Finite Group Databasebeta

← #10845514

Subgroups — #10845514

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82667C211yesmaximal, characteristic
2C2×#4763C211yesmaximal, characteristic, complemented
3C2×#4728C211yesmaximal, characteristic, complemented
4#73940C211yesmaximal, characteristic, complemented
5C3×#2034C211yesmaximal, characteristic, complemented
6#71682C211yesmaximal, characteristic, complemented
7#70577C211yesmaximal, characteristic, complemented
8#70563C211yesmaximal, characteristic, complemented
9#70207C211yesmaximal, characteristic, complemented
10#70202C211yesmaximal, characteristic, complemented
11#70152C211yesmaximal, characteristic, complemented
12#69940C211yesmaximal, characteristic, complemented
13#69513C211yesmaximal, characteristic, complemented
14#68889C211yesmaximal, characteristic, complemented
15#68805C211yesmaximal, characteristic, complemented
16#25355—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×#939—21no—
19C2×#938C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×C4×#67C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×D3×C8C2211yescharacteristic
24C2×#890C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5222—21no—
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42#4975C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4959—21no—
47#4959—21no—
48#4951—21no—
49#4942—21no—
50#4763—24no—
51#4728—24no—
52#4670—21no—
53#4658—21no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#435—31no—
64#2562—31no—
65#2537—31no—
66#2077—31no—
67#2075—31no—
68#2034—31no—
69#2034D311yescharacteristic, complemented
70#1992—31no—
71#1948—31no—
72#1895—31no—
73#1893—31no—
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×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—22no—
87C3×#121—41no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217D411yescharacteristic
98#939—44no—
99#938—22no—
100#938—41no—
101D4×Dic3—22no—
102D4×Dic3—22no—
103C4×#67—22no—
104C4×#67—22no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4D411yescharacteristic
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×#208—22no—
121C2×#208—22no—
122C2×#208—41no—
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C2×D3×C8—22no—
126C2×D3×C8—22no—
127C2×D3×C8—41no—
128#890—22no—
129#890—22no—
130C3×#106—41no—
131C3×#102—22no—
132C3×#102—22no—
133C3×C22⋊C8—22no—
134C3×#95—21no—
135#832—22no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143(C2×D3)⋊C8—22no—
144#817—22no—
145#815—22no—
146Dic3⋊C8D414yescomplemented
147#808—41no—
148#807—22no—
149C22×M16—31no—
150C23×C8—31no—
151C2×#125—31no—
152C2×#121—31no—
153C2×#121—61no—
154C2×#121C2×D311yescharacteristic
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157#478—64no—
158#435—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C22⋊C8—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170C2×#97—31no—
171#400—61no—
172#394—61no—
173#394—61no—
174#394—61no—
175#388—61no—
176#388—61no—
177#384—31no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—61no—
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#340—31no—
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—42no—
229C22×Dic3—42no—
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×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253#217—22no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—81no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271Dic3⋊C4—22no—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4—81no—
275C4×Dic3—81no—
276C4×Dic3C2×D412yes—
277C4×Dic3#4112yes—
278#208—44no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8C2×D412yes—
284C2×C3⋊C8#4112yes—
285D3×C8—44no—
286C22×D4—21no—
287C22×D4—31no—
288C22×D4—61no—
289C22×D4C22×D311yescharacteristic
290C23×C4—31no—
291C23×C4—61no—
292C23×C4C22×D311yescharacteristic
293C2×M16—62no—
294C2×M16—62no—
295C2×M16—121no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8—62no—
299C22×C8—62no—
300C22×C8—121no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303#125—62no—
304#125—62no—
305#121—22no—
306#121—41no—
307#121—62no—
308#121—121no—
309#121—124no—
310C4×D4—62no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—31no—
329C2×C22⋊C4—31no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C42—31no—
338C2×C42—61no—
339#107—62no—
340#106—41no—
341#106—62no—
342#106—121no—
343C4⋊C8—34no—
344#102—22no—
345#102—22no—
346#102—62no—
347#102—62no—
348C22⋊C8—22no—
349C22⋊C8—62no—
350#97—62no—
351#95—21no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3—21no—
359C23×C3—22no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3C22×D411yescharacteristic
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3AGL(1,ℤ/8)12yes—
