Finite Group Databasebeta

← #11864078

Subgroups — #11864078

Order 1152.

1591 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3795C211yesmaximal, characteristic, complemented
2C2×C3×#908C211yesmaximal, characteristic, complemented
3#10595022C211yesmaximal, characteristic, complemented
4C2×#63831C211yesmaximal, characteristic, complemented
5#10592819C211yesmaximal, characteristic, complemented
6C3×#5126C211yesmaximal, characteristic, complemented
7#10591086C211yesmaximal, characteristic, complemented
8#10590962C214yesmaximal, complemented
9#10590960C214yesmaximal, complemented
10#83945—31nomaximal
11#78638—31nomaximal
12C22×#634C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C22×#625—21no—
17C22×D3×Dic3—21no—
18C2×#3795—24no—
19C2×#3795—24no—
20C2×#3795—22no—
21C2×#3795C2214yes—
22C2×C32×M16C2211yescharacteristic
23#64135C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C3×#908—24no—
26C2×C3×D24C2212yescomplemented
27C2×C3×#204C2212yescomplemented
28#63976—21no—
29#63961—21no—
30#63908C2212yescomplemented
31C4×#625—22no—
32#63856C2212yescomplemented
33C2×#3719C2212yescomplemented
34#63831—24no—
35C2×#3716C2212yescomplemented
36#63643C2214yescomplemented
37C3×(C2×D3)⋊C8C2214yescomplemented
38#63563C2214yescomplemented
39C22×#237—31no—
40C22×#235—31no—
41C2×C3×AGL(1,ℤ/8)—31no—
42#5828—31no—
43C2×#951—31no—
44#5791—31no—
45C2×#909—31no—
46C2×#908D311yescharacteristic, complemented
47C3×#407—31no—
48#5215—31no—
49#5173—34no—
50#5158—34no—
51#5131—31no—
52#5126D311yescharacteristic, complemented
53#5120—31no—
54#5120—31no—
55#5120—61no—
56#4734—34no—
57#4733—34no—
58C23×C3×D3—21no—
59C23×C3×D3D411yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C22×C3⋊Dic3—21no—
62C2×#634—21no—
63C2×#634—22no—
64C2×#634D414yescomplemented
65C2×#634C2312yes—
66C2×C3×#67—21no—
67C22×C3×Dic3D411yescharacteristic
68C3×#235—22no—
69C3×#235—22no—
70C2×C3×D12—21no—
71C2×C3×D12—22no—
72C2×C3×D12D414yescomplemented
73C2×C3×D12C2311yescharacteristic
74C2×C3×D12C2312yes—
75C2×C3×C4×D3—22no—
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3D412yes—
79C2×C3×C4×D3C2312yes—
80C2×C3×Dic6C2311yescharacteristic
81C2×#625—28no—
82C2×#625—24no—
83C2×#625—22no—
84C2×D3×Dic3—28no—
85C2×D3×Dic3—24no—
86#3795—216no—
87#3795—28no—
88#3795—28no—
89C32×M16—22no—
90C2×C32×C8C2312yes—
91#3757—21no—
92C4⋊(C3⋊Dic3)C2312yes—
93C4×C3⋊Dic3—21no—
94C3×#208—22no—
95C2×C3×C3⋊C8C2312yes—
96C3×D24—24no—
97C3×#204—24no—
98#3724—22no—
99(C2×D3)⋊Dic3—22no—
100(C2×D3)⋊Dic3—22no—
101#3719—24no—
102#3716—24no—
103#3005—91noSylow
104C2×C3×#41—31no—
105C22×C3×D4—31no—
106C23×C3×C4—31no—
