Finite Group Databasebeta

← #10845503

Subgroups — #10845503

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5111C211yesmaximal, characteristic, complemented
2#82666C211yesmaximal, characteristic
3C2×#4761C211yesmaximal, characteristic, complemented
4#73937C211yesmaximal, characteristic, complemented
5C3×#1893C211yesmaximal, characteristic, complemented
6#71970C211yesmaximal, characteristic, complemented
7#71679C211yesmaximal, characteristic, complemented
8#70203C211yesmaximal, characteristic, complemented
9#70176C211yesmaximal, characteristic, complemented
10#70152C211yesmaximal, characteristic, complemented
11#69520C211yesmaximal, characteristic, complemented
12#68870C211yesmaximal, characteristic, complemented
13#68832C211yesmaximal, characteristic, complemented
14#68806C211yesmaximal, characteristic, complemented
15#68797C211yesmaximal, characteristic, complemented
16#25354—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×#938C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×C4×#67C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#885—21no—
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5223—21no—
29#5220—21no—
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5111—24no—
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#817C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4987C2211yescharacteristic
42#4987C2211yescharacteristic
43#4987C2211yescharacteristic
44#4977C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4958—21no—
49#4952—21no—
50#4952—21no—
51#4761—24no—
52#4675—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#476—31no—
63C2×#435—31no—
64#2547—31no—
65#2535—31no—
66#2077—31no—
67#2077—31no—
68#2034—31no—
69#2033—31no—
70#1993—31no—
71#1949—31no—
72#1895—31no—
73#1893D311yescharacteristic, complemented
74#1809—31no—
75#1807—31no—
76#1762—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67D411yescharacteristic
82C23×Dic3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C22×C3×C8—22no—
86C22×C3×C8—22no—
87C22×C3×C8—41no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#121—22no—
91C3×#121—41no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217D411yescharacteristic
102#938—22no—
103#938—41no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106C4×#67—22no—
107C4×#67—22no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)—22no—
124C2×ΓL(1,𝔽25)—41no—
125#888—22no—
126#888—22no—
127#885—44no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#95—21no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#830—22no—
142(C2×D3)⋊C8—22no—
143#817—22no—
144#817—41no—
145#815—22no—
146Dic3⋊C8D414yescomplemented
147#808—22no—
148#807—41no—
149C22×M16—31no—
150C23×C8C2×D311yescharacteristic
151C2×C4⋊Q8—31no—
152C2×#124—61no—
153C2×#121—31no—
154C2×#121C2×D311yescharacteristic
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157#476—64no—
158#435—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170C2×#97—31no—
171#397—61no—
172#397—61no—
173#397—61no—
174#392—61no—
175#391—61no—
176#386—61no—
177#386—61no—
178#382—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C23×D3C2×D411yescharacteristic
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—42no—
203C2×C3×D4—42no—
204C2×C3×D4—81no—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C2×#67—22no—
218C2×#67—22no—
219C2×#67—41no—
220C2×#67—42no—
221C2×#67—42no—
222C2×#67—81no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—42no—
228C22×Dic3—81no—
229C22×Dic3—81no—
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×Dic6—42no—
240C2×Dic6—42no—
241C2×Dic6—42no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—44no—
247C2×C3×C8C2×D412yes—
248C2×C3×C8#4112yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—42no—
257#217—22no—
258#217—42no—
259#217—42no—
260#217—42no—
261#217—81no—
262(C2×D3)⋊C4—22no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274Dic3⋊C4—22no—
275Dic3⋊C4—42no—
276C4×Dic3—42no—
277C4×Dic3C2×D412yes—
278C4×Dic3#4112yes—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8C2×D412yes—
285ΓL(1,𝔽25)—44no—
286C22×Q8—61no—
287C22×D4—21no—
288C22×D4—31no—
289C22×D4—61no—
290C22×D4C22×D311yescharacteristic
291C23×C4—31no—
292C23×C4—61no—
293C23×C4C22×D311yescharacteristic
294C2×M16—62no—
295C2×M16—62no—
296C2×M16—121no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—41no—
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304C4⋊Q8—62no—
305C4⋊Q8—62no—
306#124—124no—
307#121—22no—
308#121—41no—
309#121—62no—
310#121—121no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—31no—
327C2×C22⋊C4—31no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—61no—
337#107—121no—
338#106—62no—
339#106—62no—
340#106—121no—
341C4⋊C8—34no—
342#102—22no—
343#102—22no—
344#102—62no—
345#102—62no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#97—62no—
