Finite Group Databasebeta

← #10812918

Subgroups — #10812918

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4861C211yesmaximal, characteristic
2#76201C211yesmaximal, characteristic
3#73633C211yesmaximal, characteristic
4C3×#1919C211yesmaximal, characteristic, complemented
5#70558C211yesmaximal, characteristic, complemented
6#69620C211yesmaximal, characteristic, complemented
7#69481C211yesmaximal, characteristic, complemented
8#68871C211yesmaximal, characteristic
9#68870C211yesmaximal, characteristic
10#68859C211yesmaximal, characteristic, complemented
11#68820C211yesmaximal, characteristic, complemented
12#68813C211yesmaximal, characteristic
13#68784C211yesmaximal, characteristic, complemented
14#68650C211yesmaximal, characteristic
15#68621C211yesmaximal, characteristic, complemented
16#22621—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C8C2211yescharacteristic
24C2×#818C2211yescharacteristic
25C2×C8×Dic3C2211yescharacteristic
26#4987—21no—
27#4987C2211yescharacteristic
28#4987C2211yescharacteristic
29#4984C2211yescharacteristic
30#4978—21no—
31Dic3×C4⋊C4C2211yescharacteristic
32Dic3×C4⋊C4C2211yescharacteristic
33#4972C2211yescharacteristic
34C2×#808C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#807C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4940—21no—
39C4×(C2×D3)⋊C4C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4932—21no—
42#4928C2211yescharacteristic
43C2×#803C2211yescharacteristic
44#4861C418yescomplemented
45#4669—21no—
46#4666—21no—
47#4655—21no—
48#4650—21no—
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#428—31no—
62#2346—31no—
63#2345—31no—
64#1943—31no—
65#1942—31no—
66#1921—31no—
67#1920—31no—
68#1919D311yescharacteristic, complemented
69#1918—31no—
70#1825—31no—
71#1823—31no—
72#1809—31no—
73#1776—31no—
74C4×#106—31no—
75#1753—31no—
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C2×ΓL(1,𝔽25)—22no—
108C2×ΓL(1,𝔽25)—22no—
109C2×ΓL(1,𝔽25)C2×C414yes—
110#892—22no—
111#892C2×C414yes—
112D3×C4⋊C4—22no—
113D3×C4⋊C4C2×C414yes—
114C3×#106C2×C414yes—
115C3×#95—21no—
116C3×#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127(C2×D3)⋊C8—22no—
128#818C2×C414yes—
129C8×Dic3—22no—
130#808—22no—
131#808C2×C414yes—
132#807—22no—
133#807C2×C414yes—
134#803—22no—
135C22×M16—31no—
136C2×#117—31no—
137C22×C4⋊C4—31no—
138#428—38no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×#106—31no—
144C2×#106C2×D311yescharacteristic
145C2×C22⋊C8—31no—
146C2×#97—31no—
147C2×C4×C8—31no—
148#389—31no—
149#389—31no—
150#389—31no—
151#389C2×D311yescharacteristic
152#388—61no—
153#388—61no—
154#387—61no—
155#387—61no—
156#382—61no—
157#380—31no—
158#380—31no—
159#380—61no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4—31no—
162C4×C4⋊C4—31no—
163C4×C4⋊C4C2×D311yescharacteristic
164C4×C22⋊C4—31no—
165C4×C22⋊C4—61no—
166#376—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—24no—
199C2×C4×D3—81no—
200C2×C4×D3—81no—
201C2×C4×D3C22×C412yes—
202C2×C4×D3#4112yes—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8C22×C412yes—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—81no—
216C3×C4⋊C4C22×C412yes—
217C3×C4⋊C4C22×C412yes—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—42no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—22no—
223(C2×D3)⋊C4—41no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—81no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—81no—
233C4⋊Dic3C22×C412yes—
234C4⋊Dic3C22×C412yes—
235Dic3⋊C4—24no—
236Dic3⋊C4—81no—
237Dic3⋊C4—81no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3—24no—
241C4×Dic3—42no—
242C4×Dic3—81no—
243C4×Dic3—81no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8C22×C412yes—
248C2×C3⋊C8#4112yes—
249C2×C3⋊C8#4112yes—
250ΓL(1,𝔽25)—28no—
251C23×C4—31no—
252C23×C4—61no—
253C2×M16—34no—
254C2×M16—62no—
255C2×M16—62no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#117—34no—
261#117—62no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—34no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C22⋊C4—31no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C42—21no—
283C2×C42—31no—
284C2×C42—61no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#107—34no—
290#107—34no—
291#107—62no—
292#106—34no—
293#106—62no—
294#106C4×D314yes—
295C22⋊C8—62no—
296#97—62no—
297C4×C8—62no—
298#95—21no—
299#95—21no—
300#95—31no—
