Finite Group Databasebeta

← #10868348

Subgroups — #10868348

Order 768.

848 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
1C3×#2366C211yesmaximal, characteristic
2#73690C211yesmaximal, characteristic, complemented
3#71768C211yesmaximal, characteristic
4#71762C211yesmaximal, characteristic
5#71612C211yesmaximal, characteristic, complemented
6#71612C211yesmaximal, characteristic, complemented
7C4×#832C211yesmaximal, characteristic, complemented
8#71604C211yesmaximal, characteristic, complemented
9#69656C211yesmaximal, characteristic, complemented
10#69651C211yesmaximal, characteristic, complemented
11#69646C211yesmaximal, characteristic, complemented
12#69637C211yesmaximal, characteristic, complemented
13#69482C211yesmaximal, characteristic, complemented
14#69464C211yesmaximal, characteristic, complemented
15#69460C211yesmaximal, characteristic, complemented
16#27064—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C3×#390—21no—
24C3×#390C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#386—21no—
27C3×#386—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#4986—21no—
37#4980—21no—
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4935C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4929C2211yescharacteristic
45#4929C2211yescharacteristic
46#4929C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4660—21no—
51#4654—21no—
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2511—31no—
61#2366D311yescharacteristic
62#2094—31no—
63#2055—31no—
64#2041—31no—
65#2014—31no—
66#1968—31no—
67#1927—31no—
68#1926—31no—
69#1920—31no—
70#1916—31no—
71#1877—31no—
72#1877—31no—
73#1802—31no—
74C4×#102—31no—
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4—21no—
81C22×Dic6C2311yescharacteristic
82C3×#127—22no—
83C3×#127—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—41no—
87C3×C4×D4—41no—
88C3×C4×D4—44no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#214—22no—
104C2×#214—22no—
105C2×#214—22no—
106C2×#214—41no—
107C2×#214—41no—
108C2×#214—41no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C3×#95—21no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123#832—22no—
124#832—22no—
125#832—22no—
126#832—41no—
127#832—41no—
128#832—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#810—22no—
138#810—41no—
139#808—22no—
140#808—41no—
141C4⋊(C3⋊C8)—41no—
142C4×C3⋊C8—41no—
143C22×SD16—31no—
144C22×SD16—31no—
145C2×#127C2×D311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C2×D311yescharacteristic
148C2×C4×D4C2×D311yescharacteristic
149C2×#106—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×C4×C8—31no—
161#400—61no—
162#400—61no—
163#391—31no—
164#391—31no—
165#391—31no—
166#390—21no—
167#390C2×D311yescharacteristic
168#390C2×D311yescharacteristic
169#389—31no—
170#387—61no—
171#387—61no—
172#386—21no—
173#386—21no—
174#384—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4C2×D311yescharacteristic
177C4×C4⋊C4C2×D311yescharacteristic
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C24×C3—21no—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—42no—
201C2×C3×D4—42no—
202C2×C3×D4—81no—
203C2×C3×D4—81no—
204C2×C3×D4#4112yes—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4—81no—
213C22×C3×C4—81no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4SD1614yescomplemented
236C3×C4⋊C4#4112yes—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—81no—
242C3×C22⋊C4—81no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250#214—44no—
251#214—44no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3—81no—
260Dic3⋊C4—81no—
261C4×Dic3—81no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8#4112yes—
274C22×Q8—31no—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C22×D311yescharacteristic
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C23×C4—21no—
281C2×SD16—62no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
285C2×SD16—121no—
286C2×SD16—121no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8—31no—
291#127—22no—
292#127—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—41no—
296C4×D4—41no—
297C4×D4—44no—
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#6711yescharacteristic
305C2×C4⋊C4#6711yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C42—21no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—121no—
321C4⋊C8—121no—
