Finite Group Databasebeta

← #10839055

Subgroups — #10839055

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82753C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic
3C2×#4790C211yesmaximal, characteristic, complemented
4#73876C211yesmaximal, characteristic
5C3×#2035C211yesmaximal, characteristic
6#70591C211yesmaximal, characteristic, complemented
7#70588C211yesmaximal, characteristic, complemented
8#70285C211yesmaximal, characteristic, complemented
9#70281C211yesmaximal, characteristic, complemented
10#70239C211yesmaximal, characteristic, complemented
11#69963C211yesmaximal, characteristic
12#69913C211yesmaximal, characteristic
13#69518C211yesmaximal, characteristic
14#68905C211yesmaximal, characteristic, complemented
15#68850C211yesmaximal, characteristic, complemented
16#24902—31nomaximal, Sylow
17C23×Dic6C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5229C2211yescharacteristic
30#5212C2211yescharacteristic
31C4×#217C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#5050—24no—
40#4986—21no—
41#4978—21no—
42#4977C2211yescharacteristic
43#4973C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4953—21no—
50#4944—21no—
51#4942—21no—
52Dic3×C22⋊C4—21no—
53#4790—24no—
54#4646—21no—
55#4646—21no—
56C3×#340C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#451—31no—
63C2×C4×Q16—31no—
64#2421—31no—
65Q8×C22⋊C4—31no—
66#2097—31no—
67#2075—31no—
68#2056—31no—
69#2035D311yescharacteristic
70#1933—31no—
71#1923—31no—
72#1878—31no—
73#1878—31no—
74#1865—31no—
75#1805—31no—
76#1787—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6—21no—
81C22×Dic6—22no—
82C22×Dic6—22no—
83C22×Dic6—22no—
84C22×Dic6—22no—
85C22×Dic6—41no—
86C22×Dic6—41no—
87C22×Dic6C2311yescharacteristic
88C22×Dic6C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—41no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×#217—21no—
100C2×#217D411yescharacteristic
101C2×#217D411yescharacteristic
102Q8×Dic3—22no—
103Q8×Dic3—41no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—22no—
108C2×#216—41no—
109C2×#216D414yes—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#887—22no—
124#887—22no—
125#887—41no—
126#887—41no—
127C3×#106—41no—
128C3×#103—22no—
129C3×#103—41no—
130C3×C22⋊C8—41no—
131C3×#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95D411yescharacteristic
143C3⋊#95D411yescharacteristic
144#830—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148#816—41no—
149Dic3⋊C8—41no—
150C8×Dic3—22no—
151#810—22no—
152#810—22no—
153#807—22no—
154C23×Q8—31no—
155C22×Q16—31no—
156C22×Q16—31no—
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C2×C4×Q8—31no—
161#451—64no—
162C4×Q16—64no—
163C2×#107—31no—
164C2×#106C2×D311yescharacteristic
165C2×C4⋊C8—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103C2×D311yescharacteristic
173C2×#103C2×D311yescharacteristic
174C2×C22⋊C8—31no—
175C2×C22⋊C8C2×D311yescharacteristic
176C2×C4×C8—31no—
177#400—61no—
178#397—61no—
179#391—31no—
180#391—31no—
181#391—31no—
182#391—61no—
183#391—61no—
184#389—31no—
185#388—61no—
186#387—61no—
187#382—31no—
188#380—31no—
189#380—61no—
190#380C2×D311yescharacteristic
191C4×C22⋊C4—31no—
192C4×C22⋊C4—61no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340C2×D311yescharacteristic
197#336—31no—
198#336—31no—
199C24×C3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C2×D412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—24no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—42no—
224C22×Dic3—42no—
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—24no—
237C2×Dic6—21no—
238C2×Dic6—44no—
239C2×Dic6—44no—
240C2×Dic6—44no—
241C2×Dic6—42no—
242C2×Dic6—42no—
243C2×Dic6—44no—
244C2×Dic6C2×D412yes—
245C2×Dic6#4112yes—
246C2×Dic6#4112yes—
247C2×Dic6#4112yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—42no—
261C3×C22⋊C4—42no—
262C3×C22⋊C4—42no—
263#217—22no—
264#217—22no—
265#217—42no—
266#216—28no—
267#216—44no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—42no—
281C4×Dic3—42no—
282C4×Dic3—42no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8C2×D412yes—
289C22×Q8—31no—
290C22×Q8—31no—
291C22×Q8—62no—
292C22×Q8—62no—
293C22×Q8—62no—
294C22×Q8—62no—
295C22×Q8—61no—
296C22×Q8—121no—
297C22×Q8—121no—
298C22×Q8C22×D311yescharacteristic
299C23×C4—61no—
300C23×C4C22×D311yescharacteristic
301C2×Q16—34no—
302C2×Q16—62no—
303C2×Q16—62no—
304C2×Q16—62no—
305C2×Q16—62no—
306C2×Q16—121no—
307C22×C8—31no—
308C22×C8—31no—
309C22×C8C22×D311yescharacteristic
310C22×C8C22×D311yescharacteristic
311C22⋊Q8—22no—
312C22⋊Q8—41no—
313C4×Q8—62no—
314C4×Q8—62no—
315C4×Q8—62no—
316C4×Q8—121no—
317C4×Q8—121no—
318C4×Q8—121no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—61no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42—61no—
