#398—61no—
197#398—61no—
198#391—31no—
199#391—61no—
200#391C2×D311yescharacteristic
201#389—31no—
202#389—31no—
203#389—61no—
204#384—31no—
205#340—31no—
206#340C2×D311yescharacteristic
207#336—31no—
208#336—31no—
209#336—31no—
210#336C2×D311yescharacteristic
211C23×D3—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8C2×D412yes—
223C2×C3×Q8#4112yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12C2×D412yes—
244C2×C4×D3—24no—
245C2×C4×D3—42no—
246C2×C4×D3—42no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—22no—
253C2×Dic6—22no—
254C2×Dic6—22no—
255C2×Dic6C2×D412yes—
256C3×Q16—44no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4C2×D412yes—
268C3×C42—24no—
269C3×C42—21no—
270C3×C42—21no—
271#216—44no—
272#215—28no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—24no—
277(C2×D3)⋊C4—24no—
278C4⋊Dic3—22no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—42no—
284C4⋊Dic3—81no—
285Dic3⋊C4—24no—
286Dic3⋊C4—22no—
287Dic3⋊C4—22no—
288Dic3⋊C4—24no—
289Dic3⋊C4—24no—
290Dic3⋊C4—42no—
291Dic3⋊C4—81no—
292Dic3⋊C4—81no—
293C4×Dic3—24no—
294C4×Dic3—21no—
295C4×Dic3—21no—
296C4×Dic3—42no—
297C4×Dic3C2×D412yes—
298C4×Dic3C2×D412yes—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8#4112yes—
305C22×Q8—31no—
306C22×Q8—31no—
307C22×Q8C22×D311yescharacteristic
308C22×Q8C22×D311yescharacteristic
309C22×D4—31no—
310C23×C4—61no—
311C2×Q16—22no—
312C2×Q16—22no—
313C2×Q16—41no—
314C2×Q16—62no—
315C2×Q16—62no—
316C2×Q16—121no—
317C2×SD16—62no—
318C2×SD16—62no—
319C2×SD16—62no—
320C22×C8—31no—
321C22×C8—31no—
322C22×C8C22×D311yescharacteristic
323C22×C8C22×D311yescharacteristic
324C4⋊Q8—34no—
325C4⋊Q8—62no—
326#126—68no—
327#126—68no—
328#124—62no—
329#124—62no—
330#124—62no—
331#124—62no—
332C22⋊Q8—68no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—62no—
336C4×Q8—62no—
337C4×Q8—62no—
338C4×Q8—62no—
339C4×Q8—62no—
340C4×Q8—62no—
341C4×D4—34no—
342C4×D4—62no—
343#117—68no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4C22×D311yescharacteristic
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C22⋊C4—61no—
361C2×C22⋊C4—61no—
362C2×C22⋊C4—61no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365C2×C42—21no—
366C2×C42—31no—
367C2×C42—31no—
368C2×C42—61no—
369C2×C42—61no—
370C2×C42C22×D311yescharacteristic
371#106—62no—
372C4⋊C8—62no—
373#103—22no—
374#103—41no—
375#103—62no—
376#103—62no—
377#103—62no—
378#103—62no—
379#103—121no—
380#102—62no—
381#102—121no—
382#102—121no—
383C4×C8—62no—
384#95—21no—
385#95—21no—
386#95—61no—
387#95—61no—
388#95—61no—
389#95—61no—
390C23×C3C22×D411yescharacteristic
391C22×D3—22no—
392C22×D3—41no—
393C22×D3—41no—
394C22×D3—41no—
395C22×D3—41no—
396C22×D3—41no—
397C22×D3—41no—
398C3×Q8—22no—
399C3×Q8—22no—
400C3×Q8—22no—
401C3×Q8—22no—
402C3×Q8—41no—
403C3×Q8—41no—
404C3×Q8—41no—
405C3×Q8—41no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×C3×C4C2×#4111yescharacteristic
428C2×C3×C4H(5,𝔽2)11yescharacteristic
429C2×C3×C42−1+411yescharacteristic
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—22no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3#12012yes—
448C2×Dic3#12012yes—
449C2×Dic3#13512yes—
450C2×Dic3#13712yes—
451D12—22no—
452D12—22no—
453D12—41no—
454D12—41no—
455C4×D3—48no—
456Dic6—22no—
457Dic6—22no—
458Dic6—41no—
459Dic6—41no—
460Dic6—44no—
461C3×C8—41no—
462C3×C8—41no—
463C3⋊C8—22no—
464C3⋊C8—41no—
465C24—61no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—32no—
477C2×Q8—62no—
478C2×Q8—62no—
479C2×Q8—62no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8#23512yes—
486C2×Q8D3×D412yes—
487C2×D4—32no—
488C2×D4—61no—
489C2×D4—61no—
490C2×D4—61no—
491C2×D4—61no—
492C2×D4—61no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—64no—
502C22×C4—61no—
503C22×C4—61no—
504C22×C4—61no—
505C22×C4—122no—
506C22×C4—122no—
507C22×C4D3×D411yescharacteristic
