Finite Group Databasebeta

← #10845273

Subgroups — #10845273

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5166C211yesmaximal, characteristic
2#82344C211yesmaximal, characteristic
3C2×#4765C211yesmaximal, characteristic
4#76543C211yesmaximal, characteristic, complemented
5C3×#1877C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic
7#71604C211yesmaximal, characteristic
8#70281C211yesmaximal, characteristic, complemented
9#70220C211yesmaximal, characteristic, complemented
10#70165C211yesmaximal, characteristic
11#69681C211yesmaximal, characteristic, complemented
12#68904C211yesmaximal, characteristic, complemented
13#68800C211yesmaximal, characteristic, complemented
14#68636C211yesmaximal, characteristic, complemented
15#68618C211yesmaximal, characteristic, complemented
16#25336—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#936C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#391C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5210—21no—
33C4×#217—21no—
34#5166—24no—
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#810C2211yescharacteristic
43#4945—21no—
44#4944—21no—
45#4935C2211yescharacteristic
46C4×Dic3⋊C4C2211yescharacteristic
47#4929C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49#4765—24no—
50#4656—21no—
51#4656—21no—
52#4647—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#489—31no—
63C2×#464—31no—
64#2567—31no—
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2056—31no—
69#2055—31no—
70#1988—31no—
71#1940—31no—
72#1878—31no—
73#1877D311yescharacteristic
74#1803—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×Dic6—21no—
81C22×Dic6C2311yescharacteristic
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99C2×#216—22no—
100C2×#216—22no—
101C2×#216—41no—
102#936—22no—
103#936D414yes—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
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#888—22no—
124#888—22no—
125#887—22no—
126#887—41no—
127C3×#103—22no—
128C3×#102—22no—
129C3×#95—21no—
130C3×#95—21no—
131#835—22no—
132#832—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#816—22no—
142#816—22no—
143#815—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—41no—
146#810—22no—
147C4⋊(C3⋊C8)—41no—
148C22×Q16—31no—
149C22×SD16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4C2×D311yescharacteristic
155#489—64no—
156#464—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169#395—61no—
170#395—61no—
171#391—31no—
172#391—31no—
173#391C2×D311yescharacteristic
174#390—31no—
175#389—61no—
176#389—61no—
177#388—61no—
178#388—61no—
179#385—61no—
180#384—31no—
181#384—31no—
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#336C2×D311yescharacteristic
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—81no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—42no—
229C2×Dic6—81no—
230C2×Dic6—81no—
231C2×Dic6C2×D412yes—
232C3×SD16—44no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4#4112yes—
243C3×C22⋊C4—42no—
244C3×C42—21no—
245C3×C42—21no—
246#217—24no—
247#217—24no—
248#217—42no—
249#217—81no—
250#216—44no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—42no—
258C4⋊Dic3—81no—
259C4⋊Dic3—81no—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—81no—
265Dic3⋊C4—81no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C4×Dic3—81no—
271C4×Dic3C2×D412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8—31no—
280C22×Q8—61no—
281C22×Q8C22×D311yescharacteristic
282C22×D4C22×D311yescharacteristic
283C23×C4—21no—
284C2×Q16—62no—
285C2×Q16—62no—
286C2×Q16—121no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—41no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—62no—
295C4⋊Q8—62no—
296#124—34no—
297#124—62no—
298C4×Q8—62no—
299C4×Q8—62no—
300C4×Q8—62no—
301C4×Q8—121no—
302C4×D4—22no—
303C4×D4—41no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329C4⋊C8—62no—
330C4⋊C8—121no—
331C4⋊C8—121no—
332#103—22no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—22no—
338#102—62no—
339#102—62no—
340#97—62no—
341#95—21no—
342#95—21no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—41no—
365C3×D4—41no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—84no—
381C2×C3×C4#13712yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3#12112yes—
410C2×Dic3#12112yes—
411C2×Dic3#13712yes—
412Dic6—22no—
413Dic6—22no—
414Dic6—41no—
415Dic6—41no—
416Dic6—84no—
417C3×C8—41no—
418C3×C8—41no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—32no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—122no—
434C2×Q8—241no—
435C2×Q8—241no—
436C2×Q8#23512yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4D3×D412yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—81no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—124no—
461SD16—44no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—62no—
478C4⋊C4—62no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—241no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4#23712yes—
493C22⋊C4—42no—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—32no—
501C42—32no—
502C42—61no—
503C42—61no—
504C42—122no—
505C42—122no—
506C42—241no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3C2×#12111yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3#54811yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3C2×#13711yescharacteristic
519C22×C3#57611yescharacteristic
520C22×C3#57711yescharacteristic
521C3×C4—22no—
522C3×C4—42no—
523C3×C4—42no—
524C3×C4—41no—
525C3×C4—81no—
526C3×C4#52911yescharacteristic
527C3×C4#54211yescharacteristic
528C3×C4#54611yescharacteristic
529C3×C4#54711yescharacteristic
530Dic3—22no—
531Dic3—22no—
532Dic3—22no—
533Dic3—42no—
534Dic3—41no—
535Dic3—81no—
536Dic3—81no—
537Dic3—81no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×D3×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—62no—
551Q8—62no—
552Q8—121no—
553Q8—121no—
554Q8—244no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—32no—
567C2×C4—32no—
568C2×C4—32no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—84no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4#91512yes—
599C2×C4C2×#23511yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C2×C4C2×#23711yescharacteristic
602C2×C4#100411yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4#101011yescharacteristic
605C8—41no—
606C8—41no—
607C8—121no—
608C8—121no—
609C2×C3—82no—
610C2×C3#242711yescharacteristic
611C2×C3#303911yescharacteristic
612C2×C3#304511yescharacteristic
613C2×C3#331411yescharacteristic
614C2×C3#331611yescharacteristic
615C2×C3#336911yescharacteristic
616C2×C3#337511yescharacteristic
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22#555211yescharacteristic
625C22#559111yescharacteristic
626C22D3×#12111yescharacteristic
627C22C2×#91511yescharacteristic
628C22#577011yescharacteristic
629C22#577411yescharacteristic
630C22D3×#13711yescharacteristic
631C4—22no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—62no—
636C4—62no—
637C4—62no—
638C4—81no—
639C4—121no—
640C4—122no—
641C4—241no—
642C4—241no—
643C4—241no—
644C4#549011yescharacteristic
645C4#553811yescharacteristic
646C4#561111yescharacteristic
647C4#566211yescharacteristic
648C3#2533611yescharacteristic, complemented, Sylow
649C2—82no—
650C2#7469711yescharacteristic
651C2#7962611yescharacteristic
652C2#7975611yescharacteristic
653C2#8016311yescharacteristic
654C2#8025511yescharacteristic
655C2#8042911yescharacteristic
656C2#8053811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1877D3
hypocenterC3#25336
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
4. #76543C2semidirect
8. #70281C2semidirect
9. #70220C2semidirect
11. #69681C2semidirect
12. #68904C2semidirect
13. #68800C2semidirect
14. #68636C2semidirect
15. #68618C2semidirect
648. C3#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#51662
#823442
C2×#47652
#765432
C3×#18772
#720512
#716042
#702812
#702202
#701652
#696812
#689042
#688002
#686362
#686182
#253363