Finite Group Databasebeta

← #10840226

Subgroups — #10840226

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2537C211yesmaximal, characteristic
2#82335C211yesmaximal, characteristic, complemented
3#71680C211yesmaximal, characteristic, complemented
4#71680C211yesmaximal, characteristic, complemented
5#71678C211yesmaximal, characteristic, complemented
6#71677C211yesmaximal, characteristic, complemented
7#71615C211yesmaximal, characteristic, complemented
8#71609C211yesmaximal, characteristic, complemented
9#71605C211yesmaximal, characteristic, complemented
10#69575C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#69520C211yesmaximal, characteristic, complemented
13#69507C211yesmaximal, characteristic, complemented
14#69503C211yesmaximal, characteristic, complemented
15#69501C211yesmaximal, characteristic, complemented
16#24982—31nomaximal, Sylow
17C2×C3×#121—21no—
18C2×C3×#121C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C3×#394—21no—
25C3×#394—21no—
26C3×#384C2211yescharacteristic
27C3×#380—21no—
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5220—21no—
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5212C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#830C2211yescharacteristic
41#4980—21no—
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4951—21no—
45Dic3×C22⋊C4—21no—
46#4936C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×#803C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2537D311yescharacteristic
61#2352—31no—
62#2077—31no—
63#2077—31no—
64#2034—31no—
65#2034—31no—
66#1955—31no—
67#1948—31no—
68#1875—31no—
69#1874—31no—
70#1836—31no—
71#1825—31no—
72#1823—31no—
73#1804—31no—
74#1762—31no—
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4D411yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#121—41no—
86C3×#121—41no—
87C3×#121—44no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×#217—21no—
107C2×#217—21no—
108C2×#217—21no—
109#926—22no—
110#926—41no—
111C2×C4⋊Dic3—22no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—41no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C3×#95—21no—
125C3×#95—21no—
126#832—22no—
127#832—22no—
128#832—41no—
129#832—41no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#830—41no—
136#830—41no—
137#830—41no—
138#830—41no—
139#808—41no—
140#808—41no—
141C4⋊(C3⋊C8)—22no—
142#803—22no—
143C2×#121—21no—
144C2×#121C2×D311yescharacteristic
145C2×#121C2×D311yescharacteristic
146C2×C4×D4C2×D311yescharacteristic
147C2×C4×D4C2×D311yescharacteristic
148C2×#117—31no—
149C22×C4⋊C4—31no—
150C2×#106—31no—
151C2×#106—31no—
152C2×C4⋊C8—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×C22⋊C8—31no—
161C2×#97—31no—
162#394—21no—
163#394—21no—
164#388—61no—
165#387—61no—
166#386—61no—
167#384C2×D311yescharacteristic
168#382—31no—
169#380—21no—
170#380—31no—
171#380—31no—
172#380C2×D311yescharacteristic
173#380C2×D311yescharacteristic
174C4×C4⋊C4—31no—
175C4×C22⋊C4—61no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340—31no—
184#340—31no—
185#340—31no—
186C24×C3—21no—
187C24×C3C2×D411yescharacteristic
188C24×C3C2×D411yescharacteristic
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—41no—
197C2×C3×D4—42no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4—42no—
201C2×C3×D4—81no—
202C2×C3×D4—81no—
203C2×C3×D4#4112yes—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
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×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—42no—
220C22×C3×C4—42no—
221C22×C3×C4—42no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—42no—
232C22×Dic3—42no—
233C22×Dic3—81no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—81no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C42#4112yes—
251C3×C42#4112yes—
252#217—42no—
253#217—42no—
254#217—81no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263C4⋊Dic3#4112yes—
264C4×Dic3—42no—
265C4×Dic3—81no—
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—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4#6711yescharacteristic
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—61no—
284C23×C4C22×D311yescharacteristic
285C23×C4C22×D311yescharacteristic
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290#121—22no—
291#121—22no—
292#121—41no—
293#121—41no—
294#121—44no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299#117—62no—
300#117—121no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—121no—
310C2×C4⋊C4#6711yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4#6711yescharacteristic
323C2×C22⋊C4#6711yescharacteristic
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—121no—
328#106—121no—
329C4⋊C8—62no—
330#102—62no—
331#102—62no—
332#102—121no—
333#102—121no—
334C22⋊C8—121no—
335C22⋊C8—121no—
336C22⋊C8—121no—
337C22⋊C8—121no—
338#97—62no—
339#95—21no—
