Finite Group Databasebeta

← #11866135

Subgroups — #11866135

Order 1152.

1063 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic, complemented
2#10594958C211yesmaximal, characteristic, complemented
3C2×#63863C211yesmaximal, characteristic, complemented
4C2×#63823C211yesmaximal, characteristic
5#10592817C211yesmaximal, characteristic, complemented
6C3×#5131C211yesmaximal, characteristic, complemented
7#10591099C211yesmaximal, characteristic, complemented
8#10590797C214yesmaximal, complemented
9#10590750C214yesmaximal, complemented
10#83968—31nomaximal
11#78943—31nomaximal
12C2×C3×#235C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C4×C3⋊Dic3C2211yescharacteristic
17C2×#3748C2211yescharacteristic
18C3×#926C2211yescharacteristic
19C3×#909—24no—
20C2×C3×Dic12C2212yescomplemented
21C2×C3×#204C2212yescomplemented
22#67⋊Dic3—22no—
23C2×(C2×D3)⋊Dic3—22no—
24#63898C2212yescomplemented
25C4×D3×Dic3—21no—
26#63863—21no—
27#63863—21no—
28#63863C2214yes—
29#63855—21no—
30#63843C2212yescomplemented
31C2×#3715C2212yes—
32#63823—24no—
33C2×#3713C2212yes—
34#63641C2214yescomplemented
35C3×#821C2214yes—
36#63565C2214yescomplemented
37C2×C3×#137—31no—
38Dic3×#41—31no—
39C2×#949—31no—
40C2×#947—31no—
41C2×#936—31no—
42C2×#932—31no—
43C2×#909D311yescharacteristic, complemented
44#5522—31no—
45C3×#418—31no—
46#5225—31no—
47#5185—34no—
48#5165—34no—
49#5131D311yescharacteristic, complemented
50#5118—31no—
51#5118—31no—
52#5118—61no—
53#5004—31no—
54#4787—34no—
55#4767—34no—
56C23×C3×D3—21no—
57C22×C32×C4C2311yescharacteristic
58C22×C3⋊Dic3D412yes—
59C2×C3×#67—21no—
60C22×C3×Dic3—21no—
61C3×#235—22no—
62C3×#235D414yes—
63C2×C3×D12—22no—
64C2×C3×D12—22no—
65C2×C3×D12—21no—
66C2×C3×D12C2312yes—
67C2×C3×D12C2311yescharacteristic
68C2×C3×C4×D3—21no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6—22no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2312yes—
73C2×C3×Dic6C2311yescharacteristic
74C2×D3×Dic3—22no—
75C32×M16—22no—
76C2×C32×C8C2312yes—
77C4×C3⋊Dic3D411yescharacteristic
78C4×C3⋊Dic3D411yescharacteristic
79C4×C3⋊Dic3C2312yes—
80#3748—22no—
81C2×C3⋊(C3⋊C8)C2312yes—
82C3×#217—21no—
83C3×C4⋊Dic3C2312yes—
84C3×C4×Dic3—21no—
85C3×Dic12—24no—
86C3×#204—24no—
87Dic3⋊Dic3—22no—
88(C2×D3)⋊Dic3—22no—
89(C2×D3)⋊Dic3—24no—
90(C2×D3)⋊Dic3—24no—
91Dic32—22no—
92#3715—24no—
93#3713—24no—
94#3040—91noSylow
95C2×C3×#41—31no—
96C22×C3×Q8—31no—
97C22×C3×D4—31no—
98C2×#235C2×D311yescharacteristic
99C22×D12C2×D311yescharacteristic
100C22×Dic6C2×D311yescharacteristic
101C3×#137—64no—
102C2×C3×Q16—32no—
103C2×C3×SD16—32no—
104C2×C3×M16—21no—
105C2×C3×M16C2×D311yescharacteristic
106C2×C3×M16C2×D311yescharacteristic
107C3×#117—31no—
108C2×#217—62no—
109#949—64no—
110#947—34no—
111#947—61no—
112#947—61no—
113Q8×Dic3—32no—
114C2×#215—32no—
115#936—34no—
116#936—61no—
117#936—61no—
