Finite Group Databasebeta

← #11235160

Subgroups — #11235160

Order 768.

1778 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2397C211yesmaximal, characteristic
2#82384C212yesmaximal, complemented
3#82382C212yesmaximal, complemented
4#82374C211yesmaximal, characteristic, complemented
5#82373C211yesmaximal, characteristic, complemented
6#82372C212yesmaximal, complemented
7#82335C212yesmaximal, complemented
8#82335C212yesmaximal, complemented
9#78079C214yesmaximal, complemented
10#78046C212yesmaximal, complemented
11#78032C212yesmaximal, complemented
12#76734C212yesmaximal, complemented
13#76727C212yesmaximal, complemented
14#76691C214yesmaximal, complemented
15#73391C212yesmaximal, complemented
16#45433—31nomaximal, Sylow
17C2×C3×C4×D4C2212yes—
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#936C2214yes—
22C2×#935C2212yes—
23C2×#926C2212yes—
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C2×#869C2212yes—
27C3×#399C2211yescharacteristic
28C3×#399C2211yescharacteristic
29C3×#399C2212yes—
30C3×#392C2212yes—
31C3×#392C2211yescharacteristic
32C3×#392C2211yescharacteristic
33C3×#376C2211yescharacteristic
34#5222C2214yes—
35#5221C2214yes—
36#5220C2212yes—
37#5220C2214yes—
38#5220C2214yes—
39#5212C2212yes—
40#5212C2212yes—
41#5212C2212yes—
42#5212C2212yes—
43#5212C2212yes—
44#5212C2211yescharacteristic
45#5212C2211yescharacteristic
46#4986C2212yes—
47#4981C2212yes—
48#4980C2214yes—
49#4980C2214yes—
50#4978C2214yes—
51#4977C2214yes—
52#4973C2214yes—
53#4953C2214yes—
54#4953C2212yes—
55#4953C2214yes—
56#4952C2214yes—
57#4951C2214yes—
58#4951C2214yes—
59#4949C2212yes—
60#4947C2214yes—
61#4947C2214yes—
62#4946C2212yes—
63#4944C2214yes—
64#4943C2212yes—
65Dic3×C22⋊C4C2214yes—
66Dic3×C22⋊C4C2214yes—
67#4936C2212yes—
68C4×C4⋊Dic3C2212yes—
69C4×C4⋊Dic3C2212yes—
70#4928C2212yes—
71#4928C2212yes—
72#4654C2214yes—
73#4654C2214yes—
74#4646C2214yes—
75#4646C2214yes—
76#2754—34no—
77#2752—32no—
78#2712—32no—
79#2607—32no—
80#2563—32no—
81#2557—34no—
82#2547—32no—
83#2517—32no—
84#2487—31no—
85#2486—31no—
86#2485—32no—
87#2397D311yescharacteristic
88#2352—32no—
89#2352—32no—
90#2331—32no—
91C22×C3×D4C2312yes—
92C22×C3×D4C2311yescharacteristic
93C22×C3×D4C2312yes—
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2311yescharacteristic
96C23×C3×C4C2311yescharacteristic
97C23×C3×C4C2312yes—
98C23×Dic3C2312yes—
99C23×Dic3C2312yes—
100C22×Dic6C2314yes—
101C3×C4×D4—28no—
102C3×C4×D4—28no—
103C3×C4×D4—42no—
104C3×C4×D4—42no—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C22⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4C2312yes—
115C2×C3×C22⋊C4C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2312yes—
118C2×#217C2314yes—
119C2×#217C2312yes—
120C2×#217C2314yes—
121C2×#217C2312yes—
122C2×#217C2314yes—
123C2×#217C2314yes—
124C2×#217C2314yes—
125C2×#217C2314yes—
126#936—48no—
127#935—44no—
128#926—44no—
129C2×C4⋊Dic3—42no—
130C2×C4⋊Dic3—42no—
131C2×C4⋊Dic3C2314yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×C4×Dic3C2314yes—
141C2×C4×Dic3C2314yes—
142C2×C4×Dic3C2314yes—
143#869—28no—
144C3×#95C2311yescharacteristic
145C3×#95C2312yes—
146C3×#95C2311yescharacteristic
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C2×C4⋊Q8—32no—
161C2×#124—34no—
162C2×#123—32no—
163C2×C4×D4C2×D312yes—
164C2×C4×D4C2×D312yes—
165C2×C4×D4C2×D311yescharacteristic
166C2×C4×D4C2×D311yescharacteristic
167C2×#117—32no—
168C22×C4⋊C4—31no—
169C22×C4⋊C4—31no—
170#400—32no—
171#399—32no—
172#399C2×D311yescharacteristic
173#399C2×D312yes—
174#399C2×D311yescharacteristic
175#398—32no—
176#397—34no—
177#397—32no—
178#397—34no—
179#397—34no—
180#394—34no—
181#393—34no—
