Finite Group Databasebeta

← #10858193

Subgroups — #10858193

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4890C211yesmaximal, characteristic
2C2×#4852C211yesmaximal, characteristic
3#76009C211yesmaximal, characteristic, complemented
4#74228C211yesmaximal, characteristic
5C3×#2085C211yesmaximal, characteristic, complemented
6#70642C211yesmaximal, characteristic, complemented
7#70565C211yesmaximal, characteristic, complemented
8#69625C211yesmaximal, characteristic
9#69561C211yesmaximal, characteristic, complemented
10#68913C211yesmaximal, characteristic, complemented
11#68896C211yesmaximal, characteristic
12#68895C211yesmaximal, characteristic
13#68859C211yesmaximal, characteristic, complemented
14#68672C211yesmaximal, characteristic
15#68610C211yesmaximal, characteristic, complemented
16#26372—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#888C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#382C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#4984C2211yescharacteristic
32#4981—21no—
33#4973—21no—
34#4973C2211yescharacteristic
35#4972C2211yescharacteristic
36#4972C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45#4929C2211yescharacteristic
46#4890—24no—
47#4852—24no—
48#4668—21no—
49#4668—21no—
50#4667—21no—
51D3×#95—21no—
52#4656—21no—
53#4653—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#474—31no—
63C2×C4×SD16—31no—
64#2451—31no—
65#2349—31no—
66#2097—31no—
67#2085D311yescharacteristic, complemented
68#2068—31no—
69#2058—31no—
70#1942—31no—
71#1916—31no—
72#1877—31no—
73#1877—31no—
74#1864—31no—
75#1863—31no—
76#1862—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×Dic6—21no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3D411yescharacteristic
96C2×C4⋊Dic3D411yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C2×#204—22no—
109C2×#204—22no—
110C2×#204Q814yescomplemented
111#893—22no—
112#893—22no—
113#893—41no—
114#893—41no—
115#888—22no—
116#888—41no—
117#887—22no—
118#887—41no—
119C4×D12—22no—
120C4×D12—22no—
121C3×#107—22no—
122C3×#106—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#821—22no—
136#816—22no—
137Dic3⋊C8—22no—
138C8×Dic3—22no—
139#810—22no—
140#810—41no—
141#809—22no—
142#809D414yes—
143#808—41no—
144#807—22no—
145#807—22no—
146C22×SD16—31no—
147C2×C4⋊Q8—31no—
148C2×C4×Q8—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152#474—64no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C4×C8—31no—
167#398—61no—
168#395—61no—
169#391—31no—
170#391—31no—
171#391—61no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389C2×D311yescharacteristic
176#387—61no—
177#385—61no—
178#385—61no—
179#382C2×D311yescharacteristic
180#380—61no—
181C4×C4⋊C4—31no—
182C2×#95—61no—
183#340—31no—
184#340—31no—
185#340C2×D311yescharacteristic
186#340C2×D311yescharacteristic
187#336—31no—
188#336—31no—
189#336—31no—
190#336C2×D311yescharacteristic
191C23×D3—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12C2×D412yes—
210C2×D12C2×Q812yes—
211C2×D12#4112yes—
212C2×C4×D3—24no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×C4×D3—81no—
218C2×C4×D3—81no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—42no—
227C2×Dic6—42no—
228C2×Dic6—81no—
229C2×Dic6C2×Q812yes—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×Q812yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4C2×D412yes—
245C3×C42—21no—
246C3×C42—21no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250C4⋊Dic3—21no—
251C4⋊Dic3—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—81no—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259Dic3⋊C4—81no—
260Dic3⋊C4—81no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272#204—28no—
273C22×Q8—31no—
274C22×Q8—61no—
275C22×D4—31no—
276C23×C4—61no—
277C23×C4—61no—
278C23×C4—61no—
279C2×SD16—34no—
280C2×SD16—62no—
281C2×SD16—62no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C4⋊Q8—62no—
287C4⋊Q8—121no—
288C4×Q8—62no—
289C4×Q8—121no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—121no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—22no—
318#107—62no—
319#107—62no—
320#106—22no—
321#106—121no—
322C4⋊C8—62no—
323#103—62no—
324#103—62no—
325#103—121no—
326#102—34no—
327#102—62no—
328#102—62no—
329C4×C8—62no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—31no—
