Finite Group Databasebeta

← #11864147

Subgroups — #11864147

Order 1152.

1707 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic, complemented
2C2×C3×#930C211yesmaximal, characteristic, complemented
3C2×#63823C212yesmaximal, complemented
4#10592906C211yesmaximal, characteristic, complemented
5#10591171C212yesmaximal, complemented
6#10591161C214yesmaximal, complemented
7#10591160C212yesmaximal, complemented
8#10591087C212yesmaximal, complemented
9#84082—31nomaximal
10#82624—31nomaximal
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×#67—21no—
14C2×C3×#235—21no—
15C2×C3×#235—21no—
16C2×C3×#235—28no—
17C2×C3×#235C2212yes—
18C2×C3×#235C2212yescomplemented
19C22×C3×D12C2211yescharacteristic, complemented
20C22×C3×D12C2211yescharacteristic, complemented
21C22×C3×C4×D3—21no—
22C22×C3×Dic6C2211yescharacteristic, complemented
23C2×#3748C2212yescomplemented
24C3×#930—21no—
25C3×#930—21no—
26C3×#930C2212yes—
27C3×#930C2212yescomplemented
28C2×C3×(C2×D3)⋊C4—21no—
29C2×C3×C4⋊Dic3C2211yescharacteristic, complemented
30C2×#3715C2214yescomplemented
31#63823—28no—
32C2×#3713C2214yescomplemented
33#63657C2212yescomplemented
34#63657C2212yescomplemented
35#63565C2214yescomplemented
36#63562C2214yescomplemented
37C22×C3×#41—31no—
38C22×#235D311yescharacteristic, complemented
39C2×C3×#121—31no—
40C2×#949—32no—
41C2×#932—32no—
42C2×#930D311yescharacteristic, complemented
43#5238—34no—
44#5232—32no—
45#5227—32no—
46#5217—32no—
47#5207—31no—
48#5207—31no—
49#5207—61no—
50#5040—32no—
51#5037—34no—
52#5036—32no—
53#4966—32no—
54C24×C32D411yescharacteristic
55C23×C3×D3—21no—
56C23×C3×D3—21no—
57C22×C32×C4—24no—
58C22×C32×C4D411yescharacteristic
59C22×C32×C4D411yescharacteristic
60C22×C32×C4D411yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C22×C32×C4C2311yescharacteristic
64C2×C3×#67—28no—
65C2×C3×#67—22no—
66C2×C3×#67—22no—
67C2×C3×#67—22no—
68C2×C3×#67—22no—
69C2×C3×#67—44no—
70C22×C3×Dic3—21no—
71C22×C3×Dic3—21no—
72C3×#235—28no—
73C3×#235—28no—
74C3×#235—24no—
75C3×#235—24no—
76C3×#235—24no—
77C3×#235D418yescomplemented
78C2×C3×D12—21no—
79C2×C3×D12—24no—
80C2×C3×D12—24no—
81C2×C3×D12—21no—
82C2×C3×D12C2312yes—
83C2×C3×D12C2312yes—
84C2×C3×C4×D3—28no—
85C2×C3×C4×D3—22no—
86C2×C3×C4×D3—22no—
87C2×C3×Dic6—21no—
88C2×C3×Dic6—24no—
89C2×C3×Dic6C2312yes—
90#3748—24no—
91C2×C3⋊(C3⋊C8)C2314yes—
92C3×(C2×D3)⋊C4—22no—
93C3×(C2×D3)⋊C4—22no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2312yes—
96#3715—28no—
97#3713—28no—
98#3006—91noSylow
99C2×C3×#41—32no—
100C2×C3×#41—32no—
101C2×C3×#41—68no—
102C2×C3×#41—61no—
103C2×C3×#41—61no—
104C22×C3×Q8—31no—
105C22×C3×D4—31no—
106C22×C3×D4—31no—
107C22×C3×D4—61no—
108C22×C3×D4—61no—
109C23×C3×C4—21no—
110C23×C3×C4—61no—
111C23×C3×C4C2×D311yescharacteristic
112C23×C3×C4C2×D311yescharacteristic
113C22×#67—21no—
114C22×#67—21no—
115C2×#235—21no—
