Finite Group Databasebeta

← #10780607

Subgroups — #10780607

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81802C211yesmaximal, characteristic, complemented
2#80966C211yesmaximal, characteristic
3#80964C211yesmaximal, characteristic
4#79405C211yesmaximal, characteristic
5C3×#1669C211yesmaximal, characteristic, complemented
6#71308C211yesmaximal, characteristic, complemented
7#71103C211yesmaximal, characteristic, complemented
8#71101C211yesmaximal, characteristic, complemented
9#70855C211yesmaximal, characteristic, complemented
10#68591C211yesmaximal, characteristic, complemented
11#68572C211yesmaximal, characteristic, complemented
12#68471C211yesmaximal, characteristic, complemented
13#68465C211yesmaximal, characteristic, complemented
14#68423C211yesmaximal, characteristic, complemented
15D3×#332C211yesmaximal, characteristic, complemented
16#19411—31nomaximal, Sylow
17#5728C2211yescharacteristic
18D3×C4⋊Q8C2211yescharacteristic
19C4×#237C2211yescharacteristic
20C3×#494C2211yescharacteristic
21C3×#474C2211yescharacteristic
22C3×#474C2211yescharacteristic
23C3×C8×D4C2211yescharacteristic
24#5186—21no—
25#5186—21no—
26#5182—21no—
27#5182—21no—
28C8×#67—21no—
29C2×Dic3⋊C8—21no—
30#5094C2211yescharacteristic
31#5094C2211yescharacteristic
32#5092C2211yescharacteristic
33#5079C2211yescharacteristic
34D4×C3⋊C8C2211yescharacteristic
35#5008C2211yescharacteristic
36#5007C2211yescharacteristic
37#4881—21no—
38#4880—21no—
39#4877—21no—
40#4875—21no—
41#4851C2211yescharacteristic
42#4850C2211yescharacteristic
43#4837C2211yescharacteristic
44D3×C4⋊C8C2211yescharacteristic
45D3×C4⋊C8C2211yescharacteristic
46#4830C2211yescharacteristic
47#4772—21no—
48#4771—21no—
49#4766—21no—
50#4761—21no—
51#67⋊C8—21no—
52#4718—21no—
53#4685C2211yescharacteristic
54#4681C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4479C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3209—31no—
69#3079—31no—
70#3076—31no—
71#2988—31no—
72#1684—31no—
73#1684—31no—
74#1683—31no—
75#1683—31no—
76#1670—31no—
77#1670—31no—
78#1669—31no—
79#1669D311yescharacteristic, complemented
80#1571—31no—
81#1550—31no—
82C2×#332—31no—
83C2×D3×Q8—21no—
84C2×#237D411yescharacteristic
85C2×C3×Q16—21no—
86C2×C3×Q16—21no—
87C22×C3×C8—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4×D4C2311yescharacteristic
91#946—21no—
92Q8×Dic3—21no—
93#944—21no—
94C2×#216—21no—
95C2×#216—21no—
96D4×Dic3D411yescharacteristic
97C4×#67—21no—
98C2×C4×Dic3—21no—
99C22×C3⋊C8—21no—
100C2×ΓL(1,𝔽25)—21no—
101C2×D3×C8—21no—
102C2×D3×C8—21no—
103#897—21no—
104#892—21no—
105#892D411yescharacteristic
106D3×C4⋊C4—21no—
107D3×C4⋊C4—21no—
108#890—21no—
109#888—21no—
110#888—21no—
111#887—21no—
112#887D411yescharacteristic
113#67⋊C4—21no—
114#877—21no—
115C42×D3C2311yescharacteristic
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×#107—21no—
120C3×#107—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#102—21no—
125C3×#102—21no—
126C3×C22⋊C8—21no—
127C3×C4×C8C2311yescharacteristic
128#835—21no—
129#835—21no—
130#832—21no—
131#832—21no—
132#830—21no—
133(C2×D3)⋊C8D411yescharacteristic
134(C2×D3)⋊C8D411yescharacteristic
135#818—21no—
136#817—21no—
137#816—21no—
138#816—21no—
139#816—21no—
140#816—21no—
141Dic3⋊C8—21no—
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8D411yescharacteristic
147Dic3⋊C8D411yescharacteristic
148C8×Dic3—21no—
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#808—21no—
153#808—21no—
154#808—21no—
155#807—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161C2×C4⋊Q8—31no—
162C4×#41—31no—
163#494—31no—
