Finite Group Databasebeta

← #10947999

Subgroups — #10947999

Order 768.

1252 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87283C211yesmaximal, characteristic
2C3×#2883C211yesmaximal, characteristic, complemented
3#84107C211yesmaximal, characteristic, complemented
4#70060C212yesmaximal, complemented
5#70059C212yesmaximal, complemented
6#70019C214yesmaximal, complemented
7C2×#4473C214yesmaximal, complemented
8#33361—31nomaximal, Sylow
9C2×C3×H(5,𝔽2)C2211yescharacteristic
10C2×#1010C411yescharacteristic, complemented
11C3×#518—21no—
12#5832C412yescomplemented
13#5832C2211yescharacteristic
14#5831—21no—
15#5831—21no—
16Dic3×#41C411yescharacteristic, complemented
17#5825C412yescomplemented
18#5825C2211yescharacteristic
19#5806—22no—
20#5806—21no—
21#5806—22no—
22#5806—21no—
23#5805C411yescharacteristic, complemented
24#5805C2212yes—
25#5798—21no—
26C2×#936C411yescharacteristic, complemented
27C2×#936C2212yes—
28C3×#410—21no—
29C3×#410C2212yes—
30C2×C3×#99—22no—
31C2×C3×#99C2214yes—
32#5240—21no—
33#5240C2212yes—
34C2×#834—22no—
35C2×#834C2214yes—
36#4474C2214yescomplemented
37#4473C2214yescomplemented
38#4473C2218yescomplemented
39#3500—31no—
40#2883—31no—
41#2883D311yescharacteristic, complemented
42#2129—32no—
43#2127—34no—
44#2126—32no—
45C2×#354—34no—
46C3×H(5,𝔽2)—44no—
47C2×C3×#41C2×C412yes—
48C2×C3×#41C2×C411yescharacteristic
49C2×C3×#41D411yescharacteristic
50C2×C3×#41D411yescharacteristic
51C2×C3×#41C2311yescharacteristic
52C22×C3×D4C2×C411yescharacteristic
53C22×C3×D4D412yes—
54C22×C3×D4D411yescharacteristic
55C22×C3×D4D412yes—
56C22×C3×D4D411yescharacteristic
57C22×C3×D4C2312yes—
58#1010—44no—
59C2×D3×Q8—21no—
60C2×#237—22no—
61C2×#237—21no—
62C2×#235C2×C411yescharacteristic, complemented
63C2×#235C2×C412yescomplemented
64C22×Dic6C2×C411yescharacteristic, complemented
65C22×Dic6C2×C412yescomplemented
66C3×C4×D4—44no—
67C3×#117—22no—
68C2×C3×C22⋊C4—24no—
69#953—22no—
70#953—22no—
71#953—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—24no—
76#947C2×C411yescharacteristic, complemented
77Q8×Dic3—21no—
78#944C2×C411yescharacteristic, complemented
79#944C2×C412yescomplemented
80#940C2×C412yescomplemented
81#940C2311yescharacteristic
82#939—22no—
83#939—22no—
84#939—21no—
85#939—21no—
86#938—21no—
87#938—21no—
88#938—22no—
89#938—22no—
90#936—21no—
91#936—21no—
92#936—22no—
93#936—22no—
94#936C2×C412yescomplemented
95#936C2×C412yescomplemented
96#936C2×C412yescomplemented
97#936C2311yescharacteristic
98#936C2312yes—
99#935—22no—
100#935—22no—
101#935—21no—
102#935—21no—
103D4×Dic3—22no—
104D4×Dic3—21no—
105D4×Dic3—44no—
106C4×#67—22no—
107C4×#67—22no—
108C4×#67—21no—
109C4×#67—21no—
110#926—22no—
111#926C2×C411yescharacteristic, complemented
112#926C2×C412yescomplemented
113#924—22no—
114#924—22no—
115#924—21no—
116#924—21no—
117C2×C4×Dic3C2×C411yescharacteristic, complemented
118C2×C4×Dic3C2×C412yescomplemented
119C3×#99—22no—
120C3×#99—24no—
121C3×#99C2314yes—
122#834—22no—
123#834—24no—
124#834C2314yes—
125C2×2−1+4—31no—
126C2×H(5,𝔽2)C2×D311yescharacteristic
127#548—61no—
128#548—61no—
129#546—62no—
130#546—61no—
131#546—62no—
132#546—61no—
133#536—32no—
134#536—31no—
135#535—31no—
136#535—32no—
137#532—32no—
138#532—31no—
139C2×#124—31no—
140C2×#124—32no—
141#518—21no—
142#518—61no—
143C4×#41—31no—
144#410—21no—
145#410—32no—
146#410—61no—
147#410C2×D312yes—
148C2×#99—22no—
149C2×#99—34no—
150C2×#99—62no—
151C2×#99C2×D314yes—
152#354—34no—
153#354—38no—
154#353—34no—
155C24×C3C22⋊C412yes—
156C24×C3C2×D412yes—
157C24×C3C2×D412yes—
158C3×#41—24no—
159C3×#41—42no—
160C3×#41—44no—
161C3×#41—42no—
162C3×#41—42no—
163C2×C3×Q8C22⋊C411yescharacteristic
164C2×C3×Q8C22×C411yescharacteristic
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—42no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4—44no—
171C2×C3×D4C22⋊C412yes—
172C2×C3×D4C22⋊C412yes—
173C2×C3×D4C22⋊C412yes—
174C2×C3×D4C22⋊C411yescharacteristic
