Finite Group Databasebeta

← #11886912

Subgroups — #11886912

Order 1152.

1036 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1036 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×#5835C212yesmaximal, complemented
2#10595456C212yesmaximal, complemented
3#10592940C211yesmaximal, characteristic, complemented
4#10591677C212yesmaximal, complemented
5#10591673C212yesmaximal, complemented
6#10591665C212yesmaximal, complemented
7#10591522C211yesmaximal, characteristic, complemented
8#10591505C212yesmaximal, complemented
9#10591484C211yesmaximal, characteristic, complemented
10#84010—32nomaximal
11C32×H(5,𝔽2)C2211yescharacteristic, complemented
12C3×#1009C2212yescomplemented
13#64308C2211yescharacteristic, complemented
14#64158C2212yescomplemented
15#64149C2211yescharacteristic, complemented
16C3×#950—22no—
17C3×#950C2212yes—
18C3×#949—22no—
19C3×#949C2212yescomplemented
20C3×#948C2212yescomplemented
21C3×#932—22no—
22C3×#932C2212yescomplemented
23C2×C3×#213—22no—
24C2×C3×#213C2212yescomplemented
25#63949—22no—
26#63939—22no—
27#63934—22no—
28D3×#213—22no—
29#63914C2211yescharacteristic, complemented
30C2×#3714C2212yescomplemented
31#63823C2212yescomplemented
32#63814C2212yescomplemented
33#63664C2212yescomplemented
34#63660C2211yescharacteristic, complemented
35#63577C2212yescomplemented
36#63568C2212yescomplemented
37#63559C2211yescharacteristic, complemented
38#63558C2212yescomplemented
39#63553C2211yescharacteristic, complemented
40C3×#576—32no—
41#5835—32no—
42#5835D312yescomplemented
43#5769—32no—
44#5241—32no—
45#5241—32no—
46#5241—61no—
47#5199—32no—
48#5199—32no—
49#5198—32no—
50#5198—32no—
51#5196—32no—
52#5160—32no—
53#5151—32no—
54#5144—32no—
55#5142—32no—
56C32×#41—22no—
57C32×#41C2312yes—
58C2×C32×D4—22no—
59C2×C32×D4—41no—
60C2×C32×D4C2311yescharacteristic
61C2×#638—41no—
62C2×C4⋊(C3⋊D3)C2311yescharacteristic
63C3×#239—22no—
64C3×D3×D4—22no—
65C3×D3×D4—42no—
66C3×#235—22no—
67C3×#235C2312yes—
68C2×C3×D12—22no—
69C2×C3×D12C2312yes—
70#3811—41no—
71C2×#625—41no—
72D3×D12—22no—
73#3799—22no—
74#3755—22no—
75#3753—22no—
76C4×C3⋊Dic3C2311yescharacteristic
77#3748C2312yes—
78C3×#216D412yescomplemented
79C3×#215—22no—
80C3×#215D412yescomplemented
81C3×#214—22no—
82C3×#214D412yescomplemented
83C3×#213—22no—
84C3×#213—22no—
85C3×#213—22no—
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#3292—91noSylow
99C3×H(5,𝔽2)—21no—
100C3×H(5,𝔽2)—32no—
101C3×H(5,𝔽2)C2×D312yescomplemented
102#1009—32no—
103#1009C2×D312yescomplemented
104C3×AGL(1,ℤ/8)—32no—
105C3×AGL(1,ℤ/8)—32no—
106C3×AGL(1,ℤ/8)—62no—
107C3×AGL(1,ℤ/8)—62no—
108C3×#135—32no—
109C3×#135—62no—
110C2×C3×D8—32no—
111C2×C3×D8—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#940—32no—
126#940—32no—
127#940—61no—
128#932—22no—
129#932—32no—
130#932—62no—