164#494C2×D311yescharacteristic
165#487—31no—
166#487—31no—
167#479—61no—
168#478—61no—
169#477—31no—
170#477—61no—
171#476—31no—
172#476—61no—
173#475—31no—
174#475—31no—
175#475—31no—
176#474—31no—
177#474C2×D311yescharacteristic
178#474C2×D311yescharacteristic
179#462—61no—
180#462—61no—
181#451—61no—
182#451—61no—
183C8×Q8—31no—
184C8×Q8—31no—
185C8×D4—31no—
186C8×D4—61no—
187C8×D4—61no—
188C8×D4C2×D311yescharacteristic
189#431—31no—
190#431—61no—
191#428—61no—
192#427—61no—
193#423—31no—
194C2×C4⋊C8—31no—
195C2×C4⋊C8—31no—
196C2×C4⋊C8—61no—
197#417—61no—
198#416—61no—
199#335—31no—
200#335—31no—
201#335—31no—
202#335C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332C2×D311yescharacteristic
210#332C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218D3×Q8—81no—
219D3×Q8—81no—
220#237—22no—
221#237Q1614yescomplemented
222C2×C4×D3—41no—
223C2×C4×D3C2×D411yescharacteristic
224C2×C4×D3C2×D411yescharacteristic
225C2×C4×D3#4111yescharacteristic
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—41no—
229C2×Dic6C2×D411yescharacteristic
230C3×Q16—81no—
231C3×Q16—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—81no—
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C22⋊C4—21no—
244C3×C42C2411yescharacteristic
245#217—21no—
246#216—81no—
247#216—81no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—41no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—41no—
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—41no—
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3#4111yescharacteristic
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—81no—
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274ΓL(1,𝔽25)—42no—
275D3×C8—42no—
276D3×C8—81no—
277C2×#41—31no—
278C22×Q8—61no—
279C2×Q16—21no—
280C2×Q16—21no—
281C2×Q16—61no—
282C2×Q16—61no—
283C2×M16—61no—
284C22×C8—21no—
285C22×C8—61no—
286C22×C8—61no—
287C22×C8—61no—
288C4⋊Q8—31no—
289C4⋊Q8—31no—
290C4⋊Q8—61no—
291C4⋊Q8—61no—
292C4⋊Q8—61no—
293C4⋊Q8C22×D311yescharacteristic
294C4⋊Q8C22×D311yescharacteristic
295#125—61no—
296C22⋊Q8—61no—
297C22⋊Q8—61no—
298C4×Q8—31no—
299C4×Q8—31no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×D4—31no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4C22×D311yescharacteristic
306#117—31no—
307#117—61no—
308#117—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C42—31no—
312C2×C42—61no—
313#107—21no—
314#107—21no—
315#107—61no—
316#107—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320#106—61no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—61no—
327C4⋊C8—61no—
328C4⋊C8—61no—
329C4⋊C8—61no—
330C4⋊C8—61no—
331C4⋊C8C22×D311yescharacteristic
332C4⋊C8C22×D311yescharacteristic
333C4⋊C8C22×D311yescharacteristic
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#102—21no—
343#102—21no—
344#102—61no—
345#102—61no—
346C22⋊C8—21no—
347C22⋊C8—31no—
348C22⋊C8—31no—
349C22⋊C8—61no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8—61no—
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C22×D3#12111yescharacteristic
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×D4—21no—
361C3×D4C2×Q1612yes—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370#67—24no—
371C2×Dic3—21no—
372C2×Dic3—24no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—81no—
377C2×Dic3#12111yescharacteristic
378C2×Dic3#12111yescharacteristic
379C2×Dic3#12111yescharacteristic
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—41no—
383C4×D3—82no—
384C4×D3C2×Q1612yes—
385C4×D3#13512yes—
386Dic6—21no—
387Dic6—41no—
388Dic6—41no—
389Dic6—82no—
390Dic6—81no—
