Finite Group Databasebeta

← #10804530

Subgroups — #10804530

Order 768.

1398 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C2×#4719C211yesmaximal, characteristic, complemented
3#73864C211yesmaximal, characteristic, complemented
4C3×#1788C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic, complemented
6#70229C212yesmaximal, complemented
7#70227C212yesmaximal, complemented
8#70161C212yesmaximal, complemented
9#70159C212yesmaximal, complemented
10#69960C211yesmaximal, characteristic, complemented
11#69522C211yesmaximal, characteristic, complemented
12#21809—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15#5832—24no—
16#5832C418yescomplemented
17C2×#947C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C2×#940C2211yescharacteristic
20C2×#939—21no—
21C2×#939—21no—
22C2×#936C2211yescharacteristic
23C2×C4×#67—21no—
24C2×C4×#67—21no—
25C22×C4×Dic3C2211yescharacteristic
26C2×#877—21no—
27C2×C3×#103C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C2×#835C2212yes—
32C2×#832C2212yes—
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#821C2212yes—
36C2×#816C2212yes—
37C2×#815C2212yes—
38C2×Dic3⋊C8C2212yes—
39#4975C2212yes—
40Dic3×C4⋊C4C2212yes—
41C2×#810C2212yes—
42C2×#809C2212yes—
43#4942—21no—
44#4719—24no—
45#4650—21no—
46#4649—21no—
47C2×#532—31no—
48C2×#432—31no—
49#2285—31no—
50#1982—32no—
51#1981—32no—
52#1959—32no—
53#1959—32no—
54#1788—31no—
55#1788—31no—
56#1788—31no—
57#1788D311yescharacteristic, complemented
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41C2×C414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C22×#67—21no—
68C22×#67—21no—
69C22×#67—21no—
70C23×Dic3D411yescharacteristic
71C23×Dic3D411yescharacteristic
72C2×#235—22no—
73C2×#235—42no—
74C2×#235—42no—
75C2×#235C2×C414yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—21no—
85#947—22no—
86#947C2×C414yes—
87#944—22no—
88#944C2×C414yes—
89#940—22no—
90#940C2×C414yes—
91#939—28no—
92#939—28no—
93#936—22no—
94#936C2×C414yes—
95C4×#67—28no—
96C4×#67—28no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3C2312yes—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—22no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3C2×C414yes—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3D414yes—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112#877—44no—
113C3×#103—42no—
114C3×#102—42no—
115C3×C22⋊C8—22no—
116C3×#95—21no—
117#835—42no—
118#832—42no—
119C3⋊#95—21no—
120C3⋊#95—22no—
121#830—22no—
122#821—42no—
123#816—42no—
124#815—24no—
125Dic3⋊C8—24no—
126#810—42no—
127#809—42no—
128C22×#41—31no—
129C22×#41C2×D311yescharacteristic
130#532—38no—
131#532—64no—
132C2×C4⋊Q8—31no—
133C2×#127—31no—
134C2×#124—31no—
135C2×#124—31no—
136C2×#121—61no—
137C2×#121—61no—
138C2×C4×D4—61no—
139C2×C4×D4—61no—
140C2×#117—61no—
141C22×C42—31no—
142#432—64no—
143C2×C4⋊C8—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×#103C2×D312yes—
148C2×#102—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102C2×D312yes—
152C2×C22⋊C8—31no—
153C2×C22⋊C8C2×D311yescharacteristic
154C2×#97—32no—
155#384—32no—
156#382—61no—
157#380—31no—
158#380—61no—
159#380C2×D311yescharacteristic
160C4×C4⋊C4—32no—
161C4×C4⋊C4—61no—
162C24×C3—21no—
163C24×C3C2×D411yescharacteristic
164C23×D3—21no—
165C3×#41—44no—
166C3×#41—44no—
167C3×#41—48no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8C22×C412yes—
172C2×C3×D4—24no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—44no—
177C2×C3×D4C22×C412yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—44no—
