Finite Group Databasebeta

← #10876128

Subgroups — #10876128

Order 768.

912 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#391C211yesmaximal, characteristic
2#73753C211yesmaximal, characteristic
3C3×#1927C211yesmaximal, characteristic, complemented
4#70590C211yesmaximal, characteristic, complemented
5#70579C211yesmaximal, characteristic
6#70266C211yesmaximal, characteristic
7#70266C211yesmaximal, characteristic
8#70250C211yesmaximal, characteristic, complemented
9#70233C211yesmaximal, characteristic, complemented
10#69666C211yesmaximal, characteristic, complemented
11#69464C211yesmaximal, characteristic, complemented
12#68914C211yesmaximal, characteristic, complemented
13#68875C211yesmaximal, characteristic
14#68664C211yesmaximal, characteristic
15#68627C211yesmaximal, characteristic, complemented
16#27583—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C2×D3×C4⋊C4C2211yescharacteristic
19C2×#872—21no—
20C2×C42×D3C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4975C2211yescharacteristic
37#4975C2211yescharacteristic
38#4973C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44C4×Dic3⋊C4—21no—
45#4929C2211yescharacteristic
46#4667—21no—
47D3×#95—21no—
48D3×#95—21no—
49#4650—21no—
50#4646—21no—
51#4646—21no—
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60C2×#391—31no—
61#2285—31no—
62#2096—31no—
63#2096—31no—
64#2072—31no—
65#2072—31no—
66#1927—31no—
67#1927—31no—
68#1927—31no—
69#1927D311yescharacteristic, complemented
70#1853—31no—
71#1853—31no—
72#1817—31no—
73#1787—31no—
74#1787—31no—
75C22×C3×Q8C2311yescharacteristic
76C2×D3×Q8—22no—
77C2×D3×Q8—22no—
78C2×D3×Q8—22no—
79C2×D3×Q8—22no—
80C2×D3×Q8—41no—
81C2×D3×Q8—41no—
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C4×D3D411yescharacteristic
85C22×C4×D3D411yescharacteristic
86C22×C4×D3C2311yescharacteristic
87C22×Dic6—21no—
88C22×Dic6C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
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×Dic3D411yescharacteristic
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112D3×C4⋊C4—41no—
113D3×C4⋊C4—41no—
114#872—44no—
115C42×D3—22no—
116C42×D3—22no—
117C3×#106—41no—
118C3×#103—22no—
119C3×#103—41no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123#835—22no—
124#835—41no—
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⋊#95—21no—
134(C2×D3)⋊C8—22no—
135(C2×D3)⋊C8—22no—
136#817—41no—
137#816—22no—
138#816—41no—
139#810—22no—
140#810—41no—
141#808—41no—
142#808—41no—
143C23×Q8—31no—
144C2×#117—61no—
145C22×C4⋊C4—31no—
146C22×C42—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×#106C2×D311yescharacteristic
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#103—31no—
157C2×#103C2×D311yescharacteristic
158C2×#103C2×D311yescharacteristic
159C2×C22⋊C8—31no—
160C2×C22⋊C8—31no—
161#391—31no—
162#391—31no—
163#391—31no—
164#391—61no—
165#391—61no—
166#391C2×D311yescharacteristic
167#384—31no—
168#384—31no—
169#382—61no—
170#380—31no—
171#380—31no—
172#380—61no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—61no—
175C4×C4⋊C4C2×D311yescharacteristic
176C2×#95—61no—
177C2×#95—61no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C23×D3C2×D411yescharacteristic
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8#4112yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207D3×Q8—44no—
208D3×Q8—44no—
209D3×Q8—44no—
210D3×Q8—44no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—24no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—41no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3SD1614yes—
227C2×C4×D3#4112yes—
228C2×C4×D3#4112yes—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—24no—
234C2×Dic6—21no—
235C2×Dic6—42no—
236C2×Dic6—42no—
237C2×Dic6#4112yes—
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—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—22no—
269C4×Dic3—22no—
270C4×Dic3—42no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C22×Q8—31no—
278C22×Q8—61no—
279C22×Q8—62no—
280C22×Q8—62no—
281C22×Q8—62no—
282C22×Q8—62no—
283C22×Q8—121no—
284C22×Q8—121no—
285C22×Q8C22×D311yescharacteristic
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—61no—
290C23×C4—61no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295#117—124no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—121no—
305C2×C4⋊C4—121no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—21no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—62no—
316C2×C42—62no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#106—41no—
320#106—121no—
321#106—121no—
322#106—121no—
323#103—22no—
324#103—41no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329#103—121no—
330#103—121no—
