Finite Group Databasebeta

← #11069615

Subgroups — #11069615

Order 768.

2206 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5719C211yesmaximal, characteristic, complemented
2C2×#5518C211yesmaximal, characteristic, complemented
3C3×#2619C211yesmaximal, characteristic, complemented
4#83184C211yesmaximal, characteristic, complemented
5#82668C212yesmaximal, complemented
6#82647C212yesmaximal, complemented
7#82225C211yesmaximal, characteristic, complemented
8#77015C214yesmaximal, complemented
9#76934C214yesmaximal, complemented
10#76729C214yesmaximal, complemented
11#76689C214yesmaximal, complemented
12#76443C212yesmaximal, complemented
13#76307C212yesmaximal, complemented
14#74193C211yesmaximal, characteristic, complemented
15#74150C211yesmaximal, characteristic, complemented
16#39382—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121C2212yes—
19C2×C3×#117C2211yescharacteristic
20C2×#939C2214yes—
21C2×#938C2212yes—
22C2×#936C2212yes—
23C2×#936C2212yes—
24C2×#935C2214yes—
25C2×D4×Dic3C2212yes—
26C2×C4×#67C2211yescharacteristic
27C2×#924C2212yes—
28#5719—28no—
29#5518—28no—
30C2×#897C2212yes—
31C2×#889C2214yes—
32C2×#879C2214yes—
33C2×#872C2211yescharacteristic
34C2×#870C2211yescharacteristic
35C2×#869C2211yescharacteristic
36C2×C4×Dic6C2212yes—
37C3×#394C2214yes—
38C3×#392C2214yes—
39C3×#390C2212yes—
40C3×#382C2211yescharacteristic
41#5223C2214yes—
42#5222C2214yes—
43#5221C2214yes—
44#5221C2214yes—
45#5221C2214yes—
46#5220C2214yes—
47C4×#217C2211yescharacteristic
48#4990C2212yes—
49#4976C2212yes—
50Dic3×C4⋊C4C2212yes—
51#4973C2212yes—
52#4972C2212yes—
53#4959C2214yes—
54#4957C2212yes—
55#4957C2214yes—
56#4957C2214yes—
57#4952C2214yes—
58#4951C2214yes—
59#4949C2214yes—
60#4948C2214yes—
61#4946C2214yes—
62#4945C2214yes—
63#4945C2212yes—
64#4944C2214yes—
65#4942C2214yes—
66Dic3×C22⋊C4C2214yes—
67C4×(C2×D3)⋊C4C2211yescharacteristic
68C4×Dic3⋊C4C2211yescharacteristic
69C42×Dic3C2211yescharacteristic
70#4674C2214yes—
71#4661C2214yes—
72#4652C2214yes—
73#4649C2214yes—
74C2×#541—31no—
75C2×#540—31no—
76#2858—31no—
77#2714—34no—
78#2694—34no—
79#2621—31no—
80#2619D311yescharacteristic, complemented
81#2554—34no—
82#2540—32no—
83#2490—34no—
84#2427—32no—
85#2426—32no—
86#2367—32no—
87C4×#125—31no—
88#2293—31no—
89C22×C3×D4C2312yes—
90C22×C3×D4C2314yes—
91C22×C3×D4C2314yes—
92C23×C3×C4C2311yescharacteristic
93C22×#67C2314yes—
94C22×#67C2311yescharacteristic
95C23×Dic3C2314yes—
96C22×C4×D3C2311yescharacteristic
97C22×Dic6C2312yes—
98C3×#127—24no—
99C3×#121—44no—
100C3×#117—24no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C22⋊C4C2312yes—
104C2×C3×C22⋊C4C2314yes—
105C2×C3×C22⋊C4C2314yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×#217C2314yes—
110C2×#217C2314yes—
111C2×#217C2311yescharacteristic
112C2×#217C2314yes—
113C2×#217C2314yes—
114C2×#217C2314yes—
115C2×#217C2314yes—
116C2×#217C2312yes—
117C2×#217C2314yes—
118#939—216no—
119#938—28no—
120#936—28no—
121#936—44no—
122#935—216no—
123D4×Dic3—28no—
124C4×#67—24no—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2311yescharacteristic