131#932C2×D312yescomplemented
132C2×#213—22no—
133C2×#213—32no—
134C2×#213—62no—
135C2×#213C2×D312yescomplemented
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#833—32no—
147#833—32no—
148#833—61no—
149#825—32no—
150#823—32no—
151#823—32no—
152#822—32no—
153#819—32no—
154C8×Dic3—32no—
155C23×C32—21no—
156C23×C32—41no—
157C22×C3⋊D3—21no—
158C22×C3×D3—22no—
159C22×C3×D3—42no—
160C2×D32—41no—
161C32×Q8C2×D412yes—
162C32×D4—22no—
163C32×D4—22no—
164C32×D4—42no—
165C32×D4C2×D412yes—
166C2×C32×C4—22no—
167C2×C32×C4—41no—
168C2×C32×C4C2411yescharacteristic
169#638—82no—
170C2×C3⋊Dic3—21no—
171C4⋊(C3⋊D3)—22no—
172C3×#67—22no—
173C3×#67—42no—
174C3×D12—22no—
175C3×D12—22no—
176C3×D12—42no—
177C3×D12C2×D412yes—
178C3×C4×D3—22no—
179C3×C4×D3—42no—
180C3×Dic6C2×D412yes—
181#626—42no—
182#625—82no—
183#624—41no—
184C3⋊(C3⋊C8)—22no—
185C3×C3⋊C8—22no—
186C3×C3⋊C8C2×D412yes—
187C3×C3⋊C8C2×D412yes—
188#576—32no—
189#576—92no—
190#493—91no—
191#472—92no—
192#472—92no—
193#453—91no—
194#453—92no—
195#453—91no—
196#443—92no—
197C3×#41—22no—
198C3×#41—22no—
199C3×#41—22no—
200C3×#41—32no—
201C3×#41—42no—
202C3×#41—62no—
203C3×#41—62no—
204C3×#41C22×D312yes—
205C3×#41C22×D312yes—
206C2×C3×D4—22no—
207C2×C3×D4—22no—
208C2×C3×D4—21no—
209C2×C3×D4—32no—
210C2×C3×D4—42no—
211C2×C3×D4—42no—
212C2×C3×D4—42no—
213C2×C3×D4—62no—
214C2×C3×D4—62no—
215C2×C3×D4—122no—
216C2×C3×D4C22×D312yes—
217C2×#67—122no—
218C2×#67—122no—
219C2×#67—122no—
220#239—22no—
221#239—62no—
222D3×D4—22no—
223D3×D4—42no—
224D3×D4—62no—
225D3×D4—122no—
226#235—22no—
227#235—62no—
228#235C22×D312yes—
229C2×D12—22no—
230C2×D12—32no—
231C2×D12—61no—
232C2×D12—62no—
233C2×D12C22×D312yes—
234C3×Q16—32no—
235C3×SD16—32no—
236C3×SD16—32no—
237C3×SD16—62no—
238C3×SD16—62no—
239C3×D8—32no—
240C3×D8—62no—
241C3×D8—62no—
242C3×D8—62no—
243C3×D8—62no—
244C3×M16—32no—
245C3×M16—62no—
246C2×C3×C8—32no—
247C2×C3×C8—62no—
248#216#6712yescomplemented
249#215—22no—
250#215—62no—
251#215—62no—
252#215—62no—
253#215—122no—
254#215#6712yescomplemented
255#214—22no—
256#214—62no—
257#214#6712yescomplemented
258#213—22no—
259#213—22no—
260#213—22no—
261#213—22no—
262#213—62no—
263#213—62no—
264#213—62no—
265#213—62no—
266#213—62no—
267#213—62no—
268#213—122no—
269#213#6712yescomplemented
270C4×Dic3—32no—
271C4×Dic3—61no—
272#208—22no—
273#208—32no—
274#208—32no—
275#208—62no—
276#208—62no—
277#208C22×D312yes—
278C2×C3⋊C8—22no—
279C2×C3⋊C8—62no—
280C2×C3⋊C8C22×D312yes—
281D24—62no—
282D24—62no—
283ΓL(1,𝔽25)—62no—
284D3×C8—62no—
285C22×C32—22no—
286C22×C32—41no—
287C22×C32—81no—
288C22×C32—161no—
289C22×C32H(5,𝔽2)11yescharacteristic
290C2×C3⋊D3—41no—
291C2×C3⋊D3—81no—
292C2×C3×D3—22no—
293C2×C3×D3—42no—
294C2×C3×D3—82no—
295C2×C3×D3—162no—
296D32—161no—
297C32×C4—22no—
298C32×C4C22×D412yes—