118D4×Dic3—32no—
119D4×Dic3—62no—
120C2×#214—32no—
121#932—64no—
122C2×#213—32no—
123C2×#213—32no—
124C2×(C2×D3)⋊C4—62no—
125#926—61no—
126#926C2×D311yescharacteristic
127C2×C4×Dic3—31no—
128C2×C4×Dic3—31no—
129C2×C4×Dic3—61no—
130C2×C4×Dic3—61no—
131C2×#208—31no—
132C2×#208—31no—
133C2×#208—61no—
134#909—24no—
135C2×Dic12C2×D312yescomplemented
136C2×#204C2×D312yescomplemented
137#893—32no—
138#887—32no—
139#67⋊C4—62no—
140#872—61no—
141C42×D3—61no—
142C3×#102—34no—
143#835—34no—
144#821C2×D314yes—
145Dic3⋊C8—34no—
146Dic3⋊C8—34no—
147Dic3⋊C8—64no—
148#810—34no—
149C23×C32C2×D411yescharacteristic
150C22×C3×D3—21no—
151C22×C3×D3—22no—
152C22×C3×D3—44no—
153C22×C3×D3—42no—
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2×D412yes—
156C2×C32×C4#4112yes—
157C2×C32×C4C2411yescharacteristic
158C2×C3⋊Dic3—42no—
159C2×C3⋊Dic3C2×D412yes—
160C2×C3⋊Dic3C2×D412yes—
161C3×#67—24no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—42no—
166C3×D12—22no—
167C3×D12—21no—
168C3×D12—22no—
169C3×D12—42no—
170C3×D12C2×D412yes—
171C3×C4×D3—24no—
172C3×Dic6—21no—
173C3×Dic6—22no—
174C3×Dic6—22no—
175C3×Dic6—42no—
176C3×Dic6C2×D412yes—
177D3×Dic3—44no—
178C32×C8—22no—
179C3⋊(C3⋊C8)—22no—
180C2×#137—31no—
181C2×AGL(1,ℤ/8)—91no—
182C2×#124—91no—
183C4×#41—91no—
184#464—94no—
185#463—94no—
186#423—91no—
187#419—91no—
188#418—31no—
189C24×C3—61no—
190C23×D3—21no—
191C3×#41—34no—
192C3×#41—62no—
193C2×C3×Q8—31no—
194C2×C3×Q8—32no—
195C2×C3×Q8—61no—
196C2×C3×Q8—62no—
197C2×C3×Q8—62no—
198C2×C3×D4—31no—
199C2×C3×D4—32no—
200C2×C3×D4—61no—
201C2×C3×D4—61no—
202C2×C3×D4—62no—
203C2×C3×D4—62no—
204C22×C3×C4—21no—
205C22×C3×C4—61no—
206C22×C3×C4—61no—
207C22×C3×C4C22×D311yescharacteristic
208C22×C3×C4C22×D311yescharacteristic
209C2×#67—21no—
210C22×Dic3—21no—
211C22×Dic3—32no—
212C22×Dic3—32no—
213C22×Dic3—62no—
214C22×Dic3—62no—
215#235—22no—
216#235#6714yes—
217C2×D12—22no—
218C2×D12—22no—
219C2×D12—21no—
220C2×D12C22×D312yes—
221C2×D12C22×D311yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—62no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—21no—
227C2×Dic6C22×D312yes—
228C2×Dic6C22×D311yescharacteristic
229C3×Q16—64no—
230C3×SD16—64no—
231C3×M16—22no—
232C3×M16—22no—
233C3×M16—42no—
234C2×C3×C8—22no—
235C2×C3×C8C22×D312yes—
236C2×C3×C8C22×D312yes—
237C3×C4⋊C4—32no—
238C3×C22⋊C4—61no—
239C3×C42—61no—
240#217—21no—
241#217—62no—
242#217—64no—
243#217—64no—
244#215—64no—
245#214—64no—
246#213—64no—
247#213—64no—
248(C2×D3)⋊C4—62no—
249(C2×D3)⋊C4—64no—
250(C2×D3)⋊C4—64no—
251C4⋊Dic3—62no—
252C4⋊Dic3C22×D312yes—
253Dic3⋊C4—62no—
254C4×Dic3—21no—
255C4×Dic3—31no—
256C4×Dic3—31no—
257C4×Dic3—31no—
258C4×Dic3—31no—
259C4×Dic3—32no—
260C4×Dic3—32no—
261C4×Dic3—62no—
262C4×Dic3—61no—
263C4×Dic3—61no—
264C4×Dic3—62no—
265C4×Dic3—62no—
266#208—62no—
267#208—62no—
268#208—122no—
