Finite Group Databasebeta

← #11864084

Subgroups — #11864084

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×C3×#239C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic, complemented
3#10595022C211yesmaximal, characteristic, complemented
4C2×#63823C211yesmaximal, characteristic, complemented
5#10592924C211yesmaximal, characteristic, complemented
6#10591597C214yesmaximal, complemented
7#10591586C214yesmaximal, complemented
8#10591086C211yesmaximal, characteristic, complemented
9#10591069C211yesmaximal, characteristic, complemented
10#84081—31nomaximal
11#83980—31nomaximal
12C22×C32×Q8C2211yescharacteristic
13C2×C3×#239—24no—
14C2×C3×#239—24no—
15C2×C3×#239—22no—
16C2×C3×#239C2214yes—
17C2×C3×#235C2211yescharacteristic
18C22×C3×D12—21no—
19C22×C3×D12C2211yescharacteristic
20C22×C3×C4×D3—21no—
21C22×C3×C4×D3C2211yescharacteristic
22#64135C2211yescharacteristic
23C2×#3748C2211yescharacteristic
24C2×C3×#216C2212yescomplemented
25C3×#942—24no—
26C2×C3×#215C2212yescomplemented
27C2×C3×#208C2211yescharacteristic
28#63976—21no—
29#63961—21no—
30#63908C2212yescomplemented
31C4×#625—22no—
32#63856C2212yescomplemented
33C2×#3715C2212yescomplemented
34#63823—24no—
35C2×#3713C2212yescomplemented
36#63662C2214yescomplemented
37#63563C2214yescomplemented
38(C2×D3)⋊(C3⋊C8)C2214yescomplemented
39C22×C3×#41—31no—
40C22×#239D311yescharacteristic, complemented
41C2×C3×#137—31no—
42#5828—31no—
43C2×#949—31no—
44C2×#942D311yescharacteristic, complemented
45C2×#932—31no—
46#5791—31no—
47#5239—34no—
48#5238—34no—
49#5225—31no—
50#5225—31no—
51#5225—61no—
52#5215—31no—
53#5207—31no—
54#5187—34no—
55#5172—34no—
56#5131—31no—
57#5126—31no—
58C23×C3×D3—21no—
59C23×C3×D3D411yescharacteristic
60C2×C32×Q8—22no—
61C2×C32×Q8—21no—
62C2×C32×Q8D414yescomplemented
63C2×C32×Q8C2312yes—
64C22×C32×C4—21no—
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—21no—
67C22×C3×Dic3D411yescharacteristic
68C3×#239—216no—
69C3×#239—28no—
70C3×#239—28no—
71C3×#235—22no—
72C3×#235—22no—
73C2×C3×D12—28no—
74C2×C3×D12—24no—
75C2×C3×D12—21no—
76C2×C3×D12—22no—
77C2×C3×D12D414yescomplemented
78C2×C3×D12C2311yescharacteristic
79C2×C3×D12C2312yes—
80C2×C3×C4×D3—28no—
81C2×C3×C4×D3—22no—
82C2×C3×C4×D3—22no—
83C2×C3×C4×D3—24no—
84C2×C3×C4×D3—21no—
85C2×C3×C4×D3D412yes—
86C2×C3×C4×D3C2312yes—
87C2×C3×Dic6C2311yescharacteristic
88C2×#625—22no—
89#3757—21no—
90C4⋊(C3⋊Dic3)C2312yes—
91C4×C3⋊Dic3—21no—
92#3748—22no—
93C2×C3⋊(C3⋊C8)C2312yes—
94C3×#216—24no—
95C3×#215—24no—
96C3×#208—22no—
97C2×C3×C3⋊C8C2312yes—
98#3724—22no—
99(C2×D3)⋊Dic3—22no—
100(C2×D3)⋊Dic3—22no—
101#3715—24no—
102#3713—24no—
103#3005—91noSylow
104C2×C3×#41—34no—
105C2×C3×#41—31no—
106C2×C3×#41—62no—
107C2×C3×#41—64no—
108C2×C3×#41—64no—
109C22×C3×Q8—21no—
110C22×C3×Q8C2×D311yescharacteristic
111C22×C3×Q8C2×D311yescharacteristic
112C22×C3×D4—31no—
113C22×C3×D4—61no—
114C23×C3×C4—31no—
115C23×C3×C4—61no—
116C2×#239—24no—
117C2×#239—24no—
118C2×#239—22no—
119C2×#239C2×D314yes—
120C2×#235C2×D311yescharacteristic
121C22×D12—21no—
122C22×D12C2×D311yescharacteristic
123C22×C4×D3—21no—
124C22×C4×D3C2×D311yescharacteristic
