Finite Group Databasebeta

← #11886590

Subgroups — #11886590

Order 1152.

1001 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5837C212yesmaximal, complemented
2#10595456C212yesmaximal, complemented
3#10592944C211yesmaximal, characteristic, complemented
4#10591677C212yesmaximal, complemented
5#10591674C212yesmaximal, complemented
6#10591665C212yesmaximal, complemented
7#10591658C211yesmaximal, characteristic, complemented
8#10591642C212yesmaximal, complemented
9#10591621C211yesmaximal, characteristic, complemented
10#84015—32nomaximal
11C32×2−1+4C2211yescharacteristic, complemented
12C3×#1009C2212yescomplemented
13#64308C2211yescharacteristic
14#64158C2212yescomplemented
15#64156C2211yescharacteristic, complemented
16C3×#950—22no—
17C3×#950C2212yes—
18C3×#949—22no—
19C3×#949C2212yescomplemented
20C3×#948C2212yescomplemented
21C3×#942—22no—
22C3×#942C2212yescomplemented
23C2×C3×#215—22no—
24C2×C3×#215C2212yescomplemented
25#63949—22no—
26#63939—22no—
27#63934—22no—
28D3×#213—22no—
29#63914C2211yescharacteristic, complemented
30C2×#3714C2212yescomplemented
31#63823C2212yes—
32#63814C2212yescomplemented
33#63664C2212yescomplemented
34#63663C2211yescharacteristic, complemented
35#63577C2212yescomplemented
36#63568C2212yescomplemented
37#63559C2211yescharacteristic, complemented
38#63558C2212yescomplemented
39#63555C2211yescharacteristic, complemented
40C3×#577—32no—
41#5837D312yescomplemented
42#5835—32no—
43#5769—32no—
44#5245—32no—
45#5245—61no—
46#5241—32no—
47#5203—32no—
48#5202—32no—
49#5199—32no—
50#5198—32no—
51#5197—32no—
52#5177—32no—
53#5176—32no—
54#5145—32no—
55#5142—32no—
56C32×#41—22no—
57C32×#41—41no—
58C32×#41C2312yes—
59C2×C32×Q8—22no—
60C2×C32×Q8C2311yescharacteristic
61C2×C4⋊(C3⋊D3)C2311yescharacteristic
62C3×#239—22no—
63C3×D3×D4—22no—
64C3×D3×D4—42no—
65C3×#235—22no—
66C3×#235C2312yes—
67C2×C3×D12—22no—
68C2×C3×D12C2312yes—
69#3811—41no—
70C2×#625—41no—
71D3×D12—22no—
72#3799—22no—
73#3755—22no—
74#3753—22no—
75#3752—41no—
76C4×C3⋊Dic3C2311yescharacteristic
77#3748C2312yes—
78C3×#216—22no—
79C3×#216D412yescomplemented
80C3×#215—22no—
81C3×#215—22no—
82C3×#215—22no—
83C3×#215—22no—
84C3×#215D412yescomplemented
85C3×#214D412yescomplemented
86C3×#213—22no—
87C3×#213D412yescomplemented
88C3×#208—22no—
89C3×#208C2312yes—
90C2×C3×C3⋊C8—22no—
91C2×C3×C3⋊C8C2312yes—
92#3715—22no—
93#3714—22no—
94#3714—22no—
95#3713—22no—
96#3711—22no—
97D3×C3⋊C8—22no—
98#3288—91noSylow
99C3×2−1+4—21no—
100C3×2−1+4C2×D312yescomplemented
101C3×H(5,𝔽2)—32no—
102#1009—32no—
103#1009C2×D312yescomplemented
104C3×#137—32no—
105C3×#137—62no—
106C3×AGL(1,ℤ/8)—32no—
107C3×AGL(1,ℤ/8)—62no—
108C3×#135—32no—
109C3×#135—62no—
110C2×C3×SD16—32no—
111C2×C3×SD16—62no—
112C3×#131—32no—
113#950—22no—
114#950—62no—
115#950C2×D312yes—
116#949—22no—
117#949—32no—