175C2×C3×D4C22⋊C411yescharacteristic
176C2×C3×D4C22⋊C411yescharacteristic
177C2×C3×D4C22×C412yes—
178C2×C3×D4C2×D411yescharacteristic
179C2×C3×D4C2×D412yes—
180C2×C3×D4C2×D411yescharacteristic
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2411yescharacteristic
183C22×C3×C4—44no—
184C22×C3×C4C22⋊C412yes—
185C22×C3×C4C22⋊C412yes—
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C22×C412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C2×#67—21no—
191C2×#67—22no—
192C2×#67—22no—
193C2×#67—22no—
194C2×#67—21no—
195C2×#67—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—44no—
201D3×Q8—44no—
202#237—48no—
203#237—44no—
204#235—44no—
205#235—42no—
206C2×D12C22×C411yescharacteristic
207C2×C4×D3—22no—
208C2×C4×D3—21no—
209C2×Dic6—22no—
210C2×Dic6—21no—
211C2×Dic6—44no—
212C2×Dic6—42no—
213C2×Dic6C22×C412yes—
214C2×Dic6C22×C411yescharacteristic
215C3×C4⋊C4—42no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—44no—
218C3×C42—42no—
219#217—22no—
220#217—22no—
221#217—22no—
222#217—22no—
223#217—21no—
224#217—22no—
225#217—22no—
226#217—22no—
227#217—22no—
228#217—22no—
229#217—21no—
230#217—21no—
231#217—24no—
232#217—44no—
233(C2×D3)⋊C4—21no—
234(C2×D3)⋊C4—21no—
235C4⋊Dic3—22no—
236C4⋊Dic3—21no—
237C4⋊Dic3—42no—
238Dic3⋊C4—21no—
239Dic3⋊C4—21no—
240Dic3⋊C4—22no—
241Dic3⋊C4—22no—
242C4×Dic3—22no—
243C4×Dic3—21no—
244C4×Dic3—42no—
245C4×Dic3C22×C411yescharacteristic
246C4×Dic3C22×C411yescharacteristic
247C4×Dic3C22×C412yes—
2482−1+4—124no—
249H(5,𝔽2)—44no—
250C2×#41—31no—
251C2×#41—32no—
252C2×#41—62no—
253C2×#41—61no—
254C2×#41C4×D312yes—
255C2×#41C4×D311yescharacteristic
256C2×#41D1211yescharacteristic
257C2×#41#6711yescharacteristic
258C2×#41C22×D311yescharacteristic
259C22×Q8—31no—
260C22×Q8—32no—
261C22×Q8—61no—
262C22×D4C4×D311yescharacteristic
263C22×D4D1212yes—
264C22×D4D1211yescharacteristic
265C22×D4#6712yes—
266C22×D4#6711yescharacteristic
267C22×D4C22×D312yes—
268C4⋊Q8—32no—
269C4⋊Q8—31no—
270#127—32no—
271#127—31no—
272#124—31no—
273#124—32no—
274#124—32no—
275#124—32no—
276#124—31no—
277#124—32no—
278#124—61no—
279#124—61no—
280#124—62no—
281#124—62no—
282#123—62no—
283#123—62no—
284#123—61no—
285#123—61no—
286C22⋊Q8—62no—
287C22⋊Q8—62no—
288C22⋊Q8—61no—
289C22⋊Q8—61no—
290#121—62no—
291#121—62no—
292#121—61no—
293#121—61no—
294#121—61no—
295#121—62no—
296#121—61no—
297#121—62no—
298#120—62no—
299#120—62no—
300#120—62no—
301C4×Q8—61no—
302C4×D4—44no—
303C4×D4—62no—
304C4×D4—61no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—61no—
308C4×D4—61no—
309C4×D4—124no—
310#117—22no—
311#117—31no—
312#117—32no—
313#117—62no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—62no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C22⋊C4—64no—
319C2×C42—31no—
320C2×C42—32no—
321#99—22no—
322#99—24no—
323#99—34no—
324#99—64no—
325#99—62no—
326#99C22×D314yes—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—42no—
330C23×C3—82no—
331C23×C3—82no—
332C23×C3—82no—
333C23×C3C2×C22⋊C412yes—
334C23×C3C2×C22⋊C412yes—
335C23×C3C2×C22⋊C411yescharacteristic
336C23×C3C2×C22⋊C412yes—
337C23×C3C22×D411yescharacteristic
338C23×C3C22×D411yescharacteristic
339C22×D3—21no—
340C3×Q8—22no—
341C3×Q8—41no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—24no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—41no—
350C3×D4—42no—
351C3×D4—41no—
352C3×D4—41no—
353C3×D4#9914yes—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4—164no—
362C2×C3×C4C2×C22⋊C412yes—
363C2×C3×C4C2×C22⋊C411yescharacteristic
364C2×C3×C4C2×C22⋊C411yescharacteristic
365C2×C3×C4C2×C22⋊C411yescharacteristic
366C2×C3×C4C23×C411yescharacteristic
367#67—42no—
368#67—44no—
369#67—82no—
370#67—82no—
371#67—81no—
372#67—81no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—42no—
379C2×Dic3—44no—
380C2×Dic3—44no—
381C2×Dic3—82no—
382C2×Dic3—81no—
383C2×Dic3—164no—
384D12—41no—
385C4×D3—44no—
386C4×D3—42no—
387Dic6—44no—
388Dic6—42no—
389Dic6—41no—
390Dic6—42no—
