Finite Group Databasebeta

← #11872563

Subgroups — #11872563

Order 1152.

1110 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#519C211yesmaximal, characteristic, complemented
2#10596005C211yesmaximal, characteristic
3#10592927C211yesmaximal, characteristic, complemented
4#10592926C211yesmaximal, characteristic
5#10592729C211yesmaximal, characteristic, complemented
6C4×#3756C211yesmaximal, characteristic, complemented
7#10592727C211yesmaximal, characteristic, complemented
8#10592727C211yesmaximal, characteristic, complemented
9#10592725C211yesmaximal, characteristic, complemented
10C4×#3755C211yesmaximal, characteristic, complemented
11#10592720C211yesmaximal, characteristic, complemented
12#10592719C211yesmaximal, characteristic, complemented
13#10592702C211yesmaximal, characteristic
14#10592699C211yesmaximal, characteristic, complemented
15#10592695C211yesmaximal, characteristic, complemented
16#82552—34nomaximal
17C22×C32×Q8C2211yescharacteristic
18C32×C4×Q8—21no—
19C32×C4×Q8—21no—
20C32×C4×Q8C2211yescharacteristic
21C32×C4×Q8C2211yescharacteristic
22C32×C4×Q8C2211yescharacteristic
23C32×C4×Q8C2211yescharacteristic
24C32×#117—21no—
25C32×#117—21no—
26C32×#117C2211yescharacteristic
27C32×#117C2211yescharacteristic
28C2×#3756C2211yescharacteristic
29C2×#3756C2211yescharacteristic
30C2×#3755C2211yescharacteristic
31C2×#3755C2211yescharacteristic
32#64139C2211yescharacteristic
33#64133C2211yescharacteristic
34#64107—21no—
35#64105—21no—
36#64094—21no—
37#64092—21no—
38#64084C2211yescharacteristic
39C4×C4⋊(C3⋊D3)C2211yescharacteristic
40#64078C2211yescharacteristic
41C4×#634C2211yescharacteristic
42#63662C2211yescharacteristic
43#63662C2211yescharacteristic
44#63662C2211yescharacteristic
45#63662C2211yescharacteristic
46#63657C2211yescharacteristic
47#63657C2211yescharacteristic
48#63637C2211yescharacteristic
49#63637C2211yescharacteristic
50#63636C2211yescharacteristic
51#63636C2211yescharacteristic
52#63635C2211yescharacteristic
53#63635C2211yescharacteristic
54#63634C2211yescharacteristic
55#63634C2211yescharacteristic
56C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
57C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
58#63630C2211yescharacteristic
59C4×C3⋊(C3⋊C8)C2211yescharacteristic
60C3×#519D314yescomplemented
61#5534—34no—
62#5228—34no—
63#5227—34no—
64#5030—34no—
65C4×#216—34no—
66#5028—34no—
67#5028—34no—
68#5026—34no—
69C4×#215—34no—
70#5021—34no—
71#5020—34no—
72#5003—34no—
73#5000—34no—
74#4996—34no—
75C2×C32×Q8—22no—
76C2×C32×Q8—22no—
77C2×C32×Q8—21no—
78C2×C32×Q8C2311yescharacteristic
79C2×C32×Q8C2311yescharacteristic
80C22×C32×C4—21no—
81C22×C32×C4C2311yescharacteristic
82C2×#638—21no—
83C2×C4⋊(C3⋊D3)C2311yescharacteristic
84C2×C4×C3⋊D3—21no—
85C2×#634—21no—
86C2×#634C2311yescharacteristic
87C32×C4⋊C4—21no—
