Finite Group Databasebeta

← #10786154

Subgroups — #10786154

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3340C211yesmaximal, characteristic, complemented
2#83670C211yesmaximal, characteristic, complemented
3#83652C211yesmaximal, characteristic
4#79692C211yesmaximal, characteristic, complemented
5#71933C211yesmaximal, characteristic, complemented
6#71929C211yesmaximal, characteristic, complemented
7#71345C211yesmaximal, characteristic, complemented
8#71341C211yesmaximal, characteristic, complemented
9#71157C211yesmaximal, characteristic, complemented
10#71151C211yesmaximal, characteristic, complemented
11#71110C211yesmaximal, characteristic, complemented
12#71101C211yesmaximal, characteristic, complemented
13SD16⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
14#70852C211yesmaximal, characteristic, complemented
15#70841C211yesmaximal, characteristic, complemented
16#19863—31nomaximal, Sylow
17C3×#556C2211yescharacteristic
18C3×#555C2211yescharacteristic
19#5584C2211yescharacteristic
20#5551C2211yescharacteristic
21C3×#498C2211yescharacteristic
22C3×#485—21no—
23C3×#482—21no—
24C3×#477C2211yescharacteristic
25C3×#474C2211yescharacteristic
26C3×#465—21no—
27C3×C4×SD16C2211yescharacteristic
28C3×C8×D4C2211yescharacteristic
29C3×#427—21no—
30#5174—21no—
31#5093C2211yescharacteristic
32#5092C2211yescharacteristic
33#5084C2211yescharacteristic
34#5083C2211yescharacteristic
35#5079C2211yescharacteristic
36#5078C2211yescharacteristic
37#5059—21no—
38#5054—21no—
39#5041—21no—
40#5037—21no—
41C4×#216—21no—
42#5028—21no—
43Q8×C3⋊C8C2211yescharacteristic
44#5020C2211yescharacteristic
45#5019C2211yescharacteristic
46#5019C2211yescharacteristic
47C4×#213C2211yescharacteristic
48D4×C3⋊C8C2211yescharacteristic
49#5007C2211yescharacteristic
50#5002—21no—
51#4996—21no—
52#4861—21no—
53#4811—21no—
54#4784—21no—
55#4688C2211yescharacteristic
56#4682C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4535C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61D4⋊(C3⋊C8)C2211yescharacteristic
62#4486C2211yescharacteristic
63#4486C2211yescharacteristic
64#4483C2211yescharacteristic
65#4479C2211yescharacteristic
66#4458C2211yescharacteristic
67#4455C2211yescharacteristic
68#3393—31no—
69#3353—31no—
70#3340D311yescharacteristic, complemented
71#3303—31no—
72#1723—31no—
73#1691—31no—
74#1684—31no—
75#1673—31no—
76#1669—31no—
77#1650—31no—
78#1647—31no—
79#1646—31no—
80#1638—31no—
81#1581—31no—
82SD16⋊C8—31no—
83C2×#235—21no—
84C2×C3×SD16D411yescharacteristic
85C22×C3×C8—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×#126—41no—
89C3×#124—21no—
90C3×#124—41no—
91C3×C22⋊Q8—21no—
92C3×#121—21no—
93C3×#121—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8C2311yescharacteristic
96C3×C4×D4C2311yescharacteristic
97C3×C4×D4C2311yescharacteristic
98C3×#117—21no—
99C3×#117—21no—
100C3×#117—21no—
101#946—21no—
102C2×#216—21no—
103C2×#216—21no—
104C2×#215—21no—
105#938—21no—
106C2×#214—21no—
107C2×#214—21no—
108C2×#213D411yescharacteristic
109#924—21no—
110C22×C3⋊C8—21no—
111C2×ΓL(1,𝔽25)—21no—
112#898—41no—
113#892—21no—
114D3×C4⋊C4—21no—
115#883—41no—
116#875—21no—
117C4×D12C2311yescharacteristic
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C4×Dic6—21no—
121C3×#107—21no—
122C3×#107—21no—
123C3×#106—21no—
124C3×#106—21no—
125C3×#106D411yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×#103—21no—
128C3×#103D411yescharacteristic
129C3×#102—21no—
130C3×#102D411yescharacteristic
131C3×C22⋊C8—21no—
132C3×C4×C8C2311yescharacteristic
133#835—21no—
134#835—21no—
135#835—41no—
136#832—21no—
137#832D411yescharacteristic
138#830—21no—
139(C2×D3)⋊C8—21no—
140#818—21no—
141#818—21no—
142#810—21no—
143#810—21no—
144#810—21no—
145#810—21no—
146#809—21no—
147#809D411yescharacteristic
148#808—21no—