322#103—62no—
323#103—121no—
324#102—62no—
325#102—62no—
326#102—62no—
327#102—121no—
328#102—121no—
329#102—121no—
330C4×C8—121no—
331#95—21no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95#6711yescharacteristic
341#95#6711yescharacteristic
342C23×C3—22no—
343C23×C3—22no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3C22×D411yescharacteristic
357C3×D4—22no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—48no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—84no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—84no—
389C2×C3×C4—84no—
390C2×C3×C4C2×SD1612yes—
391C2×C3×C4C2×SD1612yes—
392C2×C3×C4C2×SD1612yes—
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C4H(5,𝔽2)11yescharacteristic
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—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411Dic6—41no—
412Dic6—41no—
413Dic6—41no—
414Dic6—41no—
415C3⋊C8—22no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C3⋊C8—41no—
419C24—21no—
420C24—21no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—81no—
442C2×D4—81no—
443C2×D4#23512yes—
444C2×D4#23512yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—124no—
465SD16—124no—
466C2×C8—32no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4#21414yescomplemented
492C4⋊C4#23712yes—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—241no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3#52511yescharacteristic
522C22×C3#54611yescharacteristic
523C22×C3#55011yescharacteristic
524C22×C3C22×SD1611yescharacteristic
525C22×C3#57511yescharacteristic
526C22×C3#57611yescharacteristic
527C22×C3#57711yescharacteristic
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—22no—
531C3×C4—41no—
532C3×C4—42no—
533C3×C4—41no—
534C3×C4—41no—
535C3×C4—81no—
536C3×C4#52911yescharacteristic
537C3×C4#53811yescharacteristic
538C3×C4#55111yescharacteristic
539C3×C4#55311yescharacteristic
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543Dic3—81no—
544C23—22no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C22×#6711yescharacteristic
559Q8—121no—
560Q8—121no—
561Q8—121no—
562Q8—121no—
563D4—22no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4—48no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4—84no—
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×C4C2×#21412yes—
609C2×C4C2×#21412yes—
610C2×C4C2×#21412yes—
611C2×C4C2×#23511yescharacteristic
612C2×C4C2×#23511yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4#100911yescharacteristic
617C8—62no—
618C8—121no—
619C8—121no—
620C8—121no—
621C2×C3—82no—
622C2×C3—82no—
623C2×C3#246711yescharacteristic
624C2×C3#310111yescharacteristic
625C2×C3#311811yescharacteristic
626C2×C3#329511yescharacteristic
627C2×C3#331711yescharacteristic
628C2×C3#339211yescharacteristic
629C2×C3#340311yescharacteristic
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22#579111yescharacteristic
645C22C22×#21411yescharacteristic
646C22#580611yescharacteristic
647C22#582211yescharacteristic
648C22#583411yescharacteristic
649C22#583511yescharacteristic
650C22#583611yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—241no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#553811yescharacteristic
664C4#555611yescharacteristic
665C4#556011yescharacteristic
666C4#556511yescharacteristic
667C3#2706411yescharacteristic, complemented, Sylow
668C2—82no—
669C2—82no—
670C2#8231411yescharacteristic
671C2#8237011yescharacteristic
672C2#8254111yescharacteristic
673C2#8348311yescharacteristic
674C2#8357311yescharacteristic
675C2#8358811yescharacteristic
676C2#8371611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2366C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2366D3
hypocenterC3#27064
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
2. #73690C2semidirect
5. #71612C2semidirect
6. #71612C2semidirect
7. C4×#832C2semidirect
8. #71604C2semidirect
9. #69656C2semidirect
10. #69651C2semidirect
11. #69646C2semidirect
12. #69637C2semidirect
13. #69482C2semidirect
14. #69464C2semidirect
15. #69460C2semidirect
235. C3×C4⋊C4SD16semidirect
491. C4⋊C4#214semidirect
667. C3#27064semidirect

Maximal subgroups

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

subgroupindex
C3×#23662
#736902
#717682
#717622
#716122
C4×#8322
#716042
#696562
#696512
#696462
#696372
#694822
#694642
#694602
#270643