Finite Group Databasebeta

← #10786152

Subgroups — #10786152

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
1#83520C211yesmaximal, characteristic, complemented
2#83511C211yesmaximal, characteristic
3#80987C211yesmaximal, characteristic, complemented
4#80421C211yesmaximal, characteristic, complemented
5C3×#1647C211yesmaximal, characteristic, complemented
6#71370C211yesmaximal, characteristic, complemented
7#71368C211yesmaximal, characteristic, complemented
8#71101C211yesmaximal, characteristic, complemented
9#71040C211yesmaximal, characteristic, complemented
10#68471C211yesmaximal, characteristic, complemented
11#68443C211yesmaximal, characteristic, complemented
12#68205C211yesmaximal, characteristic, complemented
13#68183C211yesmaximal, characteristic, complemented
14#68153C211yesmaximal, characteristic, complemented
15#68150C211yesmaximal, characteristic, complemented
16#19863—31nomaximal, Sylow
17C3×#547C2211yescharacteristic
18#5732C2211yescharacteristic
19#5563C2211yescharacteristic
20#5542C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#474C2211yescharacteristic
23C3×#461C2211yescharacteristic
24#5238—21no—
25#5233—21no—
26#5168—21no—
27#5166—21no—
28#5094C2211yescharacteristic
29#5092C2211yescharacteristic
30#5083C2211yescharacteristic
31#5080C2211yescharacteristic
32#5079C2211yescharacteristic
33#5078C2211yescharacteristic
34#5048—21no—
35#5045—21no—
36#5042—21no—
37#5035—21no—
38C4×#214C2211yescharacteristic
39C4×#213C2211yescharacteristic
40#5012C2211yescharacteristic
41D4×C3⋊C8C2211yescharacteristic
42#5008C2211yescharacteristic
43#5007C2211yescharacteristic
44#4970—21no—
45#4964—21no—
46#4888—21no—
47#4880—21no—
48#4850C2211yescharacteristic
49#4841C2211yescharacteristic
50#4834C2211yescharacteristic
51#4830C2211yescharacteristic
52#4798—21no—
53#4790—21no—
54#4784—21no—
55#4778—21no—
56#4681C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×D4⋊C8C2211yescharacteristic
59C3×D4⋊C8C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4481C2211yescharacteristic
63#4479C2211yescharacteristic
64#4479C2211yescharacteristic
65#4459C2211yescharacteristic
66#4456C2211yescharacteristic
67#4454C2211yescharacteristic
68#3393—31no—
69#3353—31no—
70#3340—31no—
71#3303—31no—
72#1723—31no—
73#1691—31no—
74#1684—31no—
75#1673—31no—
76#1669—31no—
77#1650—31no—
78#1647D311yescharacteristic, complemented
79#1646—31no—
80#1638—31no—
81#1581—31no—
82SD16⋊C8—31no—
83C2×C3×#41—21no—
84C2×C3×SD16—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×#123—41no—
88C3×C22⋊Q8—21no—
89C3×#121—21no—
90C3×C4×D4C2311yescharacteristic
91C3×C4×D4C2311yescharacteristic
92C2×C3×C4⋊C4—21no—
93#947—21no—
94Q8×Dic3—21no—
95C2×#216—21no—
96C2×#216—21no—
97C2×#215—21no—
98#938—21no—
99#936—21no—
100C2×#214D411yescharacteristic
101C2×#213D411yescharacteristic
102#930—21no—
103#926—21no—
104#926—21no—
105#924—21no—
106C2×#208—21no—
107C22×C3⋊C8—21no—
108C2×#204—21no—
109C2×D3×C8—21no—
110#898—41no—
111#897—21no—
112#892—21no—
113#892—21no—
114#887—21no—
115#885—41no—
116#879—41no—
117C4×D12C2311yescharacteristic
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C4×Dic6C2311yescharacteristic
121C3×#107—21no—
122C3×#107—41no—
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×#103—21no—
126C3×#102—21no—
127C3×C4×C8C2311yescharacteristic
128#835—21no—
129#835—21no—
130#835—21no—
131#832—21no—
132#832—21no—
133#832D411yescharacteristic
134#832D411yescharacteristic
135#830—21no—
136#830—21no—
137(C2×D3)⋊C8—21no—
138#818—21no—