184C22×C3×C4C22×C412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2411yescharacteristic
188C2×#67—24no—
189C2×#67—24no—
190C2×#67—24no—
191C2×#67—44no—
192C2×#67—44no—
193C2×#67—42no—
194C2×#67—42no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—42no—
200C22×Dic3C22⋊C414yes—
201C22×Dic3C22⋊C414yes—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206#235—44no—
207#235—44no—
208#235—48no—
209C2×D12—22no—
210C2×D12—22no—
211C2×D12—21no—
212C2×D12C22×C412yes—
213C2×C4×D3—24no—
214C2×C4×D3—44no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—21no—
218C2×Dic6C22×C412yes—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C22⋊C4—42no—
225#217—24no—
226#217—24no—
227#217—42no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—24no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232Dic3⋊C4—24no—
233Dic3⋊C4—24no—
234Dic3⋊C4—44no—
235Dic3⋊C4—44no—
236C4×Dic3—44no—
237C4×Dic3C22⋊C414yes—
238C4×Dic3C22⋊C414yes—
239C4×Dic3C22×C412yes—
240C4×Dic3C22×C412yes—
241C4×Dic3C2×D412yes—
242C4×Dic3C2×D412yes—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×#41—22no—
247C2×#41—34no—
248C2×#41—42no—
249C2×#41—42no—
250C2×#41—62no—
251C2×#41—122no—
252C2×#41—122no—
253C2×#41C4×D314yes—
254C22×Q8—31no—
255C22×Q8C22×D311yescharacteristic
256C22×D4—21no—
257C22×D4—31no—
258C22×D4—61no—
259C22×D4—61no—
260C22×D4—61no—
261C22×D4C22×D311yescharacteristic
262C23×C4—21no—
263C23×C4—31no—
264C23×C4—31no—
265C23×C4—61no—
266C23×C4C22×D311yescharacteristic
267C22×C8—32no—
268C22×C8C22×D312yes—
269C4⋊Q8—34no—
270C4⋊Q8—62no—
271#127—34no—
272#127—62no—
273#124—34no—
274#124—34no—
275#124—62no—
276#124—62no—
277#121—68no—
278#121—68no—
279C4×D4—68no—
280C4×D4—68no—
281#117—124no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4C22×D312yes—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—34no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—34no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—62no—
301C4⋊C8—64no—
302#103—42no—
303#103—122no—
304#103—122no—
305#103—122no—
306#102—42no—
307#102—122no—
308#102—122no—
309#102—122no—
310C22⋊C8—22no—
311C22⋊C8—62no—
312#97—64no—
313#95—21no—
314#95—62no—
315#95—61no—
316C23×C3—22no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—42no—
325C23×C3C2×C22⋊C412yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—42no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—42no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—48no—
338C3×D4—42no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—48no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4C2×C22⋊C412yes—
350C2×C3×C4C23×C411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4H(5,𝔽2)12yes—
353C2×C3×C4H(5,𝔽2)12yes—
354#67—48no—
355#67—48no—
356#67—48no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—28no—
362C2×Dic3—22no—
363C2×Dic3—44no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3C2×C22⋊C412yes—
369C2×Dic3C2×C22⋊C412yes—
370C2×Dic3C2×C22⋊C412yes—
371C2×Dic3C2×C22⋊C412yes—
372C2×Dic3AGL(1,ℤ/8)12yes—
373C2×Dic3#13712yes—
374D12—22no—
375D12—22no—
376D12—42no—
377C4×D3—48no—
378Dic6—22no—
379Dic6—22no—
380Dic6—42no—
381C3×C8—42no—
382C3⋊C8—42no—
383C24—21no—
384C24—61no—
385C24D3×D411yescharacteristic
386#41—44no—
387#41—44no—
388#41—48no—
389#41—124no—
390#41—124no—
391#41—128no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—32no—
396C2×Q8—61no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8C2×C4×D312yes—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—32no—
405C2×D4—44no—
406C2×D4—64no—
