Finite Group Databasebeta

← #10876182

Subgroups — #10876182

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#391C211yesmaximal, characteristic
2#74147C211yesmaximal, characteristic
3C3×#2074C211yesmaximal, characteristic, complemented
4#70266C211yesmaximal, characteristic
5#70266C211yesmaximal, characteristic
6#70224C211yesmaximal, characteristic, complemented
7#70223C211yesmaximal, characteristic, complemented
8#69686C211yesmaximal, characteristic, complemented
9#69560C211yesmaximal, characteristic, complemented
10#68875C211yesmaximal, characteristic
11#68874C211yesmaximal, characteristic
12#68794C211yesmaximal, characteristic, complemented
13#68793C211yesmaximal, characteristic, complemented
14#68693C211yesmaximal, characteristic, complemented
15#68664C211yesmaximal, characteristic
16#27589—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C2×D3×C4⋊C4C2211yescharacteristic
19C2×C42×D3C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C3×#391C2211yescharacteristic
23C3×#382C2211yescharacteristic
24#5229C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4936C2211yescharacteristic
43#4929C2211yescharacteristic
44#4666—21no—
45#4664—21no—
46D3×#95—21no—
47D3×#95—21no—
48#4651—21no—
49#4648—21no—
50#4646—21no—
51#4646—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60C2×#391—31no—
61#2283—31no—
62#2097—31no—
63#2097—31no—
64#2074—31no—
65#2074D311yescharacteristic, complemented
66#1928—31no—
67#1928—31no—
68#1927—31no—
69#1927—31no—
70#1853—31no—
71#1852—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×C42C2311yescharacteristic
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108D3×C4⋊C4—41no—
109D3×C4⋊C4—41no—
110C42×D3—22no—
111C42×D3—22no—
112C3×#103—22no—
113C3×#103—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118#835—41no—
119#835—41no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
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—
132(C2×D3)⋊C8—22no—
133(C2×D3)⋊C8—22no—
134#818—41no—
135#817—41no—
136#816—41no—
137#816—41no—
138#810—22no—
139#810—22no—
140#808—41no—
141#807—41no—
142C23×Q8—31no—
143C22×C4⋊C4—31no—
144C22×C42—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103C2×D311yescharacteristic
156C2×#103C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#391—31no—
160#391—31no—
161#391—31no—
162#391—61no—
163#391—61no—
164#391C2×D311yescharacteristic
165#383—61no—
166#382—31no—
167#382C2×D311yescharacteristic
168#381—61no—
169#380—31no—
170#380—31no—
171C4×C4⋊C4—31no—
172C4×C4⋊C4—31no—
173C4×C22⋊C4—61no—
174#376—61no—
175C2×#95—61no—
176C2×#95—61no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C23×D3C2×D411yescharacteristic
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8#4112yes—
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×Dic3—21no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206D3×Q8—44no—
207D3×Q8—44no—
208D3×Q8—44no—
209D3×Q8—44no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—24no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
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×D3#4112yes—
227C2×C4×D3#4112yes—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—24no—
233C2×Dic6—21no—
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6#4112yes—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C42—21no—
250C3×C42—21no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259Dic3⋊C4—42no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—22no—
263C4×Dic3—22no—
264C4×Dic3—42no—
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⋊C8—21no—
272C22×Q8—31no—
273C22×Q8—61no—
274C22×Q8—62no—
275C22×Q8—62no—
276C22×Q8—62no—
277C22×Q8—62no—
278C22×Q8—121no—
279C22×Q8—121no—
280C22×Q8C22×D311yescharacteristic
281C23×C4—31no—
282C23×C4—31no—
283C23×C4—31no—
284C23×C4—61no—
285C23×C4—61no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—121no—
295C2×C4⋊C4—121no—
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—62no—
305C2×C42—62no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—121no—
310#107—121no—
311#106—121no—
312#106—121no—
313#103—22no—
314#103—22no—
315#103—62no—
316#103—62no—
317#103—121no—
318#103—121no—
319#103—121no—
320#103—121no—
321C22⋊C8—62no—
322C22⋊C8—62no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339C23×C3C22×D411yescharacteristic
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3#12312yes—
345C22×D3#13512yes—
346C22×D3#13712yes—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—41no—
