Finite Group Databasebeta

← #11025187

Subgroups — #11025187

Order 768.

2000 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5731C211yesmaximal, characteristic, complemented
2C2×#5470C211yesmaximal, characteristic, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4#83245C211yesmaximal, characteristic, complemented
5#82814C212yesmaximal
6#82748C212yesmaximal, complemented
7#82065C211yesmaximal, characteristic
8#76633C212yesmaximal, complemented
9#76508C212yesmaximal, complemented
10#76021C214yesmaximal, complemented
11#76014C214yesmaximal, complemented
12#74344C211yesmaximal, characteristic, complemented
13#74322C211yesmaximal, characteristic, complemented
14#73965C214yesmaximal
15#73960C214yesmaximal, complemented
16#37709—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2212yes—
19C22×C3×C42C2211yescharacteristic
20C2×#946C2212yes—
21C2×Q8×Dic3C2212yes—
22C2×C4×#67C2211yescharacteristic
23C2×#926C2212yes—
24C2×#924C2214yes—
25#5731—28no—
26C2×#898C2214yes—
27C2×#889C2214yes—
28C2×#872C2211yescharacteristic
29#5470—28no—
30C2×#870C2211yescharacteristic
31C2×#870C2211yescharacteristic
32C2×#869C2212yes—
33C2×C4×Dic6C2214yes—
34C3×#391C2212yes—
35C3×#389C2214yes—
36C3×#384C2211yescharacteristic
37C3×#380C2214yes—
38#5229C2212yes—
39#5229C2212yes—
40#5213C2212yes—
41#5210C2212yes—
42C4×#217C2211yescharacteristic
43#4994C2214yes—
44#4990C2214yes—
45#4987C2212yes—
46#4985C2214yes—
47#4985C2214yes—
48#4984C2212yes—
49#4981C2214yes—
50#4980C2214yes—
51#4978C2214yes—
52#4976C2214yes—
53#4976C2214yes—
54Dic3×C4⋊C4C2212yes—
55Dic3×C4⋊C4C2214yes—
56#4972C2212yes—
57#4959C2214yes—
58#4952C2214yes—
59#4951C2214yes—
60#4951C2214yes—
61#4940C2212yes—
62C4×(C2×D3)⋊C4C2211yescharacteristic
63#4933C2212yes—
64#4932C2211yescharacteristic
65C4×Dic3⋊C4C2211yescharacteristic
66#4673C2214yes—
67#4670C2214yes—
68#4661C2214yes—
69#4660C2214yes—
70#4656C2214yes—
71#4655C2214yes—
72#4646C2214yes—
73#4646C2214yes—
74C2×#541—31no—
75C2×#533—31no—
76#2857D311yescharacteristic, complemented
77#2702—34no—
78#2690—34no—
79#2642—31no—
80#2629—31no—
81#2492—32no—
82#2469—34no—
83#2453—34no—
84#2451—32no—
85#2415—32no—
86#2362—32no—
87#2330—31no—
88#2290—31no—
89C22×C3×Q8C2312yes—
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C22×#67C2311yescharacteristic
93C22×C4×D3C2311yescharacteristic
94C22×Dic6C2314yes—
95C3×C4⋊Q8—24no—
96C3×C22⋊Q8—28no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2314yes—
99C2×C3×C4⋊C4C2314yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C22⋊C4C2314yes—
102C2×C3×C42—24no—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2312yes—
105C2×C3×C42C2311yescharacteristic
106C2×#217C2314yes—
107C2×#217C2311yescharacteristic
108C2×#217C2312yes—
109#946—28no—
110Q8×Dic3—28no—
111C4×#67—24no—
112C2×(C2×D3)⋊C4C2312yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2311yescharacteristic
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2314yes—
117#926—28no—
118C2×C4⋊Dic3C2314yes—
119C2×C4⋊Dic3C2314yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2312yes—
123#924—216no—
124C2×Dic3⋊C4C2314yes—
125C2×Dic3⋊C4C2312yes—
126C2×Dic3⋊C4C2311yescharacteristic
127C2×Dic3⋊C4C2312yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2311yescharacteristic
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137#898—216no—
138#889—216no—
139#872—24no—
140#870—24no—
141#870D418yescomplemented
142#869—28no—
143C4×Dic6—216no—
144C3×#95C2314yes—
145C3×#95C2314yes—
146C3⋊#95C2312yes—
147C3⋊#95C2312yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159#541—68no—
160#533—68no—
161C2×C4⋊Q8—32no—
162C2×C4⋊Q8C2×D311yescharacteristic
163C2×#126—34no—
164C2×#125—34no—
165C2×#125—31no—
