Finite Group Databasebeta

← #11662499

Subgroups — #11662499

Order 768.

2824 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#576C211yesmaximal, characteristic, complemented
2#87285C212yesmaximal, complemented
3C2×D3×#131C211yesmaximal, characteristic, complemented
4#84099C212yesmaximal, complemented
5#83951C216yesmaximal, complemented
6#83950C216yesmaximal, complemented
7#83917C213yesmaximal, complemented
8#83914C213yesmaximal, complemented
9#83836C213yesmaximal, complemented
10C2×#5157C213yesmaximal, complemented
11#83785C211yesmaximal, characteristic, complemented
12#60458—31nomaximal, Sylow
13C2×C3×H(5,𝔽2)C2212yes—
14C2×D3×#41C2211yescharacteristic
15C3×#576—28no—
16C2×C3×AGL(1,ℤ/8)C2216yescomplemented
17C2×C3×#135C2213yescomplemented
18C22×C3×D8C2213yescomplemented
19C2×C3×#131C2211yescharacteristic
20#5834—22no—
21#5833C2216yescomplemented
22#5831—22no—
23#5828C2216yescomplemented
24#5827C2212yes—
25C2×#948C2211yescharacteristic
26#5806—26no—
27#5805C2216yescomplemented
28D4×#67—26no—
29C2×#938C2216yescomplemented
30D3×#131—28no—
31C2×#907C2213yescomplemented
32C2×D3×M16C2213yescomplemented
33C2×#901C2213yescomplemented
34C22×D3×C8C2213yescomplemented
35#5233C2216yescomplemented
36#5232C2216yescomplemented
37#5215C2216yescomplemented
38C2×#832C2216yescomplemented
39#5188C2211yescharacteristic, complemented
40#5188C2213yescomplemented
41#5174C22112yescomplemented
42#5171C2216yescomplemented
43#5171C2216yescomplemented
44#5159C22112yescomplemented
45#5157C2213yescomplemented
46#5157C2216yescomplemented
47#5157C2216yescomplemented
48#5157C2216yescomplemented
49#5157C2213yescomplemented
50#5121C2216yescomplemented
51#5106C2213yescomplemented
52C2×#818C2213yescomplemented
53C2×#576D311yescharacteristic, complemented
54C22×#131—31no—
55#3488—32no—
56#3062—36no—
57#3061—33no—
58#3058—33no—
59#3055—36no—
60#3050—33no—
61C2×#466—33no—
62#2913—31no—
63#2897—32no—
64C3×H(5,𝔽2)—28no—
65C3×H(5,𝔽2)—28no—
66C2×C3×#41—22no—
67C2×C3×#41C2316yes—
68C2×C3×#41C2311yescharacteristic
69C22×C3×D4—26no—
70C22×C3×D4C2316yes—
71C22×#67—26no—
72D3×#41—24no—
73D3×#41—24no—
74C2×#239D411yescharacteristic
75C2×D3×Q8D411yescharacteristic
76C2×#237D413yes—
77C2×D3×D4D413yes—
78C2×#235C2313yes—
79C22×C4×D3C2313yes—
80C3×AGL(1,ℤ/8)—224no—
81C3×#135—212no—
82C2×C3×Q16C2311yescharacteristic
83C2×C3×SD16C2316yes—
84C2×C3×D8—212no—
85C2×C3×D8C2313yes—
86C2×C3×D8C2316yes—
87C3×#131D418yescomplemented
88C2×C3×M16C2313yes—
89C22×C3×C8C2313yes—
90#953—212no—
91C2×#217—26no—
92#948—24no—
93#946C2312yes—
94#946C2316yes—
95Q8×Dic3—22no—
96#940—26no—
97#939—212no—
98#938—26no—
99#938—26no—
100#938C2316yes—
101#938C23112yes—
102#938C2316yes—
103#937—26no—
104#936—26no—
105#935—26no—
106D4×Dic3—26no—
107#930—26no—
108C4×#67—26no—
109C4×#67—26no—
110#926C2316yes—
111C2×C4⋊Dic3C2316yes—
112#924—26no—
113C2×#208C2313yes—
114C22×C3⋊C8C2313yes—
115#907—212no—
116D3×M16—212no—
117#901—212no—
118C2×ΓL(1,𝔽25)C2316yes—
119C2×D3×C8—212no—
120C2×D3×C8C2311yescharacteristic
121C2×D3×C8C2313yes—
122C2×D3×C8C2313yes—
123C2×D3×C8C2313yes—
124#835C2312yes—
125#835C2316yes—
126#832C2316yes—
127#832C23112yes—
128#832C2316yes—
129#818C2316yes—
130#818C2313yes—
131#818C2311yescharacteristic
132#817C2316yes—
133C2×H(5,𝔽2)C2×D312yes—
134C22×#41—31no—
135#576—28no—
136C2×AGL(1,ℤ/8)C2×D316yescomplemented
137C2×#135C2×D313yescomplemented
138C22×D8C2×D313yescomplemented
139C2×#131—31no—
140C2×#131—33no—
141C2×#131—33no—
142C2×#131—68no—
143C2×#131C2×D311yescharacteristic
