Finite Group Databasebeta

← #10778187

Subgroups — #10778187

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3035C211yesmaximal, characteristic
2#83294C211yesmaximal, characteristic, complemented
3#82534C211yesmaximal, characteristic, complemented
4#82519C211yesmaximal, characteristic, complemented
5#71808C211yesmaximal, characteristic, complemented
6#71054C211yesmaximal, characteristic, complemented
7#71053C211yesmaximal, characteristic, complemented
8#71042C211yesmaximal, characteristic, complemented
9#71041C211yesmaximal, characteristic, complemented
10#71023C211yesmaximal, characteristic, complemented
11#71020C211yesmaximal, characteristic, complemented
12#71011C211yesmaximal, characteristic, complemented
13#71008C211yesmaximal, characteristic, complemented
14#70439C211yesmaximal, characteristic, complemented
15#70420C211yesmaximal, characteristic, complemented
16#19182—31nomaximal, Sylow
17C2×C3×#135—21no—
18C3×#532C2211yescharacteristic
19C3×C4×#41C2211yescharacteristic
20#5470C2211yescharacteristic
21C3×#464—21no—
22C3×#463—21no—
23C3×#462C2211yescharacteristic
24C3×#461C2211yescharacteristic
25C3×#460C2211yescharacteristic
26C3×#459C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#416—21no—
29#5239—21no—
30#5093C2211yescharacteristic
31#5078C2211yescharacteristic
32#5076C2211yescharacteristic
33#5070C2211yescharacteristic
34#5042—21no—
35#5042—21no—
36#5031C2211yescharacteristic
37C4×#216—21no—
38Q8×C3⋊C8C2211yescharacteristic
39Q8×C3⋊C8C2211yescharacteristic
40#5020C2211yescharacteristic
41#5019C2211yescharacteristic
42#5018—21no—
43#5018—21no—
44C4×#214—21no—
45#5012—21no—
46#5012—21no—
47#5012C2211yescharacteristic
48#5012C2211yescharacteristic
49#5007C2211yescharacteristic
50#4996—21no—
51#4926—21no—
52#4926C2211yescharacteristic
53#4925C2211yescharacteristic
54C4×#208—21no—
55C4×#208C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4544C2211yescharacteristic
59#4544C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3182—31no—
69#3107—31no—
70#3035D311yescharacteristic
71#2971—31no—
72#1701—31no—
73#1699—31no—
74#1693—31no—
75#1685—31no—
76#1682—31no—
77#1680—31no—
78#1674—31no—
79#1666—31no—
80#1565—31no—
81#1553—31no—
82#1552—31no—
83C2×C3×#41—21no—
84C2×C3×#41D411yescharacteristic
85C3×#135—44no—
86C2×C3×Q16—21no—
87C2×C3×SD16—21no—
88C2×C3×SD16—21no—
89C2×C3×D8—21no—
90C22×C3×C8—21no—
91C3×C4⋊Q8C2311yescharacteristic
92C3×#127C2311yescharacteristic
93C3×#124—21no—
94C3×#121—21no—
95C3×#121—21no—
96C3×C4×Q8D411yescharacteristic
97C3×C4×Q8C2311yescharacteristic
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4D411yescharacteristic
103C3×C4×D4C2311yescharacteristic
104C3×#117—21no—
105C3×#117D411yescharacteristic
106C2×C3×C42—21no—
107C2×C3×C42C2311yescharacteristic
108C2×#216—21no—
109C2×#214—21no—
110C2×#214—21no—
111C2×#214—21no—
112#926—21no—
113#924—21no—
114#924—21no—
115C2×#208—21no—
116C2×#208—21no—
117C2×#208D411yescharacteristic
118C2×#208D411yescharacteristic
119#870—21no—
120#869C2311yescharacteristic
121#869C2311yescharacteristic
122C4×Dic6—21no—
123C4×Dic6—21no—
124C3×C4⋊C8—21no—
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×#103—21no—
128C3×#103—21no—
129C3×#102—21no—
130C3×#102—21no—
131#835—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#830—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#810—21no—
143#808—21no—
144#807—21no—
145#807—21no—
146#807—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152C4⋊(C3⋊C8)C2311yescharacteristic
153#803—21no—
154#803D411yescharacteristic
155#803D411yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8—21no—
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—21no—
162#533—31no—
163#532C2×D311yescharacteristic
164C4×#41C2×D311yescharacteristic
165#500—31no—
166#492—31no—
167#487—31no—
168#486—31no—
169#485—61no—
170#475—31no—
171#474—31no—
172#468—61no—
173#468—61no—
174#464—21no—
175#463—21no—
176#462—31no—
177#462C2×D311yescharacteristic
178#461—31no—
179#461C2×D311yescharacteristic
180#460C2×D311yescharacteristic
181#459C2×D311yescharacteristic
182#452—61no—
183C8×Q8—31no—
184C8×Q8—31no—
185#440—61no—
186#440—61no—
187C4×Q16—61no—
188C4×SD16—61no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—31no—
194#432—61no—
195#431—31no—
196C2×C4⋊C8C2×D311yescharacteristic
197#416—21no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#331—31no—
211#331—31no—
212C3×#41—22no—
213C3×#41—41no—
214C3×#41—42no—
215C3×#41—42no—
216C2×C3×Q8—21no—
217C2×C3×Q8C2×D411yescharacteristic
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—41no—
223C2×C3×D4C2×D411yescharacteristic
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—41no—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4#4111yescharacteristic
231C2×Dic6—21no—
232C2×Dic6—21no—
233C3×Q16—42no—
234C3×SD16—42no—
235C3×SD16—42no—
236C3×D8—42no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—42no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—41no—
250C3×C22⋊C4—41no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256C3×C42#4111yescharacteristic
257C3×C42C2411yescharacteristic
258#217—41no—
259#216—42no—
260#214—42no—
261#214—42no—
262#214—42no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267Dic3⋊C4—41no—
268Dic3⋊C4—41no—
269C4×Dic3—41no—
270#208—22no—
271#208—41no—
272#208—42no—
273#208—42no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C2×D411yescharacteristic