331C22⋊C8—62no—
332C22⋊C8—62no—
333#95—21no—
334#95—21no—
335#95—21no—
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#95—61no—
345C23×C3C22×D411yescharacteristic
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3#12312yes—
351C22×D3C2×SD1612yes—
352C22×D3#13712yes—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—41no—
356C3×Q8—41no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4C2×SD1612yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
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#12312yes—
400C2×Dic3C2×SD1612yes—
401C2×Dic3#13712yes—
402C4×D3—28no—
403C4×D3—22no—
404C4×D3—22no—
405C4×D3—22no—
406C4×D3—22no—
407C4×D3—44no—
408C4×D3—48no—
409C4×D3—84no—
410Dic6—22no—
411Dic6—22no—
412Dic6—48no—
413Dic6—41no—
414Dic6—41no—
415C3×C8—41no—
416C3×C8—41no—
417C3⋊C8—41no—
418C3⋊C8—41no—
419C24—31no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×Q8—64no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—122no—
433C2×Q8—122no—
434C2×Q8—124no—
435C2×Q8—124no—
436C2×Q8—124no—
437C2×Q8—124no—
438C2×Q8#23512yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—34no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—64no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—121no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4C23×D311yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C4⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—62no—
506C42—62no—
507C42—61no—
508C42—61no—
509C42—122no—
510C22×C3C2×#12311yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3#53511yescharacteristic
513C22×C3C22×SD1611yescharacteristic
514C22×C3C2×#13711yescharacteristic
515C22×C3C2×#13711yescharacteristic
516C22×C3#57511yescharacteristic
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3#48314yes—
524C3×C4—22no—
525C3×C4—41no—
526C3×C4—42no—
527C3×C4—81no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4#55111yescharacteristic
531C3×C4#55111yescharacteristic
532C3×C4#55111yescharacteristic
533C3×C4#55111yescharacteristic
534Dic3—22no—
535Dic3—22no—
536Dic3—22no—
537Dic3—42no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541C23—32no—
542C23—32no—
543C23—32no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23C2×D3×D411yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—41no—
552Q8—41no—
553Q8—62no—
554Q8—62no—
555Q8—128no—
556Q8—121no—
557Q8—121no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—32no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—68no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—124no—
591C2×C4—128no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—244no—
604C2×C4D3×SD1612yes—
605C2×C4C2×#23511yescharacteristic
606C2×C4#100411yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C2×C4D3×#4111yescharacteristic
609C2×C4D3×#4111yescharacteristic
610C2×C4#100911yescharacteristic
611C8—41no—
612C8—41no—
613C8—121no—
614C8—121no—
615C2×C3#248211yescharacteristic
616C2×C3C2×#48311yescharacteristic
617C2×C3#309511yescharacteristic
618C2×C3#310311yescharacteristic
619C2×C3#311311yescharacteristic
620C2×C3#320611yescharacteristic
621C2×C3#321511yescharacteristic
622D3—24no—
623C22—34no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—62no—
629C22—62no—
630C22#554711yescharacteristic
631C22#560711yescharacteristic
632C22D3×#12311yescharacteristic
633C22C2×D3×SD1611yescharacteristic
634C22#576811yescharacteristic
635C22D3×#13711yescharacteristic
636C22D3×#13711yescharacteristic
637C4—22no—
638C4—42no—
639C4—41no—
640C4—62no—
641C4—62no—
642C4—62no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—122no—
647C4—241no—
648C4—241no—
649C4—241no—
650C4#552711yescharacteristic
651C4#555611yescharacteristic
652C4#567911yescharacteristic
653C4#570511yescharacteristic
654C3#2758311yescharacteristic, complemented, Sylow
655C2—64no—
656C2#7586611yescharacteristic
657C2#7965311yescharacteristic
658C2#7983311yescharacteristic
659C2#8043211yescharacteristic
660C2#8060611yescharacteristic
661C2D3×#48311yescharacteristic
662C2#8151711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1927C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1927D3
hypocenterC3#27583
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
3. C3×#1927C2semidirect
4. #70590C2semidirect
8. #70250C2semidirect
9. #70233C2semidirect
10. #69666C2semidirect
11. #69464C2semidirect
12. #68914C2semidirect
15. #68627C2semidirect
69. #1927D3semidirect
654. C3#27583semidirect

Maximal subgroups

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

subgroupindex
D3×#3912
#737532
C3×#19272
#705902
#705792
#702662
#702502
#702332
#696662
#694642
#689142
#688752
#686642
#686272
#275833