107C22×#67—61no—
108C22×#67—61no—
109C23×Dic3—61no—
110C2×#237—34no—
111C2×#237—64no—
112C2×#237—64no—
113C2×#237—62no—
114C2×#235—34no—
115C2×#235—64no—
116C2×#235—62no—
117C2×#235—64no—
118C2×#235C2×D311yescharacteristic
119C22×D12C2×D311yescharacteristic
120C22×C4×D3—61no—
121C22×C4×D3C2×D311yescharacteristic
122C22×Dic6—31no—
123C22×Dic6—31no—
124C22×Dic6—61no—
125C3×AGL(1,ℤ/8)—64no—
126C2×C3×SD16—32no—
127C2×C3×D8—32no—
128C2×C3×M16—21no—
129C2×C3×M16—31no—
130C2×C3×M16C2×D311yescharacteristic
131C2×C3×M16C2×D311yescharacteristic
132#953—61no—
133#951—64no—
134#946—32no—
135C2×#216—32no—
136#938—32no—
137#937—61no—
138#935—61no—
139C2×#214—32no—
140#930—32no—
141#929—61no—
142C4×#67—62no—
143C4×#67—62no—
144#926—31no—
145#926—31no—
146#926—61no—
147#924—32no—
148C2×#208C2×D311yescharacteristic
149#909—64no—
150#908—24no—
151C2×Dic12—32no—
152C2×D24C2×D312yescomplemented
153C2×#204—32no—
154C2×#204C2×D312yescomplemented
155C3×C22⋊C8—34no—
156#832—34no—
157#821—34no—
158(C2×D3)⋊C8C2×D314yescomplemented
159#816—34no—
160#816—34no—
161#816—64no—
162C23×C32C2×D411yescharacteristic
163C22×C3×D3—24no—
164C22×C3×D3—22no—
165C22×C3×D3—22no—
166C22×C3×D3—21no—
167C22×C3×D3—42no—
168C22×C3×D3—44no—
169C22×C3×D3C2×D412yes—
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D412yes—
172C2×C32×C4C2×D412yes—
173C2×C32×C4C2411yescharacteristic
174C2×C3⋊Dic3—24no—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—22no—
177#634—24no—
178#634—22no—
179#634C2×D414yes—
180C3×#67—42no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3C2×D412yes—
184C3×D12—24no—
185C3×D12—22no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12C2×D414yes—
189C3×C4×D3—24no—
190C3×C4×D3—22no—
191C3×C4×D3—22no—
192C3×C4×D3—42no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195#625—216no—
196#625—44no—
197D3×Dic3—216no—
198C32×C8—22no—
199C3×C3⋊C8—22no—
200C22×#41—91no—
201C2×#137—91no—
202C2×AGL(1,ℤ/8)—31no—
203#525—91no—
204#452—94no—
205#449—94no—
206#419—91no—
207#418—91no—
208#407—31no—
209C24×C3—31no—
210C24×C3—61no—
211C23×D3—21no—
212C23×D3D1211yescharacteristic
213C3×#41—62no—
214C3×#41—62no—
215C2×C3×Q8—31no—
216C2×C3×D4—34no—
217C2×C3×D4—32no—
218C2×C3×D4—31no—
219C2×C3×D4—61no—
220C2×C3×D4—61no—
221C2×C3×D4—62no—
222C22×C3×C4—21no—
223C22×C3×C4—31no—
224C22×C3×C4—32no—
225C22×C3×C4—32no—
226C22×C3×C4—62no—
227C22×C3×C4—62no—
228C22×C3×C4—61no—
229C22×C3×C4C22×D311yescharacteristic
230C22×C3×C4C22×D311yescharacteristic
231C2×#67—21no—
232C2×#67—64no—
233C2×#67—68no—
234C2×#67—68no—
235C2×#67—64no—
236C2×#67—62no—
237C2×#67—62no—
238C22×Dic3—68no—
239C22×Dic3—61no—
