Finite Group Databasebeta

← #10830062

Subgroups — #10830062

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5568C211yesmaximal, characteristic
2C2×#5185C211yesmaximal, characteristic
3C2×#4790C211yesmaximal, characteristic, complemented
4#76547C211yesmaximal, characteristic, complemented
5C3×#1878C211yesmaximal, characteristic, complemented
6#72073C211yesmaximal, characteristic, complemented
7#71628C211yesmaximal, characteristic, complemented
8#70276C211yesmaximal, characteristic
9#70246C211yesmaximal, characteristic, complemented
10#70226C211yesmaximal, characteristic
11#69689C211yesmaximal, characteristic
12#68905C211yesmaximal, characteristic, complemented
13#68894C211yesmaximal, characteristic
14#68630C211yesmaximal, characteristic
15#68619C211yesmaximal, characteristic, complemented
16#24195—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#5185—24no—
41C2×#821C2211yescharacteristic
42C2×#821C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46C2×C8×Dic3C2211yescharacteristic
47#4984C2211yescharacteristic
48#4977C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#807C2211yescharacteristic
52#4928C2211yescharacteristic
53#4790—24no—
54#4660—21no—
55#4656—21no—
56#4655—21no—
57#4654—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#464—31no—
66C2×C4×Q16—31no—
67#2581—31no—
68#2043—31no—
69#2042—31no—
70#2016—31no—
71#2015—31no—
72#1984—31no—
73#1958—31no—
74#1878D311yescharacteristic, complemented
75#1877—31no—
76#1863—31no—
77#1862—31no—
78#1805—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#947D414yes—
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101#944—22no—
102#944—22no—
103C2×#216—22no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#215—22no—
107C2×#215—22no—
108C2×#215—41no—
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⋊C4D411yescharacteristic
124C2×Dic3⋊C4D411yescharacteristic
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131#898—28no—
132#896—28no—
133#893—22no—
134#893—22no—
135#887—22no—
136#887D414yes—
137#876—28no—
138#875—22no—
139#875—22no—
140#872—28no—
141C4×Dic6—22no—
142C4×Dic6—22no—
143C3×#103—22no—
144C3×#103—41no—
145C3×#95—21no—
146C3×#95—21no—
147#835—22no—
148#835—22no—
149C3⋊#95—21no—
150C3⋊#95—21no—
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—41no—
162#809—22no—
163#807—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#464—64no—
182C4×Q16—64no—
183C2×#107—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#400—61no—
197#395—61no—
198#391—31no—
199#391C2×D311yescharacteristic
200#389—31no—
201#389—31no—
202#387—61no—
203#387—61no—
204#382—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—41no—
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—28no—
272#215—44no—
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—42no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287Dic3⋊C4—22no—
288Dic3⋊C4—22no—
289Dic3⋊C4—24no—
290Dic3⋊C4—42no—
291C4×Dic3—21no—
292C4×Dic3—21no—
293C4×Dic3—24no—
294C4×Dic3—42no—
295C4×Dic3C2×D412yes—
296C4×Dic3C2×D412yes—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8#4112yes—
303C22×Q8—31no—
304C22×Q8—31no—
305C22×Q8C22×D311yescharacteristic
306C22×Q8C22×D311yescharacteristic
307C22×D4—31no—
308C23×C4—61no—
309C2×Q16—22no—
310C2×Q16—22no—
311C2×Q16—41no—
312C2×Q16—62no—
313C2×Q16—62no—
314C2×Q16—62no—
315C2×SD16—62no—
316C2×SD16—62no—
317C2×SD16—121no—
318C22×C8—31no—
319C22×C8—31no—
320C22×C8C22×D311yescharacteristic
321C22×C8C22×D311yescharacteristic
322C4⋊Q8—62no—
323C4⋊Q8—62no—
324#126—68no—
325#126—68no—
326#124—34no—
327#124—62no—
328#124—62no—
329#124—62no—
330C22⋊Q8—68no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—34no—
334C4×Q8—62no—
335C4×Q8—62no—
336C4×Q8—62no—
337C4×Q8—62no—
338C4×Q8—62no—
339C4×D4—62no—
340C4×D4—62no—
341#117—68no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C22⋊C4—61no—
357C2×C22⋊C4—61no—
358C2×C22⋊C4—61no—
359C2×C22⋊C4—61no—
360C2×C22⋊C4—61no—
361C2×C42—21no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—61no—
365C2×C42—61no—
366C2×C42C22×D311yescharacteristic
367#107—62no—
368C4⋊C8—62no—
369#103—22no—
370#103—41no—
371#103—62no—
372#103—62no—
373#103—62no—
374#103—121no—
375#103—121no—
376#102—62no—
377#102—62no—
378#102—121no—
379C4×C8—62no—
380#95—21no—
381#95—21no—
382#95—61no—
383#95—61no—
384#95—61no—
385#95—61no—
386#95—61no—
387#95—61no—
388C23×C3C22×D411yescharacteristic
389C22×D3—22no—
390C22×D3—41no—
391C22×D3—41no—
392C22×D3—41no—
393C22×D3—41no—
394C22×D3—41no—
395C22×D3—41no—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—22no—
399C3×Q8—22no—
400C3×Q8—41no—
401C3×Q8—41no—
402C3×Q8—41no—
403C3×Q8—41no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4H(5,𝔽2)11yescharacteristic
