Finite Group Databasebeta

← #11866357

Subgroups — #11866357

Order 1152.

1147 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596301C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic, complemented
3#10594818C211yesmaximal, characteristic, complemented
4C22×#3717C211yesmaximal, characteristic, complemented
5C3×#5233C211yesmaximal, characteristic, complemented
6#10591579C214yesmaximal, complemented
7#10591444C214yesmaximal, complemented
8#10591119C211yesmaximal, characteristic, complemented
9#10591096C211yesmaximal, characteristic, complemented
10#83969—31nomaximal
11#78864—31nomaximal
12C2×C32×#41C2211yescharacteristic
13C22×C4⋊(C3⋊D3)C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15#64156C2212yes—
16#64149C2212yes—
17#64146—22no—
18#64139—22no—
19#64135C2211yescharacteristic
20C3×#951—24no—
21C2×C3×#216C2212yescomplemented
22C2×C3×#214C2212yescomplemented
23C3×#926C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26#63955—22no—
27#63843C2214yescomplemented
28Dic3×Dic6—22no—
29C2×#3717—24no—
30C2×#3717C2214yes—
31C3×#835C2212yescomplemented
32C3×#832C2212yescomplemented
33#63572C2214yescomplemented
34#63564C2214yes—
35C2×C3×#137—31no—
36#5830—31no—
37#5830—31no—
38#5830—61no—
39C2×#951D311yescharacteristic, complemented
40#5812—31no—
41C22×#215—31no—
42C22×#204—31no—
43#5516—31no—
44C3×#416—31no—
45#5233D311yescharacteristic, complemented
46#5185—34no—
47#5163—34no—
48#5131—31no—
49#5119—31no—
50#5102—31no—
51#5001—31no—
52#4798—34no—
53#4762—34no—
54C23×C3⋊D3—21no—
55C32×#41—22no—
56C32×#41—22no—
57C2×C32×Q8C2311yescharacteristic
58C2×C32×D4—21no—
59C2×C32×D4C2311yescharacteristic
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C2×#638—24no—
63C2×C4⋊(C3⋊D3)—22no—
64C2×C4⋊(C3⋊D3)—22no—
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)C2312yes—
67C22×C3×Dic3—21no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6D414yescomplemented
71C2×C3×Dic6C2312yes—
72#3757D412yes—
73#3752—24no—
74C4⋊(C3⋊Dic3)D412yes—
75C4×C3⋊Dic3C2312yes—
76C3×#217—21no—
77C3×#216—24no—
78C3×#214—24no—
79C3×C4⋊Dic3C2312yes—
80C3×C4×Dic3—21no—
81C3×#208—22no—
82C2×C3×C3⋊C8—22no—
83C2×C3×C3⋊C8C2312yes—
84C2×C3×C3⋊C8C2312yes—
85Dic3⋊Dic3—24no—
86Dic32—24no—
87#3717—28no—
88#3043—91noSylow
89C2×C3×#41—21no—
90C2×C3×#41C2×D311yescharacteristic
91C2×C3×#41C2×D311yescharacteristic
92C22×C3×Q8—31no—
93C22×D12—31no—
94C22×D12—31no—
95C22×D12—61no—
96C22×Dic6C2×D311yescharacteristic
97C3×#137—64no—
98C2×C3×Q16—32no—
99C2×C3×SD16—32no—
100C2×C3×M16—31no—
101C22×C3×C8—31no—
102C3×#117—31no—
103#951—24no—
104#947—32no—
105#947—32no—
106#947—62no—
107Q8×Dic3—34no—
108Q8×Dic3—62no—
109C2×#216C2×D312yescomplemented
110C2×#215—34no—
111C2×#215—64no—
112#940—32no—
113#940—32no—
114#940—62no—
115#937—62no—
116#937—62no—
117#937—64no—
118C2×#214C2×D312yescomplemented