339#107—62no—
340#106—41no—
341C4⋊C8—121no—
342#103—22no—
343#103—41no—
344#103—62no—
345#103—62no—
346#103—62no—
347#103—62no—
348#103—62no—
349#103—121no—
350C22⋊C8—41no—
351C22⋊C8—62no—
352C4×C8—62no—
353#95—21no—
354#95—31no—
355#95—31no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3#13512yes—
373C23×C3C22×D411yescharacteristic
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8—41no—
377C3×Q8—41no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4H(5,𝔽2)11yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C2×Dic3—22no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—48no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3#13712yes—
428Dic6—22no—
429Dic6—22no—
430Dic6—22no—
431Dic6—22no—
432Dic6—22no—
433Dic6—48no—
434Dic6—41no—
435Dic6—41no—
436Dic6#12314yes—
437C3×C8—41no—
438C3×C8—41no—
439C3⋊C8—22no—
440C3⋊C8—41no—
441C24D3×D411yescharacteristic
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—32no—
448C2×Q8—32no—
449C2×Q8—32no—
450C2×Q8—32no—
451C2×Q8—64no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—122no—
461C2×Q8—122no—
462C2×Q8—124no—
463C2×Q8—124no—
464C2×Q8—124no—
465C2×Q8—124no—
466C2×Q8D3×D412yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—61no—
482C22×C4—64no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4D3×D411yescharacteristic
489C22×C4C23×D311yescharacteristic
490Q16—68no—
491Q16—124no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—32no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—122no—
516C4⋊C4—122no—
517C4⋊C4—122no—
518C4⋊C4—122no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—62no—
522C22⋊C4—62no—
523C22⋊C4—122no—
524C42—61no—
525C42—61no—
526C42—61no—
527C42—61no—
528C42—122no—
529C42—122no—
530C42—122no—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3#52511yescharacteristic
539C22×C3D4211yescharacteristic
540C22×C3#54611yescharacteristic
541C22×C3C2×#13511yescharacteristic
542C22×C3C2×#13711yescharacteristic
543C22×C3#57811yescharacteristic
544C22×C3#57811yescharacteristic
545C3×C4—41no—
546C3×C4—42no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4C2×#12311yescharacteristic
551C3×C4#53811yescharacteristic
552C3×C4#54611yescharacteristic
553C3×C4#55111yescharacteristic
554Dic3—24no—
555Dic3—22no—
556Dic3—41no—
557Dic3—42no—
558Dic3—41no—
559Dic3—41no—
560Dic3—81no—
561Dic3—81no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23#91612yes—
569C23C2×D3×D411yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—34no—
573Q8—41no—
574Q8—41no—
575Q8—62no—
576Q8—62no—
577Q8—62no—
578Q8—62no—
579Q8—62no—
580Q8—128no—
581Q8—121no—
582Q8—121no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—32no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—62no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—128no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4C2×D3×D411yescharacteristic
628C2×C4#100411yescharacteristic
629C2×C4#100411yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—41no—
634C8—41no—
635C8—62no—
636C8—121no—
637C2×C3—42no—
638C2×C3#241811yescharacteristic
639C2×C3#310811yescharacteristic
640C2×C3#312711yescharacteristic
641C2×C3#327011yescharacteristic
642C2×C3#328511yescharacteristic
643C2×C3#330011yescharacteristic
644C2×C3#332811yescharacteristic
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22#559011yescharacteristic
653C22#561011yescharacteristic
654C22#565611yescharacteristic
655C22C2×#91611yescharacteristic
656C22#577111yescharacteristic
657C22D3×#13711yescharacteristic
658C22#577911yescharacteristic
659C4—41no—
660C4—42no—
661C4—64no—
662C4—62no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—121no—
667C4—121no—
668C4—122no—
669C4—121no—
670C4—241no—
671C4—241no—
672C4#561011yescharacteristic
673C4#561311yescharacteristic
674C4D3×#12311yescharacteristic
675C4#567911yescharacteristic
676C3#2490211yescharacteristic, complemented, Sylow
677C2—42no—
678C2#7711111yescharacteristic
679C2#8006711yescharacteristic
680C2#8024211yescharacteristic
681C2#8069011yescharacteristic
682C2#8092711yescharacteristic
683C2#8141311yescharacteristic
684C2#8149911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2035C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2035D3
hypocenterC3#24902
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. C2×#4790C2semidirect
6. #70591C2semidirect
7. #70588C2semidirect
8. #70285C2semidirect
9. #70281C2semidirect
10. #70239C2semidirect
14. #68905C2semidirect
15. #68850C2semidirect
676. C3#24902semidirect

Maximal subgroups

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

subgroupindex
#827532
C2×#50502
C2×#47902
#738762
C3×#20352
#705912
#705882
#702852
#702812
#702392
#699632
#699132
#695182
#689052
#688502
#249023