139#816—21no—
140#816—21no—
141#816—41no—
142Dic3⋊C8—21no—
143C8×Dic3—21no—
144#810—21no—
145#810D411yescharacteristic
146#809D411yescharacteristic
147#808—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#808D411yescharacteristic
152#807—21no—
153#807—21no—
154#807—21no—
155#807D411yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556—31no—
161#556—31no—
162#555—31no—
163#547C2×D311yescharacteristic
164#500—31no—
165#500C2×D311yescharacteristic
166#498—31no—
167#495—61no—
168#494—31no—
169#487—31no—
170#487—31no—
171#485—61no—
172#485—61no—
173#484—61no—
174#482—61no—
175#482—61no—
176#477—31no—
177#476—31no—
178#475—31no—
179#474—31no—
180#474C2×D311yescharacteristic
181#468—61no—
182#465—61no—
183#465—61no—
184#464—61no—
185#464—61no—
186#461—31no—
187#461—31no—
188#461C2×D311yescharacteristic
189#449—61no—
190C8×Q8—31no—
191C4×Q16—61no—
192C4×SD16—31no—
193C4×D8—31no—
194#436—31no—
195C8×D4—31no—
196C8×D4—31no—
197#428—61no—
198#427—61no—
199#416—61no—
200#335—31no—
201#335—31no—
202#333—31no—
203#332—31no—
204#332—31no—
205#332C2×D311yescharacteristic
206Q8⋊C8—31no—
207D4⋊C8—31no—
208D4⋊C8C2×D311yescharacteristic
209D4⋊C8C2×D311yescharacteristic
210#322—31no—
211C3×#41—42no—
212C3×#41—81no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×D4—41no—
216C2×C3×D4C2×D411yescharacteristic
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—41no—
221C2×#67—41no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6C2×D411yescharacteristic
227C3×SD16—81no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4C2×D411yescharacteristic
237C3×C4⋊C4C2×D411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—41no—
241C3×C42C2411yescharacteristic
242#217—41no—
243#217—41no—
244#217—41no—
245#216—42no—
246#216—81no—
247#215—81no—
248#214—22no—
249#213—41no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—41no—
252(C2×D3)⋊C4—41no—
253(C2×D3)⋊C4—41no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—41no—
259C4⋊Dic3C2×D411yescharacteristic
260C4⋊Dic3C2×D411yescharacteristic
261Dic3⋊C4—21no—
262Dic3⋊C4—41no—
263C4×Dic3—21no—
264C4×Dic3—41no—
265C4×Dic3—41no—
266C4×Dic3—41no—
267#208—42no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—81no—
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275#204—81no—
276D3×C8—42no—
277C2×#41—21no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×SD16—21no—
281C2×SD16—31no—
282C2×SD16—61no—
283C2×SD16—61no—
284C2×D8—31no—
285C2×M16—61no—
286C22×C8—61no—
287C22×C8—61no—
288C4⋊Q8—31no—
289C4⋊Q8—31no—
290C4⋊Q8C22×D311yescharacteristic
291C4⋊Q8C22×D311yescharacteristic
292#126—121no—
293#126—121no—
294#124—61no—
295#124—61no—
296#124—121no—
297#123—41no—
298C22⋊Q8—21no—
299C22⋊Q8—61no—
300C22⋊Q8—61no—
301#121—21no—
302#121—61no—
303#121—61no—
304C4×Q8—31no—
305C4×Q8—61no—
306C4×Q8—61no—
307C4×D4—31no—
308C4×D4C22×D311yescharacteristic
309C4×D4C22×D311yescharacteristic
310#117—61no—
311#117—61no—
312#117—61no—
313#117—61no—
314C2×C4⋊C4—21no—
315#107—21no—
316#107—31no—
317#107—41no—
318#107—61no—
319#107—61no—
320#107—61no—
321#107—61no—
322#106—31no—
323#106—61no—
324#106—61no—
325#106—61no—
326#106—61no—
327C4⋊C8—31no—
328C4⋊C8—31no—
329C4⋊C8—61no—
330C4⋊C8C22×D311yescharacteristic
331C4⋊C8C22×D311yescharacteristic
332#103—21no—
333#103—31no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—121no—
341#102—21no—
342#102—31no—
343#102—31no—
344#102—31no—
345#102—61no—
346#102—61no—
347C22⋊C8—61no—