391Dic6C2×Q1612yes—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402#41—34no—
403#41—62no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—31no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—121no—
410C2×Q8—241no—
411C2×Q8—241no—
412C2×D4—61no—
413C2×D4D3×D411yescharacteristic
414C22×C4—21no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—121no—
421Q16—81no—
422Q16—81no—
423Q16—241no—
424Q16—241no—
425M16—122no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—31no—
430C2×C8—31no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—81no—
435C2×C8—122no—
436C2×C8—241no—
437C2×C8—241no—
438C2×C8D3×D411yescharacteristic
439C2×C8D3×D411yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—31no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—121no—
452C4⋊C4—121no—
453C4⋊C4—121no—
454C4⋊C4—121no—
455C4⋊C4D3×D411yescharacteristic
456C22⋊C4—21no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—121no—
460C42—31no—
461C42—31no—
462C42—31no—
463C42—61no—
464C42—61no—
465C42—61no—
466C42—121no—
467C42C23×D311yescharacteristic
468C22×C3—22no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3C2×#12111yescharacteristic
472C2×D3—21no—
473C2×D3—21no—
474C2×D3#46712yes—
475C3×C4—21no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4C22×Q1611yescharacteristic
480C3×C4C2×#13511yescharacteristic
481C3×C4#57611yescharacteristic
482C3×C4#57711yescharacteristic
483Dic3—21no—
484Dic3—22no—
485Dic3—21no—
486Dic3—41no—
487Dic3—81no—
488Dic3—81no—
489Dic3#46712yes—
490C23—21no—
491C23—31no—
492Q8—32no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—61no—
498Q8—121no—
499Q8—121no—
500Q8—241no—
501Q8—242no—
502D4—21no—
503D4—64no—
504D4D3×Q1612yes—
505C2×C4—21no—
506C2×C4—31no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—64no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—241no—
524C2×C4—242no—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4D3×#4111yescharacteristic
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—61no—
534C8—61no—
535C8—121no—
536C8—121no—
537C8—121no—
538C2×C3—42no—
539C2×C3#304511yescharacteristic
540C2×C3C2×#46711yescharacteristic
541C2×C3#306911yescharacteristic
542D3—22no—
543C22—22no—
544C22—32no—
545C22—41no—
546C22—41no—
547C22—61no—
548C22—61no—
549C22D3×#12111yescharacteristic
550C4—21no—
551C4—32no—
552C4—41no—
553C4—61no—
554C4—62no—
555C4—61no—
556C4—81no—
557C4—81no—
558C4—121no—
559C4—241no—
560C4—241no—
561C4#575711yescharacteristic
562C4C2×D3×Q1611yescharacteristic
563C4D3×#13511yescharacteristic
564C4#577511yescharacteristic
565C3#1941111yescharacteristic, complemented, Sylow
566C2—42no—
567C2—62no—
568C2#8033511yescharacteristic
569C2D3×#46711yescharacteristic
570C2#8094411yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1669C2
socleC22×C3C2×#121
prime radicalC2×#237D4
hypercenter#1669D3
hypocenterC3#19411
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1669

Complements

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

NH ≅ G/Ntype
1. #81802C2semidirect
5. C3×#1669C2semidirect
6. #71308C2semidirect
7. #71103C2semidirect
8. #71101C2semidirect
9. #70855C2semidirect
10. #68591C2semidirect
11. #68572C2semidirect
12. #68471C2semidirect
13. #68465C2semidirect
14. #68423C2semidirect
15. D3×#332C2semidirect
79. #1669D3semidirect
221. #237Q16semidirect
565. C3#19411semidirect

Maximal subgroups

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

subgroupindex
#818022
#809662
#809642
#794052
C3×#16692
#713082
#711032
#711012
#708552
#685912
#685722
#684712
#684652
#684232
D3×#3322
#194113