Finite Group Databasebeta

← #10812949

Subgroups — #10812949

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×C3×#428C211yesmaximal, characteristic
2#82240C211yesmaximal, characteristic
3#82160C211yesmaximal, characteristic
4#71988C211yesmaximal, characteristic
5#71983C211yesmaximal, characteristic
6#71575C211yesmaximal, characteristic
7#71553C211yesmaximal, characteristic
8#70610C211yesmaximal, characteristic, complemented
9#70609C211yesmaximal, characteristic, complemented
10#70605C211yesmaximal, characteristic, complemented
11Dic3×#107C211yesmaximal, characteristic, complemented
12#70562C211yesmaximal, characteristic, complemented
13#70560C211yesmaximal, characteristic, complemented
14#70558C211yesmaximal, characteristic, complemented
15#70557C211yesmaximal, characteristic, complemented
16#22623—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C3×#428C418yescomplemented
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25#5212—21no—
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28C4×#217—21no—
29C4×#217C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4984C2211yescharacteristic
36#4984C2211yescharacteristic
37#4984C2211yescharacteristic
38#4984C2211yescharacteristic
39#4980—21no—
40#4980—21no—
41Dic3×C4⋊C4C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45#4951—21no—
46#4951—21no—
47#4937—21no—
48C4×C4⋊Dic3—21no—
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4550C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#428D311yescharacteristic
62#2347—31no—
63#2345—31no—
64#1944—31no—
65#1941—31no—
66#1921—31no—
67#1920—31no—
68#1919—31no—
69#1918—31no—
70#1825—31no—
71#1823—31no—
72#1809—31no—
73#1776—31no—
74C4×#107—31no—
75#1753—31no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C2×C3×M16—22no—
79C2×C3×M16—22no—
80C2×C3×M16C2×C414yes—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#117—22no—
84C3×#117C2×C414yes—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4C2×C414yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217D411yescharacteristic
97C2×#217D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C3×#107C2×C414yes—
115C3×#107C2×C414yes—
116C3×#106C2×C414yes—
117C3×#106C2×C414yes—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130#830—22no—
131#818—22no—
132#817—22no—
133#815—22no—
134C8×Dic3—22no—
135C22×M16C2×D311yescharacteristic
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4C2×D311yescharacteristic
138#428Dic318yescomplemented
139C2×#107—31no—
140C2×#107C2×D311yescharacteristic
141C2×#107C2×D311yescharacteristic
142C2×#106—31no—
143C2×#106C2×D311yescharacteristic
144C2×#106C2×D311yescharacteristic
145C2×C22⋊C8—31no—
146C2×#97—31no—
147C2×C4×C8—31no—
148#389—31no—
149#389—31no—
150#389—31no—
151#389—31no—
152#388—61no—
153#388—61no—
154#387—61no—
155#387—61no—
156#383—61no—
157#380—31no—
158#380—31no—
159#380—61no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4—31no—
162C4×C4⋊C4—31no—
163C4×C4⋊C4—31no—
164C4×C4⋊C4—61no—
165C4×C22⋊C4—31no—
166C4×C22⋊C4—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
179C24×C3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—24no—
189C22×C3×C4—81no—
190C22×C3×C4—81no—
191C22×C3×C4C22×C412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4#4112yes—
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C3×M16—28no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8C22×C412yes—
207C2×C3×C8C22×C412yes—
208C2×C3×C8#4112yes—
209C2×C3×C8#4112yes—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4C22×C412yes—
216C3×C4⋊C4C22×C412yes—
217C3×C4⋊C4C22×C412yes—
218C3×C4⋊C4C22×C412yes—
219C3×C22⋊C4—24no—
220C3×C42—24no—
221#217—22no—
222#217—41no—
223#217—42no—
224#217—81no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—42no—
234C4⋊Dic3—42no—
235C4⋊Dic3—81no—
236C4⋊Dic3—81no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—81no—
242C4×Dic3—81no—
243C4×Dic3—81no—
244C4×Dic3—81no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C23×C4—21no—
252C23×C4C22×D311yescharacteristic
253C2×M16—22no—
254C2×M16—22no—
255C2×M16C2×Dic314yes—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#117—22no—
261#117C2×Dic314yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4C2×Dic314yes—
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C42—21no—
283C2×C42—31no—
284C2×C42—31no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42—61no—