240C22×Dic3—61no—
241C22×Dic3—64no—
242C22×Dic3—121no—
243C22×Dic3D1211yescharacteristic
244#237—68no—
245#237—68no—
246#237—616no—
247#235—22no—
248#235—22no—
249#235—68no—
250#235—616no—
251#235—68no—
252C2×D12—21no—
253C2×D12—22no—
254C2×D12#6714yescomplemented
255C2×D12C22×D311yescharacteristic
256C2×D12C22×D312yes—
257C2×C4×D3—22no—
258C2×C4×D3—22no—
259C2×C4×D3—21no—
260C2×C4×D3—64no—
261C2×C4×D3—68no—
262C2×C4×D3D1212yes—
263C2×C4×D3C22×D312yes—
264C2×Dic6—34no—
265C2×Dic6—34no—
266C2×Dic6—32no—
267C2×Dic6—32no—
268C2×Dic6—62no—
269C2×Dic6—64no—
270C2×Dic6—61no—
271C2×Dic6—61no—
272C2×Dic6—62no—
273C2×Dic6—62no—
274C2×Dic6—121no—
275C2×Dic6—122no—
276C2×Dic6C22×D311yescharacteristic
277C3×SD16—64no—
278C3×D8—64no—
279C3×M16—22no—
280C3×M16—22no—
281C3×M16—42no—
282C3×M16—62no—
283C2×C3×C8—22no—
284C2×C3×C8—32no—
285C2×C3×C8C22×D312yes—
286C2×C3×C8C22×D312yes—
287#217—62no—
288#217—61no—
289#217—61no—
290#217—62no—
291#217—121no—
292#216—64no—
293#214—64no—
294(C2×D3)⋊C4—62no—
295(C2×D3)⋊C4—62no—
296C4⋊Dic3—32no—
297C4⋊Dic3—32no—
298C4⋊Dic3—62no—
299Dic3⋊C4—62no—
300Dic3⋊C4—62no—
301C4×Dic3—61no—
302C4×Dic3—61no—
303C4×Dic3—121no—
304#208—22no—
305C2×C3⋊C8C22×D312yes—
306Dic12—64no—
307D24—24no—
308#204—24no—
309#204—64no—
310C22×C32—22no—
311C22×C32#12012yes—
312C22×C32C22×D411yescharacteristic
313C2×C3×D3—24no—
314C2×C3×D3—22no—
315C2×C3×D3—22no—
316C2×C3×D3—22no—
317C2×C3×D3—44no—
318C2×C3×D3—48no—
319C2×C3×D3—44no—
320C2×C3×D3—41no—
321C2×C3×D3—42no—
322C32×C4#12012yes—
323C32×C4C22×D411yescharacteristic
324C32×C4C22×D411yescharacteristic
325C3⋊Dic3—24no—
326C3⋊Dic3—42no—
327C3×Dic3—22no—
328C3×Dic3—41no—
329C2×#41—31no—
330C2×#41—94no—
331C2×#41—182no—
332C2×#41—184no—
333C2×#41—184no—
334C22×Q8—91no—
335C22×D4—31no—
336C22×D4—181no—
337C23×C4—31no—
338C23×C4—181no—
339#137—184no—
340AGL(1,ℤ/8)—64no—
341C2×Q16—92no—
342C2×SD16—32no—
343C2×SD16—92no—
344C2×D8—32no—
345C2×M16—31no—
346C2×M16D3211yescharacteristic
347#123—181no—
348C22⋊Q8—92no—
349#121—92no—
350#120—181no—
351C4×D4—182no—
352#117—91no—
353#103—94no—
354#102—94no—
355C22⋊C8—34no—
356C23×C3—21no—
357C23×C3—32no—
358C23×C3—64no—
359C23×C3—62no—
360C23×C3—62no—
361C23×C3—61no—
362C23×C3—122no—
363C23×C3—124no—
364C23×C3C2×D1211yescharacteristic
365C23×C3D3×D411yescharacteristic
366C22×D3—22no—
367C22×D3—24no—
368C22×D3—22no—
369C22×D3—21no—
370C22×D3—42no—
371C22×D3—44no—
372C22×D3C2×D1212yes—
373C3×Q8—61no—
374C3×Q8—61no—
375C3×D4—34no—
376C3×D4—64no—
377C3×D4—62no—
378C3×D4—61no—
379C3×D4—61no—
380C3×D4—122no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—32no—
