Finite Group Databasebeta

← #11866134

Subgroups — #11866134

Order 1152.

1075 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5522C211yesmaximal, characteristic, complemented
2C2×#63860C211yesmaximal, characteristic, complemented
3C2×#63831C211yesmaximal, characteristic, complemented
4C2×#63823C211yesmaximal, characteristic
5#10592703C211yesmaximal, characteristic, complemented
6C3×#5001C211yesmaximal, characteristic, complemented
7#10591115C211yesmaximal, characteristic, complemented
8#10590532C214yesmaximal, complemented
9#10590530C214yesmaximal, complemented
10#82527—31nomaximal
11#78943—31nomaximal
12C22×#634C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×D12C2211yescharacteristic
15C32×#117C2211yescharacteristic
16C2×#3748C2211yescharacteristic
17C2×C3×C4×Dic3C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C3×#893C2212yescomplemented
20C3×#887C2212yescomplemented
21C3×#67⋊C4—22no—
22C3×#872—21no—
23C3×C42×D3—21no—
24C2×(C2×D3)⋊Dic3—22no—
25#63860—21no—
26#63860—21no—
27#63860C2214yes—
28C2×#3719C2212yescomplemented
29#63831—24no—
30C2×#3716C2212yescomplemented
31C2×#3715C2212yes—
32#63823—24no—
33C2×#3713C2212yes—
34#63637C2214yescomplemented
35C3×#809C2214yes—
36Dic3⋊(C3⋊C8)C2214yescomplemented
37C3×C4×#41—31no—
38C2×#951—31no—
39C2×#949—31no—
40C2×#936—31no—
41C2×#932—31no—
42C2×#909—31no—
43#5522D311yescharacteristic, complemented
44C2×#875—31no—
45C3×#418—31no—
46#5118—31no—
47#5028—34no—
48#5016—34no—
49#5004—31no—
50#5004—31no—
51#5004—61no—
52#5001D311yescharacteristic, complemented
53#4924—31no—
54#4787—34no—
55#4767—34no—
56C23×C3×D3—21no—
57C22×C32×C4C2311yescharacteristic
58C22×C3⋊Dic3—21no—
59C2×#634—22no—
60C2×#634—22no—
61C2×#634—21no—
62C2×#634C2312yes—
63C2×C3×#67—21no—
64C22×C3×Dic3D412yes—
65C3×#235—22no—
66C3×#235D414yes—
67C2×C3×D12—22no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12C2312yes—
71C2×C3×D12C2311yescharacteristic
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3—22no—
74C2×C3×Dic6C2311yescharacteristic
75C32×C4⋊C4C2312yes—
76C32×C22⋊C4—21no—
77C32×C42—21no—
78#3748—22no—
79C2×C3⋊(C3⋊C8)C2312yes—
80C3×(C2×D3)⋊C4—22no—
81C3×Dic3⋊C4—22no—
82C3×C4×Dic3—22no—
83C3×C4×Dic3D411yescharacteristic
84C3×C4×Dic3D411yescharacteristic
85C3×C4×Dic3C2312yes—
86C3×#208—22no—
87C2×C3×C3⋊C8C2312yes—
88(C2×D3)⋊Dic3—24no—
89(C2×D3)⋊Dic3—24no—
90#3719—24no—
91#3716—24no—
92#3715—24no—
93#3713—24no—
94#3040—91noSylow
95C2×C3×#41—31no—
96C22×C3×D4—31no—
97C2×#235C2×D311yescharacteristic
98C22×D12C2×D311yescharacteristic
99C22×Dic6—31no—
100C22×Dic6—31no—
101C22×Dic6—61no—
102C2×C3×M16—31no—
103C3×C4×Q8—32no—
104C3×C4×D4—32no—
105C3×C4×D4—62no—
106C3×#117—21no—
107C3×#117—61no—
108C3×#117C2×D311yescharacteristic
109C3×#117C2×D311yescharacteristic
110C2×C3×C42—31no—
111C2×C3×C42—61no—
112C2×#217—62no—
113#951—64no—
114#949—64no—
115C2×#216—32no—
116C2×#215—32no—