349#95—21no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—22no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3AGL(1,ℤ/8)12yes—
372C23×C3C22×D411yescharacteristic
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—21no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—84no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407#67—44no—
408#67—84no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
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—84no—
429C2×Dic3#12112yes—
430C2×Dic3#12112yes—
431C4×D3—41no—
432C4×D3—41no—
433C4×D3—41no—
434C4×D3—41no—
435Dic6—84no—
436C3×C8—22no—
437C3×C8—22no—
438C3⋊C8—22no—
439C3⋊C8—41no—
440C24—21no—
441C24—31no—
442C24D3×D411yescharacteristic
443C2×Q8—122no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—62no—
454C2×D4—62no—
455C2×D4—81no—
456C2×D4—121no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—241no—
460C2×D4#23512yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—122no—
484C22×C4—241no—
485C22×C4—241no—
486C22×C4D3×D411yescharacteristic
487C22×C4C23×D311yescharacteristic
488M16—124no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—32no—
494C2×C8—44no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8D3×D412yes—
501C2×C8#23712yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—62no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—122no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—62no—
521C22⋊C4—62no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—241no—
528C22⋊C4—241no—
529C42—32no—
530C42—32no—
531C42—122no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—22no—
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×AGL(1,ℤ/8)11yescharacteristic
550C22×C3#57511yescharacteristic
551C22×C3#57711yescharacteristic
552C22×C3#57711yescharacteristic
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—41no—
560C3×C4—41no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4—81no—
564C3×C4#52911yescharacteristic
565C3×C4#54211yescharacteristic
566C3×C4#54611yescharacteristic
567C3×C4#54811yescharacteristic
568Dic3—22no—
569Dic3—22no—
570Dic3—41no—
571Dic3—81no—
572Dic3—81no—
573Dic3—81no—
574Dic3—81no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—61no—
595C23#91112yes—
596C23C2×D3×D411yescharacteristic
597Q8—244no—
598D4—22no—
599D4—22no—
600D4—41no—
601D4—41no—
602D4—84no—
603D4—124no—
604D4—244no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—32no—
609C2×C4—32no—
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—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—244no—
647C2×C4C2×#23511yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×#23711yescharacteristic
650C2×C4#100411yescharacteristic
651C2×C4D3×#4111yescharacteristic
652C2×C4#101011yescharacteristic
653C8—22no—
654C8—22no—
655C8—62no—
656C8—121no—
657C2×C3—42no—
658C2×C3—82no—
659C2×C3#242711yescharacteristic
660C2×C3#305511yescharacteristic
661C2×C3#305711yescharacteristic
662C2×C3#330611yescharacteristic
663C2×C3#331111yescharacteristic
664C2×C3#334511yescharacteristic
665C2×C3#335011yescharacteristic
666D3—81no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—121no—
682C22—121no—
683C22—121no—
684C22—121no—
685C22—121no—
686C22—121no—
687C22—121no—
688C22#549011yescharacteristic
689C22D3×#12111yescharacteristic
690C22#561111yescharacteristic
691C22C2×#91111yescharacteristic
692C22#576211yescharacteristic
693C22#576811yescharacteristic
694C22#577011yescharacteristic
695C4—41no—
696C4—62no—
697C4—62no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—121no—
702C4—241no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4#553811yescharacteristic
707C4#555211yescharacteristic
708C4#559111yescharacteristic
709C4#566211yescharacteristic
710C3#2535411yescharacteristic, complemented, Sylow
711C2—42no—
712C2—82no—
713C2—241no—
714C2#7469711yescharacteristic
715C2#7876111yescharacteristic
716C2#7885511yescharacteristic
717C2#8065411yescharacteristic
718C2#8065911yescharacteristic
719C2#8071111yescharacteristic
720C2#8071511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
prime radical#82666C2
hypercenter#1893D3
hypocenterC3#25354
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
1. C2×#5111C2semidirect
3. C2×#4761C2semidirect
4. #73937C2semidirect
5. C3×#1893C2semidirect
6. #71970C2semidirect
7. #71679C2semidirect
8. #70203C2semidirect
9. #70176C2semidirect
10. #70152C2semidirect
11. #69520C2semidirect
12. #68870C2semidirect
13. #68832C2semidirect
14. #68806C2semidirect
15. #68797C2semidirect
73. #1893D3semidirect
146. Dic3⋊C8D4semidirect
710. C3#25354semidirect

Maximal subgroups

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

subgroupindex
C2×#51112
#826662
C2×#47612
#739372
C3×#18932
#719702
#716792
#702032
#701762
#701522
#695202
#688702
#688322
#688062
#687972
#253543