Finite Group Databasebeta

← #10595526

Subgroups — #10595526

Order 576.

974 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638C211yesmaximal, characteristic, complemented
2C22×C3×#67C211yesmaximal, characteristic, complemented
3C2×#3811C211yesmaximal, characteristic, complemented
4C2×D3×#67C211yesmaximal, characteristic, complemented
5C3×#953C211yesmaximal, characteristic, complemented
6#63983C211yesmaximal, characteristic, complemented
7#63981C211yesmaximal, characteristic, complemented
8#63980C211yesmaximal, characteristic, complemented
9#63979C211yesmaximal, characteristic, complemented
10#63978C211yesmaximal, characteristic, complemented
11#63977C211yesmaximal, characteristic, complemented
12#67⋊Dic3C211yesmaximal, characteristic, complemented
13#63971C211yesmaximal, characteristic, complemented
14#63958C211yesmaximal, characteristic, complemented
15#63952C211yesmaximal, characteristic, complemented
16D4×#67—31nomaximal
17#5590—31nomaximal
18C24×C32C2211yescharacteristic, complemented
19C23×C3⋊D3C2211yescharacteristic, complemented
20C23×C3×D3C2211yescharacteristic, complemented
21C22×D32C2211yescharacteristic, complemented
22C2×#638—24no—
23C2×#638C2211yescharacteristic
24C2×#638C2211yescharacteristic, complemented
25C2×#638C2211yescharacteristic, complemented
26C2×#638C2211yescharacteristic, complemented
27C22×C3⋊Dic3C2211yescharacteristic, complemented
28C2×C3×#67—24no—
29C2×C3×#67C2211yescharacteristic
30C2×C3×#67C2211yescharacteristic, complemented
31C2×C3×#67C2211yescharacteristic, complemented
32C2×C3×#67C2211yescharacteristic, complemented
33C2×C3×#67C2211yescharacteristic, complemented
34C2×C3×#67C2211yescharacteristic, complemented
35C2×C3×#67C2211yescharacteristic, complemented
36C22×C3×Dic3C2211yescharacteristic, complemented
37#3811—24no—
38D3×#67—24no—
39C2×#626C2211yescharacteristic, complemented
40C2×#626C2211yescharacteristic, complemented
41C2×#626C2211yescharacteristic, complemented
42C2×#626C2211yescharacteristic, complemented
43C2×#625C2211yescharacteristic, complemented
44C2×#624C2211yescharacteristic, complemented
45C2×D3×Dic3C2211yescharacteristic, complemented
46C3×#217C2211yescharacteristic, complemented
47C3×#217C2211yescharacteristic, complemented
48C3×#217C2211yescharacteristic, complemented
49#3724C2211yescharacteristic, complemented
50Dic3⋊Dic3C2211yescharacteristic, complemented
51#3722C2211yescharacteristic, complemented
52#3722C2211yescharacteristic, complemented
53(C2×D3)⋊Dic3C2211yescharacteristic, complemented
54(C2×D3)⋊Dic3C2211yescharacteristic, complemented
55(C2×D3)⋊Dic3C2211yescharacteristic, complemented
56Dic32C2211yescharacteristic, complemented
57C22×C3×D4—31no—
58C22×#67—31no—
59C22×#67—31no—
60C22×#67—31no—
61C22×#67—61no—
62C22×#67D311yescharacteristic, complemented
63C2×D3×D4—31no—
64C2×D3×D4—31no—
65C2×D3×D4—31no—
66C3×#120—31no—
67#953—31no—
68#953D311yescharacteristic, complemented
69#940—31no—
70#940—31no—
71#939—31no—
72#939—31no—
73#939—31no—
74#939—31no—
75#938—31no—
76#937—31no—
77#937—31no—
78#937—31no—
79D4×Dic3—31no—
80#930—31no—
81C4×#67—31no—
82#884—31no—
