Finite Group Databasebeta

← #10845254

Subgroups — #10845254

Order 768.

872 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
1C2×#5153C211yesmaximal, characteristic
2C2×#4765C211yesmaximal, characteristic, complemented
3#76361C211yesmaximal, characteristic, complemented
4#76307C211yesmaximal, characteristic, complemented
5C3×#2054C211yesmaximal, characteristic
6#72037C211yesmaximal, characteristic
7#70486C211yesmaximal, characteristic, complemented
8#70452C211yesmaximal, characteristic, complemented
9#70217C211yesmaximal, characteristic, complemented
10#70154C211yesmaximal, characteristic, complemented
11#70152C211yesmaximal, characteristic, complemented
12#69660C211yesmaximal, characteristic, complemented
13#69641C211yesmaximal, characteristic, complemented
14#68915C211yesmaximal, characteristic, complemented
15#68800C211yesmaximal, characteristic, complemented
16#25335—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5153—24no—
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4977C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43#4947—21no—
44#4947—21no—
45#4946—21no—
46#4945—21no—
47#4944—21no—
48#4942—21no—
49#4942—21no—
50Dic3×C22⋊C4—21no—
51#4936C2211yescharacteristic
52C4×Dic3⋊C4C2211yescharacteristic
53#4765—24no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#463—31no—
64#2563—31no—
65#2367—31no—
66#2074—31no—
67#2073—31no—
68#2055—31no—
69#2054D311yescharacteristic
70#1988—31no—
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1805—31no—
75#1804—31no—
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C23×Dic3—21no—
81C22×Dic6—21no—
82C22×Dic6C2311yescharacteristic
83C2×C3×D8—22no—
84C2×C3×D8—22no—
85C2×C3×D8—41no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101#936—22no—
102#936D414yes—
103D4×Dic3—22no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3—41no—
107C2×#214—22no—
108C2×#214—22no—
109C2×#214—41no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#888—22no—
124#888—22no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C3×C4⋊C8—41no—
128C3×#102—22no—
129C3×#102—22no—
130#832—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#816—22no—
143#816—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—41no—
147#810—22no—
148C22×SD16—31no—
149C22×D8C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155#489—64no—
156#463—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169#393—61no—
170#393—61no—
171#391—31no—
172#390—31no—
173#390C2×D311yescharacteristic
174#390C2×D311yescharacteristic
175#388—61no—
176#388—61no—
177#385—61no—
178#382—31no—
179#382—31no—
180#380—61no—
181#380—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C2×C3×D4#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—81no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—42no—
232C2×Dic6—81no—
233C2×Dic6—81no—
234C2×Dic6C2×D412yes—
235C3×D8—44no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C22⋊C4—81no—
249C3×C22⋊C4—81no—
250C3×C42—21no—
251C3×C42—21no—
252#217—24no—
253#217—24no—
254#217—42no—
255#217—42no—
256#217—81no—
257#217—81no—
258#214—44no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3#4112yes—
265Dic3⋊C4—41no—
266Dic3⋊C4—41no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C4×Dic3C2×D412yes—
274C4×Dic3#4112yes—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C22×Q8—31no—
282C22×Q8—61no—
283C22×D4C22×D311yescharacteristic
284C22×D4C22×D311yescharacteristic
285C23×C4—61no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C2×D8—22no—
291C2×D8—22no—
292C2×D8—41no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299#124—34no—
300#124—62no—
301C4×Q8—62no—
302C4×Q8—121no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—121no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331C4⋊C8—41no—
332C4⋊C8—62no—
333C4⋊C8—121no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—22no—
338#102—22no—
339#102—62no—
340#102—62no—
341#102—62no—
342#97—62no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3C22×D411yescharacteristic
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—84no—
416C2×Dic3—84no—
417C2×Dic3#12112yes—
418C2×Dic3#12112yes—
419Dic6—22no—
420Dic6—22no—
421Dic6—41no—
422Dic6—41no—
423Dic6—84no—
424C3×C8—41no—
425C3×C8—41no—
426C3⋊C8—41no—
427C3⋊C8—41no—
428C24—21no—
429C24—21no—
430C2×Q8—32no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—122no—
437C2×Q8—241no—
438C2×Q8—241no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4D3×D412yes—
450C2×D4#23712yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—241no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4C23×D311yescharacteristic
470SD16—124no—
471D8—44no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—32no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C22⋊C4—42no—
499C22⋊C4—64no—
500C22⋊C4—64no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—21no—
509C42—21no—
510C42—32no—
511C42—32no—
512C42—61no—
513C42—61no—
514C42—122no—
515C42—241no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3C2×#12111yescharacteristic
531C22×C3#54611yescharacteristic
532C22×C3#54811yescharacteristic
533C22×C3#57511yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57611yescharacteristic
536C22×C3#57711yescharacteristic
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—81no—
540C3×C4—81no—
541C3×C4#52911yescharacteristic
542C3×C4#54211yescharacteristic
543C3×C4#54611yescharacteristic
544C3×C4#54711yescharacteristic
545Dic3—22no—
546Dic3—22no—
547Dic3—42no—
548Dic3—41no—
549Dic3—42no—
550Dic3—41no—
551Dic3—81no—
552Dic3—81no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23C2×D3×D411yescharacteristic
568Q8—62no—
569Q8—62no—
570Q8—121no—
571Q8—121no—
572Q8—244no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—32no—
588C2×C4—32no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—244no—
618C2×C4—244no—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4#100411yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4#101011yescharacteristic
625C8—41no—
626C8—41no—
627C8—121no—
628C8—121no—
629C2×C3—82no—
630C2×C3—82no—
631C2×C3#242711yescharacteristic
632C2×C3#304011yescharacteristic
633C2×C3#304511yescharacteristic
634C2×C3#331911yescharacteristic
635C2×C3#332111yescharacteristic
636C2×C3#337011yescharacteristic
637C2×C3#337611yescharacteristic
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22#559811yescharacteristic
653C22D3×#12111yescharacteristic
654C22#561011yescharacteristic
655C22#575711yescharacteristic
656C22#576811yescharacteristic
657C22#577311yescharacteristic
658C22#577511yescharacteristic
659C4—41no—
660C4—41no—
661C4—62no—
662C4—62no—
663C4—81no—
664C4—81no—
665C4—121no—
666C4—122no—
667C4—122no—
668C4—121no—
669C4—241no—
670C4—241no—
671C4#559111yescharacteristic
672C4#564911yescharacteristic
673C4#565511yescharacteristic
674C4#566911yescharacteristic
675C3#2533511yescharacteristic, complemented, Sylow
676C2—82no—
677C2—82no—
678C2#7690211yescharacteristic
679C2#8032211yescharacteristic
680C2#8033511yescharacteristic
681C2#8052211yescharacteristic
682C2#8053511yescharacteristic
683C2#8073311yescharacteristic
684C2#8087711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2054C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenter#2054D3
hypocenterC3#25335
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2054

Complements

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

NH ≅ G/Ntype
2. C2×#4765C2semidirect
3. #76361C2semidirect
4. #76307C2semidirect
7. #70486C2semidirect
8. #70452C2semidirect
9. #70217C2semidirect
10. #70154C2semidirect
11. #70152C2semidirect
12. #69660C2semidirect
13. #69641C2semidirect
14. #68915C2semidirect
15. #68800C2semidirect
675. C3#25335semidirect

Maximal subgroups

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

subgroupindex
C2×#51532
C2×#47652
#763612
#763072
C3×#20542
#720372
#704862
#704522
#702172
#701542
#701522
#696602
#696412
#689152
#688002
#253353