Finite Group Databasebeta

← #10855331

Subgroups — #10855331

Order 768.

826 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5166C211yesmaximal, characteristic
2Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×C4×#216C211yesmaximal, characteristic, complemented
4#76320C211yesmaximal, characteristic
5C3×#1880C211yesmaximal, characteristic
6#72048C211yesmaximal, characteristic
7#70239C211yesmaximal, characteristic, complemented
8#70236C211yesmaximal, characteristic, complemented
9#70216C211yesmaximal, characteristic
10#70183C211yesmaximal, characteristic
11#69644C211yesmaximal, characteristic
12#68914C211yesmaximal, characteristic, complemented
13#68807C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68655C211yesmaximal, characteristic, complemented
16#26169—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#936C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C2×#835C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5166—24no—
32C2×#818C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C4×#216—24no—
37#4984C2211yescharacteristic
38#4975—21no—
39#4975C2211yescharacteristic
40#4973C2211yescharacteristic
41#4972—21no—
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4952—21no—
46#4951—21no—
47#4949—21no—
48#4945—21no—
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4928—21no—
52#4928C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×C4×Q16—31no—
64#2448—31no—
65Q8×C4⋊C4—31no—
66#2093—31no—
67#2086—31no—
68#2072—31no—
69#2042—31no—
70#1934—31no—
71#1916—31no—
72#1880—31no—
73#1880D311yescharacteristic
74#1878—31no—
75#1863—31no—
76#1863—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×SD16—22no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101Q8×Dic3—41no—
102C2×#216—22no—
103C2×#216—22no—
104C2×#216D414yes—
105#936—22no—
106#936—41no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3C2311yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C4×Dic6—22no—
128C4×Dic6—41no—
129C3×#103—22no—
130C3×#102—41no—
131#835—22no—
132#832—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#818—41no—
142#816—22no—
143#816—22no—
144Dic3⋊C8—22no—
145#810—22no—
146#807—22no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×SD16C2×D311yescharacteristic
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#464—64no—
156C4×Q16—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×C4×C8—31no—
169#399—61no—
170#397—61no—
171#391—31no—
172#390—31no—
173#390C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#389—31no—
177#389—61no—
178#389—61no—
179#388—61no—
180#388—61no—
181#384—31no—
182#384—61no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191#336—31no—
192#336C2×D311yescharacteristic
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×D412yes—
198C2×C3×Q8#4112yes—
199C2×C3×Q8#4112yes—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D412yes—
226C3×SD16—44no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—81no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—42no—
243#217—42no—
244#217—42no—
245#217—42no—
246#217—81no—
247#216—28no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3—81no—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—81no—
262Dic3⋊C4—81no—
263Dic3⋊C4—81no—
264C4×Dic3—24no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
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⋊C8C2×D412yes—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C22×D4C22×D311yescharacteristic
280C2×Q16—34no—
281C2×Q16—62no—
282C2×Q16—62no—
283C2×SD16—22no—
284C2×SD16—22no—
285C2×SD16—22no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290#124—62no—
291#124—121no—
292C4×Q8—22no—
293C4×Q8—41no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8—62no—
298C4×Q8—121no—
299C4×Q8—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—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—21no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—62no—
330#107—121no—
331C4⋊C8—62no—
332#103—22no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—41no—
338#102—62no—
339#102—62no—
340C4×C8—62no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3C22×D411yescharacteristic
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8#12314yes—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4#13712yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×C3×C42−1+411yescharacteristic
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3#13512yes—
409Dic6—22no—
410Dic6—22no—
411Dic6—41no—
412Dic6—41no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—22no—
416C3⋊C8—41no—
417C24—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—32no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8#23512yes—
428C2×Q8D3×D412yes—
429C2×Q8#23712yes—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—68no—
451SD16—44no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—32no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—42no—
470C4⋊C4—62no—
471C4⋊C4—64no—
472C4⋊C4—62no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—241no—
482C4⋊C4—241no—
483C4⋊C4—241no—
484C4⋊C4—241no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—241no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—64no—
495C42—61no—
496C42—61no—
497C42—61no—
498C42—61no—
499C42—122no—
500C42—122no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3#52511yescharacteristic
509C22×C3D4211yescharacteristic
510C22×C3#54811yescharacteristic
511C22×C3C2×#13511yescharacteristic
512C22×C3C2×#13711yescharacteristic
513C22×C3#57611yescharacteristic
514C22×C3#57811yescharacteristic
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—41no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4C2×#12311yescharacteristic
521C3×C4#53911yescharacteristic
522C3×C4#54711yescharacteristic
523C3×C4#55311yescharacteristic
524Dic3—22no—
525Dic3—41no—
526Dic3—42no—
527Dic3—42no—
528Dic3—41no—
529Dic3—41no—
530Dic3—81no—
531Dic3—81no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C2×D3×D411yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—22no—
543Q8—62no—
544Q8—62no—
545Q8—121no—
546Q8—121no—
547Q8#88314yes—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—32no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4#91512yes—
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4C2×#23711yescharacteristic
594C2×C4#100411yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#101011yescharacteristic
597C8—41no—
598C8—41no—
599C8—62no—
600C8—121no—
601C2×C3—161no—
602C2×C3#244311yescharacteristic
603C2×C3#310811yescharacteristic
604C2×C3#312411yescharacteristic
605C2×C3#327011yescharacteristic
606C2×C3#328711yescharacteristic
607C2×C3#336411yescharacteristic
608C2×C3#337511yescharacteristic
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22#554711yescharacteristic
617C22#559011yescharacteristic
618C22#559811yescharacteristic
619C22C2×#91511yescharacteristic
620C22D3×#13511yescharacteristic
621C22#577411yescharacteristic
622C22#577911yescharacteristic
623C4—22no—
624C4—24no—
625C4—41no—
626C4—62no—
627C4—81no—
628C4—81no—
629C4—121no—
630C4—121no—
631C4—121no—
632C4—122no—
633C4—122no—
634C4—241no—
635C4—241no—
636C4C2×#88311yescharacteristic
637C4#556411yescharacteristic
638C4#560111yescharacteristic
639C4#565511yescharacteristic
640C3#2616911yescharacteristic, complemented, Sylow
641C2—161no—
642C2#7470211yescharacteristic
643C2#7965611yescharacteristic
644C2#7981311yescharacteristic
645C2#8006411yescharacteristic
646C2#8013111yescharacteristic
647C2#8037411yescharacteristic
648C2#8058011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1880D3
hypocenterC3#26169
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. Q8×Dic3⋊C4C2semidirect
3. C2×C4×#216C2semidirect
7. #70239C2semidirect
8. #70236C2semidirect
12. #68914C2semidirect
13. #68807C2semidirect
14. #68683C2semidirect
15. #68655C2semidirect
640. C3#26169semidirect

Maximal subgroups

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

subgroupindex
C2×#51662
Q8×Dic3⋊C42
C2×C4×#2162
#763202
C3×#18802
#720482
#702392
#702362
#702162
#701832
#696442
#689142
#688072
#686832
#686552
#261693