391C24(C2×D3)⋊C412yes—
392C24C2×D1212yes—
393C24C2×#6712yes—
394#41—24no—
395#41—42no—
396#41—44no—
397#41—42no—
398#41—42no—
399#41—124no—
400#41—122no—
401#41—128no—
402#41—124no—
403C2×Q8—32no—
404C2×Q8—31no—
405C2×Q8—62no—
406C2×Q8—61no—
407C2×Q8—122no—
408C2×Q8—124no—
409C2×Q8—124no—
410C2×Q8(C2×D3)⋊C411yescharacteristic
411C2×Q8C2×C4×D311yescharacteristic
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—31no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—61no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4(C2×D3)⋊C412yes—
426C2×D4(C2×D3)⋊C412yes—
427C2×D4(C2×D3)⋊C412yes—
428C2×D4(C2×D3)⋊C411yescharacteristic
429C2×D4(C2×D3)⋊C411yescharacteristic
430C2×D4(C2×D3)⋊C411yescharacteristic
431C2×D4C2×C4×D312yes—
432C2×D4C2×D1211yescharacteristic
433C2×D4C2×D1212yes—
434C2×D4C2×#6711yescharacteristic
435C2×D4C2×#6712yes—
436C2×D4C23×D311yescharacteristic
437C22×C4—44no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—124no—
445C22×C4(C2×D3)⋊C412yes—
446C22×C4(C2×D3)⋊C412yes—
447C22×C4C2×C4×D311yescharacteristic
448C22×C4C2×C4×D312yes—
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C4⋊C4—42no—
452C4⋊C4—62no—
453C4⋊C4—61no—
454C4⋊C4—62no—
455C4⋊C4—61no—
456C4⋊C4—62no—
457C4⋊C4—61no—
458C4⋊C4—122no—
459C22⋊C4—24no—
460C22⋊C4—44no—
461C22⋊C4—62no—
462C22⋊C4—62no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C22⋊C4—61no—
468C22⋊C4—62no—
469C22⋊C4—61no—
470C22⋊C4—62no—
471C22⋊C4—61no—
472C22⋊C4—62no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C22⋊C4—64no—
476C22⋊C4—124no—
477C42—31no—
478C42—31no—
479C42—32no—
480C42—42no—
481C42—62no—
482C42—61no—
483C42—122no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—84no—
496C22×C3—84no—
497C22×C3—162no—
498C22×C3C2×#9914yes—
499C22×C3C22×C22⋊C411yescharacteristic
500C2×D3—41no—
501C2×D3—41no—
502C2×D3—41no—
503C3×C4—21no—
504C3×C4—22no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—84no—
508C3×C4C2×#9912yes—
509Dic3—41no—
510Dic3—41no—
511Dic3—42no—
512Dic3—41no—
513Dic3—42no—
514Dic3—84no—
515C23—24no—
516C23—24no—
517C23—42no—
518C23—61no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23C2×(C2×D3)⋊C412yes—
523C23C2×(C2×D3)⋊C412yes—
524C23C2×(C2×D3)⋊C411yescharacteristic
525C23C2×(C2×D3)⋊C412yes—
526C23C22×D1211yescharacteristic
527C23C22×#6711yescharacteristic
528Q8—22no—
529Q8—41no—
530Q8—122no—
531Q8—121no—
532Q8—124no—
533Q8—122no—
534D4—24no—
535D4—24no—
536D4—24no—
537D4—22no—
538D4—22no—
539D4—42no—
540D4—42no—
541D4—41no—
542D4—42no—
543D4—41no—
544D4—41no—
545D4—121no—
546D4—122no—
547D4—124no—
548D4—242no—
549D4—242no—
550D4—241no—
551D4—241no—
552D4#80614yes—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—124no—
566C2×C4—122no—
567C2×C4—124no—
568C2×C4—122no—
569C2×C4—124no—
570C2×C4—164no—
571C2×C4—242no—
572C2×C4—241no—
573C2×C4—484no—
574C2×C4C2×(C2×D3)⋊C412yes—
575C2×C4C2×(C2×D3)⋊C411yescharacteristic
576C2×C4C2×(C2×D3)⋊C411yescharacteristic
577C2×C4C2×(C2×D3)⋊C411yescharacteristic
578C2×C4C22×C4×D311yescharacteristic
579C2×C3—21no—
580C2×C3—24no—
581C2×C3—41no—
582C2×C3—41no—
583C2×C3—42no—
584C2×C3—42no—
585C2×C3—41no—
586C2×C3C22×#9911yescharacteristic
587D3—81no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—84no—
600C22—84no—
601C22—121no—
602C22—121no—
603C22—121no—
604C22—162no—
605C22C2×#80614yes—
606C22C22×(C2×D3)⋊C411yescharacteristic
607C4—21no—
608C4—22no—
609C4—21no—
610C4—21no—
611C4—84no—
612C4—121no—
613C4—121no—
614C4—122no—
615C4—122no—
616C4—121no—
617C4—244no—
618C4C2×#80612yes—
619C3#3336111yescharacteristic, complemented, Sylow
620C2—21no—
621C2—24no—
622C2—41no—
623C2—41no—
624C2—42no—
625C2—42no—
626C2—41no—
627C2—241no—
628C2C22×#80611yescharacteristic
subgroupG/H
centerC2C22×#806
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2883C2
socleC2×C3C22×#99
prime radical#87283C2
cosocleC2×C3×D4C24
hypercenter#2883D3
hypocenterC3#33361
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ #2883