117#936—34no—
118#936—61no—
119#936—61no—
120C2×#214—32no—
121C2×#214—32no—
122#932—64no—
123C2×#213—32no—
124C2×#213—32no—
125C2×(C2×D3)⋊C4—62no—
126C2×C4×Dic3C2×D311yescharacteristic
127C2×#208—31no—
128C2×#208—31no—
129C2×#208—61no—
130C2×#208C2×D311yescharacteristic
131#909—64no—
132C2×Dic12—32no—
133C2×#204—32no—
134#893C2×D312yescomplemented
135#887C2×D312yescomplemented
136#67⋊C4—22no—
137#875—34no—
138#875—61no—
139#875—61no—
140#872—21no—
141C42×D3—21no—
142C3×#102—34no—
143Dic3⋊C8—34no—
144#810—34no—
145#810—34no—
146#810—64no—
147#809C2×D314yes—
148C4⋊(C3⋊C8)—34no—
149C23×C32C2×D411yescharacteristic
150C22×C3×D3—21no—
151C22×C3×D3—22no—
152C22×C3×D3—44no—
153C22×C3×D3—42no—
154C2×C32×C4—22no—
155C2×C32×C4C2×D412yes—
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4#4112yes—
158C2×C32×C4C2411yescharacteristic
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3—42no—
161#634—22no—
162#634—22no—
163#634—42no—
164C3×#67—24no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—42no—
167C2×C3×Dic3C2×D412yes—
168C2×C3×Dic3C2×D412yes—
169C3×D12—22no—
170C3×D12—21no—
171C3×D12—22no—
172C3×D12—42no—
173C3×D12C2×D412yes—
174C3×C4×D3—24no—
175C3×C4×D3—44no—
176C3×Dic6—21no—
177C3×Dic6C2×D412yes—
178C3⋊(C3⋊C8)—22no—
179C3×C3⋊C8—22no—
180C2×#137—91no—
181C2×AGL(1,ℤ/8)—91no—
182C2×#124—91no—
183C4×#41—31no—
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×D4—31no—
195C2×C3×D4—32no—
196C2×C3×D4—61no—
197C2×C3×D4—62no—
198C2×C3×D4—62no—
199C2×C3×D4—61no—
200C22×C3×C4—21no—
201C22×C3×C4—32no—
202C22×C3×C4—62no—
203C22×C3×C4—61no—
204C22×C3×C4C22×D311yescharacteristic
205C22×C3×C4C22×D311yescharacteristic
206C2×#67—21no—
207C22×Dic3—61no—
208C22×Dic3—61no—
209C22×Dic3—121no—
210C22×Dic3#6712yes—
211#235—22no—
212#235D1214yes—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—21no—
216C2×D12C22×D312yes—
217C2×D12C22×D311yescharacteristic
218C2×C4×D3—21no—
219C2×C4×D3—22no—
220C2×Dic6—32no—
221C2×Dic6—32no—
222C2×Dic6—62no—
223C2×Dic6—64no—
224C2×Dic6—62no—
225C2×Dic6—62no—
226C2×Dic6—62no—
227C2×Dic6—62no—
228C2×Dic6—61no—
229C2×Dic6—61no—
230C2×Dic6—121no—
231C2×Dic6—122no—
232C2×Dic6C22×D311yescharacteristic
233C3×M16—62no—
234C2×C3×C8—32no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—62no—
237C3×C4⋊C4C22×D312yes—
238C3×C4⋊C4C22×D312yes—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—41no—
242C3×C22⋊C4—62no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—31no—
246C3×C42—31no—
247C3×C42—32no—
248C3×C42—41no—
249C3×C42—62no—
250#217—64no—
251#217—64no—
252#216—64no—
253#215—64no—
254#214—64no—
255#214—64no—
256#213—64no—
257#213—64no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—64no—
260(C2×D3)⋊C4—64no—
261Dic3⋊C4—22no—
262C4×Dic3—22no—
263C4×Dic3#6711yescharacteristic
264C4×Dic3#6711yescharacteristic
265C4×Dic3C22×D312yes—
266#208—22no—
267#208—62no—
268#208—62no—