116C2×#235—21no—
117C2×#235—28no—
118C2×#235C2×D312yes—
119C2×#235C2×D312yescomplemented
120C22×D12C2×D311yescharacteristic, complemented
121C22×D12C2×D311yescharacteristic, complemented
122C22×C4×D3—21no—
123C22×Dic6C2×D311yescharacteristic, complemented
124C3×#121—32no—
125C3×#121—32no—
126C3×#121—61no—
127C3×#121—61no—
128C2×C3×C4⋊C4—31no—
129C2×C3×C22⋊C4—61no—
130#949—68no—
131C2×#215—34no—
132C2×#214—34no—
133#932—68no—
134C2×#213—34no—
135C2×#213—34no—
136#930—21no—
137#930—21no—
138#930C2×D312yes—
139#930C2×D312yescomplemented
140C2×(C2×D3)⋊C4—21no—
141C2×C4⋊Dic3C2×D311yescharacteristic, complemented
142C2×#208—32no—
143C2×#208—32no—
144C2×#208—62no—
145#835—34no—
146#832—34no—
147#830—32no—
148#830—32no—
149#830—32no—
150#830—32no—
151#830—62no—
152#830—62no—
153#810—34no—
154#809—34no—
155C23×C32—22no—
156C23×C32—22no—
157C23×C32—41no—
158C23×C32C2×D411yescharacteristic
159C23×C32C2×D411yescharacteristic
160C23×C32C2×D411yescharacteristic
161C22×C3×D3—24no—
162C22×C3×D3—22no—
163C22×C3×D3—22no—
164C22×C3×D3—42no—
165C22×C3×D3—42no—
166C22×C3×D3—42no—
167C22×C3×D3—42no—
168C22×C3×D3—42no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—22no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4C2×D414yes—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2411yescharacteristic
179C3×#67—28no—
180C3×#67—28no—
181C3×#67—44no—
182C3×#67—44no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—42no—
187C3×D12—24no—
188C3×D12—22no—
189C3×D12—24no—
190C3×D12—42no—
191C3×D12C2×D414yescomplemented
192C3×C4×D3—28no—
193C3×C4×D3—28no—
194C3×Dic6—24no—
195C3×Dic6—22no—
196C3×Dic6C2×D414yescomplemented
197C3⋊(C3⋊C8)—24no—
198C22×#41—31no—
199C2×AGL(1,ℤ/8)—92no—
200C2×#121—31no—
201#449—94no—
202#448—92no—
203#447—92no—
204#417—92no—
205#407—91no—
206C24×C3—21no—
207C24×C3—61no—
208C24×C3—61no—
209C24×C3#6711yescharacteristic
210C24×C3#6711yescharacteristic
211C23×D3—21no—
212C23×D3—21no—
213C3×#41—38no—
214C3×#41—64no—
215C3×#41—64no—
216C3×#41—68no—
217C3×#41—68no—
218C3×#41—64no—
219C2×C3×Q8—32no—
220C2×C3×Q8—64no—
221C2×C3×Q8—61no—
222C2×C3×D4—32no—
223C2×C3×D4—32no—
224C2×C3×D4—64no—
225C2×C3×D4—68no—
226C2×C3×D4—62no—
227C2×C3×D4—62no—
228C2×C3×D4—62no—
229C2×C3×D4—62no—
230C2×C3×D4—61no—
231C2×C3×D4—64no—
232C2×C3×D4—61no—
233C2×C3×D4—124no—
234C22×C3×C4—24no—
235C22×C3×C4—24no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—44no—
243C22×C3×C4—68no—
244C22×C3×C4—61no—
245C22×C3×C4—62no—
246C22×C3×C4—62no—
247C22×C3×C4—61no—
248C22×C3×C4#6711yescharacteristic
249C22×C3×C4#6711yescharacteristic
250C22×C3×C4#6711yescharacteristic
251C22×C3×C4#6711yescharacteristic
252C22×C3×C4#6711yescharacteristic
253C22×C3×C4#6711yescharacteristic
254C22×C3×C4C22×D311yescharacteristic
255C22×C3×C4C22×D311yescharacteristic
256C22×C3×C4C22×D311yescharacteristic