144C22×M16—33no—
145C23×C8—33no—
146#550—62no—
147#548—62no—
148#546—66no—
149D42—66no—
150#537—36no—
151#535—36no—
152#525—36no—
153C2×#121—36no—
154#520—32no—
155#469—312no—
156#468—312no—
157#467—31no—
158#467—33no—
159#466—33no—
160#466—36no—
161#466—36no—
162#466—36no—
163#466—33no—
164#465—36no—
165#465—36no—
166#428—36no—
167#427—33no—
168C2×#107—33no—
169#418—36no—
170#417—36no—
171#416—36no—
172C2×#102—36no—
173C24×C3—26no—
174C23×D3C2×D413yes—
175C3×#41—212no—
176C3×#41—212no—
177C3×#41—48no—
178C3×#41C2×D414yes—
179C3×#41#4114yes—
180C2×C3×Q8C2×D411yescharacteristic
181C2×C3×Q8C2412yes—
182C2×C3×D4—26no—
183C2×C3×D4—212no—
184C2×C3×D4—26no—
185C2×C3×D4—212no—
186C2×C3×D4—424no—
187C2×C3×D4C2×D413yes—
188C2×C3×D4C2416yes—
189C22×C3×C4—26no—
190C22×C3×C4C2413yes—
191C2×#67—26no—
192C2×#67—26no—
193C2×#67—212no—
194C2×#67—812no—
195C2×#67C2×D416yes—
196C22×Dic3—26no—
197C22×Dic3C2×D413yes—
198#239—24no—
199D3×Q8—24no—
200#237—212no—
201D3×D4—212no—
202#235—26no—
203#235—26no—
204C2×D12C2×D413yes—
205C2×C4×D3—26no—
206C2×C4×D3—26no—
207C2×C4×D3C2×D413yes—
208C2×C4×D3C2×D413yes—
209C2×C4×D3C2411yescharacteristic
210C2×C4×D3C2413yes—
211C2×Dic6C2×D413yes—
212C3×Q16—22no—
213C3×SD16—212no—
214C3×D8—26no—
215C3×D8—212no—
216C3×M16C2×D4112yescomplemented
217C2×C3×C8C2×D4112yescomplemented
218C2×C3×C8C2413yes—
219C2×C3×C8C2411yescharacteristic
220#217—26no—
221#217—212no—
222#217—26no—
223#217—26no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4—26no—
226C4⋊Dic3—26no—
227C4⋊Dic3C2412yes—
228C4⋊Dic3C2416yes—
229Dic3⋊C4—26no—
230Dic3⋊C4—22no—
231C4×Dic3—26no—
232#208—26no—
233C2×C3⋊C8—26no—
234C2×C3⋊C8C2411yescharacteristic
235C2×C3⋊C8C2413yes—
236ΓL(1,𝔽25)—212no—
237D3×C8—22no—
238D3×C8—26no—
239D3×C8—26no—
240D3×C8—26no—
241H(5,𝔽2)—28no—
242H(5,𝔽2)—28no—
243C2×#41—22no—
244C2×#41—31no—
245C2×#41—33no—
246C2×#41—33no—
247C2×#41—64no—
248C2×#41—64no—
249C2×#41C22×D316yes—
250C2×#41C22×D311yescharacteristic
251C22×Q8—31no—
252C22×D4—26no—
253C22×D4—33no—
254C22×D4—66no—
255C22×D4C22×D316yes—
256C23×C4—33no—
257AGL(1,ℤ/8)—224no—
258#135—212no—
259C2×Q16C22×D311yescharacteristic
260C2×SD16C22×D316yes—
261C2×D8—212no—
262C2×D8C22×D313yes—
263C2×D8C22×D316yes—
264#131—64no—
265#131—612no—
266#131—612no—
267#131#6718yescomplemented
268C2×M16—33no—
269C2×M16—36no—
270C2×M16—612no—
271C2×M16C22×D313yes—
272C22×C8—33no—
273C22×C8—31no—
274C22×C8—33no—
275C22×C8—33no—
276C22×C8—33no—
277C22×C8—612no—
278C22×C8C22×D313yes—
279#127—66no—
280#124—66no—
281#123—66no—
282C22⋊Q8—32no—
283C22⋊Q8—36no—
284C22⋊Q8—66no—
285#121—36no—
286#121—312no—
287#121—36no—
288#121—66no—
289#121—66no—
290#121—66no—
291#121—612no—
292#120—66no—
293#120—612no—
294C4×Q8—62no—
295C4×D4—66no—
296C4×D4—66no—
297C4×D4—66no—
298#117—36no—
299C2×C4⋊C4—36no—
300C2×C22⋊C4—66no—
301#107—33no—
302#107—36no—
303#107—31no—
304#106—36no—
305#103—32no—
306#103—36no—
307#102—36no—
308#102—312no—
309#102—36no—
310C23×C3—26no—
311C23×C3—412no—
312C23×C3—86no—
313C23×C3—86no—
314C23×C3C22×D413yes—
315C22×D3—26no—
316C22×D3—26no—
317C22×D3—43no—
318C22×D3C22×D413yes—
319C22×D3C22×D411yescharacteristic
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8#12114yes—
323C3×D4—26no—
324C3×D4—26no—
325C3×D4—412no—
326C3×D4—412no—
327C3×D4#121112yes—
328C2×C3×C4—22no—
329C2×C3×C4—412no—
330C2×C3×C4C22×D413yes—
331C2×C3×C4C22×D416yes—
332C2×C3×C4C2×#4116yes—
333C2×C3×C4C2511yescharacteristic
334#67—212no—
335#67—412no—
336#67—86no—
337#67—812no—