334#95—31no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C42−1+411yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3#13512yes—
394C2×Dic3#13712yes—
395D12—22no—
396D12—22no—
397D12—22no—
398D12C22⋊Q814yes—
399C4×D3—84no—
400C4×D3—84no—
401C4×D3—84no—
402Dic6—22no—
403Dic6—22no—
404Dic6—41no—
405Dic6—41no—
406Dic6—84no—
407C3×C8—22no—
408C3×C8—41no—
409C3⋊C8—22no—
410C3⋊C8—41no—
411C24—61no—
412C2×Q8—32no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—122no—
419C2×Q8—122no—
420C2×Q8—241no—
421C2×D4—32no—
422C2×D4—32no—
423C2×D4—32no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—64no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4—241no—
444C22×C4—241no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—68no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—32no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8D3×Q812yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—32no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—81no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4D3×D412yes—
483C22⋊C4—64no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C42—21no—
487C42—21no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—61no—
492C42—122no—
493C22×C3#52511yescharacteristic
494C22×C3D4211yescharacteristic
495C22×C3D4×Q811yescharacteristic
496C22×C3C2×#13511yescharacteristic
497C22×C3C2×#13711yescharacteristic
498C22×C3#57611yescharacteristic
499C22×C3#57711yescharacteristic
500C2×D3—24no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4C2×C22⋊Q811yescharacteristic
514C3×C4#53911yescharacteristic
515C3×C4#54711yescharacteristic
516C3×C4#55411yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—42no—
520Dic3—42no—
521Dic3—41no—
522Dic3—41no—
523Dic3—81no—
524C23—62no—
525C23—62no—
526C23—62no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23C2×D3×D411yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—121no—
535Q8—121no—
536Q8—244no—
537D4—34no—
538D4—62no—
539D4—62no—
540D4—62no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—32no—
549C2×C4—32no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—244no—
581C2×C4—244no—
582C2×C4—244no—
583C2×C4C2×D3×D411yescharacteristic
584C2×C4C2×D3×Q811yescharacteristic
585C2×C4#100711yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4#100911yescharacteristic
589C8—22no—
590C8—41no—
591C8—62no—
592C8—121no—
593C2×C3#244911yescharacteristic
594C2×C3#310711yescharacteristic
595C2×C3#311811yescharacteristic
596C2×C3#326611yescharacteristic
597C2×C3#328011yescharacteristic
598C2×C3#338111yescharacteristic
599C2×C3#338511yescharacteristic
600D3—44no—
601C22—64no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—122no—
607C22—122no—
608C22#560911yescharacteristic
609C22#565611yescharacteristic
610C22#572711yescharacteristic
611C22D3×#13511yescharacteristic
612C22#577011yescharacteristic
613C22#577411yescharacteristic
614C22D3×#13711yescharacteristic
615C4—41no—
616C4—42no—
617C4—41no—
618C4—62no—
619C4—62no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—122no—
624C4—121no—
625C4—122no—
626C4—121no—
627C4—241no—
628C4D3×C22⋊Q811yescharacteristic
629C4#563111yescharacteristic
630C4#569011yescharacteristic
631C4#569211yescharacteristic
632C3#2637211yescharacteristic, complemented, Sylow
633C2—124no—
634C2#7795211yescharacteristic
635C2#8055711yescharacteristic
636C2#8062711yescharacteristic
637C2#8135211yescharacteristic
638C2#8149311yescharacteristic
639C2#8191311yescharacteristic
640C2#8196511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2085C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2085D3
hypocenterC3#26372
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2085

Complements

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

NH ≅ G/Ntype
3. #76009C2semidirect
5. C3×#2085C2semidirect
6. #70642C2semidirect
7. #70565C2semidirect
9. #69561C2semidirect
10. #68913C2semidirect
13. #68859C2semidirect
15. #68610C2semidirect
67. #2085D3semidirect
110. C2×#204Q8semidirect
632. C3#26372semidirect

Maximal subgroups

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

subgroupindex
C2×#48902
C2×#48522
#760092
#742282
C3×#20852
#706422
#705652
#696252
#695612
#689132
#688962
#688952
#688592
#686722
#686102
#263723