269C2×C3⋊C8—32no—
270C2×C3⋊C8—32no—
271C2×C3⋊C8—62no—
272Dic12—24no—
273#204—24no—
274C22×C32—22no—
275C22×C32#12112yes—
276C22×C32C22×D411yescharacteristic
277C2×C3×D3—22no—
278C2×C3×D3—42no—
279C2×C3×D3—42no—
280C2×C3×D3—44no—
281C2×C3×D3—84no—
282C32×C4#12112yes—
283C32×C4C22×D411yescharacteristic
284C32×C4C2×#4111yescharacteristic
285C3⋊Dic3—22no—
286C3⋊Dic3—22no—
287C3×Dic3—22no—
288C3×Dic3—42no—
289C3×Dic3—42no—
290C2×#41—31no—
291C22×Q8—31no—
292C22×D4—31no—
293#137—64no—
294AGL(1,ℤ/8)—184no—
295C2×Q16—32no—
296C2×SD16—32no—
297C2×SD16—92no—
298C2×D8—92no—
299C2×M16—91no—
300C2×M16D3211yescharacteristic
301#124—94no—
302#124—181no—
303#124—181no—
304C4×Q8—92no—
305C4×D4—92no—
306C4×D4—182no—
307#117—31no—
308#117—181no—
309C2×C22⋊C4—182no—
310C2×C42—91no—
311C2×C42—181no—
312C4⋊C8—94no—
313#103—94no—
314#102—34no—
315C23×C3—21no—
316C23×C3—61no—
317C23×C3—62no—
318C23×C3—124no—
319C23×C3—122no—
320C23×C3D3×D411yescharacteristic
321C23×C3D3×D411yescharacteristic
322C22×D3—21no—
323C22×D3—22no—
324C22×D3—44no—
325C22×D3—42no—
326C3×Q8—32no—
327C3×Q8—61no—
328C3×Q8—62no—
329C3×Q8—62no—
330C3×Q8—122no—
331C3×D4—32no—
332C3×D4—64no—
333C3×D4—61no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—122no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—61no—
342C2×C3×C4—64no—
343C2×C3×C4—62no—
344C2×C3×C4—62no—
345C2×C3×C4—122no—
346C2×C3×C4#23512yes—
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D412yes—
350C2×C3×C4#23712yes—
351C2×C3×C4C2×#6712yes—
352C2×C3×C4C23×D311yescharacteristic
353C2×C3×C4C23×D311yescharacteristic
354#67—24no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—42no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—124no—
366C2×Dic3—122no—
367C2×Dic3—122no—
368C2×Dic3—124no—
369D12—22no—
370D12—22no—
371D12—21no—
372D12—42no—
373D12C2×#6712yes—
374C4×D3—24no—
375C4×D3—124no—
376Dic6—21no—
377Dic6—22no—
378Dic6—22no—
379Dic6—42no—
380Dic6C2×#6712yes—
381C3×C8—22no—
382C3×C8—22no—
383C3×C8—42no—
384C3⋊C8—62no—
385C3⋊C8—62no—
386C3⋊C8—122no—
387C2×C32—22no—
388C2×C32C2×#12111yescharacteristic
389C2×C32#57512yes—
390C3×D3—42no—
391C3×D3—82no—
392C24—61no—
393#41—34no—
394#41—62no—
395C2×Q8—31no—
396C2×Q8—32no—
397C2×Q8—61no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×D4—31no—
401C2×D4—32no—
402C2×D4—61no—
403C2×D4—62no—
404C2×D4—61no—
405C2×D4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—92no—
409C22×C4—182no—
410C22×C4C2×D3211yescharacteristic
411Q16—64no—
412SD16—64no—
413SD16—184no—
414D8—184no—
415M16—22no—
416M16—182no—
417C2×C8—92no—
418C2×C8C2×D3212yes—
419C4⋊C4—32no—
420C4⋊C4—182no—
421C22⋊C4—61no—
422C22⋊C4—184no—
423C22⋊C4—184no—
424C22⋊C4—182no—
425C42—61no—
426C42—92no—
427C42—91no—
428C42—91no—
429C42—182no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—21no—
433C22×C3—22no—
434C22×C3—42no—
