Finite Group Databasebeta

← #10845266

Subgroups — #10845266

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×#4845C211yesmaximal, characteristic
2C2×#4714C211yesmaximal, characteristic
3#73721C211yesmaximal, characteristic, complemented
4#73655C211yesmaximal, characteristic
5C3×#1940C211yesmaximal, characteristic, complemented
6#70487C211yesmaximal, characteristic, complemented
7#70485C211yesmaximal, characteristic, complemented
8#70482C211yesmaximal, characteristic
9#70479C211yesmaximal, characteristic
10#70460C211yesmaximal, characteristic, complemented
11#69813C211yesmaximal, characteristic, complemented
12#68685C211yesmaximal, characteristic, complemented
13#68672C211yesmaximal, characteristic
14#68636C211yesmaximal, characteristic, complemented
15#68635C211yesmaximal, characteristic
16#25336—31nomaximal, Sylow
17C22×Dic12C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37#4990—21no—
38#4989—21no—
39#4984—21no—
40#4984—21no—
41#4939C2211yescharacteristic
42#4935C2211yescharacteristic
43#4935C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45#4929C2211yescharacteristic
46#4929C2211yescharacteristic
47#4929C2211yescharacteristic
48#4845—24no—
49#4714—24no—
50#4669—21no—
51#4666—21no—
52#4656—21no—
53#4656—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
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#1940D311yescharacteristic, complemented
72#1878—31no—
73#1877—31no—
74#1803—31no—
75#1802—31no—
76#1763—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C22×Dic6C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×Dic12—22no—
112C2×Dic12—22no—
113C2×Dic12—41no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—41no—
117#875—22no—
118#875D414yes—
119C4×D12—22no—
120C4×D12—41no—
121#869—22no—
122#869—22no—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C3×C4⋊C8—22no—
128C3×C4⋊C8—41no—
129C3×C4⋊C8—41no—
130C3×#97—22no—
131C3×#95D411yescharacteristic
132C3×#95D411yescharacteristic
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142#821—22no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#489—64no—
156#464—64no—
157C2×C4⋊C8C2×D311yescharacteristic
158C2×C4⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97C2×D311yescharacteristic
169#395—61no—
170#395—61no—
171#391—31no—
172#391—31no—
173#391—31no—
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⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193C23×D3—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12C2×D412yes—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—81no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—42no—
228C2×Dic6—81no—
229C2×Dic6—81no—
230C2×Dic6C2×D412yes—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—81no—
268C4⋊Dic3—81no—
269C4⋊Dic3—81no—
270C4⋊Dic3—81no—
271C4⋊Dic3#4112yes—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3—81no—
277Dic12—44no—
278#204—44no—
279C22×Q8—31no—
280C22×Q8—31no—
281C22×Q8—61no—
282C22×D4—31no—
283C23×C4—61no—
284C2×Q16—62no—
285C2×Q16—62no—
286C2×Q16—121no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
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—62no—
303C4×D4—121no—
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⋊C4D1211yescharacteristic
317C2×C4⋊C4D1211yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C2×C42C22×D311yescharacteristic
329C4⋊C8—22no—
330C4⋊C8—41no—
331C4⋊C8—41no—
332#103—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#97—22no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95D1211yescharacteristic
349#95D1211yescharacteristic
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4#12112yes—
373C2×C3×C4#12112yes—
374C2×C3×C4#13712yes—
375C2×C3×C4#13712yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C42−1+411yescharacteristic
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403D12—22no—
404D12—22no—
405D12—41no—
406D12—41no—
407C4×D3—84no—
408Dic6—22no—
409Dic6—22no—
410Dic6—22no—
411Dic6—22no—
412Dic6—41no—
413Dic6—41no—
414Dic6—41no—
415Dic6—41no—
416Dic6—84no—
417C3×C8—41no—
418C3×C8—41no—
419C3×C8—41no—
420C3×C8—41no—
421C24—61no—
422C2×Q8—32no—
423C2×Q8—32no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—122no—
435C2×Q8—241no—
436C2×Q8—241no—
437C2×D4—32no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—241no—
453C22×C4C2×D1211yescharacteristic
454C22×C4C2×D1211yescharacteristic
455C22×C4C2×D1211yescharacteristic
456C22×C4C2×D1211yescharacteristic
457C22×C4C2×D1211yescharacteristic
458C22×C4C2×D1211yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—124no—
461SD16—124no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—32no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—122no—
503C42—122no—
504C42—241no—
505C42D3×D412yes—
506C42#23912yes—
507C22×C3C2×#12111yescharacteristic
508C22×C3#54611yescharacteristic
509C22×C3#54811yescharacteristic
510C22×C3C2×#13711yescharacteristic
511C22×C3C2×#13711yescharacteristic
512C22×C3#57611yescharacteristic
513C22×C3#57711yescharacteristic
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—81no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4#52911yescharacteristic
528C3×C4#54211yescharacteristic
529C3×C4#54611yescharacteristic
530C3×C4#54711yescharacteristic
531Dic3—42no—
532Dic3—42no—
533Dic3—42no—
534Dic3—81no—
535Dic3—81no—
536Dic3—81no—
537Dic3—81no—
538C23—62no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23C22×D1211yescharacteristic
546Q8—62no—
547Q8—62no—
548Q8—62no—
549Q8—62no—
550Q8—121no—
551Q8—121no—
552Q8—121no—
553Q8—121no—
554Q8—244no—
555D4—62no—
556D4—62no—
557D4—121no—
558D4—121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—244no—
595C2×C4#89512yes—
596C2×C4#89512yes—
597C2×C4#90912yes—
598C2×C4#90912yes—
599C2×C4C2×D3×D411yescharacteristic
600C2×C4#100411yescharacteristic
601C2×C4C2×#23911yescharacteristic
602C2×C4#100711yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C8—41no—
606C8—41no—
607C8—41no—
608C8—41no—
609C2×C3#242711yescharacteristic
610C2×C3#303911yescharacteristic
611C2×C3#304511yescharacteristic
612C2×C3#331411yescharacteristic
613C2×C3#331611yescharacteristic
614C2×C3#336911yescharacteristic
615C2×C3#337511yescharacteristic
616D3—82no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22C2×#89511yescharacteristic
625C22#555411yescharacteristic
626C22#557611yescharacteristic
627C22C2×#90911yescharacteristic
628C22C2×#90911yescharacteristic
629C22#575111yescharacteristic
630C22#575211yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—41no—
636C4—41no—
637C4—122no—
638C4—122no—
639C4—122no—
640C4—241no—
641C4—241no—
642C4—241no—
643C4—241no—
644C4#561211yescharacteristic
645C4#563111yescharacteristic
646C4#569111yescharacteristic
647C4#573311yescharacteristic
648C3#2533611yescharacteristic, complemented, Sylow
649C2—242no—
650C2#7621911yescharacteristic
651C2#7905711yescharacteristic
652C2#7908811yescharacteristic
653C2#7950311yescharacteristic
654C2#7951311yescharacteristic
655C2#7998111yescharacteristic
656C2#7998411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1940C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1940D3
hypocenterC3#25336
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
3. #73721C2semidirect
5. C3×#1940C2semidirect
6. #70487C2semidirect
7. #70485C2semidirect
10. #70460C2semidirect
11. #69813C2semidirect
12. #68685C2semidirect
14. #68636C2semidirect
71. #1940D3semidirect
648. C3#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#48452
C2×#47142
#737212
#736552
C3×#19402
#704872
#704852
#704822
#704792
#704602
#698132
#686852
#686722
#686362
#686352
#253363