427C2×C3×C42−1+411yescharacteristic
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—22no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3#12012yes—
447C2×Dic3#12012yes—
448C2×Dic3#13512yes—
449D12—22no—
450D12—22no—
451D12—41no—
452D12—41no—
453C4×D3—48no—
454Dic6—22no—
455Dic6—22no—
456Dic6—44no—
457Dic6—41no—
458Dic6—41no—
459C3×C8—41no—
460C3×C8—41no—
461C3⋊C8—22no—
462C3⋊C8—41no—
463C24—61no—
464C2×Q8—21no—
465C2×Q8—21no—
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—32no—
475C2×Q8—62no—
476C2×Q8—62no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—121no—
483C2×Q8#23512yes—
484C2×Q8D3×D412yes—
485C2×D4—32no—
486C2×D4—61no—
487C2×D4—61no—
488C2×D4—61no—
489C2×D4—61no—
490C2×D4—61no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—31no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—64no—
500C22×C4—61no—
501C22×C4—61no—
502C22×C4—61no—
503C22×C4—122no—
504C22×C4—122no—
505C22×C4D3×D411yescharacteristic
506C22×C4D3×D411yescharacteristic
507C22×C4C23×D311yescharacteristic
508Q16—44no—
509Q16—68no—
510SD16—124no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—32no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—62no—
528C4⋊C4—62no—
529C4⋊C4—62no—
530C4⋊C4—64no—
531C4⋊C4—64no—
532C4⋊C4—64no—
533C4⋊C4—61no—
534C4⋊C4—61no—
535C4⋊C4—61no—
536C4⋊C4—61no—
537C4⋊C4—122no—
538C4⋊C4—122no—
539C4⋊C4—122no—
540C4⋊C4D3×D412yes—
541C22⋊C4—64no—
542C22⋊C4—64no—
543C22⋊C4—64no—
544C22⋊C4—64no—
545C22⋊C4—64no—
546C42—24no—
547C42—21no—
548C42—21no—
549C42—32no—
550C42—32no—
551C42—64no—
552C42—61no—
553C42—61no—
554C42—122no—
555C22×C3C2×#12011yescharacteristic
556C22×C3#54811yescharacteristic
557C22×C3#55011yescharacteristic
558C22×C3C2×#13511yescharacteristic
559C22×C3#57511yescharacteristic
560C22×C3#57811yescharacteristic
561C22×C3#57811yescharacteristic
562C2×D3—42no—
563C2×D3—42no—
564C2×D3—42no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C3×C4—42no—
570C3×C4—42no—
571C3×C4—41no—
572C3×C4—42no—
573C3×C4—41no—
574C3×C4—81no—
575C3×C4#52511yescharacteristic
576C3×C4#53711yescharacteristic
577C3×C4#54611yescharacteristic
578C3×C4#54711yescharacteristic
579Dic3—22no—
580Dic3—22no—
581Dic3—22no—
582Dic3—42no—
583Dic3—41no—
584Dic3—81no—
585Dic3—81no—
586C23—62no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23C2×D3×D411yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600Q8—41no—
601Q8—41no—
602Q8—62no—
603Q8—62no—
604Q8—124no—
605Q8—121no—
606Q8—121no—
607D4—62no—
608D4—62no—
609D4—121no—
610D4—121no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—32no—
621C2×C4—32no—
622C2×C4—32no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—62no—
642C2×C4—128no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4C2×#23511yescharacteristic
652C2×C4C2×D3×D411yescharacteristic
653C2×C4C2×D3×D411yescharacteristic
654C2×C4#100411yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4#101011yescharacteristic
657C8—41no—
658C8—41no—
659C8—62no—
660C8—121no—
661C2×C3#240711yescharacteristic
662C2×C3#300211yescharacteristic
663C2×C3#301111yescharacteristic
664C2×C3#336411yescharacteristic
665C2×C3#337711yescharacteristic
666C2×C3#339311yescharacteristic
667C2×C3#340211yescharacteristic
668D3—82no—
669C22—122no—
670C22—122no—
671C22—122no—
672C22—241no—
673C22—241no—
674C22—241no—
675C22—241no—
676C22#555211yescharacteristic
677C22D3×#12011yescharacteristic
678C22#559911yescharacteristic
679C22#576611yescharacteristic
680C22D3×#13511yescharacteristic
681C22#577711yescharacteristic
682C22#577911yescharacteristic
683C4—41no—
684C4—41no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—62no—
689C4—62no—
690C4—62no—
691C4—81no—
692C4—121no—
693C4—122no—
694C4—241no—
695C4—241no—
696C4#549011yescharacteristic
697C4#554711yescharacteristic
698C4#560811yescharacteristic
699C4#565511yescharacteristic
700C3#2419511yescharacteristic, complemented, Sylow
701C2—242no—
702C2#7468811yescharacteristic
703C2#7970011yescharacteristic
704C2#7981811yescharacteristic
705C2#8018511yescharacteristic
706C2#8022611yescharacteristic
707C2#8047111yescharacteristic
708C2#8058811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1878C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1878D3
hypocenterC3#24195
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
3. C2×#4790C2semidirect
4. #76547C2semidirect
5. C3×#1878C2semidirect
6. #72073C2semidirect
7. #71628C2semidirect
9. #70246C2semidirect
12. #68905C2semidirect
15. #68619C2semidirect
74. #1878D3semidirect
700. C3#24195semidirect

Maximal subgroups

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

subgroupindex
C2×#55682
C2×#51852
C2×#47902
#765472
C3×#18782
#720732
#716282
#702762
#702462
#702262
#696892
#689052
#688942
#686302
#686192
#241953