407C2×D4—64no—
408C2×D4—64no—
409C2×D4—61no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4C2×C4×D312yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—31no—
423C22×C4—34no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—34no—
428C22×C4—44no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—124no—
435C22×C4—122no—
436C22×C4C2×C4×D312yes—
437C22×C4D3×D411yescharacteristic
438C22×C4D3×D412yes—
439C22×C4C23×D311yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—124no—
453C4⋊C4—124no—
454C22⋊C4—42no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C22⋊C4—64no—
459C22⋊C4—122no—
460C42—34no—
461C42—34no—
462C42—32no—
463C42—32no—
464C42—32no—
465C42—32no—
466C42—124no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3C22×C22⋊C411yescharacteristic
478C22×C3#53512yes—
479C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
480C22×C3C2×#13711yescharacteristic
481C22×C3#57512yes—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—82no—
486C2×D3—82no—
487C3×C4—22no—
488C3×C4—42no—
489C3×C4—82no—
490C3×C4#51811yescharacteristic
491C3×C4#51811yescharacteristic
492C3×C4#53512yes—
493Dic3—22no—
494Dic3—22no—
495Dic3—22no—
496Dic3—42no—
497Dic3—82no—
498Dic3#41714yes—
499C23—22no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—62no—
509C23—122no—
510C23—121no—
511C23—121no—
512C23—122no—
513C23D3×C22⋊C412yes—
514C23C2×D3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—42no—
518Q8—62no—
519Q8—62no—
520Q8—122no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—42no—
525D4—62no—
526D4—62no—
527D4—128no—
528D4—128no—
529D4—128no—
530D4—122no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—32no—
541C2×C4—32no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—68no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—62no—
553C2×C4—128no—
554C2×C4—124no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4D3×C22⋊C412yes—
560C2×C4C22×C4×D311yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4#100412yes—
563C2×C4#100412yes—
564C8—42no—
565C8—122no—
566C2×C3—42no—
567C2×C3—82no—
568C2×C3#230911yescharacteristic
569C2×C3C2×#41711yescharacteristic
570C2×C3#289611yescharacteristic
571C2×C3#321412yes—
572C2×C3#321612yes—
573D3—82no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—82no—
583C22—82no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22—242no—
588C22—242no—
589C22C2×D3×C22⋊C411yescharacteristic
590C22#559212yes—
591C22#576812yes—
592C22D3×AGL(1,ℤ/8)11yescharacteristic
593C22D3×#13711yescharacteristic
594C4—22no—
595C4—34no—
596C4—42no—
597C4—62no—
598C4—62no—
599C4—62no—
600C4—82no—
601C4—122no—
602C4—242no—
603C4#548611yescharacteristic
604C4#548611yescharacteristic
605C4#559212yes—
606C3#2180911yescharacteristic, complemented, Sylow
607C2—42no—
608C2—82no—
609C2—242no—
610C2#7381911yescharacteristic
611C2#7883011yescharacteristic
612C2D3×#41711yescharacteristic
613C2#8007812yes—
614C2#8022212yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
prime radicalC2×#5832C2
cosocleC22×C3×C4C24
hypercenter#1788D3
hypocenterC3#21809
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. C2×#4719C2semidirect
3. #73864C2semidirect
4. C3×#1788C2semidirect
5. #72135C2semidirect
6. #70229C2semidirect
7. #70227C2semidirect
8. #70161C2semidirect
9. #70159C2semidirect
10. #69960C2semidirect
11. #69522C2semidirect
16. #5832C4semidirect
57. #1788D3semidirect
606. C3#21809semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
C2×#47192
#738642
C3×#17882
#721352
#702292
#702272
#701612
#701592
#699602
#695222
#218093