125C3×#137—64no—
126C2×C3×Q16—32no—
127C2×C3×SD16—32no—
128C2×C3×M16—31no—
129#953—61no—
130#949—64no—
131#946—32no—
132C2×#216C2×D312yescomplemented
133#942—24no—
134C2×#215—32no—
135C2×#215C2×D312yescomplemented
136#938—32no—
137#937—61no—
138#935—61no—
139C2×#214—32no—
140#932—64no—
141C2×#213—32no—
142C2×#213—32no—
143#930—32no—
144#929—61no—
145C4×#67—62no—
146C4×#67—62no—
147#926—31no—
148#926—31no—
149#926—61no—
150#924—32no—
151C2×#208—31no—
152C2×#208—31no—
153C2×#208—61no—
154C2×#208C2×D311yescharacteristic
155#835—34no—
156#835—34no—
157#835—64no—
158#832—34no—
159#830—34no—
160#821—34no—
161(C2×D3)⋊C8—34no—
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—
170C32×Q8—24no—
171C32×Q8—22no—
172C32×Q8C2×D414yes—
173C2×C32×C4—24no—
174C2×C32×C4—22no—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2411yescharacteristic
179C2×C3⋊Dic3—22no—
180C3×#67—42no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3C2×D412yes—
184C3×D12—216no—
185C3×D12—24no—
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12C2×D414yes—
190C3×C4×D3—216no—
191C3×C4×D3—24no—
192C3×C4×D3—22no—
193C3×C4×D3—22no—
194C3×C4×D3—42no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197#625—44no—
198C3⋊(C3⋊C8)—22no—
199C3×C3⋊C8—22no—
200C22×#41—31no—
201C2×#137—31no—
202C2×AGL(1,ℤ/8)—91no—
203#525—91no—
204#452—94no—
205#449—94no—
206#419—91no—
207#418—91no—
208#407—91no—
209C24×C3—31no—
210C24×C3—61no—
211C23×D3—21no—
212C23×D3#6711yescharacteristic
213C3×#41—68no—
214C3×#41—616no—
215C3×#41—68no—
216C3×#41—62no—
217C3×#41—62no—
218C2×C3×Q8—22no—
219C2×C3×Q8—21no—
220C2×C3×Q8—22no—
221C2×C3×Q8—21no—
222C2×C3×Q8—24no—
223C2×C3×Q8—22no—
224C2×C3×Q8—31no—
225C2×C3×Q8—42no—
226C2×C3×Q8—41no—
227C2×C3×Q8#6714yescomplemented
228C2×C3×Q8#6714yescomplemented
229C2×C3×Q8C22×D312yes—
230C2×C3×Q8C22×D312yes—
231C2×C3×D4—34no—
232C2×C3×D4—32no—
233C2×C3×D4—31no—
234C2×C3×D4—64no—
235C2×C3×D4—68no—
236C2×C3×D4—61no—
237C2×C3×D4—61no—
238C2×C3×D4—62no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—31no—
243C22×C3×C4—32no—
244C22×C3×C4—32no—
245C22×C3×C4—41no—
246C22×C3×C4—64no—
247C22×C3×C4—68no—
248C22×C3×C4—62no—
249C22×C3×C4—62no—
250C22×C3×C4—61no—
251C22×C3×C4C22×D311yescharacteristic
252C22×C3×C4C22×D311yescharacteristic
253C2×#67—21no—
254C2×#67—62no—
255C2×#67—62no—
256C22×Dic3#6711yescharacteristic
257#239—216no—
258#239—28no—
259#239—28no—
260#235—22no—
261#235—22no—
262C2×D12—28no—
263C2×D12—24no—
264C2×D12—22no—
265C2×D12—21no—
266C2×D12#6714yescomplemented
267C2×D12C22×D311yescharacteristic
268C2×D12C22×D312yes—
269C2×C4×D3—28no—
270C2×C4×D3—22no—
271C2×C4×D3—22no—
272C2×C4×D3—24no—
273C2×C4×D3—21no—
274C2×C4×D3#6712yes—
275C2×C4×D3C22×D312yes—
276C2×Dic6C22×D311yescharacteristic
277C3×Q16—64no—
278C3×SD16—64no—
279C3×M16—62no—
280C2×C3×C8—32no—
281#217—62no—
282#217—61no—
283#217—61no—
284#217—62no—
285#217—121no—
286#216—24no—
287#215—24no—
288#215—64no—
289#214—64no—
290#213—64no—
291#213—64no—
292(C2×D3)⋊C4—62no—
293(C2×D3)⋊C4—62no—
294C4⋊Dic3—32no—
295C4⋊Dic3—32no—
296C4⋊Dic3—62no—
297Dic3⋊C4—62no—
298Dic3⋊C4—62no—
299C4×Dic3—61no—
