Finite Group Databasebeta

← #11864081

Subgroups — #11864081

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
2C3×#5524C211yesmaximal, characteristic, complemented
3C2×#63831C211yesmaximal, characteristic, complemented
4C2×#63823C211yesmaximal, characteristic, complemented
5#10592703C211yesmaximal, characteristic, complemented
6C3×#5001C211yesmaximal, characteristic, complemented
7#10591069C211yesmaximal, characteristic, complemented
8#10590614C214yesmaximal, complemented
9#10590613C214yesmaximal, complemented
10#82928—31nomaximal
11#78927—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—
22C32×#117C2211yescharacteristic
23C2×#3748C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C3×#897C2212yescomplemented
26C3×#895C2212yescomplemented
27C3×#886—21no—
28C3×#882—21no—
29C3×C4×D12—22no—
30C2×#3719C2212yescomplemented
31#63831—24no—
32C2×#3716C2212yescomplemented
33C2×#3715C2212yescomplemented
34#63823—24no—
35C2×#3713C2212yescomplemented
36#63637C2214yescomplemented
37C3×#809C2214yescomplemented
38(C2×D3)⋊(C3⋊C8)C2214yescomplemented
39C22×#237—31no—
40C22×#235—31no—
41C3×#525—31no—
42C2×#951—31no—
43C2×#949—31no—
44C2×#932—31no—
45C2×#909—31no—
46#5524D311yescharacteristic, complemented
47C3×#418—31no—
48#5207—31no—
49#5126—31no—
50#5047—34no—
51#5037—34no—
52#5004—31no—
53#5004—31no—
54#5004—61no—
55#5001D311yescharacteristic, complemented
56#4783—34no—
57#4781—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×D12—22no—
73C2×C3×D12D414yescomplemented
74C2×C3×D12C2311yescharacteristic
75C2×C3×D12C2312yes—
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3D412yes—
80C2×C3×C4×D3C2312yes—
81C2×C3×Dic6C2311yescharacteristic
82C2×#625—28no—
83C2×#625—24no—
84C2×D3×Dic3—28no—
85C2×D3×Dic3—24no—
86#3795—216no—
87#3795—28no—
88#3795—28no—
89C32×C4⋊C4C2312yes—
90C32×C22⋊C4—21no—
91C32×C42—21no—
92#3748—22no—
93C2×C3⋊(C3⋊C8)C2312yes—
94C3×(C2×D3)⋊C4—22no—
95C3×(C2×D3)⋊C4—22no—
96C3×C4⋊Dic3—22no—
97C3×#208—22no—
98C2×C3×C3⋊C8C2312yes—
99#3719—24no—
100#3716—24no—
101#3715—24no—
102#3713—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—
125C2×C3×M16—31no—
126C3×#123—61no—
127C3×C22⋊Q8—32no—
128C3×#121—32no—
129C3×#120—61no—
130C3×C4×D4—62no—
131C3×#117—21no—
132C3×#117C2×D311yescharacteristic
133C3×#117C2×D311yescharacteristic
134#951—64no—
135#949—64no—
136C2×#216—32no—
137C2×#215—32no—
138C2×#214—32no—
139C2×#214—32no—
140#932—64no—
141C2×#213—32no—
142C2×#213—32no—
143C2×#208—31no—
144C2×#208—31no—
145C2×#208—61no—
146C2×#208C2×D311yescharacteristic
147#909—64no—
148C2×Dic12—32no—
149C2×#204—32no—
150#897C2×D312yescomplemented
151#895C2×D312yescomplemented
152#886—21no—
153#882—21no—
154C4×D12—22no—
155C3×#102—34no—
156#830—34no—
157(C2×D3)⋊C8—34no—
158#810—34no—
159#810—34no—
160#810—64no—
161#809C2×D314yescomplemented
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×C4—22no—
171C2×C32×C4C2×D412yes—
172C2×C32×C4C2×D412yes—
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2411yescharacteristic
175C2×C3⋊Dic3—24no—
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×D12—44no—
189C3×D12C2×D414yes—
190C3×C4×D3—24no—
191C3×C4×D3—22no—
192C3×C4×D3—22no—
193C3×C4×D3—42no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196#625—216no—
197D3×Dic3—216no—
198C3⋊(C3⋊C8)—22no—
199C3×C3⋊C8—22no—
200C22×#41—91no—
201C2×#137—91no—
202C2×AGL(1,ℤ/8)—91no—
203#525—31no—
204#452—94no—
205#449—94no—
206#419—91no—
207#418—31no—
208#407—91no—
209C24×C3—31no—
210C24×C3—61no—
211C23×D3—21no—
212C23×D3#6711yescharacteristic
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—62no—