166C2×#125—31no—
167C2×C22⋊Q8—32no—
168C2×C22⋊Q8—34no—
169C2×C22⋊Q8C2×D312yes—
170C2×C4×Q8—32no—
171C2×C4×Q8—34no—
172C2×C4×D4—31no—
173C2×#117—32no—
174C2×#117—31no—
175C22×C42C2×D311yescharacteristic
176#400—34no—
177#400—34no—
178#400—34no—
179#398—34no—
180#398—34no—
181#397—34no—
182#396—34no—
183#395—34no—
184#394—34no—
185#393—34no—
186#391—32no—
187#391—32no—
188#391—34no—
189#391—34no—
190#391C2×D312yes—
191#389—32no—
192#389—34no—
193#389—32no—
194#389—34no—
195#389C2×D314yes—
196#388—34no—
197#388—34no—
198#388—34no—
199#388—32no—
200#388—34no—
201#387—34no—
202#387—34no—
203#387—32no—
204#387—34no—
205#386—32no—
206#385—32no—
207#384C2×D311yescharacteristic
208#380—32no—
209#380C2×D314yes—
210C4×C4⋊C4—32no—
211C4×C4⋊C4—34no—
212C4×C4⋊C4—31no—
213C4×C22⋊C4—31no—
214C4×C22⋊C4—31no—
215#376—31no—
216C24×C3C2411yescharacteristic
217C23×D3C2411yescharacteristic
218C2×C3×Q8—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8—24no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—24no—
224C22×C3×C4—24no—
225C22×C3×C4—42no—
226C22×C3×C4#4114yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—22no—
234C2×#67—42no—
235C22×Dic3C2412yes—
236C22×Dic3C2411yescharacteristic
237C22×Dic3C2411yescharacteristic
238C22×Dic3C2414yes—
239C22×Dic3C2411yescharacteristic
240C22×Dic3C2412yes—
241C22×Dic3C2414yes—
242C2×C4×D3—22no—
243C2×C4×D3—22no—
244C2×C4×D3—22no—
245C2×Dic6—28no—
246C2×Dic6—28no—
247C2×Dic6—44no—
248C3×C4⋊C4—28no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—28no—
251C3×C4⋊C4—44no—
252C3×C22⋊C4—28no—
253C3×C42—24no—
254C3×C42—22no—
255C3×C42C2×D414yes—
256#217—22no—
257#217—28no—
258#217—42no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—28no—
263(C2×D3)⋊C4—44no—
264C4⋊Dic3—24no—
265C4⋊Dic3—28no—
266C4⋊Dic3—28no—
267C4⋊Dic3—44no—
268C4⋊Dic3C2×D418yes—
269Dic3⋊C4—28no—
270Dic3⋊C4—24no—
271Dic3⋊C4—22no—
272Dic3⋊C4—24no—
273Dic3⋊C4—28no—
274Dic3⋊C4—44no—
275Dic3⋊C4C2×D4116yes—
276C4×Dic3—22no—
277C4×Dic3—22no—
278C4×Dic3—22no—
279C4×Dic3—28no—
280C4×Dic3—28no—
281C4×Dic3—44no—
282C22×Q8—34no—
283C22×Q8C22×D312yes—
284C22×D4—31no—
285C23×C4—31no—
286C23×C4C22×D312yes—
287C23×C4C22×D311yescharacteristic
288C4⋊Q8—24no—
289C4⋊Q8—68no—
290#126—616no—
291#125—38no—
292#125—616no—
293#125—64no—
294C22⋊Q8—28no—
295C22⋊Q8—616no—
296C22⋊Q8—68no—
297C4×Q8—616no—
298C4×Q8—68no—
299C4×D4—64no—
300#117—68no—
301#117—64no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4C22×D312yes—
315C2×C4⋊C4C22×D314yes—
316C2×C4⋊C4C22×D314yes—
317C2×C4⋊C4C22×D314yes—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—31no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—32no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4C22×D314yes—
327C2×C42—24no—
328C2×C42—31no—
329C2×C42—34no—
330C2×C42—31no—
331C2×C42—34no—
332C2×C42—31no—
333C2×C42—34no—
334C2×C42C22×D312yes—
335C2×C42C22×D311yescharacteristic
336C2×C42C22×D311yescharacteristic
337#95—32no—
338#95—32no—
339#95—32no—
340#95—32no—
341#95—32no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95—34no—
346#95—34no—
347#95—34no—
348#95—34no—
349#95—34no—
350#95C22×D314yes—
351#95C22×D314yes—
352C23×C3—21no—
353C23×C3—22no—
354C23×C3—21no—
355C23×C3—22no—
356C23×C3C2×#4112yes—
357C23×C3C2511yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—22no—
361C22×D3—21no—
362C22×D3—22no—
363C3×Q8—48no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—28no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—48no—
378C2×C3×C4C22×D414yes—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C42−1+414yes—
383C2×C3×C42−1+412yes—
384#67—44no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—24no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—24no—