300C4×Dic3—61no—
301C4×Dic3—121no—
302#208—22no—
303#208—62no—
304#208—62no—
305#208—122no—
306C2×C3⋊C8—32no—
307C2×C3⋊C8—32no—
308C2×C3⋊C8—62no—
309C2×C3⋊C8C22×D312yes—
310C22×C32—22no—
311C22×C32#12012yes—
312C22×C32C22×D411yescharacteristic
313C2×C3×D3—22no—
314C2×C3×D3—24no—
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—24no—
323C32×C4#12012yes—
324C32×C4C22×D411yescharacteristic
325C32×C4C22×D411yescharacteristic
326C3⋊Dic3—42no—
327C3×Dic3—22no—
328C3×Dic3—41no—
329C2×#41—34no—
330C2×#41—31no—
331C2×#41—62no—
332C2×#41—64no—
333C2×#41—64no—
334C22×Q8D3211yescharacteristic
335C22×D4—31no—
336C22×D4—61no—
337C23×C4—31no—
338C23×C4—61no—
339#137—64no—
340AGL(1,ℤ/8)—184no—
341C2×Q16—32no—
342C2×SD16—32no—
343C2×SD16—92no—
344C2×D8—92no—
345C2×M16—31no—
346C2×M16—91no—
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—94no—
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×C3D3×D411yescharacteristic
365C23×C3C2×#6711yescharacteristic
366C22×D3—22no—
367C22×D3—24no—
368C22×D3—22no—
369C22×D3—21no—
370C22×D3—42no—
371C22×D3—44no—
372C22×D3C2×#6712yes—
373C3×Q8—24no—
374C3×Q8—24no—
375C3×Q8—22no—
376C3×Q8—22no—
377C3×Q8—24no—
378C3×Q8—42no—
379C3×Q8—44no—
380C3×Q8—61no—
381C3×Q8—61no—
382C3×Q8C2×#6714yes—
383C3×Q8C2×#6714yes—
384C3×D4—34no—
385C3×D4—616no—
386C3×D4—64no—
387C3×D4—62no—
388C3×D4—61no—
389C3×D4—61no—
390C3×D4—122no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—32no—
400C2×C3×C4—42no—
401C2×C3×C4—44no—
402C2×C3×C4—616no—
403C2×C3×C4—62no—
404C2×C3×C4—64no—
405C2×C3×C4—62no—
406C2×C3×C4—62no—
407C2×C3×C4—61no—
408C2×C3×C4—122no—
409C2×C3×C4D3×D411yescharacteristic
410C2×C3×C4D3×D412yes—
411C2×C3×C4C2×#6712yes—
412C2×C3×C4C2×#6712yes—
413C2×C3×C4C2×#6711yescharacteristic
414C2×C3×C4C2×#6712yes—
415C2×C3×C4C23×D311yescharacteristic
416C2×C3×C4C23×D311yescharacteristic
417#67—42no—
418#67—124no—
419#67—124no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—62no—
423C2×Dic3—62no—
424C2×Dic3—122no—
425C2×Dic3C2×#6712yes—
426D12—216no—
427D12—24no—
428D12—21no—
429D12—22no—
430D12—21no—
431D12C2×#6714yes—
432C4×D3—216no—
433C4×D3—24no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—42no—
437Dic6—21no—
438Dic6—21no—
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—68no—
453#41—616no—
454#41—68no—
455#41—62no—
456#41—62no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—31no—
460C2×Q8#62514yescomplemented
461C2×Q8C2×D3212yes—
462C2×D4—32no—
463C2×D4—31no—
464C2×D4—34no—
465C2×D4—64no—
466C2×D4—68no—
467C2×D4—61no—
468C2×D4—62no—
469C2×D4—61no—
470C2×D4—182no—
471C22×C4—21no—
472C22×C4—31no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—64no—
476C22×C4—68no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—61no—
480C22×C4C2×D3211yescharacteristic
481Q16—64no—
482SD16—64no—
483SD16—184no—
484D8—184no—
485M16—62no—
486M16—182no—
487C2×C8—32no—
488C2×C8—92no—
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—62no—
501C22×C3—62no—
502C22×C3—64no—
503C22×C3—62no—
504C22×C3—124no—
505C22×C3—128no—
506C22×C3—124no—
507C22×C3—122no—
508C22×C3—121no—
509C22×C3#93712yes—