301#95—31no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310C23×C3C22×D411yescharacteristic
311C22×D3—21no—
312C22×D3—21no—
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3C4×D412yes—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4C23×C411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C42−1+411yescharacteristic
341C2×C3×C42−1+411yescharacteristic
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3C4×D412yes—
362C2×Dic3#13712yes—
363C4×D3—22no—
364C4×D3—22no—
365C4×D3—22no—
366C4×D3—22no—
367C4×D3—84no—
368C3×C8—22no—
369C3×C8—41no—
370C3⋊C8—41no—
371C3⋊C8#11714yes—
372C24—31no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—31no—
382C22×C4—64no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—241no—
392C22×C4—241no—
393C22×C4D3×D411yescharacteristic
394C22×C4D3×D411yescharacteristic
395C22×C4C23×D311yescharacteristic
396M16—68no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—32no—
403C2×C8—32no—
404C2×C8—32no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8C2×C4×D312yes—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—32no—
416C4⋊C4—32no—
417C4⋊C4—64no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—81no—
425C4⋊C4—241no—
426C4⋊C4—241no—
427C4⋊C4—241no—
428C4⋊C4C2×C4×D312yes—
429C4⋊C4C2×C4×D312yes—
430C22⋊C4—62no—
431C22⋊C4—64no—
432C22⋊C4—121no—
433C22⋊C4—122no—
434C22⋊C4—241no—
435C42—21no—
436C42—21no—
437C42—42no—
438C42—64no—
439C42—61no—
440C42—61no—
441C42—122no—
442C42—241no—
443C42—241no—
444C22×C3C2×C4×D411yescharacteristic
445C22×C3#54711yescharacteristic
446C22×C3#54711yescharacteristic
447C22×C3C2×#13711yescharacteristic
448C22×C3#57511yescharacteristic
449C22×C3#57611yescharacteristic
450C22×C3#57711yescharacteristic
451C2×D3—22no—
452C2×D3—22no—
453C2×D3—22no—
454C2×D3—41no—
455C2×D3—41no—
456C2×D3—41no—
457C2×D3—41no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—81no—
463C3×C4—81no—
464C3×C4C2×#11711yescharacteristic
465C3×C4#51911yescharacteristic
466C3×C4#55211yescharacteristic
467C3×C4#55211yescharacteristic
468Dic3—22no—
469Dic3—22no—
470Dic3—41no—
471Dic3—42no—
472Dic3—42no—
473Dic3—81no—
474Dic3—81no—
475C23—32no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23C2×D3×D411yescharacteristic
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—32no—
492C2×C4—32no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—244no—
527C2×C4C22×C4×D311yescharacteristic
528C2×C4#100711yescharacteristic
529C2×C4D3×#4111yescharacteristic
530C2×C4D3×#4111yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C2×C4#101011yescharacteristic
533C8—22no—
534C8—34no—
535C8—41no—
536C8—121no—
537C2×C3#234411yescharacteristic
538C2×C3C4×#13711yescharacteristic
539C2×C3#294611yescharacteristic
540C2×C3#334411yescharacteristic
541C2×C3#334711yescharacteristic
542C2×C3#335111yescharacteristic
543C2×C3#335711yescharacteristic
544D3—42no—
545C22—62no—
546C22—62no—
547C22—62no—
548C22—121no—
549C22—121no—
550C22—121no—
551C22—121no—
552C22C4×D3×D411yescharacteristic
553C22#563111yescharacteristic
554C22#565511yescharacteristic
555C22#576211yescharacteristic
556C22#576911yescharacteristic
557C22#577311yescharacteristic
558C22D3×#13711yescharacteristic
559C4—42no—
560C4—41no—
561C4—42no—
562C4—41no—
563C4—62no—
564C4—62no—
565C4—81no—
566C4—81no—
567C4—122no—
568C4—122no—
569C4—121no—
570C4—241no—
571C4—241no—
572C4D3×#11711yescharacteristic
573C4#557211yescharacteristic
574C4#569311yescharacteristic
575C4#570811yescharacteristic
576C3#2262111yescharacteristic, complemented, Sylow
577C2—122no—
578C2#7496111yescharacteristic
579C2#7964711yescharacteristic
580C2#7977411yescharacteristic
581C2#8103011yescharacteristic
582C2#8119811yescharacteristic
583C2#8133711yescharacteristic
584C2#8151311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1919C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1919D3
hypocenterC3#22621
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. C3×#1919C2semidirect
5. #70558C2semidirect
6. #69620C2semidirect
7. #69481C2semidirect
10. #68859C2semidirect
11. #68820C2semidirect
13. #68784C2semidirect
15. #68621C2semidirect
44. #4861C4semidirect
68. #1919D3semidirect
576. C3#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#48612
#762012
#736332
C3×#19192
#705582
#696202
#694812
#688712
#688702
#688592
#688202
#688132
#687842
#686502
#686212
#226213