403C2×Dic3C22×D418yes—
404C4×D3—44no—
405Dic6—416no—
406C24—31no—
407C24C23×D311yescharacteristic
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—68no—
412C2×Q8—68no—
413C2×Q8—124no—
414C2×D4—62no—
415C2×D4—122no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—34no—
421C22×C4—32no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—34no—
425C22×C4—31no—
426C22×C4—32no—
427C22×C4—42no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4#23714yes—
432C22×C4C23×D314yes—
433C22×C4C23×D314yes—
434C22×C4C23×D312yes—
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—38no—
442C4⋊C4—316no—
443C4⋊C4—44no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—68no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—68no—
450C4⋊C4—62no—
451C4⋊C4—64no—
452C4⋊C4—124no—
453C4⋊C4—124no—
454C22⋊C4—28no—
455C22⋊C4—68no—
456C22⋊C4—68no—
457C22⋊C4—62no—
458C22⋊C4—64no—
459C22⋊C4—62no—
460C22⋊C4—64no—
461C22⋊C4—124no—
462C22⋊C4—122no—
463C42—24no—
464C42—22no—
465C42—68no—
466C42—68no—
467C42—62no—
468C42—62no—
469C42—62no—
470C42—124no—
471C42D3×D414yes—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3C23×D411yescharacteristic
478C22×C3C22×#4111yescharacteristic
479C22×C3C2×2−1+412yes—
480C22×C3C2×2−1+411yescharacteristic
481C22×C3#58512yes—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—42no—
485C2×D3—41no—
486C2×D3—42no—
487C3×C4—22no—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—44no—
491C3×C4—44no—
492C3×C4#54814yes—
493Dic3—28no—
494Dic3—41no—
495Dic3—41no—
496Dic3—44no—
497Dic3—42no—
498Dic3—41no—
499Dic3—42no—
500C23—21no—
501C23—22no—
502C23—21no—
503C23—22no—
504C23—61no—
505C23—61no—
506C23—62no—
507C23—61no—
508C23—62no—
509C23C2×#23712yes—
510C23C24×D311yescharacteristic
511Q8—48no—
512Q8—1216no—
513D4—124no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—38no—
528C2×C4—48no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—62no—
532C2×C4—64no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—62no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—124no—
548C2×C4C2×D3×D414yes—
549C2×C4C2×D3×D412yes—
550C2×C4C2×#23712yes—
551C2×C4C2×#23712yes—
552C2×C4#100714yes—
553C2×C4#101012yes—
554C2×C3—42no—
555C2×C3#345511yescharacteristic
556C2×C3C2×#54811yescharacteristic
557C2×C3#348912yes—
558C2×C3#350911yescharacteristic
559C2×C3#355512yes—
560D3—81no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—121no—
567C22—121no—
568C22—121no—
569C22—122no—
570C22—122no—
571C22C22×D3×D411yescharacteristic
572C22C22×#23711yescharacteristic
573C22C2×#100712yes—
574C22C2×#101011yescharacteristic
575C22#596412yes—
576C4—22no—
577C4—24no—
578C4—24no—
579C4—44no—
580C4—44no—
581C4—68no—
582C4—121no—
583C4—121no—
584C4—121no—
585C4—124no—
586C4—122no—
587C4—122no—
588C4#559814yes—
589C3#3770911yescharacteristic, complemented, Sylow
590C2—42no—
591C2—241no—
592C2C2×#559811yescharacteristic
593C2#8598512yes—
594C2#8620411yescharacteristic
595C2#8623611yescharacteristic
596C2#8626812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2857C2
prime radicalC22×#67C23
hypercenter#2857D3
hypocenterC3#37709
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×#5470C2semidirect
3. C3×#2857C2semidirect
4. #83245C2semidirect
6. #82748C2semidirect
8. #76633C2semidirect
9. #76508C2semidirect
10. #76021C2semidirect
11. #76014C2semidirect
12. #74344C2semidirect
13. #74322C2semidirect
15. #73960C2semidirect
76. #2857D3semidirect
141. #870D4semidirect
589. C3#37709semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×#54702
C3×#28572
#832452
#828142
#827482
#820652
#766332
#765082
#760212
#760142
#743442
#743222
#739652
#739602
#377093