119#930—62no—
120#930—62no—
121#930—64no—
122#926—31no—
123#926—31no—
124#926—61no—
125#926—62no—
126#926C2×D311yescharacteristic
127C2×#208C2×D311yescharacteristic
128C22×C3⋊C8C2×D311yescharacteristic
129C2×#204—34no—
130C2×#204—64no—
131#887—34no—
132#877—62no—
133C4×Dic6—62no—
134C3×#103—32no—
135C3×#102—32no—
136#835C2×D312yescomplemented
137#832C2×D312yescomplemented
138#821—34no—
139Dic3⋊C8—34no—
140Dic3⋊C8—34no—
141#809—34no—
142C23×C32—21no—
143C23×C32C2×D411yescharacteristic
144C22×C3⋊D3—22no—
145C22×C3⋊D3—42no—
146C22×C3⋊D3—82no—
147C32×Q8—21no—
148C32×Q8—21no—
149C32×D4—21no—
150C32×D4—21no—
151C32×D4—42no—
152C2×C32×C4—21no—
153C2×C32×C4—42no—
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2×D412yes—
156C2×C32×C4#4112yes—
157C2×C32×C4C2411yescharacteristic
158#638—84no—
159C2×C3⋊Dic3C2×D414yes—
160C4⋊(C3⋊D3)—22no—
161C4⋊(C3⋊D3)—22no—
162C4⋊(C3⋊D3)—42no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C3×Dic6—24no—
167C3×Dic6—22no—
168C3×Dic6C2×D414yes—
169C3×C3⋊C8—22no—
170C3×C3⋊C8—22no—
171C2×#137—31no—
172C22×SD16—91no—
173#535—91no—
174#519—91no—
175#464—94no—
176#460—94no—
177#424—91no—
178#418—91no—
179#416—31no—
180C23×D3—62no—
181C23×D3—61no—
182C23×D3—61no—
183C3×#41—22no—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—42no—
188C3×#41—42no—
189C2×C3×Q8—21no—
190C2×C3×Q8—34no—
191C2×C3×Q8—32no—
192C2×C3×Q8—62no—
193C2×C3×Q8—61no—
194C2×C3×Q8C22×D311yescharacteristic
195C2×C3×Q8C22×D311yescharacteristic
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C22×D311yescharacteristic
201C2×C3×D4C22×D311yescharacteristic
202C22×C3×C4—22no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—61no—
207C22×C3×C4C22×D311yescharacteristic
208C22×C3×C4C22×D311yescharacteristic
209C2×#67—64no—
210C2×#67—64no—
211C2×#67—68no—
212C22×Dic3—21no—
213C2×D12—32no—
214C2×D12—32no—
215C2×D12—62no—
216C2×D12—62no—
217C2×D12—62no—
218C2×D12—62no—
219C2×D12—62no—
220C2×D12—62no—
221C2×D12—61no—
222C2×D12—61no—
223C2×D12—122no—
224C2×D12—122no—
225C2×Dic6—21no—
226C2×Dic6—22no—
227C2×Dic6#6714yescomplemented
228C2×Dic6C22×D312yes—
229C3×Q16—64no—
230C3×SD16—64no—
231C3×M16—62no—
232C2×C3×C8—32no—
233C2×C3×C8—32no—
234C2×C3×C8—62no—
235C3×C4⋊C4—32no—
236C3×C22⋊C4—61no—
237C3×C42—61no—
238#217—21no—
239#217—32no—
240#217—32no—
241#217—62no—
242#216—24no—
243#215—68no—
244#214—24no—
245(C2×D3)⋊C4—64no—
246(C2×D3)⋊C4—64no—
247(C2×D3)⋊C4—68no—
248C4⋊Dic3—32no—
249C4⋊Dic3—32no—
250C4⋊Dic3—62no—
251C4⋊Dic3—64no—
252C4⋊Dic3C22×D312yes—
253Dic3⋊C4—64no—
254C4×Dic3—21no—
255C4×Dic3—32no—
256C4×Dic3—32no—
257C4×Dic3—62no—
258C4×Dic3—64no—
259C4×Dic3—64no—
260#208—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8C22×D312yes—
263C2×C3⋊C8C22×D312yes—
264#204—68no—
265C22×C32—22no—
266C22×C32—41no—
267C22×C32—42no—
268C22×C32#12112yes—