508C22×C4D3×D411yescharacteristic
509C22×C4C23×D311yescharacteristic
510Q16—44no—
511Q16—124no—
512SD16—68no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—32no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C2×C8—61no—
524C2×C8—61no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—62no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—64no—
533C4⋊C4—64no—
534C4⋊C4—64no—
535C4⋊C4—61no—
536C4⋊C4—61no—
537C4⋊C4—61no—
538C4⋊C4—61no—
539C4⋊C4—122no—
540C4⋊C4—122no—
541C4⋊C4—241no—
542C4⋊C4—241no—
543C4⋊C4—241no—
544C4⋊C4D3×D412yes—
545C22⋊C4—64no—
546C22⋊C4—64no—
547C22⋊C4—64no—
548C22⋊C4—64no—
549C22⋊C4—64no—
550C42—24no—
551C42—21no—
552C42—21no—
553C42—32no—
554C42—32no—
555C42—64no—
556C42—61no—
557C42—61no—
558C42—122no—
559C22×C3C2×#12011yescharacteristic
560C22×C3#54811yescharacteristic
561C22×C3#55011yescharacteristic
562C22×C3C2×#13511yescharacteristic
563C22×C3C2×#13711yescharacteristic
564C22×C3#57711yescharacteristic
565C22×C3#57811yescharacteristic
566C2×D3—42no—
567C2×D3—42no—
568C2×D3—42no—
569C2×D3—81no—
570C2×D3—81no—
571C2×D3—81no—
572C2×D3—81no—
573C3×C4—42no—
574C3×C4—42no—
575C3×C4—41no—
576C3×C4—42no—
577C3×C4—41no—
578C3×C4—81no—
579C3×C4#52511yescharacteristic
580C3×C4#53711yescharacteristic
581C3×C4#54611yescharacteristic
582C3×C4#54711yescharacteristic
583Dic3—22no—
584Dic3—42no—
585Dic3—41no—
586Dic3—41no—
587Dic3—81no—
588Dic3—81no—
589Dic3#45214yes—
590C23—62no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23C2×D3×D411yescharacteristic
598Q8—22no—
599Q8—22no—
600Q8—22no—
601Q8—22no—
602Q8—41no—
603Q8—41no—
604Q8—41no—
605Q8—41no—
606Q8—62no—
607Q8—62no—
608Q8—124no—
609Q8—121no—
610Q8—121no—
611D4—62no—
612D4—62no—
613D4—121no—
614D4—121no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—32no—
625C2×C4—32no—
626C2×C4—32no—
627C2×C4—32no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—62no—
646C2×C4—128no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4C2×#23511yescharacteristic
656C2×C4C2×D3×D411yescharacteristic
657C2×C4C2×D3×D411yescharacteristic
658C2×C4#100411yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4#101011yescharacteristic
661C8—41no—
662C8—41no—
663C8—62no—
664C8—121no—
665C2×C3#240711yescharacteristic
666C2×C3C2×#45211yescharacteristic
667C2×C3#301311yescharacteristic
668C2×C3#336411yescharacteristic
669C2×C3#336911yescharacteristic
670C2×C3#339411yescharacteristic
671C2×C3#339811yescharacteristic
672D3—82no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22#555211yescharacteristic
681C22D3×#12011yescharacteristic
682C22#559911yescharacteristic
683C22#576611yescharacteristic
684C22D3×#13511yescharacteristic
685C22D3×#13711yescharacteristic
686C22#577811yescharacteristic
687C4—34no—
688C4—41no—
689C4—41no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—62no—
694C4—81no—
695C4—121no—
696C4—121no—
697C4—122no—
698C4—241no—
699C4—241no—
700C4#549011yescharacteristic
701C4#554711yescharacteristic
702C4#560811yescharacteristic
703C4#565511yescharacteristic
704C3#2418911yescharacteristic, complemented, Sylow
705C2—242no—
706C2#7468811yescharacteristic
707C2#7970011yescharacteristic
708C2#7981511yescharacteristic
709C2#8018611yescharacteristic
710C2D3×#45211yescharacteristic
711C2#8046911yescharacteristic
712C2#8058911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1878C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1878D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
2. C2×#5182C2semidirect
4. #76548C2semidirect
5. C3×#1878C2semidirect
7. #71629C2semidirect
8. #70281C2semidirect
10. #70225C2semidirect
12. #68907C2semidirect
15. #68619C2semidirect
75. #1878D3semidirect
704. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#55682
C2×#51822
C2×#47882
#765482
C3×#18782
#720722
#716292
#702812
#702452
#702252
#696892
#689072
#688922
#686302
#686192
#241893