149#808—21no—
150#807—21no—
151#807—41no—
152#807D411yescharacteristic
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556—31no—
161#556C2×D311yescharacteristic
162#555C2×D311yescharacteristic
163#547—31no—
164#500—31no—
165#500—31no—
166#498C2×D311yescharacteristic
167#495—61no—
168#494—31no—
169#487—31no—
170#487—31no—
171#485—21no—
172#485—61no—
173#484—61no—
174#482—21no—
175#482—61no—
176#477C2×D311yescharacteristic
177#476—31no—
178#475—31no—
179#474—31no—
180#474C2×D311yescharacteristic
181#468—61no—
182#465—21no—
183#465—61no—
184#464—61no—
185#464—61no—
186#461—31no—
187#461—31no—
188#461—31no—
189#449—61no—
190C8×Q8—31no—
191C4×Q16—61no—
192C4×SD16C2×D311yescharacteristic
193C4×D8—31no—
194#436—31no—
195C8×D4—31no—
196C8×D4C2×D311yescharacteristic
197#428—61no—
198#427—21no—
199#416—61no—
200#335—31no—
201#335—31no—
202#333—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8—31no—
207D4⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4—41no—
215C2×C3×D4C2×D411yescharacteristic
216C2×C3×D4C2×D411yescharacteristic
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C2×#67—41no—
220#235—42no—
221#235—81no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—41no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C3×SD16—22no—
228C2×C3×C8—21no—
229C2×C3×C8—42no—
230C2×C3×C8C2×D411yescharacteristic
231C2×C3×C8C2×D411yescharacteristic
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4C2×D411yescharacteristic
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C4⋊C4C2×D411yescharacteristic
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—41no—
244C3×C22⋊C4—41no—
245C3×C22⋊C4—41no—
246C3×C22⋊C4—41no—
247C3×C42—21no—
248C3×C42—41no—
249C3×C42—41no—
250C3×C42C2411yescharacteristic
251#217—41no—
252#216—42no—
253#216—81no—
254#215—81no—
255#214—81no—
256#214—81no—
257#213—41no—
258(C2×D3)⋊C4—21no—
259(C2×D3)⋊C4—41no—
260(C2×D3)⋊C4—41no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3C2×D411yescharacteristic
266Dic3⋊C4—41no—
267Dic3⋊C4—41no—
268C4×Dic3—41no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—81no—
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276ΓL(1,𝔽25)—42no—
277C2×#41—61no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×SD16—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×SD16#6711yescharacteristic
284C2×D8—31no—
285C2×M16—61no—
286C22×C8—21no—
287C22×C8—61no—
288C4⋊Q8—31no—
289C4⋊Q8—31no—
290C4⋊Q8C22×D311yescharacteristic
291C4⋊Q8C22×D311yescharacteristic
292#126—41no—
293#126—121no—
294#124—21no—
295#124—41no—
296#124—61no—
297#123—121no—
298C22⋊Q8—21no—
299C22⋊Q8—61no—
300C22⋊Q8—61no—
301#121—21no—
302#121—21no—
303#121—61no—
304C4×Q8—21no—
305C4×Q8—61no—
306C4×Q8C22×D311yescharacteristic
307C4×D4—31no—
308C4×D4C22×D311yescharacteristic
309C4×D4C22×D311yescharacteristic
310#117—21no—
311#117—21no—
312#117—21no—
313#117—61no—
314C2×C4⋊C4—61no—
315#107—21no—
316#107—21no—
317#107—31no—
318#107—61no—
319#107—61no—
320#107—61no—
321#107—121no—
322#106—21no—
323#106—21no—
324#106—61no—
325#106—61no—
326#106#6711yescharacteristic
327C4⋊C8—31no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—61no—
331C4⋊C8C22×D311yescharacteristic
332#103—21no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—121no—
340#103#6711yescharacteristic
341#102—21no—
342#102—31no—
343#102—31no—
344#102—61no—
345#102—61no—
346#102#6711yescharacteristic
347C22⋊C8—21no—
348C22⋊C8—61no—
349C22⋊C8—61no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—61no—
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C23×C3—21no—
356C22×D3—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—81no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—21no—
366C3×D4—161no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380#67—82no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384D12—21no—
