Finite Group Databasebeta

← #10855110

Subgroups — #10855110

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5121C211yesmaximal, characteristic, complemented
2#82814C211yesmaximal, characteristic
3C2×#4815C211yesmaximal, characteristic, complemented
4#73883C211yesmaximal, characteristic
5C3×#1894C211yesmaximal, characteristic, complemented
6#71996C211yesmaximal, characteristic
7#71692C211yesmaximal, characteristic, complemented
8#70251C211yesmaximal, characteristic, complemented
9#70243C211yesmaximal, characteristic, complemented
10#70224C211yesmaximal, characteristic, complemented
11#69506C211yesmaximal, characteristic, complemented
12#68877C211yesmaximal, characteristic, complemented
13#68836C211yesmaximal, characteristic, complemented
14#68831C211yesmaximal, characteristic, complemented
15#68818C211yesmaximal, characteristic, complemented
16#26154—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×#946C2211yescharacteristic
20C2×C4×#67C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C2×#892C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#380C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#5121—24no—
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#817C2211yescharacteristic
38#4990—21no—
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4977C2211yescharacteristic
42#4976—21no—
43Dic3×C4⋊C4C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4959—21no—
48#4958—21no—
49#4952—21no—
50Dic3×C22⋊C4—21no—
51#4815—24no—
52#4670—21no—
53#4655—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#485—31no—
63C2×#428—31no—
64#2415—31no—
65#2352—31no—
66#2097—31no—
67#2085—31no—
68#2076—31no—
69#2035—31no—
70#1928—31no—
71#1920—31no—
72#1894D311yescharacteristic, complemented
73#1872—31no—
74#1870—31no—
75#1854—31no—
76#1854—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×#67D411yescharacteristic
80C23×Dic3—21no—
81C22×C4×D3C2311yescharacteristic
82C2×C3×M16—22no—
83C2×C3×M16—22no—
84C2×C3×M16—22no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×#217—21no—
96C2×#217D411yescharacteristic
97#946—22no—
98#946—41no—
99C4×#67—22no—
100C4×#67—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106#926—22no—
107#926—41no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—41no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#892—22no—
125#892—41no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#95—21no—
129#835—22no—
130#835—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#830—22no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142#818—22no—
143#818—22no—
144#817—22no—
145#817D414yescomplemented
146#808—41no—
147#807—22no—
148C22×M16C2×D311yescharacteristic
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×C4×D4—31no—
152C2×#117—31no—
153C2×#117—31no—
154C22×C4⋊C4—31no—
155#485—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170#397—61no—
171#394—61no—
172#391—31no—
173#389—31no—
174#389—61no—
175#388—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#382—31no—
180#380—31no—
181#380—31no—
182#380C2×D311yescharacteristic
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C24×C3C2×D411yescharacteristic
194C23×D3C2×D411yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4#4112yes—
211C22×C3×C4#4112yes—
212C22×C3×C4C2411yescharacteristic
213C2×#67—22no—
214C2×#67—41no—
215C2×#67—41no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—42no—
221C22×Dic3—42no—
222C22×Dic3—81no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C3×M16—28no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C2×C3×C8#4112yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248#217—22no—
249#217—42no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—42no—
262C4⋊Dic3C2×D412yes—
263C4⋊Dic3C2×D412yes—
264Dic3⋊C4—22no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3—81no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C22×Q8C22×D311yescharacteristic
279C22×D4—31no—
280C23×C4—31no—
281C23×C4—61no—
282C23×C4C22×D311yescharacteristic
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—62no—
293C22⋊Q8—121no—
294C4×D4—62no—
295C4×D4—62no—
296#117—62no—
297#117—62no—
298#117—121no—
299#117—121no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—62no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—121no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42—61no—
327#107—62no—
328#107—62no—
329#107—62no—
330#106—34no—