386C2×C3×C4—62no—
387C2×C3×C4—64no—
388C2×C3×C4—62no—
389C2×C3×C4—62no—
390C2×C3×C4—61no—
391C2×C3×C4—122no—
392C2×C3×C4C2×D1211yescharacteristic
393C2×C3×C4D3×D411yescharacteristic
394C2×C3×C4D3×D412yes—
395C2×C3×C4D3×D412yes—
396C2×C3×C4D3×D412yes—
397C2×C3×C4C2×#6712yes—
398C2×C3×C4C23×D311yescharacteristic
399C2×C3×C4C23×D311yescharacteristic
400#67—42no—
401#67—616no—
402#67—616no—
403#67—124no—
404#67—124no—
405C2×Dic3—22no—
406C2×Dic3—21no—
407C2×Dic3—616no—
408C2×Dic3—64no—
409C2×Dic3—64no—
410C2×Dic3—62no—
411C2×Dic3—62no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—122no—
415C2×Dic3—124no—
416C2×Dic3—122no—
417C2×Dic3C2×D1212yes—
418D12—24no—
419D12—22no—
420D12—21no—
421D12—21no—
422D12C2×#6714yes—
423C4×D3—24no—
424C4×D3—22no—
425C4×D3—22no—
426C4×D3—42no—
427C4×D3—616no—
428Dic6—21no—
429Dic6—21no—
430Dic6—34no—
431Dic6—34no—
432Dic6—64no—
433Dic6—64no—
434Dic6—64no—
435Dic6—62no—
436Dic6—62no—
437Dic6—122no—
438Dic6—124no—
439C3×C8—22no—
440C3×C8—22no—
441C3×C8—42no—
442C3×C8—62no—
443C3⋊C8—22no—
444C2×C32—22no—
445C2×C32C2×#12011yescharacteristic
446C2×C32#57512yes—
447C3×D3—42no—
448C3×D3—44no—
449C3×D3—81no—
450C24—31no—
451C24—61no—
452#41—62no—
453#41—62no—
454#41—188no—
455#41—188no—
456#41—1816no—
457C2×Q8—31no—
458C2×Q8—92no—
459C2×Q8—94no—
460C2×Q8—181no—
461C2×Q8—182no—
462C2×D4—32no—
463C2×D4—31no—
464C2×D4—34no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—62no—
468C2×D4—184no—
469C2×D4—188no—
470C2×D4—182no—
471C22×C4—31no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—61no—
477C22×C4—184no—
478C22×C4—188no—
479C22×C4—181no—
480C22×C4C2×D3211yescharacteristic
481Q16—184no—
482SD16—64no—
483SD16—184no—
484D8—64no—
485M16—22no—
486M16—62no—
487C2×C8—32no—
488C2×C8C2×D3212yes—
489C4⋊C4—92no—
490C4⋊C4—182no—
491C22⋊C4—182no—
492C22⋊C4—182no—
493C22⋊C4—181no—
494C42—181no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—42no—
500C22×C3—64no—
501C22×C3—62no—
502C22×C3—62no—
503C22×C3—62no—
504C22×C3—128no—
505C22×C3—124no—
506C22×C3—124no—
507C22×C3—121no—
508C22×C3—122no—
509C22×C3#88212yes—
510C22×C3#93712yes—
511C22×C3C22×D1211yescharacteristic
512C22×C3C2×D3×D411yescharacteristic
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—24no—
517C2×D3—44no—
518C2×D3—48no—
519C2×D3—44no—
520C2×D3—42no—
521C2×D3—41no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—22no—
525C3×C4—62no—
526C3×C4—121no—
527C3×C4#88212yes—
528C3×C4#93712yes—
529C3×C4C2×D3×D411yescharacteristic
530C3×C4C2×D3×D411yescharacteristic
531C3×C4C2×D3×D411yescharacteristic
532C3×C4C22×#6711yescharacteristic
533Dic3—22no—
534Dic3—41no—