338C2×Dic3—26no—
339C2×Dic3—22no—
340C2×Dic3—26no—
341C2×Dic3—86no—
342C2×Dic3C22×D413yes—
343C2×Dic3C22×D411yescharacteristic
344D12—26no—
345C4×D3—26no—
346C4×D3—26no—
347C4×D3—21no—
348C4×D3—23no—
349C4×D3—21no—
350C4×D3—23no—
351Dic6—26no—
352C3×C8C22×D412yes—
353C3×C8C22×D416yes—
354C3⋊C8—21no—
355C3⋊C8—23no—
356C24—26no—
357C24—33no—
358#41—212no—
359#41—212no—
360#41—48no—
361#41—64no—
362#41—612no—
363#41—66no—
364#41—66no—
365#41#23714yes—
366#41C2×#6714yes—
367C2×Q8—33no—
368C2×Q8—64no—
369C2×Q8D3×D411yescharacteristic
370C2×Q8C23×D312yes—
371C2×D4—26no—
372C2×D4—26no—
373C2×D4—212no—
374C2×D4—212no—
375C2×D4—36no—
376C2×D4—33no—
377C2×D4—424no—
378C2×D4—612no—
379C2×D4—66no—
380C2×D4—66no—
381C2×D4—612no—
382C2×D4—2412no—
383C2×D4D3×D413yes—
384C2×D4C23×D316yes—
385C22×C4—26no—
386C22×C4—33no—
387C22×C4—33no—
388C22×C4—33no—
389C22×C4—31no—
390C22×C4—33no—
391C22×C4—66no—
392C22×C4—66no—
393C22×C4—66no—
394C22×C4C23×D313yes—
395Q16—22no—
396SD16—212no—
397D8—26no—
398D8—212no—
399M16—66no—
400M16—612no—
401M16C2×#67112yescomplemented
402C2×C8—31no—
403C2×C8—33no—
404C2×C8—66no—
405C2×C8—62no—
406C2×C8—66no—
407C2×C8—66no—
408C2×C8—66no—
409C2×C8C2×#67112yescomplemented
410C2×C8C23×D313yes—
411C2×C8C23×D311yescharacteristic
412C4⋊C4—32no—
413C4⋊C4—36no—
414C4⋊C4—62no—
415C4⋊C4—66no—
416C4⋊C4—66no—
417C22⋊C4—62no—
418C22⋊C4—66no—
419C22⋊C4—66no—
420C22⋊C4—612no—
421C22⋊C4—66no—
422C22⋊C4—66no—
423C42—66no—
424C22×C3—23no—
425C22×C3—23no—
426C22×C3—46no—
427C22×C3—46no—
428C22×C3—46no—
429C22×C3—1612no—
430C22×C3C2×#12116yes—
431C22×C3C23×D411yescharacteristic
432C2×D3—21no—
433C2×D3—21no—
434C2×D3—23no—
435C2×D3—23no—
436C2×D3—23no—
437C2×D3—412no—
438C2×D3#57612yescomplemented
439C3×C4—42no—
440C3×C4C2×#12116yes—
441C3×C4C23×D411yescharacteristic
442C3×C4C22×#4111yescharacteristic
443Dic3—23no—
444Dic3—21no—
445Dic3—42no—
446Dic3—46no—
447C23—26no—
448C23—31no—
449C23—33no—
450C23—412no—
451C23—66no—
452C23—66no—
453C23—86no—
454C23—86no—
455C23—123no—
456C23C2×D3×D413yes—
457Q8—22no—
458Q8—22no—
459Q8—66no—
460Q8#93814yes—
461D4—26no—
462D4—26no—
463D4—412no—
464D4—412no—
465D4—612no—
466D4—66no—
467D4—1212no—
468D4—2412no—
469D4—246no—
470D4#938112yes—
471C2×C4—22no—
472C2×C4—31no—
473C2×C4—33no—
474C2×C4—412no—
475C2×C4—66no—
476C2×C4—61no—
477C2×C4—63no—
478C2×C4—61no—
479C2×C4—63no—
480C2×C4—66no—
481C2×C4—66no—
482C2×C4—62no—
483C2×C4—66no—
484C2×C4—246no—
485C2×C4C2×D3×D413yes—
486C2×C4C2×#23716yes—
487C2×C4C22×#6716yes—
488C2×C4C24×D311yescharacteristic
489C8—61no—
490C8—63no—
491C8C22×#6712yes—
492C8C22×#6716yes—
493C2×C3—26no—
494C2×C3—86no—
495C2×C3C22×#12111yescharacteristic
496C2×C3C2×#57611yescharacteristic
497C2×C3#358411yescharacteristic
498D3—22no—
499D3—43no—
500C22—23no—
501C22—23no—
502C22—32no—
503C22—46no—
504C22—46no—
505C22—46no—
506C22—61no—
507C22—61no—
508C22—63no—
509C22—63no—
510C22—63no—
511C22—1212no—
512C22—1612no—
513C22C2×#93816yes—
514C22C22×D3×D411yescharacteristic
515C4—42no—
516C4—61no—
517C4—63no—
518C4—126no—
519C4—122no—
520C4C2×#93816yes—
521C4C22×#23711yescharacteristic
522C4C23×#6711yescharacteristic
523C3#6045811yescharacteristic, complemented, Sylow
524C2—26no—
525C2—62no—
526C2—86no—
527C2—123no—
528C2D3×#57611yescharacteristic
529C2#8720311yescharacteristic
530C2C22×#93811yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#576C2
socleC22×C3C23×D4
hypercenterC2×#576D3
hypocenterC3#60458
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#576