83#884—31no—
84#882—31no—
85#882—31no—
86#67⋊C4—31no—
87#67⋊C4—31no—
88C23×C32—22no—
89C23×C32—22no—
90C23×C32—41no—
91C23×C32C2311yescharacteristic
92C23×C32C2311yescharacteristic
93C23×C32C2311yescharacteristic
94C22×C3⋊D3—22no—
95C22×C3⋊D3—22no—
96C22×C3⋊D3—41no—
97C22×C3⋊D3C2311yescharacteristic
98C22×C3⋊D3C2311yescharacteristic
99C22×C3×D3—22no—
100C22×C3×D3—22no—
101C22×C3×D3—41no—
102C22×C3×D3C2311yescharacteristic
103C22×C3×D3C2311yescharacteristic
104C22×C3×D3C2311yescharacteristic
105C2×D32—22no—
106C2×D32—22no—
107C2×D32—41no—
108#638—22no—
109#638—22no—
110#638—22no—
111#638D414yescomplemented
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3C2311yescharacteristic
114C2×C3⋊Dic3C2311yescharacteristic
115C3×#67—22no—
116C3×#67—22no—
117C3×#67—22no—
118C3×#67—22no—
119C3×#67—22no—
120C3×#67—41no—
121C3×#67D414yescomplemented
122C2×C3×Dic3—22no—
123C2×C3×Dic3C2311yescharacteristic
124C2×C3×Dic3C2311yescharacteristic
125C2×C3×Dic3C2311yescharacteristic
126C2×C3×Dic3C2311yescharacteristic
127C2×C3×Dic3C2311yescharacteristic
128#626—22no—
129#626—22no—
130#626—22no—
131#626—41no—
132#625—22no—
133#624—22no—
134D3×Dic3—22no—
135D42—91noSylow
136C24×C3—21no—
137C24×C3—31no—
138C24×C3C2×D311yescharacteristic, complemented
139C24×C3C2×D311yescharacteristic, complemented
140C23×D3—31no—
141C23×D3—31no—
142C23×D3—31no—
143C23×D3—31no—
144C23×D3—61no—
145C23×D3C2×D311yescharacteristic, complemented
146C2×C3×D4—31no—
147C2×C3×D4—31no—
148C2×C3×D4—31no—
149C2×C3×D4—31no—
150C2×C3×D4—31no—
151C2×C3×D4—31no—
152C2×C3×D4—31no—
153C2×C3×D4—64no—
154C22×C3×C4—31no—
155C2×#67—24no—
156C2×#67—31no—
157C2×#67—31no—
158C2×#67—31no—
159C2×#67—31no—
160C2×#67—31no—
161C2×#67—31no—
162C2×#67—31no—
163C2×#67—31no—
164C2×#67—31no—
165C2×#67—31no—
166C2×#67—31no—
167C2×#67—31no—
168C2×#67—31no—
169C2×#67—31no—
170C2×#67—61no—
171C2×#67—61no—
172C2×#67—61no—
173C2×#67—61no—
174C2×#67—64no—
175C2×#67—64no—
176C2×#67—64no—
177C2×#67—68no—
178C2×#67C2×D311yescharacteristic
179C2×#67C2×D311yescharacteristic, complemented
180C2×#67C2×D311yescharacteristic, complemented
181C2×#67C2×D311yescharacteristic, complemented
182C2×#67C2×D311yescharacteristic, complemented
183C2×#67C2×D311yescharacteristic, complemented
184C2×#67C2×D311yescharacteristic, complemented
185C22×Dic3—31no—
186C22×Dic3—31no—
187C22×Dic3—31no—
188C22×Dic3—61no—
189C22×Dic3C2×D311yescharacteristic, complemented
190D3×D4—64no—
191D3×D4—64no—
192D3×D4—64no—
193C2×D12—31no—
194C2×D12—31no—
195C2×D12—31no—
196C2×D12—31no—
197C2×C4×D3—31no—
198C2×C4×D3—31no—
199C2×C4×D3—31no—
200C3×C22⋊C4—31no—
201C3×C22⋊C4—31no—
202C3×C22⋊C4—31no—
203#217—31no—
204#217—31no—
205#217—31no—
206#217C2×D311yescharacteristic, complemented
207#217C2×D311yescharacteristic, complemented
208#217C2×D311yescharacteristic, complemented
209(C2×D3)⋊C4—31no—
210(C2×D3)⋊C4—31no—
211(C2×D3)⋊C4—31no—
212(C2×D3)⋊C4—31no—
213(C2×D3)⋊C4—31no—
214(C2×D3)⋊C4—31no—
215(C2×D3)⋊C4—31no—
216C4⋊Dic3—31no—
217Dic3⋊C4—31no—
218Dic3⋊C4—31no—
219Dic3⋊C4—31no—
220C4×Dic3—31no—
221C4×Dic3—31no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—44no—
