Finite Group Databasebeta

← #10845346

Subgroups — #10845346

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2535C211yesmaximal, characteristic
2#82383C211yesmaximal, characteristic, complemented
3C2×D4×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#5009C211yesmaximal, characteristic, complemented
5#71681C211yesmaximal, characteristic, complemented
6#71681C211yesmaximal, characteristic, complemented
7#71680C211yesmaximal, characteristic, complemented
8#71679C211yesmaximal, characteristic, complemented
9C4×#832C211yesmaximal, characteristic, complemented
10#69554C211yesmaximal, characteristic, complemented
11#69540C211yesmaximal, characteristic, complemented
12#69519C211yesmaximal, characteristic, complemented
13#69512C211yesmaximal, characteristic, complemented
14#69511C211yesmaximal, characteristic, complemented
15#69507C211yesmaximal, characteristic, complemented
16#25342—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C3×#397—21no—
25C3×#392—21no—
26C3×#386—21no—
27C3×#386—21no—
28C3×#382C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×#380C2211yescharacteristic
31#5222—21no—
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36#5212C2211yescharacteristic
37#5212C2211yescharacteristic
38#5212C2211yescharacteristic
39#5212C2211yescharacteristic
40C2×#830C2211yescharacteristic
41C2×#830C2211yescharacteristic
42D4×C3⋊C8—24no—
43#5009—24no—
44#4985—21no—
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4945—21no—
50#4945—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×C8×D4—31no—
64#2562—31no—
65#2535D311yescharacteristic
66#2077—31no—
67#2076—31no—
68#2034—31no—
69#2033—31no—
70#1991—31no—
71#1949—31no—
72#1893—31no—
73#1893—31no—
74#1808—31no—
75#1807—31no—
76C4×#102—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4D411yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4C2311yescharacteristic
83C23×C3×C4C2311yescharacteristic
84C3×#121—22no—
85C3×#121—22no—
86C3×#121—41no—
87C3×#121—41no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107C2×#217—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8—22no—
116C22×C3⋊C8—22no—
117C22×C3⋊C8—22no—
118C22×C3⋊C8—22no—
119C22×C3⋊C8—41no—
120C22×C3⋊C8—41no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#870—22no—
126#870—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131#832—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#830—22no—
141#830—22no—
142#808—22no—
143#808—41no—
144#807—22no—
145#807—41no—
146C4⋊(C3⋊C8)D414yescomplemented
147C4×C3⋊C8—22no—
148C23×C8—31no—
149C23×C8—31no—
150C2×#125—31no—
151C2×#121C2×D311yescharacteristic
152C2×#121C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154C2×C4×D4C2×D311yescharacteristic
155#478—64no—
156C8×D4—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×C4×C8—31no—
169#400—61no—
170#397—21no—
171#394—61no—
172#392—21no—
173#388—61no—
174#388—61no—
175#386—21no—
176#386—21no—
177#382C2×D311yescharacteristic
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380C2×D311yescharacteristic
183#380C2×D311yescharacteristic
184C4×C4⋊C4—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3—21no—
194C24×C3C2×D411yescharacteristic
195C24×C3C2×D411yescharacteristic
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—42no—
205C2×C3×D4—42no—
206C2×C3×D4—42no—
207C2×C3×D4—42no—
208C2×C3×D4—81no—
209C2×C3×D4—81no—
210C2×C3×D4#4112yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—42no—
227C22×C3×C4—81no—
228C22×C3×C4—81no—
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C22×Dic3—21no—
235C22×Dic3—21no—
236C22×Dic3—21no—
237C22×Dic3—21no—
238C3×C4⋊C4—22no—
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×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—81no—
251C3×C22⋊C4—81no—
252C3×C42C2×D412yes—
253C3×C42#4112yes—
254#217—42no—
255#217—42no—
256#217—81no—
257#217—81no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268C4×Dic3—81no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—44no—
279C2×C3⋊C8—44no—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8C2×D412yes—
282C2×C3⋊C8#4112yes—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4#6711yescharacteristic
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4C22×D311yescharacteristic
289C23×C4C22×D311yescharacteristic
290C22×C8—31no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8—62no—
295C22×C8—62no—
296C22×C8—62no—
297C22×C8—62no—
298C22×C8—121no—
299C22×C8—121no—
300#125—62no—
301#125—62no—
302#121—22no—
303#121—22no—
304#121—41no—
305#121—41no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4#6711yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4#6711yescharacteristic