Complements

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

NH ≅ G/Ntype
2. C3×#2883C2semidirect
3. #84107C2semidirect
4. #70060C2semidirect
5. #70059C2semidirect
6. #70019C2semidirect
7. C2×#4473C2semidirect
10. C2×#1010C4semidirect
12. #5832C4semidirect
16. Dic3×#41C4semidirect
17. #5825C4semidirect
23. #5805C4semidirect
26. C2×#936C4semidirect
36. #4474C22semidirect
37. #4473C22semidirect
38. #4473C22semidirect
41. #2883D3semidirect
62. C2×#235C2×C4semidirect
63. C2×#235C2×C4semidirect
64. C22×Dic6C2×C4semidirect
65. C22×Dic6C2×C4semidirect
76. #947C2×C4semidirect
78. #944C2×C4semidirect
79. #944C2×C4semidirect
80. #940C2×C4semidirect
94. #936C2×C4semidirect
95. #936C2×C4semidirect
96. #936C2×C4semidirect
111. #926C2×C4semidirect
112. #926C2×C4semidirect
117. C2×C4×Dic3C2×C4semidirect
118. C2×C4×Dic3C2×C4semidirect
619. C3#33361semidirect

Maximal subgroups

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

subgroupindex
#872832
C3×#28832
#841072
#700602
#700592
#700192
C2×#44732
#333613