230C22×C32C2×D412yescomplemented
231C22×C32C2×D412yescomplemented
232C22×C32C2411yescharacteristic
233C2×C3⋊D3—21no—
234C2×C3⋊D3—21no—
235C2×C3⋊D3—21no—
236C2×C3⋊D3—21no—
237C2×C3⋊D3—21no—
238C2×C3⋊D3—44no—
239C2×C3⋊D3C2×D412yescomplemented
240C2×C3×D3—21no—
241C2×C3×D3—21no—
242C2×C3×D3—21no—
243C2×C3×D3—21no—
244C2×C3×D3—21no—
245C2×C3×D3—21no—
246C2×C3×D3—21no—
247C2×C3×D3—21no—
248C2×C3×D3—44no—
249C2×C3×D3C2×D412yescomplemented
250D32—44no—
251C3⋊Dic3—21no—
252C3⋊Dic3C2×D412yescomplemented
253C3×Dic3—21no—
254C3×Dic3—21no—
255C3×Dic3—21no—
256C3×Dic3—21no—
257C3×Dic3C2×D412yescomplemented
258C22×D4—31no—
259C22×D4—91no—
260C22×D4—91no—
261C22×D4—91no—
262#127—91no—
263#121—91no—
264#121—91no—
265#121—91no—
266#121—91no—
267#120—31no—
268#120—91no—
269#120—91no—
270#120—91no—
271C4×D4—91no—
272C4×D4—91no—
273C23×C3—24no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—31no—
284C23×C3—42no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—42no—
288C23×C3—62no—
289C23×C3—62no—
290C23×C3—121no—
291C23×C3C22×D311yescharacteristic
292C23×C3C22×D311yescharacteristic
293C23×C3C22×D311yescharacteristic
294C23×C3C22×D311yescharacteristic
295C23×C3C22×D311yescharacteristic
296C23×C3C22×D311yescharacteristic
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—31no—
300C22×D3—31no—
301C22×D3—31no—
302C22×D3—31no—
303C22×D3—41no—
304C22×D3—61no—
305C22×D3—61no—
306C22×D3—62no—
307C22×D3—62no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—64no—
312C22×D3—62no—
313C22×D3—62no—
314C22×D3—62no—
315C22×D3—122no—
316C22×D3—121no—
317C22×D3—121no—
318C22×D3—121no—
319C22×D3—121no—
320C22×D3—122no—
321C22×D3C22×D311yescharacteristic
322C22×D3C22×D311yescharacteristic
323C22×D3C22×D311yescharacteristic
324C3×D4—34no—
325C3×D4—62no—
326C3×D4—62no—
327C3×D4—62no—
328C3×D4—62no—
329C3×D4—62no—
330C3×D4—121no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—62no—
337#67—22no—
338#67—22no—
339#67—22no—
340#67—22no—
341#67—22no—
342#67—34no—
343#67—34no—
344#67—41no—
345#67—64no—
346#67—62no—
347#67—62no—
348#67—62no—
349#67—62no—
350#67—62no—
351#67—62no—
352#67—62no—
353#67—62no—
354#67—64no—
355#67—64no—
356#67—64no—
357#67—62no—
358#67—62no—
359#67—62no—
360#67—121no—
361#67#6714yescomplemented
362C2×Dic3—22no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—62no—
370C2×Dic3—62no—
371C2×Dic3—62no—
372C2×Dic3—64no—
373C2×Dic3C22×D311yescharacteristic
374C2×Dic3C22×D311yescharacteristic
375C2×Dic3C22×D311yescharacteristic
376C2×Dic3C22×D311yescharacteristic
377C2×Dic3C22×D311yescharacteristic
378D12—62no—
379D12—62no—
380D12—62no—
381D12—121no—
382C4×D3—62no—
383C4×D3—62no—
384C4×D3—62no—
385C2×C32—22no—
386C2×C32—22no—
387C2×C32—41no—
388C2×C32C22×D411yescharacteristic
389C2×C32C22×D411yescharacteristic
390C2×C32H(5,𝔽2)11yescharacteristic
391C3⋊D3—22no—
392C3⋊D3—41no—
393C3×D3—22no—
394C3×D3—41no—
395C3×D3—41no—
396C24—31no—
397C24—91no—
398C24—91no—
399C24D3211yescharacteristic, complemented
400C2×D4—31no—
401C2×D4—31no—
402C2×D4—31no—
403C2×D4—31no—
404C2×D4—31no—
405C2×D4—31no—
406C2×D4—31no—
407C2×D4—64no—
408C2×D4—91no—