118#949—32no—
119#949—32no—
120#949—62no—
121#949—62no—
122#949C2×D312yescomplemented
123#948—32no—
124#948C2×D312yescomplemented
125#947—32no—
126#947—61no—
127#942—22no—
128#942C2×D312yescomplemented
129C2×#215—22no—
130C2×#215C2×D312yescomplemented
131#940—32no—
132#932—32no—
133#932—62no—
134C2×#213—32no—
135C2×#213—62no—
136#919—62no—
137#914—62no—
138#914—62no—
139D3×D8—62no—
140C2×D24—32no—
141#901—32no—
142#837—32no—
143#837—32no—
144#837—32no—
145#837—62no—
146#836—32no—
147#836—61no—
148#833—32no—
149#825—32no—
150#823—32no—
151#823—32no—
152#822—32no—
153#819—32no—
154#815—32no—
155C22×C3⋊D3—21no—
156C22×C3×D3—22no—
157C22×C3×D3—42no—
158C2×D32—41no—
159C32×Q8—22no—
160C32×Q8—41no—
161C32×Q8C2×D412yes—
162C32×D4—22no—
163C32×D4—41no—
164C32×D4C2×D412yes—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—42no—
168C2×C32×C4C2411yescharacteristic
169C2×C3⋊Dic3—21no—
170C4⋊(C3⋊D3)—22no—
171C3×#67—22no—
172C3×#67—42no—
173C3×D12—22no—
174C3×D12—22no—
175C3×D12—42no—
176C3×D12C2×D412yes—
177C3×C4×D3—22no—
178C3×C4×D3—42no—
179C3×Dic6C2×D412yes—
180#626—42no—
181#625—82no—
182#624—41no—
183C3⋊(C3⋊C8)—22no—
184C3×C3⋊C8—22no—
185C3×C3⋊C8C2×D412yes—
186C3×C3⋊C8C2×D412yes—
187#577—32no—
188#576—92no—
189#496—91no—
190#472—92no—
191#471—92no—
192#455—91no—
193#454—91no—
194#453—92no—
195#444—92no—
196C3×#41—22no—
197C3×#41—22no—
198C3×#41—22no—
199C3×#41—32no—
200C3×#41—42no—
201C3×#41—42no—
202C3×#41—42no—
203C3×#41—62no—
204C3×#41—62no—
205C3×#41C22×D312yes—
206C3×#41C22×D312yes—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C2×C3×Q8—21no—
210C2×C3×Q8—42no—
211C2×C3×Q8C22×D312yes—
212C2×C3×D4—32no—
213C2×C3×D4—62no—
214C2×C3×D4—62no—
215C2×C3×D4—122no—
216C2×#67—122no—
217#239—22no—
218#239—62no—
219D3×D4—22no—
220D3×D4—42no—
221D3×D4—62no—
222D3×D4—122no—
223#235—22no—
224#235—62no—
225#235C22×D312yes—
226C2×D12—22no—
227C2×D12—32no—
228C2×D12—61no—
229C2×D12—62no—
230C2×D12C22×D312yes—
231C3×Q16—32no—
232C3×Q16—62no—
233C3×SD16—32no—
234C3×SD16—32no—
235C3×SD16—62no—
236C3×SD16—62no—
237C3×SD16—62no—
238C3×SD16—62no—
239C3×D8—32no—
240C3×D8—62no—
241C3×M16—32no—
242C3×M16—62no—
243C2×C3×C8—32no—
244C2×C3×C8—62no—
245#216—22no—
246#216#6712yescomplemented
247#215—22no—
248#215—22no—
249#215—22no—
250#215—22no—
251#215—62no—
252#215—62no—
253#215—62no—
254#215—122no—
255#215#6712yescomplemented
256#214—62no—
257#214#6712yescomplemented
258#213—22no—
259#213—62no—
260#213—62no—
261#213—62no—
262#213—62no—
263#213—62no—
264#213—62no—
265#213—122no—
266#213#6712yescomplemented
267(C2×D3)⋊C4—122no—
268(C2×D3)⋊C4—122no—
269C4×Dic3—32no—
270C4×Dic3—61no—
271#208—22no—
272#208—32no—
273#208—32no—
274#208—62no—
275#208—62no—
276#208C22×D312yes—
277C2×C3⋊C8—22no—
278C2×C3⋊C8—62no—
279C2×C3⋊C8C22×D312yes—
280D24—62no—
281D24—62no—
