Finite Group Databasebeta

← #11069657

Subgroups — #11069657

Order 768.

2030 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5731C211yesmaximal, characteristic, complemented
2C2×#5518C211yesmaximal, characteristic, complemented
3C3×#2621C211yesmaximal, characteristic, complemented
4#83253C211yesmaximal, characteristic, complemented
5#82780C212yesmaximal
6#82758C212yesmaximal, complemented
7#82225C211yesmaximal, characteristic
8#77863C214yesmaximal, complemented
9#77787C214yesmaximal, complemented
10#77590C214yesmaximal, complemented
11#77493C214yesmaximal
12#76625C212yesmaximal, complemented
13#76492C212yesmaximal, complemented
14#74193C211yesmaximal, characteristic, complemented
15#74150C211yesmaximal, characteristic, complemented
16#39385—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×#117C2211yescharacteristic
20C2×#946C2212yes—
21C2×Q8×Dic3C2212yes—
22C2×#944C2212yes—
23C2×#944C2212yes—
24C2×C4×#67C2211yescharacteristic
25C2×#924C2212yes—
26#5731—28no—
27#5518—28no—
28C2×#898C2214yes—
29C2×#897C2212yes—
30C2×#889C2214yes—
31C2×#889C2214yes—
32C2×#879C2214yes—
33C2×#872C2211yescharacteristic
34C2×#870C2211yescharacteristic
35C2×#869C2211yescharacteristic
36C2×C4×Dic6C2212yes—
37C3×#398C2214yes—
38C3×#397C2214yes—
39C3×#391C2212yes—
40C3×#382C2211yescharacteristic
41C4×#217C2211yescharacteristic
42#4993C2214yes—
43#4990C2212yes—
44#4982C2214yes—
45#4976C2214yes—
46#4976C2212yes—
47#4975C2214yes—
48#4975C2214yes—
49#4975C2214yes—
50Dic3×C4⋊C4C2212yes—
51#4973C2212yes—
52#4972C2212yes—
53#4957C2212yes—
54#4957C2214yes—
55#4949C2214yes—
56#4948C2214yes—
57#4945C2212yes—
58#4945C2214yes—
59C4×(C2×D3)⋊C4C2211yescharacteristic
60C4×Dic3⋊C4C2211yescharacteristic
61C42×Dic3C2211yescharacteristic
62#4676C2214yes—
63#4674C2214yes—
64#4669C2214yes—
65#4661C2214yes—
66#4657C2214yes—
67#4655C2214yes—
68#4652C2214yes—
69#4652C2214yes—
70#4649C2214yes—
71#4649C2214yes—
72#4647C2214yes—
73#4647C2214yes—
74C2×#541—31no—
75C2×#541—31no—
76#2868—31no—
77#2721—34no—
78#2721—34no—
79#2621—31no—
80#2621D311yescharacteristic, complemented
81#2552—34no—
82#2542—32no—
83#2512—34no—
84#2429—32no—
85#2429—32no—
86#2365—32no—
87C4×#125—31no—
88#2293—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2311yescharacteristic
92C22×C4×D3C2311yescharacteristic
93C22×Dic6C2312yes—
94C3×C4⋊Q8—24no—
95C3×C22⋊Q8—44no—
96C3×#117—24no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2314yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C22⋊C4C2312yes—
102C2×C3×C22⋊C4C2314yes—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×#217C2312yes—
106C2×#217C2311yescharacteristic
107C2×#217C2314yes—
108#946—28no—
109Q8×Dic3—28no—
110#944—28no—
111#944—44no—
112C4×#67—24no—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2311yescharacteristic
117C2×(C2×D3)⋊C4C2314yes—
118C2×C4⋊Dic3C2314yes—
119C2×C4⋊Dic3C2314yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2312yes—
122#924—28no—
123C2×Dic3⋊C4C2312yes—
124C2×Dic3⋊C4C2314yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2311yescharacteristic
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2314yes—
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143#898—216no—
144#897—44no—
145#889—216no—
146#889—216no—
147#879—216no—
148#872—24no—
149#870D418yescomplemented
150#869—24no—
151C4×Dic6—28no—
152C3×#95C2314yes—
153C3×#95C2314yes—
154C3×#95C2314yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162#541—68no—
163#541—68no—
164C2×C4⋊Q8—32no—
165C2×C4⋊Q8—32no—
166C2×C4⋊Q8—31no—
167C2×C4⋊Q8C2×D311yescharacteristic
168C2×#126—34no—
169C2×#126—34no—
170C2×#125—34no—
171C2×#125—34no—
172C2×#125—31no—
173C2×C22⋊Q8—32no—
174C2×C22⋊Q8—32no—
175C2×C22⋊Q8—32no—
176C2×C22⋊Q8C2×D312yes—
177C2×C4×Q8—32no—
178C2×C4×Q8—32no—
179C2×C4×D4—31no—
180C2×#117—31no—
181C2×#117C2×D311yescharacteristic
182#399—34no—
183#399—34no—