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2312yes—
132#924—28no—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2311yescharacteristic
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2314yes—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2311yescharacteristic
147#897—44no—
148#889—216no—
149#879—216no—
150#872—24no—
151#870D418yescomplemented
152#869—24no—
153C4×Dic6—28no—
154C3×#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162#541—68no—
163#540—68no—
164C2×C4⋊Q8—31no—
165C2×#127C2×D311yescharacteristic
166C2×#126—34no—
167C2×#125—34no—
168C2×#125—31no—
169C2×#124—32no—
170C2×#124—32no—
171C2×#123—34no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—32no—
174C2×#121—32no—
175C2×#121—34no—
176C2×#121C2×D312yes—
177C2×C4×Q8—32no—
178C2×C4×D4—32no—
179C2×C4×D4—31no—
180C2×#117—31no—
181C2×#117C2×D311yescharacteristic
182#399—34no—
183#398—34no—
184#397—34no—
185#397—34no—
186#396—34no—
187#394—34no—
188#394—34no—
189#394C2×D314yes—
190#392—34no—
191#392C2×D314yes—
192#391—32no—
193#390—34no—
194#390—34no—
195#390—34no—
196#390—34no—
197#390—32no—
198#390—34no—
199#390C2×D312yes—
200#389—32no—
201#389—32no—
202#388—32no—
203#388—34no—
204#388—34no—
205#388—32no—
206#388—34no—
207#387—34no—
208#386—34no—
209#385—34no—
210#382C2×D311yescharacteristic
211#380—34no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—31no—
214C4×C4⋊C4—34no—
215C4×C22⋊C4—31no—
216C4×C22⋊C4—34no—
217C4×C22⋊C4—31no—
218C43—31no—
219C24×C3C2411yescharacteristic
220C24×C3C2414yes—
221C23×D3C2411yescharacteristic
222C2×C3×D4—24no—
223C2×C3×D4—28no—
224C2×C3×D4—24no—
225C2×C3×D4—24no—
226C2×C3×D4—44no—
227C2×C3×D4—44no—
228C2×C3×D4—44no—
229C2×C3×D4—44no—
230C2×C3×D4—44no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4C2414yes—
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2412yes—
238C22×C3×C4C2411yescharacteristic
239C2×#67—28no—
240C2×#67—22no—
241C2×#67—42no—
242C2×#67—44no—
243C2×#67—44no—
244C22×Dic3—28no—
245C22×Dic3—44no—
246C22×Dic3—44no—
247C22×Dic3C2412yes—
248C22×Dic3C2412yes—
249C22×Dic3C2414yes—
250C22×Dic3C2411yescharacteristic
251C22×Dic3C2414yes—
252C22×Dic3C2411yescharacteristic
253C22×Dic3C2411yescharacteristic
254C2×C4×D3—22no—
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×Dic6—24no—
258C2×Dic6—24no—
259C2×Dic6—24no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—24no—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—44no—
264C3×C22⋊C4—44no—
265C3×C22⋊C4—44no—
266C3×C42—22no—
267C3×C42C2×D414yes—
268#217—28no—
269#217—28no—
270#217—28no—
271#217—22no—
272#217—28no—
273#217—28no—
274#217—44no—
275#217—44no—
276#217—42no—
277(C2×D3)⋊C4—24no—
278(C2×D3)⋊C4—22no—
279(C2×D3)⋊C4—44no—
280(C2×D3)⋊C4—42no—
281C4⋊Dic3—28no—
282C4⋊Dic3—44no—
283C4⋊Dic3C2×D418yes—
284Dic3⋊C4—24no—
285Dic3⋊C4—28no—
286Dic3⋊C4—28no—
287Dic3⋊C4—22no—
288Dic3⋊C4—42no—
289Dic3⋊C4—44no—
290Dic3⋊C4C2×D4116yes—
291C4×Dic3—22no—
292C4×Dic3—22no—
293C4×Dic3—28no—
294C4×Dic3—44no—
295C4×Dic3—42no—
296C4×Dic3—42no—
297C4×Dic3#4114yes—
298C22×Q8—32no—
299C22×D4—31no—
300C22×D4—34no—
301C22×D4C22×D312yes—