269#208—122no—
270C2×C3⋊C8—32no—
271C2×C3⋊C8—32no—
272C2×C3⋊C8—62no—
273C2×C3⋊C8C22×D312yes—
274Dic12—64no—
275#204—64no—
276C22×C32—22no—
277C22×C32#12112yes—
278C22×C32C22×D411yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—42no—
281C2×C3×D3—42no—
282C2×C3×D3—44no—
283C2×C3×D3—84no—
284C32×C4—42no—
285C32×C4#12112yes—
286C32×C4C22×D411yescharacteristic
287C32×C4C2×#4111yescharacteristic
288C3⋊Dic3—42no—
289C3×Dic3—22no—
290C3×Dic3—22no—
291C3×Dic3—22no—
292C2×#41—31no—
293C22×Q8—91no—
294C22×D4—31no—
295#137—184no—
296AGL(1,ℤ/8)—184no—
297C2×Q16—92no—
298C2×SD16—92no—
299C2×SD16—92no—
300C2×D8—92no—
301C2×M16—31no—
302C2×M16—91no—
303#124—94no—
304#124—181no—
305#124—181no—
306C4×Q8—32no—
307C4×D4—32no—
308C4×D4—62no—
309#117—61no—
310#117D3211yescharacteristic
311C2×C22⋊C4—182no—
312C2×C42—31no—
313C2×C42—61no—
314C4⋊C8—94no—
315#103—94no—
316#102—34no—
317C23×C3—21no—
318C23×C3—61no—
319C23×C3—62no—
320C23×C3—124no—
321C23×C3—122no—
322C23×C3D3×D411yescharacteristic
323C23×C3C2×#6711yescharacteristic
324C22×D3—21no—
325C22×D3—22no—
326C22×D3—44no—
327C22×D3—42no—
328C3×Q8—32no—
329C3×Q8—61no—
330C3×D4—32no—
331C3×D4—64no—
332C3×D4—62no—
333C3×D4—61no—
334C3×D4—62no—
335C3×D4—122no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—42no—
345C2×C3×C4—61no—
346C2×C3×C4—64no—
347C2×C3×C4—124no—
348C2×C3×C4—122no—
349C2×C3×C4C2×D1212yes—
350C2×C3×C4#23512yes—
351C2×C3×C4#23512yes—
352C2×C3×C4D3×D412yes—
353C2×C3×C4D3×D411yescharacteristic
354C2×C3×C4C2×#6711yescharacteristic
355C2×C3×C4C23×D311yescharacteristic
356C2×C3×C4C23×D311yescharacteristic
357#67—24no—
358C2×Dic3—21no—
359C2×Dic3—42no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C2×Dic3—122no—
363C2×Dic3—124no—
364C2×Dic3—122no—
365C2×Dic3—122no—
366C2×Dic3C2×#6712yes—
367C2×Dic3C2×#6712yes—
368D12—21no—
369D12—22no—
370D12—22no—
371D12—42no—
372D12C2×D1212yes—
373C4×D3—24no—
374C4×D3—44no—
375Dic6—21no—
376Dic6—64no—
377Dic6—62no—
378Dic6—62no—
379Dic6—62no—
380Dic6—62no—
381Dic6—122no—
382Dic6—124no—
383Dic6—122no—
384Dic6—122no—
385Dic6C2×D1212yes—
386C3×C8—62no—
387C3⋊C8—22no—
388C3⋊C8—62no—
389C3⋊C8—62no—
390C3⋊C8—122no—
391C2×C32—22no—
392C2×C32C2×#12111yescharacteristic
393C2×C32#57512yes—
394C3×D3—42no—
395C3×D3—82no—
396C24—61no—
397#41—34no—
398#41—62no—
399C2×Q8—31no—
400C2×Q8—92no—
401C2×Q8—181no—
402C2×Q8—182no—
403C2×Q8—182no—
404C2×D4—31no—
405C2×D4—32no—
406C2×D4—61no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—61no—
410C22×C4—32no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—181no—
414C22×C4C2×D3211yescharacteristic
415Q16—184no—
416SD16—184no—
417SD16—184no—
418D8—184no—
419M16—62no—
420M16—182no—
421C2×C8—32no—
422C2×C8—92no—
423C4⋊C4—62no—
424C4⋊C4C2×D3212yes—
425C22⋊C4—21no—
426C22⋊C4—62no—
427C22⋊C4—184no—
428C22⋊C4—184no—
429C42—21no—
430C42—32no—
431C42—31no—
432C42—31no—