220C2×C3×D4—61no—
221C2×C3×D4—62no—
222C2×C3×D4—61no—
223C22×C3×C4—21no—
224C22×C3×C4—31no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—62no—
228C22×C3×C4—62no—
229C22×C3×C4—61no—
230C22×C3×C4C22×D311yescharacteristic
231C22×C3×C4C22×D311yescharacteristic
232C2×#67—21no—
233C2×#67—64no—
234C2×#67—68no—
235C2×#67—68no—
236C2×#67—64no—
237C22×Dic3—68no—
238C22×Dic3—61no—
239C22×Dic3—61no—
240C22×Dic3—64no—
241C22×Dic3—121no—
242C22×Dic3#6711yescharacteristic
243#237—68no—
244#237—68no—
245#237—616no—
246#235—22no—
247#235—22no—
248#235—68no—
249#235—616no—
250#235—68no—
251C2×D12—22no—
252C2×D12—21no—
253C2×D12—22no—
254C2×D12D1214yescomplemented
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×D3#6712yes—
263C2×C4×D3C22×D312yes—
264C2×Dic6—34no—
265C2×Dic6—34no—
266C2×Dic6—32no—
267C2×Dic6—32no—
268C2×Dic6—62no—
269C2×Dic6—64no—
270C2×Dic6—62no—
271C2×Dic6—61no—
272C2×Dic6—62no—
273C2×Dic6—61no—
274C2×Dic6—122no—
275C2×Dic6—121no—
276C2×Dic6C22×D311yescharacteristic
277C3×M16—62no—
278C2×C3×C8—32no—
279C3×C4⋊C4—22no—
280C3×C4⋊C4—62no—
281C3×C4⋊C4C22×D312yes—
282C3×C4⋊C4C22×D312yes—
283C3×C22⋊C4—21no—
284C3×C22⋊C4—21no—
285C3×C22⋊C4—41no—
286C3×C22⋊C4—62no—
287C3×C22⋊C4—62no—
288C3×C42—21no—
289C3×C42—21no—
290C3×C42—41no—
291#216—64no—
292#215—64no—
293#214—64no—
294#214—64no—
295#213—64no—
296#213—64no—
297(C2×D3)⋊C4—22no—
298(C2×D3)⋊C4—22no—
299C4⋊Dic3—22no—
300#208—22no—
301#208—62no—
302#208—62no—
303#208—122no—
304C2×C3⋊C8—32no—
305C2×C3⋊C8—32no—
306C2×C3⋊C8—62no—
307C2×C3⋊C8C22×D312yes—
308Dic12—64no—
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—42no—
323C32×C4#12012yes—
324C32×C4C22×D411yescharacteristic
325C32×C4C22×D411yescharacteristic
326C3⋊Dic3—24no—
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)—184no—
341C2×Q16—92no—
342C2×SD16—92no—
343C2×SD16—92no—
344C2×D8—92no—
345C2×M16—31no—
346C2×M16—91no—
347#123—61no—
348C22⋊Q8—32no—
349#121—32no—
350#120—61no—
351C4×D4—62no—
352#117D3211yescharacteristic
353#103—94no—
354#102—34no—
355C22⋊C8—94no—
356C23×C3—21no—
357C23×C3—32no—
358C23×C3—62no—
359C23×C3—64no—
360C23×C3—62no—
361C23×C3—61no—
362C23×C3—122no—
363C23×C3—124no—
364C23×C3D3×D411yescharacteristic
365C23×C3C2×#6711yescharacteristic
366C22×D3—24no—
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—21no—
370C22×D3—42no—
371C22×D3—44no—
372C22×D3C2×#6712yes—
373C3×Q8—61no—
374C3×Q8—61no—
375C3×D4—34no—
376C3×D4—64no—
377C3×D4—61no—
378C3×D4—62no—
379C3×D4—61no—
380C3×D4—124no—
381C3×D4—122no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—32no—
389C2×C3×C4—42no—
390C2×C3×C4—62no—
391C2×C3×C4—64no—
392C2×C3×C4—62no—
393C2×C3×C4—62no—
394C2×C3×C4—61no—
395C2×C3×C4—122no—
396C2×C3×C4C2×D1212yes—
397C2×C3×C4D3×D412yes—
398C2×C3×C4D3×D412yes—
399C2×C3×C4D3×D412yes—
400C2×C3×C4D3×D411yescharacteristic
401C2×C3×C4C2×#6711yescharacteristic
402C2×C3×C4C23×D311yescharacteristic
403C2×C3×C4C23×D311yescharacteristic
404#67—42no—
405#67—616no—
406#67—616no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—616no—
410C2×Dic3—64no—
411C2×Dic3—64no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—122no—
415C2×Dic3—124no—
416C2×Dic3C2×#6712yes—
417D12—24no—
418D12—22no—
419D12—21no—
420D12—21no—
421D12—44no—
422D12C2×D1214yes—
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—62no—
440C3⋊C8—22no—
441C3⋊C8—62no—
442C3⋊C8—62no—
443C3⋊C8—122no—
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—182no—
461C2×Q8—181no—
462C2×D4—32no—
463C2×D4—31no—
464C2×D4—34no—
465C2×D4—62no—
466C2×D4—61no—
467C2×D4—62no—