282ΓL(1,𝔽25)—62no—
283D3×C8—62no—
284C22×C32—22no—
285C22×C32H(5,𝔽2)11yescharacteristic
286C2×C3⋊D3—41no—
287C2×C3⋊D3—81no—
288C2×C3×D3—22no—
289C2×C3×D3—42no—
290C2×C3×D3—82no—
291C2×C3×D3—162no—
292D32—161no—
293C32×C4—22no—
294C32×C4—41no—
295C32×C4C22×D412yes—
296C3⋊Dic3—41no—
297C3×Dic3—22no—
2982−1+4D3211yescharacteristic, complemented
299H(5,𝔽2)—32no—
300H(5,𝔽2)—91no—
301#137—32no—
302#137—62no—
303AGL(1,ℤ/8)—32no—
304AGL(1,ℤ/8)—62no—
305AGL(1,ℤ/8)—92no—
306AGL(1,ℤ/8)—92no—
307AGL(1,ℤ/8)—182no—
308AGL(1,ℤ/8)—182no—
309#135—32no—
310#135—62no—
311#135—182no—
312C2×SD16—32no—
313C2×SD16—62no—
314C2×D8—92no—
315C2×D8—182no—
316#131—32no—
317#131—92no—
318#127—91no—
319#124—91no—
320#108—92no—
321#104—92no—
322#104—92no—
323#101—91no—
324#100—91no—
325#100—92no—
326#97—91no—
327C23×C3—62no—
328C23×C3—122no—
329C22×D3—22no—
330C22×D3—42no—
331C22×D3—62no—
332C22×D3—121no—
333C22×D3—122no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—22no—
337C3×Q8—32no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—42no—
341C3×Q8D3×D412yes—
342C3×Q8C2×#6712yes—
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—32no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—62no—
351C3×D4—62no—
352C3×D4—62no—
353C3×D4—122no—
354C3×D4—122no—
355C3×D4D3×D412yes—
356C3×D4C2×#6712yes—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—44no—
366C2×C3×C4—62no—
367C2×C3×C4—122no—
368C2×C3×C4C23×D312yes—
369#67—22no—
370#67—42no—
371#67—122no—
372#67—244no—
373C2×Dic3—62no—
374C2×Dic3—121no—
375D12—22no—
376D12—22no—
377D12—42no—
378D12—62no—
379D12—62no—
380D12—122no—
381D12—122no—
382D12C2×#6712yes—
383C4×D3—22no—
384C4×D3—42no—
385C4×D3—122no—
386Dic6C2×#6712yes—
387C3×C8—32no—
388C3×C8—32no—
389C3×C8—62no—
390C3⋊C8—22no—
391C3⋊C8—62no—
392C3⋊C8—62no—
393C3⋊C8—122no—
394C3⋊C8C2×#6712yes—
395C3⋊C8C2×#6712yes—
396C2×C32—21no—
397C2×C32—42no—
398C2×C32D4211yescharacteristic
399C3⋊D3—81no—
400C3×D3—42no—
401C3×D3—82no—
402#41—22no—
403#41—32no—
404#41—41no—
405#41—62no—
406#41—62no—
407#41—182no—
408#41C2×D3212yes—
409C2×Q8—22no—
410C2×Q8C2×D3211yescharacteristic
411C2×D4—32no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—91no—
415C2×D4—122no—
416C2×D4—182no—
417C2×D4—361no—
418C2×D4—361no—
419Q16—32no—
420Q16—62no—
421SD16—32no—
422SD16—32no—
423SD16—62no—
424SD16—62no—
425SD16—62no—
426SD16—62no—
427SD16—182no—
428SD16—182no—
429D8—32no—
430D8—62no—
431D8—182no—
432D8—182no—
433D8—182no—
434D8—182no—
435M16—32no—
436M16—62no—
437M16—92no—
438M16—182no—
439C2×C8—32no—
440C2×C8—62no—
441C2×C8—182no—
442C22⋊C4—361no—
443C42—91no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—21no—
447C22×C3—42no—
448C22×C3—62no—
449C22×C3—122no—
450C22×C3—242no—
451C22×C3—482no—
452C22×C3#100412yes—
453C2×D3—22no—
454C2×D3—42no—
455C2×D3—82no—
456C2×D3—122no—