409C2×D4—91no—
410C2×D4—91no—
411C2×D4—91no—
412C2×D4—91no—
413C2×D4—91no—
414C2×D4—91no—
415C2×D4—91no—
416C2×D4—91no—
417C2×D4—184no—
418C2×D4—184no—
419C2×D4—184no—
420C22×C4—31no—
421C22×C4—91no—
422C22×C4—91no—
423C22×C4—91no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C22⋊C4—31no—
427C22⋊C4—31no—
428C22⋊C4—31no—
429C22⋊C4—91no—
430C22⋊C4—91no—
431C22⋊C4—91no—
432C22⋊C4—91no—
433C22⋊C4—91no—
434C42—91no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—22no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—32no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—48no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—124no—
472C22×C3D3×D412yescomplemented
473C22×C3D3×D412yescomplemented
474C22×C3D3×D412yescomplemented
475C22×C3C2×#6712yescomplemented
476C22×C3C23×D311yescharacteristic
477C22×C3C23×D311yescharacteristic
478C2×D3—21no—
479C2×D3—21no—
480C2×D3—21no—
481C2×D3—21no—
482C2×D3—21no—
483C2×D3—21no—
484C2×D3—21no—
485C2×D3—21no—
486C2×D3—32no—
487C2×D3—32no—
488C2×D3—44no—
489C2×D3—62no—
490C2×D3—61no—
491C2×D3—61no—
492C2×D3—61no—
493C2×D3—61no—
494C2×D3—62no—
495C2×D3—61no—
496C2×D3—61no—
497C2×D3—61no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—121no—
502C2×D3—121no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—124no—
506C2×D3—124no—
507C2×D3—124no—
508C2×D3—124no—
509C2×D3—128no—
510C2×D3C2×#6712yescomplemented
511C3×C4—32no—
512C3×C4—61no—
513C3×C4—61no—
514C3×C4—61no—
515C3×C4—61no—
516Dic3—21no—
517Dic3—21no—
518Dic3—21no—
519Dic3—21no—
520Dic3—32no—
521Dic3—32no—
522Dic3—62no—
523Dic3—62no—
524Dic3—61no—
525Dic3—61no—
526Dic3C2×#6712yescomplemented
527C32D4211yescharacteristic, complemented, Sylow
528C23—22no—
529C23—22no—
530C23—31no—
531C23—31no—
532C23—31no—
533C23—41no—
534C23—62no—
535C23—62no—
536C23—91no—
537C23—91no—
538C23—121no—
539C23—182no—
540C23—182no—
541C23—182no—
542C23—182no—
543C23—361no—
544C23—361no—
545C23C2×D3211yescharacteristic
546C23C2×D3211yescharacteristic
547C23C2×D3211yescharacteristic
548D4—34no—
549D4—62no—
550D4—62no—
551D4—62no—
552D4—62no—
553D4—62no—
554D4—94no—
555D4—121no—
556D4—182no—
557D4—182no—
558D4—182no—
559D4—182no—
560D4—182no—
561D4—182no—
562D4—182no—
563D4—361no—
564C2×C4—31no—
565C2×C4—31no—
566C2×C4—31no—
567C2×C4—31no—
568C2×C4—31no—
569C2×C4—62no—
570C2×C4—91no—
571C2×C4—91no—
572C2×C4—182no—
573C2×C4—182no—
574C2×C4—182no—
575C2×C3—24no—
576C2×C3—22no—
577C2×C3—22no—
578C2×C3—22no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—42no—
584C2×C3—42no—
585C2×C3—41no—
586C2×C3—41no—
587C2×C3—62no—
588C2×C3—121no—
589C2×C3—121no—
590C2×C3C2×D3×D411yescharacteristic
591C2×C3C2×D3×D411yescharacteristic
592C2×C3C2×D3×D411yescharacteristic
593C2×C3#100411yescharacteristic
594C2×C3#100411yescharacteristic
595C2×C3C22×#6711yescharacteristic
596D3—22no—
597D3—41no—
598D3—41no—
599D3—62no—
600D3—62no—
601D3—122no—
602D3—122no—
603D3—121no—
604D3—121no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—32no—
613C22—44no—
614C22—61no—
615C22—61no—
616C22—61no—
617C22—61no—
618C22—61no—
619C22—61no—
620C22—61no—
621C22—61no—
622C22—92no—