182#393—34no—
183#392C2×D311yescharacteristic
184#392C2×D311yescharacteristic
185#392C2×D312yes—
186#391—34no—
187#391—34no—
188#391—34no—
189#390—34no—
190#389—32no—
191#389—32no—
192#388—34no—
193#388—34no—
194#388—34no—
195#387—34no—
196#387—34no—
197#387—34no—
198#387—34no—
199#387—34no—
200#386—34no—
201#386—34no—
202#386—32no—
203#385—32no—
204#382—34no—
205#382—32no—
206#381—32no—
207#380—31no—
208#380—32no—
209#380—32no—
210#380—32no—
211#380—32no—
212#380—32no—
213#380—31no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4—32no—
216C4×C22⋊C4—34no—
217C4×C22⋊C4—34no—
218#376C2×D311yescharacteristic
219C24×C3C2412yes—
220C24×C3C2411yescharacteristic
221C24×C3C2411yescharacteristic
222C2×C3×D4—22no—
223C2×C3×D4—24no—
224C2×C3×D4—24no—
225C2×C3×D4—22no—
226C2×C3×D4—24no—
227C2×C3×D4—24no—
228C2×C3×D4—41no—
229C2×C3×D4—41no—
230C2×C3×D4—42no—
231C2×C3×D4—42no—
232C2×C3×D4—42no—
233C22×C3×C4—24no—
234C22×C3×C4—24no—
235C22×C3×C4—24no—
236C22×C3×C4—24no—
237C22×C3×C4—22no—
238C22×C3×C4—22no—
239C22×C3×C4—24no—
240C22×C3×C4C2412yes—
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2411yescharacteristic
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2412yes—
246C22×C3×C4C2412yes—
247C22×C3×C4C2411yescharacteristic
248C22×Dic3—24no—
249C22×Dic3—42no—
250C22×Dic3—42no—
251C22×Dic3—42no—
252C22×Dic3—44no—
253C22×Dic3C2414yes—
254C22×Dic3C2414yes—
255C22×Dic3C2414yes—
256C22×Dic3C2414yes—
257C2×Dic6—28no—
258C2×Dic6—44no—
259C2×Dic6—44no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—24no—
262C3×C4⋊C4—41no—
263C3×C4⋊C4—41no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—42no—
269C3×C22⋊C4—41no—
270C3×C22⋊C4—41no—
271C3×C42—22no—
272C3×C42#4118yes—
273#217—28no—
274#217—24no—
275#217—44no—
276#217—42no—
277#217—44no—
278#217—44no—
279#217—44no—
280#217—44no—
281C4⋊Dic3—28no—
282C4⋊Dic3—24no—
283C4⋊Dic3—24no—
284C4⋊Dic3—24no—
285C4⋊Dic3—24no—
286C4⋊Dic3—42no—
287C4⋊Dic3—42no—
288Dic3⋊C4—44no—
289Dic3⋊C4—44no—
290C4×Dic3—44no—
291C4×Dic3—44no—
292C4×Dic3—44no—
293C22×Q8—34no—
294C22×D4C22×D311yescharacteristic
295C22×D4C22×D312yes—
296C22×D4C22×D312yes—
297C22×D4C22×D311yescharacteristic
298C23×C4—32no—
299C23×C4—32no—
300C23×C4C22×D311yescharacteristic
301C23×C4C22×D311yescharacteristic
302C23×C4C22×D312yes—
303C4⋊Q8—68no—
304#124—128no—
305#123—124no—
306C4×D4—28no—
307C4×D4—28no—
308C4×D4—42no—
309C4×D4—42no—
310#117—124no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—122no—
321C2×C4⋊C4—122no—
322C2×C4⋊C4C22×D312yes—
323C2×C4⋊C4C22×D312yes—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—34no—
327C2×C22⋊C4—32no—
328C2×C22⋊C4—34no—
329C2×C22⋊C4—32no—
330C2×C22⋊C4—34no—
331C2×C22⋊C4—34no—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—34no—
334C2×C22⋊C4C22×D312yes—
335C2×C22⋊C4C22×D312yes—
336C2×C22⋊C4C22×D312yes—
337C2×C22⋊C4C22×D312yes—
338C2×C22⋊C4C22×D311yescharacteristic
339C2×C22⋊C4C22×D311yescharacteristic
340C2×C22⋊C4C22×D312yes—
341C2×C42—34no—
342C2×C42—34no—
343C2×C42—34no—
344C2×C42C22×D311yescharacteristic
345C2×C42C22×D312yes—
346#95—34no—
347#95—34no—
348#95—34no—
349#95—34no—
350#95—34no—
351#95—34no—
352#95—34no—
353#95—32no—
354#95—34no—
355#95—34no—
356#95—32no—
357#95—34no—
358#95—34no—
359#95C22×D311yescharacteristic
360#95C22×D312yes—
361#95C22×D311yescharacteristic
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3H(5,𝔽2)12yes—
378C23×C32−1+412yes—
379C23×C3C2511yescharacteristic
380C3×D4—48no—
381C3×D4—48no—
382C3×D4—44no—
383C3×D4—44no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—44no—
401C2×C3×C4—44no—
402C2×C3×C4—48no—
403C2×C3×C4C2×#4114yes—
404C2×C3×C4C2×#4114yes—