279C2×C3⋊C8C2×D411yescharacteristic
280C2×C3⋊C8C2×D411yescharacteristic
281C2×C3⋊C8C2×D411yescharacteristic
282C2×#41—21no—
283C2×#41#6711yescharacteristic
284#135—44no—
285C2×Q16—21no—
286C2×Q16—61no—
287C2×SD16—21no—
288C2×SD16—21no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—21no—
293C2×M16—31no—
294C2×M16—31no—
295C2×M16—61no—
296C2×M16—61no—
297C22×C8—21no—
298C4⋊Q8—31no—
299C4⋊Q8—31no—
300C4⋊Q8C22×D311yescharacteristic
301#127C22×D311yescharacteristic
302#125—61no—
303#124—21no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306#121—21no—
307#121—21no—
308C4×Q8—61no—
309C4×Q8—61no—
310C4×Q8#6711yescharacteristic
311C4×Q8C22×D311yescharacteristic
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4#6711yescharacteristic
317C4×D4C22×D311yescharacteristic
318#117—21no—
319#117—61no—
320#117#6711yescharacteristic
321C2×C42—21no—
322C2×C42C22×D311yescharacteristic
323#107—61no—
324#107—61no—
325#107—61no—
326#106—61no—
327C4⋊C8—21no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—61no—
333C4⋊C8—61no—
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—21no—
337#103—21no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#102—21no—
344#102—21no—
345#102—61no—
346#102—61no—
347#102—61no—
348C22⋊C8—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352#97—31no—
353#97—31no—
354#97—61no—
355C4×C8—31no—
356C4×C8—31no—
357C4×C8—61no—
358C4×C8—61no—
359C4×C8—61no—
360C23×C3—21no—
361C23×C3—41no—
362C23×C3#12111yescharacteristic
363C3×Q8—21no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8AGL(1,ℤ/8)12yes—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—42no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—81no—
375C3×D4—82no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—42no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—81no—
389C2×C3×C4—82no—
390C2×C3×C4#12111yescharacteristic
391C2×C3×C4#12111yescharacteristic
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×Dic3—41no—
397C2×Dic3—41no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8#13512yescomplemented
412#41—22no—
413#41—41no—
414#41—42no—
415#41—42no—
416C2×Q8—21no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8C2×#6711yescharacteristic
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—41no—
425C2×D4C2×#6711yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—41no—
430C22×C4D3×D411yescharacteristic
431C22×C4#23711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433Q16—42no—
434Q16—122no—
435SD16—42no—
436SD16—42no—
437SD16—122no—
438SD16—122no—
439SD16—122no—
440D8—42no—
441M16—62no—
442M16—122no—
443M16—121no—
444M16—122no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—31no—
451C2×C8—42no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—121no—
465C4⋊C4—121no—
466C4⋊C4C2×#6711yescharacteristic
467C4⋊C4C2×#6711yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—121no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—121no—
478C42D3×D411yescharacteristic
479C42#23711yescharacteristic
480C42C2×#6711yescharacteristic
481C42C23×D311yescharacteristic
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3C2×#12111yescharacteristic
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—81no—
500C3×C4C2×#13511yescharacteristic
501C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
502C3×C4#57511yescharacteristic
503C3×C4#57511yescharacteristic
504Dic3—81no—
505Dic3—81no—
506C23—21no—
507C23—41no—
508C23#93911yescharacteristic
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—121no—
513Q8—121no—
514Q8—121no—
515Q8—121no—
516Q8#93212yes—
517D4—21no—
518D4—21no—
519D4—42no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—81no—
525D4—82no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4#93911yescharacteristic
543C2×C4#93911yescharacteristic
544C2×C4#93911yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×#23711yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—32no—
549C8—41no—
550C8—41no—
551C8—61no—
552C8—61no—
553C8—61no—
554C8—121no—
555C8—121no—
556C8—121no—
557C8—121no—
558C2×C3—41no—
559C2×C3—81no—
560C2×C3—161no—
561C2×C3#303511yescharacteristic
562C2×C3#305511yescharacteristic
563C2×C3#306911yescharacteristic
564C22—21no—
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#93911yescharacteristic
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—241no—
583C4—241no—
584C4D3×#13511yescharacteristic
585C4#576811yescharacteristic
586C4C2×#93211yescharacteristic
587C4#582211yescharacteristic
588C3#1918211yescharacteristic, complemented, Sylow
589C2—41no—
590C2—81no—
591C2—161no—
592C2#8262111yescharacteristic
593C2#8390011yescharacteristic
594C2#8400311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
Fitting, prime radicalC3×#3035C2
socleC22×C3C2×#121
hypercenter#3035D3
hypocenterC3#19182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
2. #83294C2semidirect
3. #82534C2semidirect
4. #82519C2semidirect
5. #71808C2semidirect
6. #71054C2semidirect
7. #71053C2semidirect
8. #71042C2semidirect
9. #71041C2semidirect
10. #71023C2semidirect
11. #71020C2semidirect
12. #71011C2semidirect
13. #71008C2semidirect
14. #70439C2semidirect
15. #70420C2semidirect
411. C3⋊C8#135semidirect
588. C3#19182semidirect

Maximal subgroups

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

subgroupindex
C3×#30352
#832942
#825342
#825192
#718082
#710542
#710532
#710422
#710412
#710232
#710202
#710112
#710082
#704392
#704202
#191823