468C2×D4—61no—
469C2×D4—184no—
470C2×D4—188no—
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—184no—
483SD16—184no—
484D8—184no—
485M16—62no—
486M16—182no—
487C2×C8—32no—
488C2×C8—92no—
489C4⋊C4—62no—
490C4⋊C4C2×D3212yes—
491C22⋊C4—21no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C42—21no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—42no—
500C22×C3—62no—
501C22×C3—62no—
502C22×C3—62no—
503C22×C3—64no—
504C22×C3—124no—
505C22×C3—128no—
506C22×C3—124no—
507C22×C3—121no—
508C22×C3—122no—
509C22×C3#88212yes—
510C22×C3#93712yes—
511C22×C3C2×D3×D411yescharacteristic
512C22×C3C22×#6711yescharacteristic
513C2×D3—24no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—48no—
518C2×D3—44no—
519C2×D3—44no—
520C2×D3—41no—
521C2×D3—42no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—22no—
525C3×C4—42no—
526C3×C4—42no—
527C3×C4—62no—
528C3×C4—82no—
529C3×C4—121no—
530C3×C4#88212yes—
531C3×C4#93712yes—
532C3×C4C22×D1211yescharacteristic
533C3×C4C2×D3×D411yescharacteristic
534C3×C4C2×D3×D411yescharacteristic
535C3×C4C2×D3×D411yescharacteristic
536Dic3—22no—
537Dic3—41no—
538Dic3—64no—
539Dic3—64no—
540Dic3—124no—
541C32#300511yescharacteristic, complemented, Sylow
542C23—32no—
543C23—62no—
544C23—62no—
545C23—64no—
546C23—61no—
547C23—122no—
548C23—124no—
549C23D3×#6711yescharacteristic
550Q8—61no—
551Q8—61no—
552Q8—94no—
553Q8—182no—
554Q8—184no—
555D4—34no—
556D4—61no—
557D4—62no—
558D4—61no—
559D4—64no—
560D4—124no—
561D4—122no—
562D4—1816no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—61no—
570C2×C4—122no—
571C2×C4—1816no—
572C2×C4—182no—
573C2×C4—184no—
574C2×C4D3×D1212yes—
575C2×C4#380212yes—
576C2×C4D3×#6711yescharacteristic
577C2×C4C22×D3211yescharacteristic
578C8—62no—
579C8—182no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—22no—
583C2×C3—21no—
584C2×C3—42no—
585C2×C3—122no—
586C2×C3—124no—
587C2×C3—241no—
588C2×C3C2×#88211yescharacteristic
589C2×C3#577312yes—
590C2×C3C2×#93711yescharacteristic
591C2×C3#582212yes—
592D3—44no—
593D3—42no—
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×#6711yescharacteristic
607C4—42no—
608C4—62no—
609C4—121no—
610C4—184no—
611C4#6392212yes—
612C4C2×D3×D1211yescharacteristic
613C4C2×#380211yescharacteristic
614C3—21no—
615C3#7892711yescharacteristic, complemented
616C3#8292811yescharacteristic, complemented
617C2—22no—
618C2—122no—
619C2—124no—
620C2—241no—
621C2C2×#6392211yescharacteristic
622C2#1059546912yes—
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#117C22
socleC22×C32C22×D4
hypercenter#117D32
hypocenterC32#3005
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. C22×#3795C2semidirect
2. C3×#5524C2semidirect
3. C2×#63831C2semidirect
4. C2×#63823C2semidirect
5. #10592703C2semidirect
6. C3×#5001C2semidirect
7. #10591069C2semidirect
8. #10590614C2semidirect
9. #10590613C2semidirect
25. C3×#897C22semidirect
26. C3×#895C22semidirect
30. C2×#3719C22semidirect
32. C2×#3716C22semidirect
33. C2×#3715C22semidirect
35. C2×#3713C22semidirect
36. #63637C22semidirect
37. C3×#809C22semidirect
38. (C2×D3)⋊(C3⋊C8)C22semidirect
46. #5524D3semidirect
55. #5001D3semidirect
64. C2×#634D4semidirect
73. C2×C3×D12D4semidirect
150. #897C2×D3semidirect
151. #895C2×D3semidirect
161. #809C2×D3semidirect
254. C2×D12D12semidirect
541. C32#3005semidirect
615. C3#78927semidirect
616. C3#82928semidirect

Maximal subgroups

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

subgroupindex
C22×#37952
C3×#55242
C2×#638312
C2×#638232
#105927032
C3×#50012
#105910692
#105906142
#105906132
#829283
#789273