350C3×Q8—41no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
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×C4—41no—
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—21no—
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#12312yes—
395C2×Dic3#13712yes—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—44no—
401C4×D3—48no—
402C4×D3—44no—
403C4×D3—84no—
404Dic6—22no—
405Dic6—22no—
406Dic6—48no—
407Dic6—41no—
408Dic6—41no—
409C3×C8—41no—
410C3×C8—41no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C24—31no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—32no—
420C2×Q8—64no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—122no—
427C2×Q8—122no—
428C2×Q8—124no—
429C2×Q8—124no—
430C2×Q8—124no—
431C2×Q8—124no—
432C2×Q8#23712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—31no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—64no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—121no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4D3×D411yescharacteristic
463C22×C4D3×D411yescharacteristic
464C22×C4C23×D311yescharacteristic
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C42—21no—
493C42—21no—
494C42—62no—
495C42—62no—
496C42—61no—
497C42—61no—
498C42—122no—
499C42—122no—
500C22×C3C2×#12311yescharacteristic
501C22×C3#52511yescharacteristic
502C22×C3#53511yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×#13711yescharacteristic
505C22×C3C2×#13711yescharacteristic
506C22×C3#57511yescharacteristic
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—22no—
510C2×D3—22no—
511C2×D3—22no—
512C2×D3—22no—
513C2×D3#48514yes—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4—81no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4#55111yescharacteristic
521C3×C4#55111yescharacteristic
522C3×C4#55111yescharacteristic
523C3×C4#55111yescharacteristic
524Dic3—22no—
525Dic3—22no—
526Dic3—41no—
527Dic3—42no—
528Dic3—81no—
529Dic3—81no—
530Dic3—81no—
531C23—32no—
532C23—32no—
533C23—32no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—61no—
538C23C2×D3×D411yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—62no—
544Q8—62no—
545Q8—128no—
546Q8—121no—
547Q8—121no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—124no—
581C2×C4—124no—
582C2×C4—128no—
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×C4C2×#23711yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C2×C4D3×#4111yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4#100911yescharacteristic
600C2×C4#100911yescharacteristic
601C8—41no—
602C8—41no—
603C8—121no—
604C8—121no—
605C2×C3#248211yescharacteristic
606C2×C3C2×#48511yescharacteristic
607C2×C3#309511yescharacteristic
608C2×C3#310811yescharacteristic
609C2×C3#311711yescharacteristic
610C2×C3#321111yescharacteristic
611C2×C3#321811yescharacteristic
612D3—24no—
613C22—34no—
614C22—62no—
615C22—62no—
616C22—62no—
617C22—62no—
618C22—62no—
619C22—62no—
620C22#560611yescharacteristic
621C22D3×#12311yescharacteristic
622C22#567411yescharacteristic
623C22D3×#13511yescharacteristic
624C22#576811yescharacteristic
625C22D3×#13711yescharacteristic
626C22D3×#13711yescharacteristic
627C4—41no—
628C4—42no—
629C4—41no—
630C4—62no—
631C4—62no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—121no—
636C4—122no—
637C4—241no—
638C4—241no—
639C4—241no—
640C4#565811yescharacteristic
641C4#565811yescharacteristic
642C4#570511yescharacteristic
643C4#570511yescharacteristic
644C3#2758911yescharacteristic, complemented, Sylow
645C2—64no—
646C2#7767611yescharacteristic
647C2#8056911yescharacteristic
648C2#8057111yescharacteristic
649C2D3×#48511yescharacteristic
650C2#8151711yescharacteristic
651C2#8164211yescharacteristic
652C2#8174111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2074C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2074D3
hypocenterC3#27589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2074

Complements

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

NH ≅ G/Ntype
3. C3×#2074C2semidirect
6. #70224C2semidirect
7. #70223C2semidirect
8. #69686C2semidirect
9. #69560C2semidirect
12. #68794C2semidirect
13. #68793C2semidirect
14. #68693C2semidirect
65. #2074D3semidirect
644. C3#27589semidirect

Maximal subgroups

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

subgroupindex
D3×#3912
#741472
C3×#20742
#702662
#702242
#702232
#696862
#695602
#688752
#688742
#687942
#687932
#686932
#686642
#275893