88C32×C4⋊C4—21no—
89C32×C4⋊C4—21no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4D411yescharacteristic
92C32×C4⋊C4D411yescharacteristic
93C32×C4⋊C4C2311yescharacteristic
94C32×C4⋊C4C2311yescharacteristic
95C32×C22⋊C4—21no—
96C32×C22⋊C4D411yescharacteristic
97C32×C22⋊C4D411yescharacteristic
98C32×C42—21no—
99C32×C42—21no—
100C32×C42—21no—
101C32×C42—21no—
102C32×C42—21no—
103C32×C42C2311yescharacteristic
104C32×C42C2311yescharacteristic
105#3757—21no—
106#3756—22no—
107#3756—22no—
108#3755—22no—
109#3755—22no—
110#3752—21no—
111#3752—21no—
112#3752—21no—
113#3752—21no—
114C4⋊(C3⋊Dic3)—21no—
115C4⋊(C3⋊Dic3)C2311yescharacteristic
116C4⋊(C3⋊Dic3)C2311yescharacteristic
117#3750—21no—
118#3750—21no—
119C4×C3⋊Dic3—21no—
120C2×C3⋊(C3⋊C8)C2311yescharacteristic
121C2×C3⋊(C3⋊C8)C2311yescharacteristic
122C2×C3⋊(C3⋊C8)C2311yescharacteristic
123C2×C3⋊(C3⋊C8)C2311yescharacteristic
124#3121—91noSylow
125C22×C3×Q8C2×D314yes—
126C3×C4×Q8—24no—
127C3×C4×Q8—24no—
128C3×C4×Q8C2×D314yes—
129C3×C4×Q8C2×D314yes—
130C3×C4×Q8C2×D314yes—
131C3×C4×Q8C2×D314yes—
132C3×#117—24no—
133C3×#117—24no—
134C3×#117C2×D314yes—
135C3×#117C2×D314yes—
136C2×#216—34no—
137C2×#216—34no—
138C2×#215—34no—
139C2×#215—34no—
140#930—34no—
141#924—34no—
142#898—64no—
143#896—64no—
144#885—64no—
145#883—64no—
146#875—34no—
147C4×D12—34no—
148#869—34no—
149C4×Dic6—34no—
150#835—34no—
151#835—34no—
152#835—34no—
153#835—34no—
154#830—34no—
155#830—34no—
156#810—34no—
157#810—34no—
158#809—34no—
159#809—34no—
160#808—34no—
161#808—34no—
162#807—34no—
163#807—34no—
164C4⋊(C3⋊C8)—34no—
165C4⋊(C3⋊C8)—34no—
166#803—34no—
167C4×C3⋊C8—34no—
168C23×C32C2×D411yescharacteristic
169C22×C3⋊D3—21no—
170C32×Q8—24no—
171C32×Q8—21no—
172C32×Q8—21no—
173C32×Q8#4112yes—
174C32×Q8#4112yes—
175C2×C32×C4—24no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4C2×D411yescharacteristic
183C2×C32×C4C2×D411yescharacteristic
184C2×C32×C4C2×D411yescharacteristic
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2411yescharacteristic
188#638—81no—
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—21no—
191C2×C3⋊Dic3—21no—
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)—21no—
194C4×C3⋊D3—81no—
195#634—21no—
196#634—21no—
197#634—81no—
198C3⋊(C3⋊C8)—21no—
199C3⋊(C3⋊C8)—21no—
200C3⋊(C3⋊C8)—21no—
201C3⋊(C3⋊C8)—21no—
202#542—91no—
203#519C3⋊D311yescharacteristic, complemented
204#485—91no—
205#482—91no—
206#477—91no—
207#475—91no—
208#464—91no—
209#464—91no—
210#451—91no—
211#448—91no—
212#441—91no—
213#440—91no—
214C4×Q16—91no—
215C4×SD16—91no—
216#433—91no—
217C2×C3×Q8—28no—
218C2×C3×Q8—28no—
219C2×C3×Q8—24no—
220C2×C3×Q8C22×D314yes—
221C2×C3×Q8C22×D314yes—
222C22×C3×C4—24no—
223C22×C3×C4C22×D314yes—
224C2×#67—64no—
225C2×D12—34no—
226C2×C4×D3—64no—
227C2×Dic6—34no—