340#95—21no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3AGL(1,ℤ/8)12yes—
367C23×C3C22×D411yescharacteristic
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—44no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—84no—
374C3×D4—84no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
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—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
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×C4—41no—
402C2×C3×C4—84no—
403C2×C3×C4#12312yes—
404C2×C3×C4#12312yes—
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4H(5,𝔽2)11yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×Dic3—22no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—84no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C3⋊C8—41no—
428C24—21no—
429C24C2×#6711yescharacteristic
430C24C2×#6711yescharacteristic
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—41no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—81no—
444C2×D4—81no—
445C2×D4#23712yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
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—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—241no—
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C23×D311yescharacteristic
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—32no—
496C4⋊C4—42no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—81no—
505C4⋊C4—122no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—241no—
516C42—122no—
517C42—241no—
518C42#23712yes—
519C42#23712yes—
520C22×C3—22no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—41no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3C2×#12311yescharacteristic
542C22×C3#54611yescharacteristic
543C22×C3#54611yescharacteristic
544C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
545C22×C3#57511yescharacteristic
546C22×C3#57711yescharacteristic
547C22×C3#57711yescharacteristic
548C3×C4—22no—
549C3×C4—22no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—81no—
553C3×C4—81no—
554C3×C4—81no—
555C3×C4#52911yescharacteristic
556C3×C4#54011yescharacteristic
557C3×C4#55311yescharacteristic
558C3×C4#55511yescharacteristic
559Dic3—42no—
560Dic3—81no—
561Dic3—81no—
562Dic3—81no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—22no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23#93212yes—
584C23C22×#6711yescharacteristic
585D4—22no—
586D4—22no—
587D4—44no—
588D4—41no—
589D4—41no—
590D4—84no—
591D4—84no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
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—62no—
620C2×C4—84no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—244no—
633C2×C4#93512yes—
634C2×C4#93512yes—
635C2×C4C2×#23711yescharacteristic
636C2×C4C2×#23711yescharacteristic
637C2×C4C2×#23711yescharacteristic
638C2×C4#100411yescharacteristic
639C2×C4D3×#4111yescharacteristic
640C2×C4#100911yescharacteristic
641C8—121no—
642C8—121no—
643C8—121no—
644C8—121no—
645C2×C3—42no—
646C2×C3—81no—
647C2×C3—82no—
648C2×C3#242011yescharacteristic
649C2×C3#309411yescharacteristic
650C2×C3#309911yescharacteristic
651C2×C3#330611yescharacteristic
652C2×C3#330711yescharacteristic
653C2×C3#331011yescharacteristic
654C2×C3#332111yescharacteristic
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—42no—
664C22—41no—
665C22—42no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—42no—
671C22—41no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22C2×#93211yescharacteristic
677C22C2×#93511yescharacteristic
678C22#580611yescharacteristic
679C22#582211yescharacteristic
680C22#582911yescharacteristic
681C22#583611yescharacteristic
682C22#583711yescharacteristic
683C4—22no—
684C4—22no—
685C4—41no—
686C4—41no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—122no—
691C4—241no—
692C4—241no—
693C4—241no—
694C4#560011yescharacteristic
695C4#561611yescharacteristic
696C4#561811yescharacteristic
697C4#561911yescharacteristic
698C3#2498211yescharacteristic, complemented, Sylow
699C2—42no—
700C2—81no—
701C2—82no—
702C2#8261611yescharacteristic
703C2#8264311yescharacteristic
704C2#8341911yescharacteristic
705C2#8348411yescharacteristic
706C2#8351211yescharacteristic
707C2#8352611yescharacteristic
708C2#8365611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3C22×D4
hypercenter#2537D3
hypocenterC3#24982
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. #82335C2semidirect
3. #71680C2semidirect
4. #71680C2semidirect
5. #71678C2semidirect
6. #71677C2semidirect
7. #71615C2semidirect
8. #71609C2semidirect
9. #71605C2semidirect
10. #69575C2semidirect
11. #69568C2semidirect
12. #69520C2semidirect
13. #69507C2semidirect
14. #69503C2semidirect
15. #69501C2semidirect
698. C3#24982semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
#823352
#716802
#716782
#716772
#716152
#716092
#716052
#695752
#695682
#695202
#695072
#695032
#695012
#249823