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—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—82no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—82no—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378#67—161no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384D12—21no—
385D12—21no—
386C4×D3—41no—
387C4×D3—41no—
388Dic6—21no—
389Dic6—21no—
390Dic6—41no—
391Dic6—41no—
392Dic6—81no—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8#13512yes—
402#41—42no—
403#41—81no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—31no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×D4—31no—
410C2×D4—41no—
411C2×D4—121no—
412C2×D4D3×D411yescharacteristic
413C2×D4D3×D411yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—61no—
418Q16—122no—
419Q16—241no—
420SD16—62no—
421SD16—81no—
422SD16—241no—
423SD16—241no—
424D8—121no—
425M16—122no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—31no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—122no—
436C2×C8—241no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—31no—
442C4⋊C4—31no—
443C4⋊C4—41no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C4⋊C4D3×D411yescharacteristic
452C4⋊C4D3×D411yescharacteristic
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—41no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C22⋊C4—121no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C42—61no—
464C42—121no—
465C42—121no—
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—81no—
482C3×C4—81no—
483C3×C4C2×#13511yescharacteristic
484C3×C4#57511yescharacteristic
485C3×C4#57611yescharacteristic
486C3×C4#57811yescharacteristic
487Dic3—41no—
488Dic3—41no—
489Dic3—41no—
490Dic3—81no—
491Dic3—81no—
492C23—21no—
493C23—21no—
494C23—61no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—61no—
500Q8—61no—
501Q8—121no—
502Q8—121no—
503Q8—241no—
504D4—21no—
505D4—21no—
506D4—21no—
507D4—21no—
508D4—61no—
509D4—61no—
510D4—82no—
511D4—481no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—82no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4#100411yescharacteristic
531C8—32no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—61no—
536C8—61no—
537C8—61no—
538C8—121no—
539C8—121no—
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—
555C22#559011yescharacteristic
556C4—21no—
557C4—41no—
558C4—41no—
559C4—81no—
560C4—81no—
561C4—121no—
562C4—121no—
563C4—121no—
564C4—241no—
565C4—241no—
566C4#575711yescharacteristic
567C4D3×#13511yescharacteristic
568C4#577311yescharacteristic
569C4#577911yescharacteristic
570C3#1986311yescharacteristic, complemented, Sylow
571C2—81no—
572C2—81no—
573C2—241no—
574C2#8001311yescharacteristic
575C2#8026911yescharacteristic
576C2#8030111yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1647C2
socleC22×C3D42
prime radical#83511C2
hypercenter#1647D3
hypocenterC3#19863
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1647

Complements

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

NH ≅ G/Ntype
1. #83520C2semidirect
3. #80987C2semidirect
4. #80421C2semidirect
5. C3×#1647C2semidirect
6. #71370C2semidirect
7. #71368C2semidirect
8. #71101C2semidirect
9. #71040C2semidirect
10. #68471C2semidirect
11. #68443C2semidirect
12. #68205C2semidirect
13. #68183C2semidirect
14. #68153C2semidirect
15. #68150C2semidirect
78. #1647D3semidirect
570. C3#19863semidirect

Maximal subgroups

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

subgroupindex
#835202
#835112
#809872
#804212
C3×#16472
#713702
#713682
#711012
#710402
#684712
#684432
#682052
#681832
#681532
#681502
#198633