435C22×C3—62no—
436C22×C3—122no—
437C22×C3—124no—
438C22×C3—122no—
439C22×C3—244no—
440C22×C3#88412yes—
441C22×C3#93912yes—
442C22×C3C2×D3×D411yescharacteristic
443C22×C3C2×D3×D411yescharacteristic
444C2×D3—22no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—44no—
448C2×D3—84no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—62no—
453C3×C4—122no—
454C3×C4—122no—
455C3×C4#88412yes—
456C3×C4#93912yes—
457C3×C4C2×#23511yescharacteristic
458C3×C4C2×D3×D411yescharacteristic
459C3×C4C2×#23711yescharacteristic
460C3×C4C22×#6711yescharacteristic
461Dic3—22no—
462Dic3—42no—
463Dic3—42no—
464Dic3—64no—
465Dic3—62no—
466Dic3—62no—
467Dic3—62no—
468Dic3—62no—
469Dic3—122no—
470C32#304011yescharacteristic, complemented, Sylow
471C23—61no—
472C23—62no—
473C23—124no—
474C23—122no—
475C23#380211yescharacteristic
476Q8—32no—
477Q8—61no—
478Q8—62no—
479Q8—62no—
480Q8—122no—
481D4—32no—
482D4—64no—
483D4—61no—
484D4—62no—
485D4—62no—
486D4—122no—
487C2×C4—61no—
488C2×C4—64no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—122no—
494C2×C4—364no—
495C2×C4—362no—
496C2×C4#379512yes—
497C2×C4#380211yescharacteristic
498C2×C4D3×#6712yes—
499C2×C4C22×D3211yescharacteristic
500C8—22no—
501C8—182no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—21no—
505C2×C3—22no—
506C2×C3—42no—
507C2×C3—122no—
508C2×C3—242no—
509C2×C3C2×#88411yescharacteristic
510C2×C3#577312yes—
511C2×C3#577712yes—
512C2×C3C2×#93911yescharacteristic
513D3—42no—
514D3—82no—
515C22—22no—
516C22—62no—
517C22—122no—
518C22—122no—
519C22—124no—
520C22—244no—
521C22#6388512yes—
522C22C2×#380211yescharacteristic
523C4—62no—
524C4—122no—
525C4—122no—
526C4—182no—
527C4—182no—
528C4#6388512yes—
529C4C2×#379511yescharacteristic
530C4C2×D3×#6711yescharacteristic
531C3—21no—
532C3#7894311yescharacteristic, complemented
533C3#8396811yescharacteristic, complemented
534C2—22no—
535C2—122no—
536C2—242no—
537C2#1059471812yes—
538C2C2×#6388511yescharacteristic
subgroupG/H
centerC22C2×#3802
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radicalC2×#63823C2
hypercenterC2×M16D32
hypocenterC32#3040
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#909C2semidirect
2. #10594958C2semidirect
3. C2×#63863C2semidirect
5. #10592817C2semidirect
6. C3×#5131C2semidirect
7. #10591099C2semidirect
8. #10590797C2semidirect
9. #10590750C2semidirect
20. C2×C3×Dic12C22semidirect
21. C2×C3×#204C22semidirect
24. #63898C22semidirect
30. #63843C22semidirect
34. #63641C22semidirect
36. #63565C22semidirect
43. C2×#909D3semidirect
49. #5131D3semidirect
135. C2×Dic12C2×D3semidirect
136. C2×#204C2×D3semidirect
470. C32#3040semidirect
532. C3#78943semidirect
533. C3#83968semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
#105949582
C2×#638632
C2×#638232
#105928172
C3×#51312
#105910992
#105907972
#105907502
#839683
#789433