184#398—34no—
185#398—34no—
186#398—34no—
187#398C2×D314yes—
188#397—34no—
189#397C2×D314yes—
190#396—34no—
191#396—34no—
192#391—32no—
193#391C2×D312yes—
194#390—34no—
195#390—32no—
196#389—32no—
197#389—34no—
198#389—32no—
199#389—34no—
200#389—34no—
201#388—32no—
202#388—34no—
203#388—32no—
204#388—34no—
205#387—34no—
206#387—34no—
207#387—34no—
208#384—34no—
209#384—34no—
210#384—34no—
211#382C2×D311yescharacteristic
212C4×C4⋊C4—32no—
213C4×C4⋊C4—31no—
214C4×C4⋊C4—34no—
215C4×C4⋊C4—34no—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218C43—31no—
219C24×C3C2411yescharacteristic
220C23×D3C2411yescharacteristic
221C2×C3×Q8—24no—
222C2×C3×Q8—24no—
223C2×C3×Q8—24no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4C2414yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2414yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—22no—
234C2×#67—24no—
235C22×Dic3C2412yes—
236C22×Dic3C2414yes—
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2414yes—
239C22×Dic3C2411yescharacteristic
240C22×Dic3C2412yes—
241C22×Dic3C2411yescharacteristic
242C2×C4×D3—22no—
243C2×C4×D3—22no—
244C2×C4×D3—22no—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—28no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—44no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—44no—
254C3×C42—22no—
255C3×C42C2×D414yes—
256#217—28no—
257#217—22no—
258#217—24no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—28no—
261(C2×D3)⋊C4—22no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—44no—
264C4⋊Dic3—28no—
265C4⋊Dic3—28no—
266C4⋊Dic3—44no—
267C4⋊Dic3C2×D418yes—
268Dic3⋊C4—24no—
269Dic3⋊C4—28no—
270Dic3⋊C4—28no—
271Dic3⋊C4—28no—
272Dic3⋊C4—22no—
273Dic3⋊C4—28no—
274Dic3⋊C4—28no—
275Dic3⋊C4—28no—
276Dic3⋊C4—24no—
277Dic3⋊C4—44no—
278Dic3⋊C4—44no—
279Dic3⋊C4C2×D4116yes—
280C4×Dic3—22no—
281C4×Dic3—22no—
282C4×Dic3—28no—
283C4×Dic3—28no—
284C4×Dic3—24no—
285C4×Dic3—24no—
286C4×Dic3—28no—
287C4×Dic3#4114yes—
288C22×Q8—32no—
289C22×Q8C22×D312yes—
290C22×D4—31no—
291C23×C4—31no—
292C23×C4C22×D311yescharacteristic
293C4⋊Q8—24no—
294C4⋊Q8—68no—
295C4⋊Q8—64no—
296C4⋊Q8—124no—
297#126—616no—
298#126—616no—
299#125—38no—
300#125—616no—
301#125—616no—
302C22⋊Q8—44no—
303C22⋊Q8—68no—
304C22⋊Q8—68no—
305C22⋊Q8—124no—
306C4×Q8—68no—
307C4×Q8—68no—
308C4×D4—64no—
309#117—24no—
310#117—64no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—34no—
317C2×C4⋊C4—34no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4C22×D312yes—
328C2×C4⋊C4C22×D312yes—
329C2×C4⋊C4C22×D314yes—
330C2×C4⋊C4C22×D314yes—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—31no—
336C2×C22⋊C4—34no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4C22×D312yes—
340C2×C22⋊C4C22×D314yes—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—34no—
344C2×C42—34no—
345C2×C42—31no—
346C2×C42—32no—
347C2×C42—32no—
348C2×C42—34no—
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D311yescharacteristic
351#95—32no—
352#95—32no—
353#95—32no—
354#95—34no—
355#95—34no—
356#95—34no—
357#95—34no—
358#95C22×D314yes—
359#95C22×D314yes—
360#95C22×D314yes—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3C2511yescharacteristic
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—22no—
371C22×D3—22no—
372C3×Q8—48no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—24no—
377C2×C3×C4—21no—
378C2×C3×C4—24no—
379C2×C3×C4—21no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—21no—
383C2×C3×C4—24no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4—44no—
388C2×C3×C4C22×D414yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C2×#4112yes—
391C2×C3×C4H(5,𝔽2)12yes—
392#67—44no—
393C2×Dic3—22no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—21no—
401C2×Dic3—24no—
402C2×Dic3—21no—
403C2×Dic3—24no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3C22×D418yes—
408C2×Dic3C2×#4112yes—
409C2×Dic3C2×#4112yes—