510C22×C3#95312yes—
511C22×C3C2×D3×D411yescharacteristic
512C22×C3C22×#6711yescharacteristic
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—24no—
523C3×C4—24no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—22no—
527C3×C4—44no—
528C3×C4—62no—
529C3×C4—121no—
530C3×C4#93712yes—
531C3×C4#95312yes—
532C3×C4C2×D3×D411yescharacteristic
533C3×C4C22×#6711yescharacteristic
534C3×C4C22×#6711yescharacteristic
535C3×C4C22×#6711yescharacteristic
536Dic3—22no—
537Dic3—41no—
538Dic3—122no—
539Dic3—122no—
540Dic3—242no—
541C32#300511yescharacteristic, complemented, Sylow
542C23—32no—
543C23—62no—
544C23—64no—
545C23—62no—
546C23—61no—
547C23—122no—
548C23—124no—
549C23D3×#6711yescharacteristic
550Q8—22no—
551Q8—24no—
552Q8—61no—
553Q8—61no—
554Q8C2×#62514yes—
555D4—34no—
556D4—616no—
557D4—62no—
558D4—61no—
559D4—61no—
560D4—64no—
561D4—122no—
562D4—364no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—32no—
566C2×C4—616no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—64no—
571C2×C4—61no—
572C2×C4—122no—
573C2×C4—182no—
574C2×C4C2×#62512yes—
575C2×C4D3×#6712yes—
576C2×C4D3×#6711yescharacteristic
577C2×C4C22×D3211yescharacteristic
578C8—62no—
579C8—182no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—21no—
583C2×C3—22no—
584C2×C3—42no—
585C2×C3—122no—
586C2×C3—124no—
587C2×C3—241no—
588C2×C3#577712yes—
589C2×C3C2×#93711yescharacteristic
590C2×C3#582212yes—
591C2×C3C2×#95311yescharacteristic
592D3—42no—
593D3—44no—
594D3—81no—
595C22—22no—
596C22—64no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—124no—
601C22—128no—
602C22—124no—
603C22—121no—
604C22—122no—
605C22#6397612yes—
606C22C2×D3×#6711yescharacteristic
607C4—24no—
608C4—62no—
609C4—121no—
610C4—362no—
611C4#6397612yes—
612C4C22×#62511yescharacteristic
613C4C2×D3×#6711yescharacteristic
614C3—21no—
615C3#8398011yescharacteristic, complemented
616C3#8408111yescharacteristic, complemented
617C2—22no—
618C2—122no—
619C2—124no—
620C2—241no—
621C2#1059531712yes—
622C2C2×#6397611yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×Q8C22
socleC22×C32C22×D4
hypercenterC22×Q8D32
hypocenterC32#3005
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×#239C2semidirect
2. C2×C3×#942C2semidirect
3. #10595022C2semidirect
4. C2×#63823C2semidirect
5. #10592924C2semidirect
6. #10591597C2semidirect
7. #10591586C2semidirect
8. #10591086C2semidirect
9. #10591069C2semidirect
24. C2×C3×#216C22semidirect
26. C2×C3×#215C22semidirect
30. #63908C22semidirect
32. #63856C22semidirect
33. C2×#3715C22semidirect
35. C2×#3713C22semidirect
36. #63662C22semidirect
37. #63563C22semidirect
38. (C2×D3)⋊(C3⋊C8)C22semidirect
40. C22×#239D3semidirect
44. C2×#942D3semidirect
62. C2×C32×Q8D4semidirect
77. C2×C3×D12D4semidirect
132. C2×#216C2×D3semidirect
135. C2×#215C2×D3semidirect
227. C2×C3×Q8#67semidirect
228. C2×C3×Q8#67semidirect
266. C2×D12#67semidirect
460. C2×Q8#625semidirect
541. C32#3005semidirect
615. C3#83980semidirect
616. C3#84081semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C2×C3×#9422
#105950222
C2×#638232
#105929242
#105915972
#105915862
#105910862
#105910692
#840813
#839803