623C22—124no—
624C22—181no—
625C22—181no—
626C22—181no—
627C22—181no—
628C22—181no—
629C22—364no—
630C22—364no—
631C22D3×#6712yescomplemented
632C22#381112yescomplemented
633C22C22×D3211yescharacteristic
634C4—32no—
635C4—61no—
636C4—61no—
637C4—61no—
638C4—61no—
639C4—92no—
640C4—181no—
641C3—21no—
642C3#559011yescharacteristic, complemented
643C3D4×#6711yescharacteristic, complemented
644C2—22no—
645C2—22no—
646C2—41no—
647C2—62no—
648C2—121no—
649C2—121no—
650C2—182no—
651C2—361no—
652C2C2×D3×#6711yescharacteristic
653C2C2×#381111yescharacteristic
654C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC24×C32C22
hypercenterC24D32
hypocenterC32D42
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#638C2semidirect
2. C22×C3×#67C2semidirect
3. C2×#3811C2semidirect
4. C2×D3×#67C2semidirect
5. C3×#953C2semidirect
6. #63983C2semidirect
7. #63981C2semidirect
8. #63980C2semidirect
9. #63979C2semidirect
10. #63978C2semidirect
11. #63977C2semidirect
12. #67⋊Dic3C2semidirect
13. #63971C2semidirect
14. #63958C2semidirect
15. #63952C2semidirect
18. C24×C32C22semidirect
19. C23×C3⋊D3C22semidirect
20. C23×C3×D3C22semidirect
21. C22×D32C22semidirect
24. C2×#638C22semidirect
25. C2×#638C22semidirect
26. C2×#638C22semidirect
27. C22×C3⋊Dic3C22semidirect
30. C2×C3×#67C22semidirect
31. C2×C3×#67C22semidirect
32. C2×C3×#67C22semidirect
33. C2×C3×#67C22semidirect
34. C2×C3×#67C22semidirect
35. C2×C3×#67C22semidirect
36. C22×C3×Dic3C22semidirect
39. C2×#626C22semidirect
40. C2×#626C22semidirect
41. C2×#626C22semidirect
42. C2×#626C22semidirect
43. C2×#625C22semidirect
44. C2×#624C22semidirect
45. C2×D3×Dic3C22semidirect
46. C3×#217C22semidirect
47. C3×#217C22semidirect
48. C3×#217C22semidirect
49. #3724C22semidirect
50. Dic3⋊Dic3C22semidirect
51. #3722C22semidirect
52. #3722C22semidirect
53. (C2×D3)⋊Dic3C22semidirect
54. (C2×D3)⋊Dic3C22semidirect
55. (C2×D3)⋊Dic3C22semidirect
56. Dic32C22semidirect
62. C22×#67D3semidirect
68. #953D3semidirect
111. #638D4semidirect
121. C3×#67D4semidirect
138. C24×C3C2×D3semidirect
139. C24×C3C2×D3semidirect
145. C23×D3C2×D3semidirect
179. C2×#67C2×D3semidirect
180. C2×#67C2×D3semidirect
181. C2×#67C2×D3semidirect
182. C2×#67C2×D3semidirect
183. C2×#67C2×D3semidirect
184. C2×#67C2×D3semidirect
189. C22×Dic3C2×D3semidirect
206. #217C2×D3semidirect
207. #217C2×D3semidirect
208. #217C2×D3semidirect
230. C22×C32C2×D4semidirect
231. C22×C32C2×D4semidirect
239. C2×C3⋊D3C2×D4semidirect
249. C2×C3×D3C2×D4semidirect
252. C3⋊Dic3C2×D4semidirect
257. C3×Dic3C2×D4semidirect
361. #67#67semidirect
399. C24D32semidirect
472. C22×C3D3×D4semidirect
473. C22×C3D3×D4semidirect
474. C22×C3D3×D4semidirect
475. C22×C3C2×#67semidirect
510. C2×D3C2×#67semidirect
526. Dic3C2×#67semidirect
527. C32D42semidirect
631. C22D3×#67semidirect
632. C22#3811semidirect
642. C3#5590semidirect
643. C3D4×#67semidirect

Maximal subgroups

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

subgroupindex
C22×#6382
C22×C3×#672
C2×#38112
C2×D3×#672
C3×#9532
#639832
#639812
#639802
#639792
#639782
#639772
#67⋊Dic32
#639712
#639582
#639522
D4×#673
#55903