410C2×Dic32−1+414yes—
411C2×Dic32−1+414yes—
412C4×D3—44no—
413Dic6—48no—
414C24—31no—
415C24C23×D311yescharacteristic
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—64no—
420C2×Q8—64no—
421C2×Q8—64no—
422C2×D4—62no—
423C2×D4—64no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—34no—
431C22×C4—32no—
432C22×C4—34no—
433C22×C4—31no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4C23×D314yes—
438C22×C4C23×D312yes—
439C22×C4C23×D311yescharacteristic
440C22×C4C23×D314yes—
441C22×C4C23×D312yes—
442C22×C4C23×D311yescharacteristic
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—24no—
446C4⋊C4—38no—
447C4⋊C4—316no—
448C4⋊C4—44no—
449C4⋊C4—62no—
450C4⋊C4—68no—
451C4⋊C4—68no—
452C4⋊C4—68no—
453C4⋊C4—68no—
454C4⋊C4—68no—
455C4⋊C4—68no—
456C4⋊C4—64no—
457C4⋊C4—68no—
458C4⋊C4—64no—
459C4⋊C4—68no—
460C4⋊C4—124no—
461C4⋊C4—124no—
462C4⋊C4—124no—
463C22⋊C4—24no—
464C22⋊C4—44no—
465C22⋊C4—62no—
466C22⋊C4—68no—
467C22⋊C4—64no—
468C22⋊C4—62no—
469C22⋊C4—68no—
470C22⋊C4—64no—
471C22⋊C4—64no—
472C22⋊C4—124no—
473C42—22no—
474C42—34no—
475C42—68no—
476C42—68no—
477C42—64no—
478C42—64no—
479C42—68no—
480C42—62no—
481C42—62no—
482C42D3×D414yes—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—42no—
486C22×C3—41no—
487C22×C3—42no—
488C22×C3C23×D411yescharacteristic
489C22×C3C22×#4111yescharacteristic
490C22×C3C2×H(5,𝔽2)11yescharacteristic
491C22×C3C2×2−1+412yes—
492C22×C3C2×2−1+412yes—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—42no—
496C2×D3—41no—
497C2×D3—42no—
498C3×C4—24no—
499C3×C4—41no—
500C3×C4—44no—
501C3×C4—42no—
502C3×C4—44no—
503C3×C4#55014yes—
504Dic3—28no—
505Dic3—22no—
506Dic3—24no—
507Dic3—44no—
508Dic3—41no—
509Dic3—42no—
510Dic3#54114yescomplemented
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—62no—
520C23—62no—
521C23C24×D311yescharacteristic
522Q8—48no—
523Q8—128no—
524D4—124no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—32no—
540C2×C4—34no—
541C2×C4—38no—
542C2×C4—32no—
543C2×C4—34no—
544C2×C4—44no—
545C2×C4—62no—
546C2×C4—64no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—64no—
552C2×C4—61no—
553C2×C4—64no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—64no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—124no—
560C2×C4C2×D3×D414yes—
561C2×C4C2×D3×D412yes—
562C2×C4#100412yes—
563C2×C4D3×#4112yes—
564C2×C3—81no—
565C2×C3C2×#54111yescharacteristic
566C2×C3C2×#55011yescharacteristic
567C2×C3#349912yes—
568C2×C3#350911yescharacteristic
569C2×C3#355212yes—
570D3—81no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—122no—
577C22—122no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22C22×D3×D411yescharacteristic
582C22C2×#100411yescharacteristic
583C22C2×D3×#4111yescharacteristic
584C22D3×2−1+412yes—
585C22D3×2−1+412yes—
586C4—24no—
587C4—34no—
588C4—42no—
589C4—44no—
590C4—41no—
591C4—44no—
592C4—68no—
593C4—64no—
594C4—62no—
595C4—122no—
596C4—121no—
597C4—124no—
598C4#559914yes—
599C3#3938511yescharacteristic, complemented, Sylow
600C2—81no—
601C2—241no—
602C2C2×#559911yescharacteristic
603C2#8634012yes—
604C2#8636112yes—
605C2D3×#54111yescharacteristic
606C2#8651011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2621C2
prime radicalC22×#67C23
hypercenter#2621D3
hypocenterC3#39385
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2621

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×#5518C2semidirect
3. C3×#2621C2semidirect
4. #83253C2semidirect
6. #82758C2semidirect
8. #77863C2semidirect
9. #77787C2semidirect
10. #77590C2semidirect
12. #76625C2semidirect
13. #76492C2semidirect
14. #74193C2semidirect
15. #74150C2semidirect
80. #2621D3semidirect
149. #870D4semidirect
510. Dic3#541semidirect
599. C3#39385semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×#55182
C3×#26212
#832532
#827802
#827582
#822252
#778632
#777872
#775902
#774932
#766252
#764922
#741932
#741502
#393853