288C2×C42—61no—
289#107—62no—
290#107C2×Dic314yes—
291#107C2×Dic314yes—
292#106—62no—
293#106C2×Dic314yes—
294#106C2×Dic314yes—
295C22⋊C8—62no—
296#97—62no—
297C4×C8—62no—
298#95—31no—
299#95—31no—
300#95—61no—
301#95—61no—
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×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3C4×D412yes—
317C23×C3C22×D411yescharacteristic
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
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×C4—84no—
337C2×C3×C4C4×D412yes—
338C2×C3×C4C23×C411yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C42−1+411yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3#13712yes—
368C3×C8—22no—
369C3×C8#11714yes—
370C3⋊C8—41no—
371C3⋊C8—41no—
372C24D3×D411yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4D3×D411yescharacteristic
393C22×C4#23712yes—
394C22×C4C22×Dic312yes—
395C22×C4C23×D311yescharacteristic
396M16—28no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8#23512yes—
406C2×C8#23512yes—
407C2×C8C22×Dic312yes—
408C2×C8C22×Dic312yes—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—122no—
423C4⋊C4—122no—
424C4⋊C4—241no—
425C4⋊C4—241no—
426C4⋊C4C22×Dic312yes—
427C4⋊C4C22×Dic312yes—
428C4⋊C4C22×Dic312yes—
429C4⋊C4C22×Dic312yes—
430C22⋊C4—24no—
431C22⋊C4—62no—
432C22⋊C4—121no—
433C22⋊C4—122no—
434C22⋊C4—241no—
435C42—24no—
436C42—61no—
437C42—61no—
438C42—61no—
439C42—61no—
440C42—241no—
441C42—241no—
442C42—241no—
443C42—241no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3C2×C4×D411yescharacteristic
452C22×C3#54711yescharacteristic
453C22×C3#54711yescharacteristic
454C22×C3C2×#13711yescharacteristic
455C22×C3#57511yescharacteristic
456C22×C3#57611yescharacteristic
457C22×C3#57711yescharacteristic
458C3×C4—22no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4C2×#11711yescharacteristic
464C3×C4#51911yescharacteristic
465C3×C4#55211yescharacteristic
466C3×C4#55211yescharacteristic
467Dic3—22no—
468Dic3—41no—
469Dic3—41no—
470Dic3—41no—
471Dic3—81no—
472Dic3—81no—
473Dic3—81no—
474Dic3—81no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23D4×Dic312yes—
482C23C2×D3×D411yescharacteristic
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—32no—
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—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—84no—
514C2×C4—121no—
515C2×C4—121no—
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×C4D4×Dic312yes—
527C2×C4C2×#23511yescharacteristic
528C2×C4C2×#23511yescharacteristic
529C2×C4C2×#23711yescharacteristic
530C2×C4#100711yescharacteristic
531C2×C4#100711yescharacteristic
532C2×C4C23×Dic311yescharacteristic
533C8—22no—
534C8—121no—
535C8—121no—
536C8#92614yes—
537C2×C3—42no—
538C2×C3#234411yescharacteristic
539C2×C3C4×#13711yescharacteristic
540C2×C3#294611yescharacteristic
541C2×C3#334611yescharacteristic
542C2×C3#335011yescharacteristic
543C2×C3#335511yescharacteristic
544C2×C3#335611yescharacteristic
545C22—22no—
546C22—22no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22#550711yescharacteristic
553C22#550711yescharacteristic
554C22#577411yescharacteristic
555C22#577511yescharacteristic
556C22D3×#13711yescharacteristic
557C22#577711yescharacteristic
558C22C2×D4×Dic311yescharacteristic
559C4—22no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—62no—
565C4—121no—
566C4—121no—
567C4—121no—
568C4—241no—
569C4—241no—
570C4—241no—
571C4—241no—
572C4#551311yescharacteristic
573C4#551311yescharacteristic
574C4C2×#92611yescharacteristic
575C4#581211yescharacteristic
576C3#2262311yescharacteristic, complemented, Sylow
577C2—42no—
578C2#7922611yescharacteristic
579C2#7923011yescharacteristic
580C2#7923611yescharacteristic
581C2#7923711yescharacteristic
582C2#8215911yescharacteristic
583C2#8393011yescharacteristic
584C2Dic3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenterC2×#428D3
hypocenterC3#22623
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
8. #70610C2semidirect
9. #70609C2semidirect
10. #70605C2semidirect
11. Dic3×#107C2semidirect
12. #70562C2semidirect
13. #70560C2semidirect
14. #70558C2semidirect
15. #70557C2semidirect
20. C3×#428C4semidirect
138. #428Dic3semidirect
576. C3#22623semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4282
#822402
#821602
#719882
#719832
#715752
#715532
#706102
#706092
#706052
Dic3×#1072
#705622
#705602
#705582
#705572
#226233