Finite Group Databasebeta

← #11896114

Subgroups — #11896114

Order 1152.

1074 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#559C211yesmaximal, characteristic, complemented
2#10596050C211yesmaximal, characteristic, complemented
3#10592771C211yesmaximal, characteristic
4#10592770C211yesmaximal, characteristic, complemented
5#10592756C211yesmaximal, characteristic, complemented
6#10592756C211yesmaximal, characteristic, complemented
7#10592755C211yesmaximal, characteristic, complemented
8#10592754C211yesmaximal, characteristic, complemented
9#10592725C211yesmaximal, characteristic, complemented
10#10592723C211yesmaximal, characteristic, complemented
11#10592721C211yesmaximal, characteristic, complemented
12#10592713C211yesmaximal, characteristic, complemented
13C4×#3753C211yesmaximal, characteristic, complemented
14#10592708C211yesmaximal, characteristic, complemented
15#10592706C211yesmaximal, characteristic, complemented
16#83714—34nomaximal
17C32×#127C2211yescharacteristic
18C32×#124—21no—
19C32×#124C2211yescharacteristic
20C32×#124C2211yescharacteristic
21C32×#123—21no—
22C32×#123—21no—
23C32×C4×Q8C2211yescharacteristic
24C32×C4×Q8C2211yescharacteristic
25C32×C4×D4—21no—
26C32×C4×D4C2211yescharacteristic
27C32×C4×D4C2211yescharacteristic
28#64155—21no—
29C2×#3755C2211yescharacteristic
30C2×#3753C2211yescharacteristic
31C2×#3753C2211yescharacteristic
32#64105—21no—
33#64105—21no—
34C4⋊C4×C3⋊D3—21no—
35C4×C4⋊(C3⋊D3)C2211yescharacteristic
36C4×C4⋊(C3⋊D3)C2211yescharacteristic
37C4×C4⋊(C3⋊D3)C2211yescharacteristic
38#64079C2211yescharacteristic
39#64079C2211yescharacteristic
40#64078C2211yescharacteristic
41#63662C2211yescharacteristic
42#63662C2211yescharacteristic
43#63662C2211yescharacteristic
44#63659C2211yescharacteristic
45#63659C2211yescharacteristic
46#63659C2211yescharacteristic
47#63659C2211yescharacteristic
48#63659C2211yescharacteristic
49#63659C2211yescharacteristic
50#63636C2211yescharacteristic
51#63635C2211yescharacteristic
52#63634C2211yescharacteristic
53#63634C2211yescharacteristic
54C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
55C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
56C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
57#63630C2211yescharacteristic
58#63630C2211yescharacteristic
59C4×C3⋊(C3⋊C8)C2211yescharacteristic
60C3×#559D314yescomplemented
61#5579—34no—
62#5072—34no—
63#5071—34no—
64#5057—34no—
65#5057—34no—
66#5056—34no—
67#5055—34no—
68#5026—34no—
69#5024—34no—
70#5022—34no—
71#5014—34no—
72C4×#213—34no—
73#5009—34no—
74#5007—34no—
75C2×C32×Q8D411yescharacteristic
76C2×C32×Q8C2311yescharacteristic
77C2×C32×D4—21no—
78C2×C32×D4—21no—
79C2×C32×D4C2311yescharacteristic
80C2×C32×D4C2311yescharacteristic
81C22×C32×C4—21no—
82C22×C32×C4—21no—
83C2×C4⋊(C3⋊D3)C2311yescharacteristic
84C2×C4×C3⋊D3—21no—
85C2×C4×C3⋊D3—21no—
86C2×C4×C3⋊D3—21no—