299C3⋊Dic3—22no—
300C3×Dic3—22no—
301H(5,𝔽2)—32no—
302H(5,𝔽2)—91no—
303H(5,𝔽2)D3211yescharacteristic, complemented
304AGL(1,ℤ/8)—32no—
305AGL(1,ℤ/8)—32no—
306AGL(1,ℤ/8)—62no—
307AGL(1,ℤ/8)—62no—
308AGL(1,ℤ/8)—92no—
309AGL(1,ℤ/8)—92no—
310AGL(1,ℤ/8)—182no—
311AGL(1,ℤ/8)—182no—
312#135—32no—
313#135—62no—
314#135—182no—
315C2×D8—32no—
316C2×D8—62no—
317C2×D8—92no—
318C2×D8—182no—
319#131—32no—
320#131—92no—
321#127—91no—
322#127—91no—
323#108—92no—
324#104—92no—
325#104—92no—
326#100—91no—
327#100—92no—
328#100—91no—
329C4×C8—91no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—62no—
335C23×C3—122no—
336C22×D3—22no—
337C22×D3—42no—
338C22×D3—62no—
339C22×D3—121no—
340C22×D3—122no—
341C3×Q8—22no—
342C3×Q8—32no—
343C3×Q8D3×D412yes—
344C3×Q8C2×#6712yes—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—22no—
349C3×D4—22no—
350C3×D4—32no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—42no—
355C3×D4—44no—
356C3×D4—62no—
357C3×D4—62no—
358C3×D4—62no—
359C3×D4—122no—
360C3×D4—122no—
361C3×D4D3×D412yes—
362C3×D4C2×#6712yes—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—62no—
370C2×C3×C4—122no—
371C2×C3×C4C23×D312yes—
372#67—22no—
373#67—42no—
374#67—122no—
375#67—244no—
376#67—244no—
377#67—244no—
378C2×Dic3—62no—
379C2×Dic3—121no—
380D12—22no—
381D12—22no—
382D12—42no—
383D12—62no—
384D12—62no—
385D12—122no—
386D12—122no—
387D12C2×#6712yes—
388C4×D3—22no—
389C4×D3—42no—
390C4×D3—122no—
391Dic6C2×#6712yes—
392C3×C8—32no—
393C3×C8—32no—
394C3×C8—62no—
395C3⋊C8—22no—
396C3⋊C8—62no—
397C3⋊C8—62no—
398C3⋊C8—122no—
399C3⋊C8C2×#6712yes—
400C3⋊C8C2×#6712yes—
401C2×C32—21no—
402C2×C32—42no—
403C2×C32—81no—
404C2×C32D4211yescharacteristic
405C3⋊D3—81no—
406C3×D3—42no—
407C3×D3—82no—
408#41—22no—
409#41—32no—
410#41—62no—
411#41—62no—
412#41—182no—
413#41C2×D3212yes—
414C2×D4—22no—
415C2×D4—32no—
416C2×D4—41no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—91no—
420C2×D4—122no—
421C2×D4—182no—
422C2×D4—361no—
423C2×D4—361no—
424C2×D4—361no—
425C2×D4C2×D3211yescharacteristic
426Q16—32no—
427SD16—32no—
428SD16—32no—
429SD16—62no—
430SD16—62no—
431SD16—182no—
432SD16—182no—
433D8—32no—
434D8—62no—
435D8—62no—
436D8—62no—
437D8—62no—
438D8—182no—
439D8—182no—
440D8—182no—
441D8—182no—
442M16—32no—
443M16—62no—
444M16—92no—
445M16—182no—
446C2×C8—32no—
447C2×C8—62no—
448C2×C8—182no—
449C42—91no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—62no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—122no—
460C22×C3—162no—
461C22×C3—162no—
462C22×C3—242no—
463C22×C3—482no—
464C22×C3#100412yes—
465C2×D3—22no—
466C2×D3—42no—
467C2×D3—82no—
468C2×D3—122no—
469C2×D3—162no—
470C2×D3—241no—
471C2×D3—242no—
472C2×D3—242no—
473C2×D3—482no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—42no—
478C3×C4—62no—
479C3×C4C2×D3×D412yes—