433C42—62no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—42no—
439C22×C3—62no—
440C22×C3—122no—
441C22×C3—122no—
442C22×C3—124no—
443C22×C3—244no—
444C22×C3#88412yes—
445C22×C3#93012yes—
446C22×C3C2×D3×D411yescharacteristic
447C22×C3C22×#6711yescharacteristic
448C2×D3—22no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—44no—
452C2×D3—84no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—62no—
461C3×C4—82no—
462C3×C4#88412yes—
463C3×C4#93012yes—
464C3×C4C22×D1211yescharacteristic
465C3×C4C2×#23511yescharacteristic
466C3×C4C2×#23511yescharacteristic
467C3×C4C2×D3×D411yescharacteristic
468Dic3—22no—
469Dic3—22no—
470Dic3—22no—
471Dic3—124no—
472Dic3—122no—
473Dic3—122no—
474C32#304011yescharacteristic, complemented, Sylow
475C23—61no—
476C23—62no—
477C23—124no—
478C23—122no—
479C23D3×#6711yescharacteristic
480Q8—32no—
481Q8—61no—
482Q8—182no—
483Q8—182no—
484Q8—362no—
485D4—32no—
486D4—64no—
487D4—61no—
488D4—62no—
489D4—62no—
490D4—122no—
491C2×C4—22no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—61no—
495C2×C4—64no—
496C2×C4—124no—
497C2×C4—122no—
498C2×C4—182no—
499C2×C4—362no—
500C2×C4#379812yes—
501C2×C4D3×D1212yes—
502C2×C4D3×#6711yescharacteristic
503C2×C4C22×D3211yescharacteristic
504C8—62no—
505C8—182no—
506C2×C3—22no—
507C2×C3—22no—
508C2×C3—22no—
509C2×C3—21no—
510C2×C3—42no—
511C2×C3—122no—
512C2×C3—242no—
513C2×C3C2×#88411yescharacteristic
514C2×C3#577312yes—
515C2×C3C2×#93011yescharacteristic
516C2×C3#582212yes—
517D3—42no—
518D3—82no—
519C22—22no—
520C22—62no—
521C22—122no—
522C22—124no—
523C22—122no—
524C22—244no—
525C22#6390612yes—
526C22C2×D3×#6711yescharacteristic
527C4—42no—
528C4—62no—
529C4—62no—
530C4—62no—
531C4—362no—
532C4#6390612yes—
533C4C2×#379811yescharacteristic
534C4C2×D3×D1211yescharacteristic
535C3—21no—
536C3#7894311yescharacteristic, complemented
537C3#8252711yescharacteristic, complemented
538C2—22no—
539C2—122no—
540C2—242no—
541C2C2×#6390611yescharacteristic
542C2#1059546912yes—
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#117C22
socleC22×C32C22×D4
prime radicalC2×#63823C2
hypercenter#117D32
hypocenterC32#3040
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C3×#5522C2semidirect
2. C2×#63860C2semidirect
3. C2×#63831C2semidirect
5. #10592703C2semidirect
6. C3×#5001C2semidirect
7. #10591115C2semidirect
8. #10590532C2semidirect
9. #10590530C2semidirect
19. C3×#893C22semidirect
20. C3×#887C22semidirect
28. C2×#3719C22semidirect
30. C2×#3716C22semidirect
34. #63637C22semidirect
36. Dic3⋊(C3⋊C8)C22semidirect
43. #5522D3semidirect
52. #5001D3semidirect
134. #893C2×D3semidirect
135. #887C2×D3semidirect
474. C32#3040semidirect
536. C3#78943semidirect
537. C3#82527semidirect

Maximal subgroups

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

subgroupindex
C3×#55222
C2×#638602
C2×#638312
C2×#638232
#105927032
C3×#50012
#105911152
#105905322
#105905302
#825273
#789433