330C2×C22⋊C4#6711yescharacteristic
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#107—121no—
335#106—62no—
336#106—121no—
337C4⋊C8—34no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342C22⋊C8—62no—
343C22⋊C8—62no—
344C4×C8—62no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—22no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3#13512yes—
374C23×C3#13512yes—
375C23×C3C22×D411yescharacteristic
376C3×D4—28no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—84no—
382C3×D4—84no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—84no—
411C2×C3×C4#12112yes—
412C2×C3×C4#12112yes—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4H(5,𝔽2)11yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C3⋊C8—22no—
432C3⋊C8—22no—
433C3⋊C8—22no—
434C3⋊C8—41no—
435C24—21no—
436C24C2×#6711yescharacteristic
437C24C2×#6711yescharacteristic
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—81no—
451C2×D4—81no—
452C2×D4#23712yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C23×D311yescharacteristic
480C2×C8—32no—
481C2×C8—32no—
482C2×C8—32no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—124no—
493C2×C8—124no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—122no—
510C4⋊C4—122no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C42—241no—
523C42D3×D412yes—
524C42#23712yes—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—22no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3C2×#12111yescharacteristic
547C22×C3#54611yescharacteristic
548C22×C3#54711yescharacteristic
549C22×C3C2×#13511yescharacteristic
550C22×C3C2×#13511yescharacteristic
551C22×C3#57611yescharacteristic
552C22×C3#57711yescharacteristic
553C3×C4—22no—
554C3×C4—22no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—81no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4#52911yescharacteristic
561C3×C4#54211yescharacteristic
562C3×C4#54611yescharacteristic
563C3×C4#54811yescharacteristic
564Dic3—81no—
565Dic3—81no—
566Dic3—81no—
567Dic3—81no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—22no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23#95012yes—
588C23#95012yes—
589C23C22×#6711yescharacteristic
590D4—28no—
591D4—22no—
592D4—22no—
593D4—41no—
594D4—41no—
595D4—84no—
596D4—84no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—84no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4#93912yes—
638C2×C4#93912yes—
639C2×C4C2×D3×D411yescharacteristic
640C2×C4C2×#23711yescharacteristic
641C2×C4C2×#23711yescharacteristic
642C2×C4#100411yescharacteristic
643C2×C4D3×#4111yescharacteristic
644C2×C4#101011yescharacteristic
645C8—62no—
646C8—62no—
647C8—62no—
648C8—121no—
649C2×C3—42no—
650C2×C3—42no—
651C2×C3—82no—
652C2×C3#242711yescharacteristic
653C2×C3#305011yescharacteristic
654C2×C3#306211yescharacteristic
655C2×C3#329611yescharacteristic
656C2×C3#329711yescharacteristic
657C2×C3#333711yescharacteristic
658C2×C3#333911yescharacteristic
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22C2×#93911yescharacteristic
681C22#580611yescharacteristic
682C22C2×#95011yescharacteristic
683C22C2×#95011yescharacteristic
684C22#582411yescharacteristic
685C22#583511yescharacteristic
686C22#583611yescharacteristic
687C4—22no—
688C4—22no—
689C4—41no—
690C4—41no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—241no—
695C4—241no—
696C4—241no—
697C4—241no—
698C4#559811yescharacteristic
699C4#560011yescharacteristic
700C4#560311yescharacteristic
701C4#561211yescharacteristic
702C3#2534211yescharacteristic, complemented, Sylow
703C2—42no—
704C2—42no—
705C2—82no—
706C2#8264711yescharacteristic
707C2#8292211yescharacteristic
708C2#8337911yescharacteristic
709C2#8349411yescharacteristic
710C2#8349511yescharacteristic
711C2#8360611yescharacteristic
712C2#8360711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2535C2
socleC23×C3C22×D4
hypercenter#2535D3
hypocenterC3#25342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
2. #82383C2semidirect
3. C2×D4×C3⋊C8C2semidirect
4. C2×#5009C2semidirect
5. #71681C2semidirect
6. #71681C2semidirect
7. #71680C2semidirect
8. #71679C2semidirect
9. C4×#832C2semidirect
10. #69554C2semidirect
11. #69540C2semidirect
12. #69519C2semidirect
13. #69512C2semidirect
14. #69511C2semidirect
15. #69507C2semidirect
146. C4⋊(C3⋊C8)D4semidirect
702. C3#25342semidirect

Maximal subgroups

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

subgroupindex
C3×#25352
#823832
C2×D4×C3⋊C82
C2×#50092
#716812
#716802
#716792
C4×#8322
#695542
#695402
#695192
#695122
#695112
#695072
#253423