228C2×Dic6—64no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4#6714yes—
234C3×C4⋊C4#6714yes—
235C3×C4⋊C4C22×D314yes—
236C3×C4⋊C4C22×D314yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4#6714yes—
239C3×C22⋊C4#6714yes—
240C3×C42—24no—
241C3×C42—24no—
242C3×C42—24no—
243C3×C42—24no—
244C3×C42—24no—
245C3×C42C22×D314yes—
246C3×C42C22×D314yes—
247#217—64no—
248#216—68no—
249#216—68no—
250#215—68no—
251#215—68no—
252(C2×D3)⋊C4—64no—
253(C2×D3)⋊C4—64no—
254(C2×D3)⋊C4—64no—
255(C2×D3)⋊C4—64no—
256C4⋊Dic3—34no—
257C4⋊Dic3—34no—
258C4⋊Dic3—64no—
259Dic3⋊C4—64no—
260Dic3⋊C4—64no—
261C4×Dic3—64no—
262C2×C3⋊C8—34no—
263C2×C3⋊C8—34no—
264C2×C3⋊C8—34no—
265C2×C3⋊C8—34no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32C22×D411yescharacteristic
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4C2×#4111yescharacteristic
283C32×C4C2×#4111yescharacteristic
284C3⋊Dic3—41no—
285C3⋊Dic3—41no—
286C3⋊Dic3—41no—
287C22×Q8C2×C3⋊D311yescharacteristic
288C2×Q16—91no—
289C2×Q16—91no—
290C2×SD16—91no—
291C2×SD16—91no—
292C4⋊Q8—91no—
293#126—181no—
294#124—91no—
295#124—181no—
296#123—181no—
297C22⋊Q8—91no—
298C22⋊Q8—181no—
299#121—91no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—91no—
303C4×Q8C2×C3⋊D311yescharacteristic
304C4×Q8C2×C3⋊D311yescharacteristic
305C4×Q8C2×C3⋊D311yescharacteristic
306C4×Q8C2×C3⋊D311yescharacteristic
307C4×D4—91no—
308#117—21no—
309#117—21no—
310#117C2×C3⋊D311yescharacteristic
311#117C2×C3⋊D311yescharacteristic
312#107—91no—
313#107—91no—
314#106—91no—
315#106—91no—
316C4⋊C8—91no—
317C4⋊C8—91no—
318#103—91no—
319#103—91no—
320#103—91no—
321#103—91no—
322#103—91no—
323#103—91no—
324#102—91no—
325#102—91no—
326C22⋊C8—91no—
327C22⋊C8—91no—
328#97—91no—
329C4×C8—91no—
330C23×C3C2×#6714yes—
331C22×D3—64no—
332C3×Q8—216no—
333C3×Q8—24no—
334C3×Q8—24no—
335C3×Q8#23518yes—
336C3×Q8#23518yes—
337C2×C3×C4—216no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4C2×#6714yes—
345C2×C3×C4C2×#6714yes—
346C2×C3×C4C2×#6714yes—
347C2×C3×C4C2×#6714yes—
348C2×C3×C4C2×#6714yes—
349C2×C3×C4C23×D314yes—
350#67—244no—
351C2×Dic3—64no—
352C2×Dic3—64no—
353C2×Dic3—64no—
354D12—64no—
355D12—64no—
356C4×D3—244no—
357Dic6—64no—
358Dic6—64no—
359Dic6—244no—
360C3⋊C8—64no—
361C3⋊C8—64no—
362C3⋊C8—64no—
363C3⋊C8—64no—
364C2×C32—41no—
365C2×C32#52511yescharacteristic
366C2×C32#57711yescharacteristic
367C2×C32#57811yescharacteristic
368C3⋊D3—81no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—91no—
373C2×Q8—181no—
374C2×Q8C22×C3⋊D311yescharacteristic
375C2×Q8C22×C3⋊D311yescharacteristic
376C2×D4—91no—
377C2×D4—181no—
378C22×C4—21no—
379C22×C4—181no—
380C22×C4C22×C3⋊D311yescharacteristic
381Q16—182no—
382Q16—182no—
383SD16—182no—
384SD16—182no—
385C2×C8—91no—
386C2×C8—91no—
387C2×C8—91no—
388C2×C8—91no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—91no—
394C4⋊C4—91no—