257C22×C3×C4C22×D311yescharacteristic
258C22×C3×C4C22×D311yescharacteristic
259C22×C3×C4C22×D311yescharacteristic
260C2×#67—28no—
261C2×#67—22no—
262C2×#67—22no—
263C2×#67—22no—
264C2×#67—22no—
265C2×#67—44no—
266C22×Dic3—21no—
267C22×Dic3—21no—
268#235—28no—
269#235—28no—
270#235—24no—
271#235—24no—
272#235—24no—
273#235#6718yescomplemented
274C2×D12—21no—
275C2×D12—24no—
276C2×D12—24no—
277C2×D12—21no—
278C2×D12C22×D312yes—
279C2×D12C22×D312yes—
280C2×C4×D3—28no—
281C2×C4×D3—22no—
282C2×C4×D3—22no—
283C2×Dic6—21no—
284C2×Dic6—24no—
285C2×Dic6C22×D312yes—
286C3×C4⋊C4—32no—
287C3×C4⋊C4—61no—
288C3×C22⋊C4—62no—
289C3×C22⋊C4—62no—
290#215—68no—
291#214—68no—
292#213—68no—
293#213—68no—
294(C2×D3)⋊C4—22no—
295(C2×D3)⋊C4—22no—
296C4⋊Dic3—21no—
297C4⋊Dic3C22×D312yes—
298#208—64no—
299#208—64no—
300#208—124no—
301C2×C3⋊C8—34no—
302C2×C3⋊C8—34no—
303C2×C3⋊C8—64no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—44no—
312C22×C32#12012yes—
313C22×C32AGL(1,ℤ/8)12yescomplemented
314C22×C32C22×D411yescharacteristic
315C2×C3×D3—24no—
316C2×C3×D3—48no—
317C2×C3×D3—42no—
318C2×C3×D3—42no—
319C2×C3×D3—42no—
320C2×C3×D3—42no—
321C2×C3×D3—42no—
322C2×C3×D3—84no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4#12012yes—
326C32×C4C22×D412yes—
327C3×Dic3—24no—
328C3×Dic3—42no—
329C2×#41—32no—
330C2×#41—32no—
331C2×#41—68no—
332C2×#41—61no—
333C2×#41—61no—
334C22×Q8—31no—
335C22×D4—31no—
336C22×D4—31no—
337C22×D4—61no—
338C22×D4—61no—
339C23×C4—61no—
340C23×C4D3211yescharacteristic
341AGL(1,ℤ/8)—188no—
342C2×SD16—94no—
343C2×D8—94no—
344C2×M16—92no—
345#121—32no—
346#121—32no—
347#121—61no—
348#121—61no—
349C2×C4⋊C4—31no—
350C2×C22⋊C4—61no—
351#103—94no—
352#102—94no—
353C22⋊C8—92no—
354C22⋊C8—92no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—42no—
363C23×C3—42no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—42no—
367C23×C3—64no—
368C23×C3—62no—
369C23×C3—62no—
370C23×C3—122no—
371C23×C3—122no—
372C23×C3—122no—
373C23×C3—122no—
374C23×C3—122no—
375C23×C3C2×#6711yescharacteristic
376C23×C3C2×#6711yescharacteristic
377C23×C3C2×#6711yescharacteristic
378C23×C3C2×#6711yescharacteristic
379C23×C3C2×#6711yescharacteristic
380C23×C3C2×#6711yescharacteristic
381C22×D3—24no—
382C22×D3—22no—
383C22×D3—22no—
384C22×D3—42no—
385C22×D3—42no—
386C22×D3—42no—
387C22×D3—42no—
388C22×D3—42no—
389C3×Q8—34no—
390C3×Q8—64no—
391C3×Q8—62no—
392C3×D4—34no—
393C3×D4—64no—
394C3×D4—68no—
395C3×D4—68no—
396C3×D4—62no—
397C3×D4—64no—
398C3×D4—124no—
399C3×D4—124no—
400C3×D4—122no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—24no—
416C2×C3×C4—42no—
417C2×C3×C4—42no—
418C2×C3×C4—42no—
419C2×C3×C4—42no—
420C2×C3×C4—42no—
421C2×C3×C4—68no—
422C2×C3×C4—64no—
423C2×C3×C4—62no—
424C2×C3×C4—68no—
425C2×C3×C4—62no—
426C2×C3×C4—122no—
427C2×C3×C4D3×D414yes—
428C2×C3×C4C2×#6711yescharacteristic