269C22×C32C22×D411yescharacteristic
270C2×C3⋊D3—42no—
271C2×C3⋊D3—44no—
272C2×C3⋊D3—84no—
273C32×C4—41no—
274C32×C4#12112yes—
275C32×C4C22×D411yescharacteristic
276C32×C4C2×#4111yescharacteristic
277C3⋊Dic3—24no—
278C3×Dic3—24no—
279C3×Dic3—42no—
280C2×#41D3211yescharacteristic
281C22×Q8—31no—
282C22×D4—91no—
283#137—64no—
284C2×Q16—32no—
285C2×SD16—32no—
286C2×SD16—94no—
287C2×SD16—184no—
288C2×M16—31no—
289C22×C8—31no—
290#127—92no—
291#124—92no—
292#121—182no—
293#120—182no—
294C4×Q8—94no—
295C4×Q8—182no—
296#117—31no—
297#117—91no—
298#117—182no—
299C4⋊C8—94no—
300#103—32no—
301#102—32no—
302#102—94no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3D3×D411yescharacteristic
308C23×C3D3×D411yescharacteristic
309C22×D3—64no—
310C22×D3—62no—
311C22×D3—62no—
312C22×D3—124no—
313C22×D3—122no—
314C22×D3—122no—
315C22×D3—242no—
316C22×D3—242no—
317C22×D3—244no—
318C3×Q8—21no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×Q8—34no—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—64no—
326C3×Q8—62no—
327C3×D4—21no—
328C3×D4—21no—
329C3×D4—21no—
330C3×D4—21no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—44no—
334C3×D4—42no—
335C3×D4—42no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—44no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—64no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4#23512yes—
350C2×C3×C4D3×D412yes—
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4#23712yes—
354C2×C3×C4C2×#6712yes—
355C2×C3×C4C23×D311yescharacteristic
356C2×C3×C4C23×D311yescharacteristic
357#67—244no—
358#67—244no—
359#67—248no—
360C2×Dic3—24no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—34no—
364C2×Dic3—34no—
365C2×Dic3—64no—
366D12—62no—
367D12—62no—
368D12—62no—
369D12—62no—
370D12—122no—
371D12—122no—
372D12—124no—
373D12—122no—
374D12—122no—
375Dic6—24no—
376Dic6—22no—
377Dic6C2×#6714yes—
378C3×C8—62no—
379C3×C8—62no—
380C3⋊C8—22no—
381C3⋊C8—22no—
382C2×C32—22no—
383C2×C32—81no—
384C2×C32C2×#12111yescharacteristic
385C2×C32#57712yes—
386C3⋊D3—82no—
387C24—181no—
388#41—22no—
389#41—22no—
390C2×Q8—32no—
391C2×Q8—34no—
392C2×Q8—62no—
393C2×Q8—61no—
394C2×Q8C2×D3211yescharacteristic
395C2×D4—21no—
396C2×D4—92no—
397C2×D4—184no—
398C2×D4—182no—
399C2×D4—182no—
400C2×D4—181no—
401C2×D4C2×D3211yescharacteristic
402C22×C4—21no—
403C22×C4—61no—
404C22×C4C2×D3211yescharacteristic
405Q16—64no—
406SD16—64no—
407SD16—188no—
408M16—62no—
409C2×C8—32no—
410C2×C8—32no—
411C2×C8—62no—
412C4⋊C4—32no—
413C4⋊C4—92no—
414C4⋊C4—184no—
415C22⋊C4—61no—
416C22⋊C4—92no—
417C22⋊C4—184no—
418C42—61no—
419C42—92no—
420C42—184no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—21no—
424C22×C3—22no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—41no—
432C22×C3#88412yes—
433C22×C3#93912yes—
434C22×C3C2×D3×D411yescharacteristic