302C22×D4C22×D314yes—
303C22×D4C22×D314yes—
304C23×C4—34no—
305C23×C4—31no—
306C23×C4C22×D311yescharacteristic
307C4⋊Q8—64no—
308#127—24no—
309#126—616no—
310#125—38no—
311#125—616no—
312#124—68no—
313#124—124no—
314#123—616no—
315C22⋊Q8—68no—
316C22⋊Q8—124no—
317#121—44no—
318#121—68no—
319#121—616no—
320C4×Q8—68no—
321C4×D4—64no—
322C4×D4—68no—
323#117—24no—
324#117—64no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D312yes—
337C2×C22⋊C4—34no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4—31no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—32no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—32no—
350C2×C22⋊C4C22×D312yes—
351C2×C22⋊C4C22×D314yes—
352C2×C22⋊C4C22×D314yes—
353C2×C22⋊C4C22×D314yes—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—34no—
357C2×C42—31no—
358C2×C42—32no—
359C2×C42—32no—
360C2×C42—34no—
361C2×C42C22×D311yescharacteristic
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—34no—
366#95—32no—
367#95—34no—
368#95—34no—
369#95—34no—
370#95C22×D314yes—
371C23×C3—21no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3—21no—
375C23×C3—24no—
376C23×C3—21no—
377C23×C3—22no—
378C23×C3—24no—
379C23×C3—24no—
380C23×C3—24no—
381C23×C3—22no—
382C23×C3—24no—
383C23×C3C2511yescharacteristic
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—22no—
388C22×D3—22no—
389C3×D4—48no—
390C3×D4—88no—
391C3×D4—88no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—24no—
398C2×C3×C4—21no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—24no—
402C2×C3×C4—22no—
403C2×C3×C4—44no—
404C2×C3×C4C22×D414yes—
405C2×C3×C4C22×D412yes—
406C2×C3×C4C2×#4112yes—
407C2×C3×C4H(5,𝔽2)12yes—
408#67—44no—
409#67—88no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—24no—
413C2×Dic3—24no—
414C2×Dic3—24no—
415C2×Dic3—22no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—24no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—88no—
427C2×Dic3C22×D418yes—
428C2×Dic3C2×#4112yes—
429C2×Dic3C2×#4112yes—
430C4×D3—44no—
431Dic6—48no—
432C24—31no—
433C24C23×D311yescharacteristic
434C24C23×D314yes—
435C2×Q8—64no—
436C2×Q8—64no—
437C2×Q8—64no—
438C2×D4—24no—
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—62no—
448C2×D4—68no—
449C2×D4—122no—
450C2×D4—124no—
451C2×D4—124no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—34no—
458C22×C4—32no—
459C22×C4—32no—
460C22×C4—34no—
461C22×C4—31no—
462C22×C4—68no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—124no—
467C22×C4—124no—
468C22×C4C23×D314yes—
469C22×C4C23×D312yes—
470C22×C4C23×D311yescharacteristic
471C22×C4C23×D312yes—
472C22×C4C23×D311yescharacteristic
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—38no—
476C4⋊C4—316no—
477C4⋊C4—62no—
478C4⋊C4—68no—
479C4⋊C4—68no—
480C4⋊C4—64no—
481C4⋊C4—68no—
482C4⋊C4—124no—
483C4⋊C4—122no—
484C4⋊C4—124no—
485C22⋊C4—24no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—62no—
490C22⋊C4—68no—
491C22⋊C4—62no—
492C22⋊C4—68no—
493C22⋊C4—68no—
494C22⋊C4—68no—
495C22⋊C4—68no—
496C22⋊C4—64no—
497C22⋊C4—124no—
498C22⋊C4—122no—
499C22⋊C4—124no—
500C22⋊C4—124no—
501C22⋊C4—122no—