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—
435C3×C8—22no—
436C3×C8—41no—
437C3⋊C8—22no—
438C3⋊C8—22no—
439C24—21no—
440C24—31no—
441C24D3×D411yescharacteristic
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4—81no—
452C2×D4—121no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—241no—
456C2×D4#23712yes—
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—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—122no—
480C22×C4—122no—
481C22×C4—122no—
482C22×C4D3×D411yescharacteristic
483C22×C4C23×D311yescharacteristic
484M16—124no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—32no—
491C2×C8—32no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—124no—
497C2×C8D3×D412yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—62no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4—122no—
516C4⋊C4—241no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—62no—
520C22⋊C4—62no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—32no—
529C42—32no—
530C42—241no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—41no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3C2×#12111yescharacteristic
546C22×C3#54611yescharacteristic
547C22×C3#54711yescharacteristic
548C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
549C22×C3#57511yescharacteristic
550C22×C3#57711yescharacteristic
551C22×C3#57711yescharacteristic
552C2×D3—22no—
553C2×D3—22no—
554C2×D3—22no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C3×C4—41no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4#52911yescharacteristic
564C3×C4#54211yescharacteristic
565C3×C4#54611yescharacteristic
566C3×C4#54811yescharacteristic
567Dic3—22no—
568Dic3—22no—
569Dic3—41no—
570Dic3—81no—
571Dic3—81no—
572Dic3—81no—
573Dic3—81no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—22no—
580C23—21no—
581C23—21no—
582C23—32no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—61no—
595C23C2×D3×D411yescharacteristic
596D4—22no—
597D4—22no—
598D4—41no—
599D4—41no—
600D4—84no—
601D4—124no—
602D4—244no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—32no—
607C2×C4—32no—
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—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
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—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—244no—
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4#100411yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4#101011yescharacteristic
651C8—22no—
652C8—41no—
653C8—62no—
654C8—62no—
655C2×C3—81no—
656C2×C3—82no—
657C2×C3#242711yescharacteristic
658C2×C3#305511yescharacteristic
659C2×C3#306011yescharacteristic
660C2×C3#330711yescharacteristic
661C2×C3#331011yescharacteristic
662C2×C3#334511yescharacteristic
663C2×C3#334811yescharacteristic
664D3—42no—
665C22—41no—
666C22—41no—
667C22—42no—
668C22—41no—
669C22—42no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—62no—
676C22—62no—
677C22—62no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—121no—
683C22—121no—
684C22—121no—
685C22—121no—
686C22#559111yescharacteristic
687C22D3×#12111yescharacteristic
688C22#565511yescharacteristic
689C22#576211yescharacteristic
690C22#577011yescharacteristic
691C22D3×AGL(1,ℤ/8)11yescharacteristic
692C22#577311yescharacteristic
693C4—41no—
694C4—62no—
695C4—62no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—121no—
700C4—241no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4#559811yescharacteristic
705C4#561011yescharacteristic
706C4#564911yescharacteristic
707C4#566911yescharacteristic
708C3#2535511yescharacteristic, complemented, Sylow
709C2—81no—
710C2—82no—
711C2—122no—
712C2#7690211yescharacteristic
713C2#8010811yescharacteristic
714C2#8016111yescharacteristic
715C2#8066011yescharacteristic
716C2#8086411yescharacteristic
717C2#8133611yescharacteristic
718C2#8139111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2034C2
socleC23×C3C22×D4
prime radical#82667C2
hypercenter#2034D3
hypocenterC3#25355
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
2. C2×#4763C2semidirect
3. C2×#4728C2semidirect
4. #73940C2semidirect
5. C3×#2034C2semidirect
6. #71682C2semidirect
7. #70577C2semidirect
8. #70563C2semidirect
9. #70207C2semidirect
10. #70202C2semidirect
11. #70152C2semidirect
12. #69940C2semidirect
13. #69513C2semidirect
14. #68889C2semidirect
15. #68805C2semidirect
69. #2034D3semidirect
146. Dic3⋊C8D4semidirect
708. C3#25355semidirect

Maximal subgroups

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

subgroupindex
#826672
C2×#47632
C2×#47282
#739402
C3×#20342
#716822
#705772
#705632
#702072
#702022
#701522
#699402
#695132
#688892
#688052
#253553