429C2×C3×C4C2×#6711yescharacteristic
430C2×C3×C4C2×#6711yescharacteristic
431C2×C3×C4C2×#6711yescharacteristic
432C2×C3×C4C2×#6711yescharacteristic
433C2×C3×C4C2×#6711yescharacteristic
434C2×C3×C4C2×#6714yes—
435C2×C3×C4C23×D311yescharacteristic
436C2×C3×C4C23×D311yescharacteristic
437#67—28no—
438#67—28no—
439#67—44no—
440#67—44no—
441C2×Dic3—24no—
442C2×Dic3—22no—
443C2×Dic3—22no—
444C2×Dic3—42no—
445D12—24no—
446D12—24no—
447D12—22no—
448D12—42no—
449D12C2×#6714yescomplemented
450C4×D3—28no—
451C4×D3—28no—
452Dic6—24no—
453Dic6—22no—
454Dic6C2×#6714yescomplemented
455C3⋊C8—64no—
456C3⋊C8—64no—
457C3⋊C8—124no—
458C2×C32—22no—
459C2×C32—22no—
460C2×C32—41no—
461C2×C32C2×#12011yescharacteristic
462C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C32#57511yescharacteristic
464C3×D3—44no—
465C3×D3—82no—
466C24—61no—
467C24—61no—
468C24#62511yescharacteristic
469#41—38no—
470#41—64no—
471#41—64no—
472#41—68no—
473#41—68no—
474#41—64no—
475C2×Q8—32no—
476C2×Q8—64no—
477C2×Q8—61no—
478C2×D4—32no—
479C2×D4—32no—
480C2×D4—64no—
481C2×D4—68no—
482C2×D4—62no—
483C2×D4—62no—
484C2×D4—62no—
485C2×D4—62no—
486C2×D4—61no—
487C2×D4—61no—
488C2×D4—64no—
489C2×D4—124no—
490C22×C4—24no—
491C22×C4—68no—
492C22×C4—61no—
493C22×C4—62no—
494C22×C4—62no—
495C22×C4—61no—
496C22×C4#62511yescharacteristic
497C22×C4#62511yescharacteristic
498C22×C4#62511yescharacteristic
499C22×C4C2×D3211yescharacteristic
500C22×C4C2×D3211yescharacteristic
501C22×C4C2×D3211yescharacteristic
502SD16—188no—
503D8—188no—
504M16—184no—
505C2×C8—94no—
506C4⋊C4—32no—
507C4⋊C4—61no—
508C22⋊C4—62no—
509C22⋊C4—62no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—21no—
518C22×C3—22no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—44no—
535C22×C3—44no—
536C22×C3—48no—
537C22×C3—64no—
538C22×C3—128no—
539C22×C3—122no—
540C22×C3—122no—
541C22×C3—122no—
542C22×C3—122no—
543C22×C3—122no—
544C22×C3—244no—
545C22×C3#93212yescomplemented
546C22×C3#93712yes—
547C22×C3#94912yescomplemented
548C22×C3#95312yes—
549C22×C3C22×#6711yescharacteristic
550C22×C3C22×#6711yescharacteristic
551C2×D3—24no—
552C2×D3—48no—
553C2×D3—42no—
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—42no—
557C2×D3—42no—
558C2×D3—84no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—22no—
564C3×C4—22no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—64no—
568C3×C4—122no—
569C3×C4#93712yes—
570C3×C4#95312yes—
571C3×C4C2×D3×D412yes—
572C3×C4C22×#6712yes—
573Dic3—24no—
574Dic3—42no—
575C32#300611yescharacteristic, complemented, Sylow
576C23—22no—
577C23—22no—
578C23—41no—
579C23—64no—
580C23—62no—
581C23—62no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23C2×#62511yescharacteristic
588C23C2×#62511yescharacteristic
589C23C2×#62511yescharacteristic
590Q8—34no—
591Q8—64no—
592Q8—62no—
593D4—34no—
594D4—64no—
595D4—68no—
596D4—68no—
597D4—62no—