502C42—22no—
503C42—34no—
504C42—68no—
505C42—62no—
506C42—62no—
507C42—122no—
508C42—122no—
509C42—124no—
510C42D3×D414yes—
511C22×C3—41no—
512C22×C3—44no—
513C22×C3—41no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—42no—
518C22×C3—44no—
519C22×C3—41no—
520C22×C3—42no—
521C22×C3—44no—
522C22×C3—44no—
523C22×C3C23×D411yescharacteristic
524C22×C3C22×#4111yescharacteristic
525C22×C3C2×H(5,𝔽2)11yescharacteristic
526C22×C3#58512yes—
527C22×C3#58512yes—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—42no—
531C2×D3—41no—
532C2×D3—42no—
533C3×C4—24no—
534C3×C4—41no—
535C3×C4—42no—
536C3×C4—44no—
537C3×C4#55014yes—
538Dic3—22no—
539Dic3—28no—
540Dic3—22no—
541Dic3—44no—
542Dic3—41no—
543Dic3—42no—
544Dic3—42no—
545C23—21no—
546C23—24no—
547C23—24no—
548C23—21no—
549C23—24no—
550C23—21no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—62no—
561C23—62no—
562C23C24×D311yescharacteristic
563Q8—128no—
564D4—48no—
565D4—88no—
566D4—88no—
567D4—124no—
568D4—248no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—38no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—44no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—62no—
589C2×C4—64no—
590C2×C4—61no—
591C2×C4—62no—
592C2×C4—64no—
593C2×C4—61no—
594C2×C4—64no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—124no—
601C2×C4—248no—
602C2×C4C2×D3×D414yes—
603C2×C4C2×D3×D412yes—
604C2×C4#100412yes—
605C2×C4D3×#4112yes—
606C2×C3—81no—
607C2×C3—84no—
608C2×C3#345511yescharacteristic
609C2×C3C2×#55011yescharacteristic
610C2×C3#350012yes—
611C2×C3#351211yescharacteristic
612C2×C3#356212yes—
613D3—81no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—41no—
621C22—44no—
622C22—41no—
623C22—44no—
624C22—44no—
625C22—41no—
626C22—122no—
627C22—122no—
628C22—121no—
629C22—121no—
630C22—121no—
631C22C22×D3×D411yescharacteristic
632C22C2×#100411yescharacteristic
633C22C2×D3×#4111yescharacteristic
634C22#596612yes—
635C22#596612yes—
636C4—24no—
637C4—42no—
638C4—44no—
639C4—41no—
640C4—68no—
641C4—62no—
642C4—62no—
643C4—122no—
644C4—122no—
645C4—121no—
646C4—124no—
647C4#559914yes—
648C3#3938211yescharacteristic, complemented, Sylow
649C2—84no—
650C2—81no—
651C2—241no—
652C2C2×#559911yescharacteristic
653C2#8631712yes—
654C2#8632812yes—
655C2#8646711yescharacteristic
656C2#8646911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2619C2
hypercenter#2619D3
hypocenterC3#39382
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#5719C2semidirect
2. C2×#5518C2semidirect
3. C3×#2619C2semidirect
4. #83184C2semidirect
5. #82668C2semidirect
6. #82647C2semidirect
7. #82225C2semidirect
8. #77015C2semidirect
9. #76934C2semidirect
10. #76729C2semidirect
11. #76689C2semidirect
12. #76443C2semidirect
13. #76307C2semidirect
14. #74193C2semidirect
15. #74150C2semidirect
80. #2619D3semidirect
151. #870D4semidirect
648. C3#39382semidirect

Maximal subgroups

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

subgroupindex
C2×#57192
C2×#55182
C3×#26192
#831842
#826682
#826472
#822252
#770152
#769342
#767292
#766892
#764432
#763072
#741932
#741502
#393823