Complements

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

NH ≅ G/Ntype
1. C2×C3×#576C2semidirect
2. #87285C2semidirect
3. C2×D3×#131C2semidirect
4. #84099C2semidirect
5. #83951C2semidirect
6. #83950C2semidirect
7. #83917C2semidirect
8. #83914C2semidirect
9. #83836C2semidirect
10. C2×#5157C2semidirect
11. #83785C2semidirect
16. C2×C3×AGL(1,ℤ/8)C22semidirect
17. C2×C3×#135C22semidirect
18. C22×C3×D8C22semidirect
21. #5833C22semidirect
23. #5828C22semidirect
27. #5805C22semidirect
29. C2×#938C22semidirect
31. C2×#907C22semidirect
32. C2×D3×M16C22semidirect
33. C2×#901C22semidirect
34. C22×D3×C8C22semidirect
35. #5233C22semidirect
36. #5232C22semidirect
37. #5215C22semidirect
38. C2×#832C22semidirect
39. #5188C22semidirect
40. #5188C22semidirect
41. #5174C22semidirect
42. #5171C22semidirect
43. #5171C22semidirect
44. #5159C22semidirect
45. #5157C22semidirect
46. #5157C22semidirect
47. #5157C22semidirect
48. #5157C22semidirect
49. #5157C22semidirect
50. #5121C22semidirect
51. #5106C22semidirect
52. C2×#818C22semidirect
53. C2×#576D3semidirect
87. C3×#131D4semidirect
136. C2×AGL(1,ℤ/8)C2×D3semidirect
137. C2×#135C2×D3semidirect
138. C22×D8C2×D3semidirect
216. C3×M16C2×D4semidirect
217. C2×C3×C8C2×D4semidirect
267. #131#67semidirect
401. M16C2×#67semidirect
409. C2×C8C2×#67semidirect
438. C2×D3#576semidirect
523. C3#60458semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5762
#872852
C2×D3×#1312
#840992
#839512
#839502
#839172
#839142
#838362
C2×#51572
#837852
#604583