395C4⋊C4—181no—
396C4⋊C4—181no—
397C4⋊C4—181no—
398C4⋊C4#63811yescharacteristic
399C4⋊C4#63811yescharacteristic
400C4⋊C4C22×C3⋊D311yescharacteristic
401C4⋊C4C22×C3⋊D311yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—181no—
404C22⋊C4—181no—
405C22⋊C4—181no—
406C22⋊C4—181no—
407C22⋊C4—181no—
408C22⋊C4#63811yescharacteristic
409C22⋊C4#63811yescharacteristic
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—181no—
416C42C22×C3⋊D311yescharacteristic
417C42C22×C3⋊D311yescharacteristic
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—24no—
421C22×C3C22×#6714yes—
422C2×D3—124no—
423C2×D3—124no—
424C2×D3—124no—
425C3×C4—28no—
426C3×C4—28no—
427C3×C4—24no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—44no—
433C3×C4—44no—
434C3×C4C2×#23514yes—
435C3×C4C2×#23514yes—
436Dic3—124no—
437Dic3—124no—
438Dic3—124no—
439C32#312111yescharacteristic, complemented, Sylow
440C23—181no—
441C23C2×#63811yescharacteristic
442Q8—24no—
443Q8—21no—
444Q8—21no—
445Q8—181no—
446Q8—181no—
447Q8—721no—
448Q8#382812yes—
449Q8#382812yes—
450D4—181no—
451D4—181no—
452D4—721no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4—721no—
464C2×C4C2×#63811yescharacteristic
465C2×C4C2×#63811yescharacteristic
466C2×C4C2×#63811yescharacteristic
467C2×C4C2×#63811yescharacteristic
468C2×C4C2×#63811yescharacteristic
469C2×C4C23×C3⋊D311yescharacteristic
470C8—181no—
471C8—181no—
472C8—181no—
473C8—181no—
474C2×C3—44no—
475C2×C3#579114yes—
476C2×C3#583714yes—
477C2×C3#583814yes—
478D3—244no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—361no—
483C22—361no—
484C22—361no—
485C22C22×#63811yescharacteristic
486C4—22no—
487C4—22no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—41no—
494C4—41no—
495C4—361no—
496C4—361no—
497C4—361no—
498C4C2×#382811yescharacteristic
499C4C2×#382811yescharacteristic
500C3#8255214yescomplemented
501C2—41no—
502C2—721no—
503C2#1059626211yescharacteristic
504C2#1059630811yescharacteristic
505C2#1059630911yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#519C2
socleC22×C32C22×D4
prime radical#64139C22
hypercenter#519C3⋊D3
hypocenterC32#3121
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C32×#519C2semidirect
3. #10592927C2semidirect
5. #10592729C2semidirect
6. C4×#3756C2semidirect
7. #10592727C2semidirect
8. #10592727C2semidirect
9. #10592725C2semidirect
10. C4×#3755C2semidirect
11. #10592720C2semidirect
12. #10592719C2semidirect
14. #10592699C2semidirect
15. #10592695C2semidirect
60. C3×#519D3semidirect
203. #519C3⋊D3semidirect
439. C32#3121semidirect
500. C3#82552semidirect

Maximal subgroups

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

subgroupindex
C32×#5192
#105960052
#105929272
#105929262
#105927292
C4×#37562
#105927272
#105927252
C4×#37552
#105927202
#105927192
#105927022
#105926992
#105926952
#825523