457C2×D3—162no—
458C2×D3—241no—
459C2×D3—242no—
460C2×D3—242no—
461C2×D3—482no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—62no—
469C3×C4C2×D3×D412yes—
470C3×C4C22×#6712yes—
471Dic3—22no—
472Dic3—122no—
473Dic3—241no—
474C32#328811yescharacteristic, complemented, Sylow
475C23—62no—
476C23—122no—
477C23—181no—
478C23—361no—
479Q8—22no—
480Q8—32no—
481Q8—41no—
482Q8D3×#6712yes—
483D4—22no—
484D4—32no—
485D4—41no—
486D4—62no—
487D4—62no—
488D4—62no—
489D4—122no—
490D4—122no—
491D4—182no—
492D4—362no—
493D4—722no—
494D4D3×#6712yes—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—42no—
498C2×C4—62no—
499C2×C4—122no—
500C2×C4—181no—
501C2×C4—361no—
502C2×C4C22×D3211yescharacteristic
503C8—32no—
504C8—32no—
505C8—62no—
506C8—182no—
507C2×C3—22no—
508C2×C3—21no—
509C2×C3—41no—
510C2×C3—42no—
511C2×C3—42no—
512C2×C3—82no—
513C2×C3—122no—
514C2×C3—242no—
515C2×C3D4×#6712yes—
516D3—42no—
517D3—82no—
518D3—242no—
519D3—481no—
520C22—22no—
521C22—62no—
522C22—122no—
523C22—242no—
524C22—361no—
525C22—482no—
526C22—721no—
527C22—1441no—
528C22#6433011yescharacteristic
529C4—22no—
530C4—41no—
531C4—62no—
532C4—361no—
533C4C2×D3×#6712yes—
534C3—21no—
535C3#8401512yescomplemented
536C2—21no—
537C2—42no—
538C2—122no—
539C2—242no—
540C2—721no—
541C2#67211yescharacteristic
subgroupG/H
centerC2#672
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×2−1+4C22
socleC2×C32D42
hypercenter2−1+4D32
hypocenterC32#3288
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×#5837C2semidirect
2. #10595456C2semidirect
3. #10592944C2semidirect
4. #10591677C2semidirect
5. #10591674C2semidirect
6. #10591665C2semidirect
7. #10591658C2semidirect
8. #10591642C2semidirect
9. #10591621C2semidirect
11. C32×2−1+4C22semidirect
12. C3×#1009C22semidirect
14. #64158C22semidirect
15. #64156C22semidirect
19. C3×#949C22semidirect
20. C3×#948C22semidirect
22. C3×#942C22semidirect
24. C2×C3×#215C22semidirect
29. #63914C22semidirect
30. C2×#3714C22semidirect
32. #63814C22semidirect
33. #63664C22semidirect
34. #63663C22semidirect
35. #63577C22semidirect
36. #63568C22semidirect
37. #63559C22semidirect
38. #63558C22semidirect
39. #63555C22semidirect
41. #5837D3semidirect
79. C3×#216D4semidirect
84. C3×#215D4semidirect
85. C3×#214D4semidirect
87. C3×#213D4semidirect
100. C3×2−1+4C2×D3semidirect
103. #1009C2×D3semidirect
122. #949C2×D3semidirect
124. #948C2×D3semidirect
128. #942C2×D3semidirect
130. C2×#215C2×D3semidirect
246. #216#67semidirect
255. #215#67semidirect
257. #214#67semidirect
266. #213#67semidirect
298. 2−1+4D32semidirect
474. C32#3288semidirect
535. C3#84015semidirect

Maximal subgroups

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

subgroupindex
C3×#58372
#105954562
#105929442
#105916772
#105916742
#105916652
#105916582
#105916422
#105916212
#840153