87C2×#634—21no—
88C32×C4⋊C4—21no—
89C32×C4⋊C4D411yescharacteristic
90C32×C4⋊C4D411yescharacteristic
91C32×C4⋊C4D411yescharacteristic
92C32×C4⋊C4C2311yescharacteristic
93C32×C22⋊C4—21no—
94C32×C22⋊C4—21no—
95C32×C22⋊C4—21no—
96C32×C22⋊C4—21no—
97C32×C22⋊C4—21no—
98C32×C22⋊C4—21no—
99C32×C42—21no—
100C32×C42C2311yescharacteristic
101C32×C42C2311yescharacteristic
102C32×C42C2311yescharacteristic
103#3755—22no—
104#3753—22no—
105#3753—22no—
106#3752—21no—
107#3752—21no—
108#3752—21no—
109C4⋊(C3⋊Dic3)—21no—
110C4⋊(C3⋊Dic3)C2311yescharacteristic
111C4⋊(C3⋊Dic3)C2311yescharacteristic
112C4⋊(C3⋊Dic3)C2311yescharacteristic
113#3750—21no—
114#3750—21no—
115#3750—21no—
116#3750—21no—
117C2×C3⋊(C3⋊C8)C2311yescharacteristic
118C2×C3⋊(C3⋊C8)C2311yescharacteristic
119C2×C3⋊(C3⋊C8)C2311yescharacteristic
120C2×C3⋊(C3⋊C8)C2311yescharacteristic
121#3370—91noSylow
122C3×#127C2×D314yes—
123C3×#124—24no—
124C3×#124C2×D314yes—
125C3×#124C2×D314yes—
126C3×#123—24no—
127C3×#123—24no—
128C3×C4×Q8C2×D314yes—
129C3×C4×Q8C2×D314yes—
130C3×C4×D4—24no—
131C3×C4×D4C2×D314yes—
132C3×C4×D4C2×D314yes—
133#946—64no—
134C2×#215—34no—
135C2×#213—34no—
136C2×#213—34no—
137#896—64no—
138#896—64no—
139D3×C4⋊C4—64no—
140C4×D12—34no—
141C4×D12—34no—
142C4×D12—34no—
143#870—34no—
144#870—34no—
145#869—34no—
146#835—34no—
147#835—34no—
148#835—34no—
149#832—34no—
150#832—34no—
151#832—34no—
152#832—34no—
153#832—34no—
154#832—34no—
155#809—34no—
156#808—34no—
157#807—34no—
158#807—34no—
159C4⋊(C3⋊C8)—34no—
160C4⋊(C3⋊C8)—34no—
161C4⋊(C3⋊C8)—34no—
162#803—34no—
163#803—34no—
164C4×C3⋊C8—34no—
165C23×C32—21no—
166C23×C32—21no—
167C22×C3⋊D3—21no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×Q8—41no—
171C32×D4—21no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—42no—
176C32×D4—81no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—81no—
180C2×C32×C4—81no—
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2×D411yescharacteristic
183C2×C32×C4C2×D411yescharacteristic
184C2×C32×C4C2×D411yescharacteristic
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2411yescharacteristic
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—21no—
191C4⋊(C3⋊D3)—21no—
192C4⋊(C3⋊D3)#4112yes—
193C4×C3⋊D3—42no—
194C4×C3⋊D3—81no—
195C4×C3⋊D3—81no—
196#634—81no—
197C3⋊(C3⋊C8)—21no—
198C3⋊(C3⋊C8)—21no—
199C3⋊(C3⋊C8)—21no—
200C3⋊(C3⋊C8)—21no—
201#559C3⋊D311yescharacteristic, complemented
202#554—91no—
203#490—91no—
204#489—91no—
205#488—91no—
206#479—91no—
207#478—91no—
208#474—91no—
209#463—91no—
210#463—91no—
211#459—91no—
212#446—91no—
213#442—91no—
214#440—91no—
215C4×D8—91no—
216C2×C3×Q8#6714yes—
217C2×C3×Q8C22×D314yes—
218C2×C3×D4—24no—
219C2×C3×D4—24no—
220C2×C3×D4C22×D314yes—
221C2×C3×D4C22×D314yes—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C2×D12—34no—
225C2×C4×D3—64no—
226C2×C4×D3—64no—
227C2×C4×D3—64no—