385D12—21no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—82no—
389Dic6—41no—
390Dic6—41no—
391Dic6—41no—
392Dic6—41no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8#13512yes—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402#41—122no—
403#41—241no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8C2×#6711yescharacteristic
409C2×D4—31no—
410C2×D4—41no—
411C2×D4—121no—
412C2×D4D3×D411yescharacteristic
413C2×D4C2×#6711yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—61no—
417C22×C4—121no—
418Q16—122no—
419Q16—241no—
420SD16—22no—
421SD16—241no—
422SD16—241no—
423SD16—241no—
424D8—121no—
425M16—122no—
426C2×C8—21no—
427C2×C8—31no—
428C2×C8—31no—
429C2×C8—42no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—241no—
435C2×C8C2×#6711yescharacteristic
436C2×C8C2×#6711yescharacteristic
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—31no—
443C4⋊C4—41no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—121no—
449C4⋊C4—121no—
450C4⋊C4D3×D411yescharacteristic
451C4⋊C4C2×#6711yescharacteristic
452C4⋊C4C2×#6711yescharacteristic
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—61no—
460C22⋊C4—121no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C42—21no—
464C42—41no—
465C42—41no—
466C42—121no—
467C42C23×D311yescharacteristic
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3D4211yescharacteristic
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C3×C4—21no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4C2×#13511yescharacteristic
486C3×C4#57511yescharacteristic
487C3×C4#57611yescharacteristic
488C3×C4#57811yescharacteristic
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492C23—21no—
493C23—21no—
494C23—61no—
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—81no—
500Q8—121no—
501Q8—121no—
502Q8—121no—
503Q8—121no—
504D4—21no—
505D4—21no—
506D4—21no—
507D4—21no—
508D4—61no—
509D4—61no—
510D4—161no—
511D4—242no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—61no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—242no—
528C2×C4C2×D3×D411yescharacteristic
529C2×C4#100411yescharacteristic
530C2×C4C22×#6711yescharacteristic
531C8—21no—
532C8—41no—
533C8—61no—
534C8—61no—
535C8—121no—
536C8—121no—
537C8—121no—
538C8—121no—
539C8#95012yes—
540C2×C3—81no—
541C2×C3—81no—
542C2×C3#326811yescharacteristic
543C2×C3#328311yescharacteristic
544C2×C3#328911yescharacteristic
545D3—81no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22D4×#6711yescharacteristic
556C4—21no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—121no—
564C4—241no—
565C4—241no—
566C4#576811yescharacteristic
567C4#576911yescharacteristic
568C4C2×#95011yescharacteristic
569C4#583811yescharacteristic
570C3#1986311yescharacteristic, complemented, Sylow
571C2—81no—
572C2—81no—
573C2—241no—
574C2#8352411yescharacteristic
575C2#8392211yescharacteristic
576C2#8401311yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3340C2
socleC22×C3D42
prime radical#83652C2
hypercenter#3340D3
hypocenterC3#19863
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #3340

Complements

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

NH ≅ G/Ntype
1. C3×#3340C2semidirect
2. #83670C2semidirect
4. #79692C2semidirect
5. #71933C2semidirect
6. #71929C2semidirect
7. #71345C2semidirect
8. #71341C2semidirect
9. #71157C2semidirect
10. #71151C2semidirect
11. #71110C2semidirect
12. #71101C2semidirect
13. SD16⋊(C3⋊C8)C2semidirect
14. #70852C2semidirect
15. #70841C2semidirect
70. #3340D3semidirect
570. C3#19863semidirect

Maximal subgroups

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

subgroupindex
C3×#33402
#836702
#836522
#796922
#719332
#719292
#713452
#713412
#711572
#711512
#711102
#711012
SD16⋊(C3⋊C8)2
#708522
#708412
#198633