535Dic3—64no—
536Dic3—64no—
537Dic3—124no—
538Dic3—122no—
539Dic3—122no—
540Dic3—242no—
541C32#300511yescharacteristic, complemented, Sylow
542C23—32no—
543C23—62no—
544C23—62no—
545C23—64no—
546C23—61no—
547C23—122no—
548C23—124no—
549C23D3×D1211yescharacteristic
550Q8—61no—
551Q8—61no—
552Q8—94no—
553Q8—182no—
554Q8—184no—
555D4—34no—
556D4—62no—
557D4—61no—
558D4—61no—
559D4—64no—
560D4—122no—
561D4—1816no—
562D4—364no—
563C2×C4—32no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—61no—
569C2×C4—122no—
570C2×C4—1816no—
571C2×C4—182no—
572C2×C4—184no—
573C2×C4—182no—
574C2×C4D3×D1211yescharacteristic
575C2×C4#380212yes—
576C2×C4D3×#6712yes—
577C2×C4C22×D3211yescharacteristic
578C8—22no—
579C8—62no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—22no—
583C2×C3—21no—
584C2×C3—42no—
585C2×C3—124no—
586C2×C3—122no—
587C2×C3—241no—
588C2×C3C2×#88211yescharacteristic
589C2×C3#574812yes—
590C2×C3#577312yes—
591C2×C3C2×#93711yescharacteristic
592D3—42no—
593D3—44no—
594D3—81no—
595C22—22no—
596C22—62no—
597C22—62no—
598C22—62no—
599C22—64no—
600C22—124no—
601C22—128no—
602C22—124no—
603C22—122no—
604C22—121no—
605C22#6392212yes—
606C22C2×D3×D1211yescharacteristic
607C4—62no—
608C4—121no—
609C4—184no—
610C4—362no—
611C4#6392212yes—
612C4C2×#380211yescharacteristic
613C4C2×D3×#6711yescharacteristic
614C3—21no—
615C3#7863811yescharacteristic, complemented
616C3#8394511yescharacteristic, complemented
617C2—22no—
618C2—122no—
619C2—124no—
620C2—241no—
621C2#1059471212yes—
622C2C2×#6392211yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
hypercenterC2×M16D32
hypocenterC32#3005
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. C22×#3795C2semidirect
2. C2×C3×#908C2semidirect
3. #10595022C2semidirect
4. C2×#63831C2semidirect
5. #10592819C2semidirect
6. C3×#5126C2semidirect
7. #10591086C2semidirect
8. #10590962C2semidirect
9. #10590960C2semidirect
26. C2×C3×D24C22semidirect
27. C2×C3×#204C22semidirect
30. #63908C22semidirect
32. #63856C22semidirect
33. C2×#3719C22semidirect
35. C2×#3716C22semidirect
36. #63643C22semidirect
37. C3×(C2×D3)⋊C8C22semidirect
38. #63563C22semidirect
46. C2×#908D3semidirect
52. #5126D3semidirect
64. C2×#634D4semidirect
72. C2×C3×D12D4semidirect
152. C2×D24C2×D3semidirect
154. C2×#204C2×D3semidirect
158. (C2×D3)⋊C8C2×D3semidirect
254. C2×D12#67semidirect
541. C32#3005semidirect
615. C3#78638semidirect
616. C3#83945semidirect

Maximal subgroups

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

subgroupindex
C22×#37952
C2×C3×#9082
#105950222
C2×#638312
#105928192
C3×#51262
#105910862
#105909622
#105909602
#839453
#786383