228C2×Dic6—64no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4#6714yes—
231C3×C4⋊C4#6714yes—
232C3×C4⋊C4#6714yes—
233C3×C4⋊C4C22×D314yes—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C42—24no—
241C3×C42C22×D314yes—
242C3×C42C22×D314yes—
243C3×C42C22×D314yes—
244#215—68no—
245#213—68no—
246#213—68no—
247(C2×D3)⋊C4—64no—
248(C2×D3)⋊C4—64no—
249(C2×D3)⋊C4—64no—
250C4⋊Dic3—34no—
251C4⋊Dic3—34no—
252C4⋊Dic3—34no—
253C4⋊Dic3—64no—
254Dic3⋊C4—64no—
255Dic3⋊C4—64no—
256Dic3⋊C4—64no—
257Dic3⋊C4—64no—
258C2×C3⋊C8—34no—
259C2×C3⋊C8—34no—
260C2×C3⋊C8—34no—
261C2×C3⋊C8—34no—
262C22×C32—41no—
263C22×C32—41no—
264C22×C32—41no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32C22×D411yescharacteristic
269C2×C3⋊D3—22no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4C2×#4111yescharacteristic
281C32×C42−1+411yescharacteristic
282C3⋊Dic3—41no—
283C3⋊Dic3—41no—
284C3⋊Dic3—41no—
285C2×SD16—91no—
286C2×D8—91no—
287C2×D8—91no—
288C4⋊Q8—91no—
289#127C2×C3⋊D311yescharacteristic
290#125—91no—
291#125—91no—
292#124—21no—
293#124C2×C3⋊D311yescharacteristic
294#124C2×C3⋊D311yescharacteristic
295#123—21no—
296#123—21no—
297C22⋊Q8—181no—
298C22⋊Q8—181no—
299C22⋊Q8—181no—
300C4×Q8C2×C3⋊D311yescharacteristic
301C4×Q8C2×C3⋊D311yescharacteristic
302C4×D4—21no—
303C4×D4—91no—
304C4×D4—91no—
305C4×D4—91no—
306C4×D4C2×C3⋊D311yescharacteristic
307C4×D4C2×C3⋊D311yescharacteristic
308C2×C4⋊C4—181no—
309#107—91no—
310#107—91no—
311#106—91no—
312C4⋊C8—91no—
313C4⋊C8—91no—
314C4⋊C8—91no—
315#103—91no—
316#103—91no—
317#103—91no—
318#102—91no—
319#102—91no—
320#102—91no—
321#102—91no—
322#102—91no—
323#102—91no—
324#102—91no—
325#97—91no—
326#97—91no—
327C4×C8—91no—
328C23×C3—24no—
329C23×C3—24no—
330C22×D3—64no—
331C3×Q8—24no—
332C3×Q8—24no—
333C3×Q8—44no—
334C3×D4—24no—
335C3×D4—24no—
336C3×D4—24no—
337C3×D4—24no—
338C3×D4—48no—
339C3×D4—84no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—84no—
343C2×C3×C4—84no—
344C2×C3×C4C2×#6714yes—
345C2×C3×C4C2×#6714yes—
346C2×C3×C4C2×#6714yes—
347C2×C3×C4C2×#6714yes—
348C2×C3×C4C2×#6714yes—
349C2×C3×C4C2×#6714yes—
350C2×C3×C4C23×D314yes—
351C2×Dic3—64no—
352C2×Dic3—64no—
353C2×Dic3—64no—
354D12—38no—
355D12—64no—
356C4×D3—128no—
357C4×D3—244no—
358C4×D3—244no—
359Dic6—244no—
360C3⋊C8—64no—
361C3⋊C8—64no—
362C3⋊C8—64no—
363C3⋊C8—64no—
364C2×C32—81no—
365C2×C32—81no—
366C2×C32#54811yescharacteristic
367C2×C32#57511yescharacteristic
368C2×C32#57611yescharacteristic
369C3⋊D3—42no—
370C2×Q8—181no—
371C2×Q8#63811yescharacteristic
372C2×Q8C22×C3⋊D311yescharacteristic
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—91no—
376C2×D4C22×C3⋊D311yescharacteristic
377C2×D4C22×C3⋊D311yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—181no—
381C22×C4—181no—
382C22×C4—181no—
383SD16—182no—
384D8—182no—
385D8—182no—
386C2×C8—91no—
387C2×C8—91no—
388C2×C8—91no—
389C2×C8—91no—