480C3×C4C22×#6712yes—
481Dic3—22no—
482Dic3—64no—
483Dic3—122no—
484C32#329211yescharacteristic, complemented, Sylow
485C23—21no—
486C23—41no—
487C23—62no—
488C23—122no—
489C23—181no—
490C23—361no—
491Q8—32no—
492Q8D3×#6712yes—
493D4—22no—
494D4—22no—
495D4—32no—
496D4—42no—
497D4—62no—
498D4—62no—
499D4—62no—
500D4—122no—
501D4—122no—
502D4—182no—
503D4—362no—
504D4—722no—
505D4—722no—
506D4D3×#6712yes—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—62no—
510C2×C4—122no—
511C2×C4—181no—
512C2×C4—361no—
513C2×C4C22×D3211yescharacteristic
514C8—32no—
515C8—32no—
516C8—62no—
517C8—182no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—41no—
521C2×C3—42no—
522C2×C3—42no—
523C2×C3—82no—
524C2×C3—82no—
525C2×C3—82no—
526C2×C3—122no—
527C2×C3—242no—
528C2×C3D4×#6712yes—
529D3—42no—
530D3—82no—
531D3—242no—
532D3—481no—
533C22—22no—
534C22—41no—
535C22—62no—
536C22—81no—
537C22—122no—
538C22—161no—
539C22—242no—
540C22—361no—
541C22—482no—
542C22—721no—
543C22—1441no—
544C22#6433011yescharacteristic
545C4—22no—
546C4—62no—
547C4—182no—
548C4C2×D3×#6712yes—
549C3—21no—
550C3#8401012yescomplemented
551C2—21no—
552C2—42no—
553C2—81no—
554C2—122no—
555C2—242no—
556C2—721no—
557C2#67211yescharacteristic
subgroupG/H
centerC2#672
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×H(5,𝔽2)C22
socleC2×C32D42
hypercenterH(5,𝔽2)D32
hypocenterC32#3292
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×#5835C2semidirect
2. #10595456C2semidirect
3. #10592940C2semidirect
4. #10591677C2semidirect
5. #10591673C2semidirect
6. #10591665C2semidirect
7. #10591522C2semidirect
8. #10591505C2semidirect
9. #10591484C2semidirect
11. C32×H(5,𝔽2)C22semidirect
12. C3×#1009C22semidirect
13. #64308C22semidirect
14. #64158C22semidirect
15. #64149C22semidirect
19. C3×#949C22semidirect
20. C3×#948C22semidirect
22. C3×#932C22semidirect
24. C2×C3×#213C22semidirect
29. #63914C22semidirect
30. C2×#3714C22semidirect
31. #63823C22semidirect
32. #63814C22semidirect
33. #63664C22semidirect
34. #63660C22semidirect
35. #63577C22semidirect
36. #63568C22semidirect
37. #63559C22semidirect
38. #63558C22semidirect
39. #63553C22semidirect
42. #5835D3semidirect
78. C3×#216D4semidirect
80. C3×#215D4semidirect
82. C3×#214D4semidirect
87. C3×#213D4semidirect
101. C3×H(5,𝔽2)C2×D3semidirect
103. #1009C2×D3semidirect
122. #949C2×D3semidirect
124. #948C2×D3semidirect
131. #932C2×D3semidirect
135. C2×#213C2×D3semidirect
248. #216#67semidirect
254. #215#67semidirect
257. #214#67semidirect
269. #213#67semidirect
303. H(5,𝔽2)D32semidirect
484. C32#3292semidirect
550. C3#84010semidirect

Maximal subgroups

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

subgroupindex
C3×#58352
#105954562
#105929402
#105916772
#105916732
#105916652
#105915222
#105915052
#105914842
#840103