Finite Group Databasebeta

← #10855274

Subgroups — #10855274

Order 768.

832 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4845C211yesmaximal, characteristic
2C2×C4×#204C211yesmaximal, characteristic
3#73781C211yesmaximal, characteristic
4#73637C211yesmaximal, characteristic, complemented
5C3×#1916C211yesmaximal, characteristic, complemented
6#70464C211yesmaximal, characteristic, complemented
7#69767C211yesmaximal, characteristic, complemented
8#68693C211yesmaximal, characteristic, complemented
9#68686C211yesmaximal, characteristic
10#68685C211yesmaximal, characteristic, complemented
11#68685C211yesmaximal, characteristic, complemented
12#68683C211yesmaximal, characteristic, complemented
13#68670C211yesmaximal, characteristic
14#68637C211yesmaximal, characteristic
15#68630C211yesmaximal, characteristic
16#26165—31nomaximal, Sylow
17C22×Dic12C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37#4992—21no—
38#4989—21no—
39#4976—21no—
40#4976—21no—
41#4939C2211yescharacteristic
42#4936C2211yescharacteristic
43#4929C2211yescharacteristic
44#4929C2211yescharacteristic
45#4928C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4845—24no—
49C4×#204—24no—
50#4675—21no—
51#4664—21no—
52#4650—21no—
53#4649—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#464—31no—
63C2×C4×SD16—31no—
64#2447—31no—
65#2351—31no—
66#2097—31no—
67#2088—31no—
68#2073—31no—
69#2042—31no—
70#1933—31no—
71#1916D311yescharacteristic, complemented
72#1880—31no—
73#1878—31no—
74#1878—31no—
75#1863—31no—
76#1862—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×Dic12—22no—
110C2×Dic12—22no—
111C2×Dic12—22no—
112C2×#204—22no—
113C2×#204—22no—
114C2×#204D414yes—
115#875—22no—
116#875—41no—
117C4×D12—22no—
118C4×D12—41no—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C4×Dic6—41no—
124C4×Dic6—41no—
125C3×C4⋊C8—22no—
126C3×C4×C8—22no—
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#821—22no—
139#821—22no—
140#821—22no—
141#818—41no—
142#817—22no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—41no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#464—64no—
156C4×SD16—64no—
157C2×#107—31no—
158C2×#106—31no—
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×C4×C8C2×D311yescharacteristic
169#397—61no—
170#397—61no—
171#391—31no—
172#391—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389—61no—
178#389—61no—
179#385—61no—
180#382—31no—
181#382—61no—
182#381—61no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#336C2×D311yescharacteristic
190#336C2×D311yescharacteristic
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×Dic6C2×D412yes—
227C2×Dic6#4112yes—
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
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×C8C2×D412yes—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—41no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—81no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3#4112yes—
264Dic3⋊C4—24no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—81no—
270C4×Dic3—24no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273Dic12—44no—
274#204—28no—
275C22×Q8—31no—
276C22×Q8—31no—
277C22×D4—31no—
278C23×C4—61no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—62no—
282C2×SD16—34no—
283C2×SD16—62no—
284C2×SD16—62no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#124—62no—
290#124—121no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—121no—
296C4×Q8—121no—
297C4×D4—62no—
298C4×D4—121no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4D1211yescharacteristic
311C2×C4⋊C4D1211yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—121no—
324#106—62no—
325C4⋊C8—22no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—121no—
331#102—62no—
332#102—62no—
333#102—62no—
334C4×C8—22no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95D1211yescharacteristic
345#95D1211yescharacteristic
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
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#13512yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C42−1+411yescharacteristic
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
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—
399D12—22no—
400D12—22no—
401D12—41no—
402D12—41no—
403C4×D3—84no—
404Dic6—22no—
405Dic6—22no—
406Dic6—22no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6#12314yes—
412C3×C8—22no—
413C3×C8—41no—
414C3×C8—41no—
415C3×C8—41no—
416C24—61no—
417C2×Q8—32no—
418C2×Q8—32no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×D4—32no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—241no—
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—124no—
453SD16—68no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8D3×D412yes—
466C4⋊C4—22no—
467C4⋊C4—32no—
468C4⋊C4—41no—
469C4⋊C4—64no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—241no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—241no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—64no—
496C42—122no—
497C42—122no—
498C22×C3#52511yescharacteristic
499C22×C3D4211yescharacteristic
500C22×C3#54811yescharacteristic
501C22×C3C2×#13511yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57711yescharacteristic
504C22×C3#57811yescharacteristic
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4C2×#12311yescharacteristic
519C3×C4#53911yescharacteristic
520C3×C4#54711yescharacteristic
521C3×C4#55311yescharacteristic
522Dic3—24no—
523Dic3—42no—
524Dic3—81no—
525Dic3—81no—
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C22×D1211yescharacteristic
537Q8—34no—
538Q8—62no—
539Q8—62no—
540Q8—62no—
541Q8—121no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545D4—62no—
546D4—62no—
547D4—121no—
548D4—121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
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—244no—
588C2×C4#90412yes—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4#100711yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#100911yescharacteristic
595C8—22no—
596C8—41no—
597C8—41no—
598C8—41no—
599C2×C3#244311yescharacteristic
600C2×C3#310211yescharacteristic
601C2×C3#312111yescharacteristic
602C2×C3#326911yescharacteristic
603C2×C3#328211yescharacteristic
604C2×C3#336411yescharacteristic
605C2×C3#337111yescharacteristic
606D3—82no—
607C22—122no—
608C22—122no—
609C22—122no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#552411yescharacteristic
615C22D4×D1211yescharacteristic
616C22#557611yescharacteristic
617C22C2×#90411yescharacteristic
618C22#574811yescharacteristic
619C22#575111yescharacteristic
620C22#575311yescharacteristic
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—64no—
628C4—122no—
629C4—241no—
630C4—241no—
631C4—241no—
632C4—241no—
633C4—241no—
634C4#563111yescharacteristic
635C4D3×#12311yescharacteristic
636C4#569211yescharacteristic
637C4#570911yescharacteristic
638C3#2616511yescharacteristic, complemented, Sylow
639C2—242no—
640C2#7531911yescharacteristic
641C2#7929911yescharacteristic
642C2#7939411yescharacteristic
643C2#7943511yescharacteristic
644C2#7949811yescharacteristic
645C2#7993411yescharacteristic
646C2#7998211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1916C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1916D3
hypocenterC3#26165
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
4. #73637C2semidirect
5. C3×#1916C2semidirect
6. #70464C2semidirect
7. #69767C2semidirect
8. #68693C2semidirect
10. #68685C2semidirect
11. #68685C2semidirect
12. #68683C2semidirect
71. #1916D3semidirect
638. C3#26165semidirect

Maximal subgroups

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

subgroupindex
C2×#48452
C2×C4×#2042
#737812
#736372
C3×#19162
#704642
#697672
#686932
#686862
#686852
#686832
#686702
#686372
#686302
#261653