331#106—62no—
332#106—121no—
333#103—22no—
334#103—22no—
335#103—62no—
336#103—121no—
337C22⋊C8—62no—
338C22⋊C8—62no—
339C22⋊C8—62no—
340#95—21no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3#12312yes—
356C23×C3C22×D411yescharacteristic
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4#12312yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393#67—44no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—84no—
416C4×D3—41no—
417C4×D3—41no—
418C4×D3—41no—
419C4×D3—41no—
420C3×C8—22no—
421C3×C8—22no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C24—31no—
425C24D3×D411yescharacteristic
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×D4—62no—
433C2×D4—121no—
434C2×D4—121no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—61no—
447C22×C4—61no—
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—122no—
457C22×C4—122no—
458C22×C4—241no—
459C22×C4#23512yes—
460C22×C4D3×D411yescharacteristic
461C22×C4#23712yes—
462C22×C4C23×D311yescharacteristic
463M16—28no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8D3×D412yes—
475C2×C8#23712yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—32no—
484C4⋊C4—42no—
485C4⋊C4—62no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—62no—
498C22⋊C4—62no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C42—61no—
505C42—61no—
506C42—122no—
507C42—122no—
508C42—241no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3C2×#12311yescharacteristic
517C22×C3D4211yescharacteristic
518C22×C3#54711yescharacteristic
519C22×C3#57511yescharacteristic
520C22×C3#57511yescharacteristic
521C22×C3#57711yescharacteristic
522C22×C3#57811yescharacteristic
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C3×C4—22no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4—81no—
535C3×C4#52511yescharacteristic
536C3×C4#53911yescharacteristic
537C3×C4#54811yescharacteristic
538C3×C4#55311yescharacteristic
539Dic3—41no—
540Dic3—41no—
541Dic3—41no—
542Dic3—42no—
543Dic3—42no—
544Dic3—81no—
545Dic3—81no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23#88312yes—
560C23C2×D3×D411yescharacteristic
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565D4—124no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—61no—
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—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—244no—
610C2×C4#88312yes—
611C2×C4C2×#23511yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4#100911yescharacteristic
616C2×C4#101011yescharacteristic
617C8—22no—
618C8—22no—
619C8—121no—
620C8—121no—
621C2×C3—42no—
622C2×C3#244311yescharacteristic
623C2×C3#309311yescharacteristic
624C2×C3#309711yescharacteristic
625C2×C3#326911yescharacteristic
626C2×C3#327511yescharacteristic
627C2×C3#335011yescharacteristic
628C2×C3#335211yescharacteristic
629D3—81no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22C2×#88311yescharacteristic
645C22#560911yescharacteristic
646C22#561111yescharacteristic
647C22#577311yescharacteristic
648C22#577711yescharacteristic
649C22#577811yescharacteristic
650C22#577911yescharacteristic
651C4—22no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—122no—
657C4—121no—
658C4—121no—
659C4—122no—
660C4—121no—
661C4—241no—
662C4—241no—
663C4#555211yescharacteristic
664C4#556511yescharacteristic
665C4#558611yescharacteristic
666C4#566511yescharacteristic
667C3#2615411yescharacteristic, complemented, Sylow
668C2—42no—
669C2—241no—
670C2#7470411yescharacteristic
671C2#7889311yescharacteristic
672C2#7895111yescharacteristic
673C2#8083211yescharacteristic
674C2#8083711yescharacteristic
675C2#8090011yescharacteristic
676C2#8092211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1894C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#1894D3
hypocenterC3#26154
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
1. C2×#5121C2semidirect
3. C2×#4815C2semidirect
5. C3×#1894C2semidirect
7. #71692C2semidirect
8. #70251C2semidirect
9. #70243C2semidirect
10. #70224C2semidirect
11. #69506C2semidirect
12. #68877C2semidirect
13. #68836C2semidirect
14. #68831C2semidirect
15. #68818C2semidirect
72. #1894D3semidirect
145. #817D4semidirect
667. C3#26154semidirect

Maximal subgroups

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

subgroupindex
C2×#51212
#828142
C2×#48152
#738832
C3×#18942
#719962
#716922
#702512
#702432
#702242
#695062
#688772
#688362
#688312
#688182
#261543