405C2×C3×C4C2×#4114yes—
406C2×C3×C4H(5,𝔽2)12yes—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3—24no—
410C2×Dic3—24no—
411C2×Dic3—24no—
412C2×Dic3—24no—
413C2×Dic3—24no—
414C2×Dic3—24no—
415C2×Dic3—24no—
416C2×Dic3—24no—
417C2×Dic3—24no—
418C2×Dic3—24no—
419C2×Dic3—84no—
420C2×Dic3—84no—
421Dic6—88no—
422C24C23×D312yes—
423C24C23×D311yescharacteristic
424C24C23×D311yescharacteristic
425C2×Q8—68no—
426C2×Q8—124no—
427C2×Q8—124no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—34no—
447C22×C4—34no—
448C22×C4—34no—
449C22×C4—34no—
450C22×C4—64no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—124no—
455C22×C4C23×D312yes—
456C22×C4C23×D311yescharacteristic
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D311yescharacteristic
459C22×C4C23×D311yescharacteristic
460C22×C4C23×D312yes—
461C22×C4C23×D312yes—
462C22×C4C23×D311yescharacteristic
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—68no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—64no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—124no—
475C4⋊C4—124no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—68no—
484C22⋊C4—64no—
485C22⋊C4—124no—
486C22⋊C4—122no—
487C22⋊C4—124no—
488C22⋊C4—124no—
489C22⋊C4—124no—
490C22⋊C4—124no—
491C42—22no—
492C42—124no—
493C42—124no—
494C42—124no—
495C42#23718yes—
496C22×C3—24no—
497C22×C3—24no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3C22×#4112yes—
514C22×C3C2×H(5,𝔽2)11yescharacteristic
515C22×C3C2×H(5,𝔽2)11yescharacteristic
516C22×C3C2×2−1+411yescharacteristic
517C22×C3#58512yes—
518C3×C4—24no—
519C3×C4—24no—
520C3×C4—42no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4#55914yes—
526Dic3—44no—
527Dic3—44no—
528Dic3—44no—
529Dic3—44no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23#100912yes—
546C23#101012yes—
547C23C24×D311yescharacteristic
548Q8—248no—
549D4—48no—
550D4—48no—
551D4—44no—
552D4—44no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—244no—
585C2×C4—244no—
586C2×C4C2×#23714yes—
587C2×C4C2×#23714yes—
588C2×C4C2×#23714yes—
589C2×C4#100412yes—
590C2×C3—42no—
591C2×C3—42no—
592C2×C3—82no—
593C2×C3C2×#55911yescharacteristic
594C2×C3#349412yes—
595C2×C3#349512yes—
596C2×C3#353611yescharacteristic
597C2×C3#354011yescharacteristic
598C22—24no—
599C22—24no—
600C22—22no—
601C22—22no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22C22×#23712yes—
616C22C2×#100411yescharacteristic
617C22C2×#100911yescharacteristic
618C22C2×#101011yescharacteristic
619C22#596412yes—
620C4—24no—
621C4—24no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—41no—
626C4—41no—
627C4—124no—
628C4—124no—
629C4—124no—
630C4—124no—
631C4#571314yes—
632C3#4543311yescharacteristic, complemented, Sylow
633C2—42no—
634C2—42no—
635C2—82no—
636C2#8602712yes—
637C2#8602812yes—
638C2#8605111yescharacteristic
639C2#8605511yescharacteristic
640C2C2×#571311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC3×#2397C2
hypercenter#2397D3
hypocenterC3#45433
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2397

Complements

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

NH ≅ G/Ntype
2. #82384C2semidirect
3. #82382C2semidirect
4. #82374C2semidirect
5. #82373C2semidirect
6. #82372C2semidirect
7. #82335C2semidirect
8. #82335C2semidirect
9. #78079C2semidirect
10. #78046C2semidirect
11. #78032C2semidirect
12. #76734C2semidirect
13. #76727C2semidirect
14. #76691C2semidirect
15. #73391C2semidirect
632. C3#45433semidirect

Maximal subgroups

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

subgroupindex
C3×#23972
#823842
#823822
#823742
#823732
#823722
#823352
#780792
#780462
#780322
#767342
#767272
#766912
#733912
#454333