598D4—64no—
599D4—124no—
600D4—124no—
601D4—122no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—68no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—68no—
611C2×C4—62no—
612C2×C4—122no—
613C2×C4C2×#62511yescharacteristic
614C2×C4C2×#62511yescharacteristic
615C2×C4C2×#62511yescharacteristic
616C2×C4D3×#6714yes—
617C2×C4C22×D3211yescharacteristic
618C8—184no—
619C2×C3—22no—
620C2×C3—22no—
621C2×C3—22no—
622C2×C3—22no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—42no—
627C2×C3—42no—
628C2×C3—42no—
629C2×C3—41no—
630C2×C3—41no—
631C2×C3—124no—
632C2×C3—242no—
633C2×C3C2×#93211yescharacteristic
634C2×C3C2×#93711yescharacteristic
635C2×C3C2×#94911yescharacteristic
636C2×C3#582211yescharacteristic
637C2×C3#582211yescharacteristic
638C2×C3C2×#95311yescharacteristic
639D3—44no—
640D3—82no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—44no—
649C22—64no—
650C22—128no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—244no—
657C22#6382312yescomplemented
658C22#6397612yes—
659C22C22×#62511yescharacteristic
660C4—21no—
661C4—21no—
662C4—64no—
663C4—122no—
664C4#6397612yes—
665C4C2×D3×#6712yes—
666C3—21no—
667C3#8262411yescharacteristic, complemented
668C3#8408211yescharacteristic, complemented
669C2—22no—
670C2—22no—
671C2—41no—
672C2—124no—
673C2—242no—
674C2C2×#6382311yescharacteristic
675C2#1059478611yescharacteristic
676C2C2×#6397611yescharacteristic
subgroupG/H
centerC22C22×#625
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC23×C32×C4C22
socleC22×C32C22×D4
hypercenterC23×C4D32
hypocenterC32#3006
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#235C2semidirect
2. C2×C3×#930C2semidirect
3. C2×#63823C2semidirect
4. #10592906C2semidirect
5. #10591171C2semidirect
6. #10591161C2semidirect
7. #10591160C2semidirect
8. #10591087C2semidirect
18. C2×C3×#235C22semidirect
19. C22×C3×D12C22semidirect
20. C22×C3×D12C22semidirect
22. C22×C3×Dic6C22semidirect
23. C2×#3748C22semidirect
27. C3×#930C22semidirect
29. C2×C3×C4⋊Dic3C22semidirect
30. C2×#3715C22semidirect
32. C2×#3713C22semidirect
33. #63657C22semidirect
34. #63657C22semidirect
35. #63565C22semidirect
36. #63562C22semidirect
38. C22×#235D3semidirect
42. C2×#930D3semidirect
77. C3×#235D4semidirect
119. C2×#235C2×D3semidirect
120. C22×D12C2×D3semidirect
121. C22×D12C2×D3semidirect
123. C22×Dic6C2×D3semidirect
139. #930C2×D3semidirect
141. C2×C4⋊Dic3C2×D3semidirect
191. C3×D12C2×D4semidirect
196. C3×Dic6C2×D4semidirect
273. #235#67semidirect
313. C22×C32AGL(1,ℤ/8)semidirect
449. D12C2×#67semidirect
454. Dic6C2×#67semidirect
545. C22×C3#932semidirect
547. C22×C3#949semidirect
575. C32#3006semidirect
657. C22#63823semidirect
667. C3#82624semidirect
668. C3#84082semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C2×C3×#9302
C2×#638232
#105929062
#105911712
#105911612
#105911602
#105910872
#840823
#826243