435C22×C3C2×D3×D411yescharacteristic
436C2×D3—124no—
437C2×D3—128no—
438C2×D3—122no—
439C2×D3—124no—
440C2×D3—124no—
441C2×D3—122no—
442C2×D3—244no—
443C2×D3—244no—
444C2×D3—248no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—22no—
448C3×C4—42no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—64no—
452C3×C4—122no—
453C3×C4#88412yes—
454C3×C4#93912yes—
455C3×C4C2×#23511yescharacteristic
456C3×C4C2×D3×D411yescharacteristic
457C3×C4C2×#23711yescharacteristic
458C3×C4C22×#6711yescharacteristic
459Dic3—24no—
460Dic3—42no—
461Dic3—64no—
462Dic3—64no—
463Dic3—124no—
464C32#304311yescharacteristic, complemented, Sylow
465C23—21no—
466C23—182no—
467C23—362no—
468C23—722no—
469C23#381111yescharacteristic
470Q8—21no—
471Q8—21no—
472Q8—34no—
473Q8—62no—
474Q8—64no—
475D4—21no—
476D4—21no—
477D4—42no—
478D4—182no—
479D4—182no—
480D4—362no—
481D4—724no—
482C2×C4—21no—
483C2×C4—42no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—94no—
488C2×C4#380412yes—
489C2×C4D3×#6712yes—
490C2×C4#381111yescharacteristic
491C2×C4C22×D3211yescharacteristic
492C8—62no—
493C8—62no—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—82no—
500C2×C3—81no—
501C2×C3—81no—
502C2×C3C2×#88411yescharacteristic
503C2×C3#577012yes—
504C2×C3#577812yes—
505C2×C3C2×#93911yescharacteristic
506D3—244no—
507D3—242no—
508D3—242no—
509C22—22no—
510C22—42no—
511C22—41no—
512C22—364no—
513C22—362no—
514C22—724no—
515C22#6397112yes—
516C22C2×#381111yescharacteristic
517C4—41no—
518C4—64no—
519C4—122no—
520C4—184no—
521C4#6397112yes—
522C4C2×#380411yescharacteristic
523C4C2×D3×#6711yescharacteristic
524C3—21no—
525C3#7886411yescharacteristic, complemented
526C3#8396911yescharacteristic, complemented
527C2—22no—
528C2—81no—
529C2—722no—
530C2#1059547312yes—
531C2C2×#6397111yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×#41C22
socleC22×C32C22×D4
prime radical#10596301C2
hypercenterC2×#41D32
hypocenterC32#3043
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#951C2semidirect
3. #10594818C2semidirect
4. C22×#3717C2semidirect
5. C3×#5233C2semidirect
6. #10591579C2semidirect
7. #10591444C2semidirect
8. #10591119C2semidirect
9. #10591096C2semidirect
21. C2×C3×#216C22semidirect
22. C2×C3×#214C22semidirect
27. #63843C22semidirect
31. C3×#835C22semidirect
32. C3×#832C22semidirect
33. #63572C22semidirect
39. C2×#951D3semidirect
45. #5233D3semidirect
70. C2×C3×Dic6D4semidirect
109. C2×#216C2×D3semidirect
118. C2×#214C2×D3semidirect
136. #835C2×D3semidirect
137. #832C2×D3semidirect
227. C2×Dic6#67semidirect
464. C32#3043semidirect
525. C3#78864semidirect
526. C3#83969semidirect

Maximal subgroups

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

subgroupindex
#105963012
C2×C3×#9512
#105948182
C22×#37172
C3×#52332
#105915792
#105914442
#105911192
#105910962
#839693
#788643