390C4⋊C4—21no—
391C4⋊C4—91no—
392C4⋊C4—91no—
393C4⋊C4—91no—
394C4⋊C4—181no—
395C4⋊C4—181no—
396C4⋊C4—181no—
397C4⋊C4—181no—
398C4⋊C4—181no—
399C4⋊C4#63811yescharacteristic
400C4⋊C4#63811yescharacteristic
401C4⋊C4#63811yescharacteristic
402C4⋊C4C22×C3⋊D311yescharacteristic
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—181no—
410C22⋊C4—181no—
411C22⋊C4—181no—
412C42—21no—
413C42C22×C3⋊D311yescharacteristic
414C42C22×C3⋊D311yescharacteristic
415C42C22×C3⋊D311yescharacteristic
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3C22×#6714yes—
423C2×D3—68no—
424C2×D3—124no—
425C2×D3—124no—
426C3×C4—24no—
427C3×C4—24no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—44no—
433C3×C4—44no—
434C3×C4D3×#4114yes—
435C3×C4#101014yes—
436Dic3—124no—
437Dic3—124no—
438Dic3—124no—
439C32#337011yescharacteristic, complemented, Sylow
440C23—21no—
441C23—21no—
442C23—181no—
443Q8—21no—
444Q8—21no—
445Q8—41no—
446Q8—721no—
447D4—21no—
448D4—21no—
449D4—21no—
450D4—21no—
451D4—42no—
452D4—81no—
453D4—92no—
454D4—181no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—362no—
463C2×C4—721no—
464C2×C4—721no—
465C2×C4C2×#63811yescharacteristic
466C2×C4C2×#63811yescharacteristic
467C2×C4C2×#63811yescharacteristic
468C2×C4C2×#63811yescharacteristic
469C2×C4C2×#63811yescharacteristic
470C2×C4C2×#63811yescharacteristic
471C2×C4C23×C3⋊D311yescharacteristic
472C8—181no—
473C8—181no—
474C8—181no—
475C8—181no—
476C2×C3—84no—
477C2×C3—84no—
478C2×C3#582214yes—
479C2×C3#583114yes—
480C2×C3#583514yes—
481D3—128no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—182no—
489C22—361no—
490C22—361no—
491C22C22×#63811yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—41no—
499C4—41no—
500C4—361no—
501C4—361no—
502C4—361no—
503C4#41×C3⋊D311yescharacteristic
504C4#6436711yescharacteristic
505C3#8371414yescomplemented
506C2—81no—
507C2—81no—
508C2—362no—
509C2#1059629311yescharacteristic
510C2#1059630211yescharacteristic
511C2#1059630611yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#559C2
socleC22×C32C22×D4
prime radical#10592771C2
hypercenter#559C3⋊D3
hypocenterC32#3370
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. C32×#559C2semidirect
2. #10596050C2semidirect
4. #10592770C2semidirect
5. #10592756C2semidirect
6. #10592756C2semidirect
7. #10592755C2semidirect
8. #10592754C2semidirect
9. #10592725C2semidirect
10. #10592723C2semidirect
11. #10592721C2semidirect
12. #10592713C2semidirect
13. C4×#3753C2semidirect
14. #10592708C2semidirect
15. #10592706C2semidirect
60. C3×#559D3semidirect
201. #559C3⋊D3semidirect
439. C32#3370semidirect
505. C3#83714semidirect

Maximal subgroups

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

subgroupindex
C32×#5592
#105960502
#105927712
#105927702
#105927562
#105927552
#105927542
#105927252
#105927232
#105927212
#105927132
C4×#37532
#105927082
#105927062
#837143