Finite Group Databasebeta

← #10944110

Subgroups — #10944110

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5546C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C2×#5173C212yesmaximal
4#83261C211yesmaximal, characteristic, complemented
5C2×#5090C211yesmaximal, characteristic, complemented
6#72047C212yesmaximal, complemented
7#71605C211yesmaximal, characteristic, complemented
8#70486C211yesmaximal, characteristic, complemented
9#70476C211yesmaximal, characteristic, complemented
10#70281C212yesmaximal, complemented
11#70268C212yesmaximal, complemented
12#32916—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#946C2212yes—
18C22×#216C2212yes—
19C2×D4×Dic3C2211yescharacteristic
20C2×C4×#67—21no—
21C22×Dic3⋊C4—21no—
22#5546—24no—
23#5546—24no—
24C2×D3×C4⋊C4C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×#67⋊C4—21no—
27C2×#877—21no—
28C2×#872C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#461—24no—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2212yes—
33#5229C2212yes—
34C2×#835C2212yes—
35C2×#832C2212yes—
36#5173—28no—
37C2×(C2×D3)⋊C8C2212yes—
38C2×#816C2212yes—
39#5090—24no—
40#4992—21no—
41#4992—21no—
42#4983—21no—
43#4983—21no—
44#4936C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×#520—31no—
50C2×#462—31no—
51C2×#461D311yescharacteristic, complemented
52C2×#452—32no—
53#2621—31no—
54#1878—32no—
55#1877—32no—
56#1872—32no—
57#1805—31no—
58#1804—31no—
59#1798—31no—
60C22×C3×Q8C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C23×Dic3—21no—
66C2×#237—22no—
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—22no—
70C2×#237—22no—
71C2×#237D414yes—
72C22×C4×D3—21no—
73C22×C4×D3C2311yescharacteristic
74C22×Dic6C2311yescharacteristic
75C2×C3×SD16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C22×C3×C8C2312yes—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8—22no—
81C3×C4×D4—22no—
82C3×C4×D4D414yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89#946—24no—
90#946—24no—
91C2×#216—24no—
92C2×#216—24no—
93C2×#216—24no—
94D4×Dic3—22no—
95D4×Dic3—22no—
96C4×#67—28no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—28no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113D3×C4⋊C4—22no—
114D3×C4⋊C4—22no—
115#887—22no—
116#887—22no—
117#67⋊C4—28no—
118#877—28no—
119#872—22no—
120#872—22no—
121C4×Dic6—22no—
122C4×Dic6D414yes—
123C3×C4⋊C8—22no—
124C3×#103—24no—
125#835—24no—
126#832—24no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—22no—
130C3⋊#95—22no—
131(C2×D3)⋊C8—24no—
132#816—24no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×Q16—32no—
136C22×SD16C2×D312yes—
137C2×C4⋊Q8C2×D311yescharacteristic
138C2×C22⋊Q8—32no—
139#520—64no—
140#520—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—61no—
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#462—64no—
152#461—24no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—32no—
158C2×#103C2×D312yes—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#398—61no—
162#398—61no—
163#397—61no—
164#397—61no—
165#391—32no—
166#382—31no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—21no—
171C23×D3C2×D411yescharacteristic
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2×D412yes—
183C2×C3×D4#4112yes—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—82no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×#67—24no—
192C2×#67—24no—
193C2×#67—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—21no—
197C22×Dic3—24no—
198C22×Dic3—21no—
199C22×Dic3—24no—
200C22×Dic3—22no—
201C22×Dic3—82no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205#237—28no—
206#237—28no—
207#237—28no—
208#237—28no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—24no—
214C2×C4×D3—82no—
215C2×C4×D3C2×D412yes—
216C2×C4×D3#4112yes—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C2×D412yes—
221C2×Dic6C2×D412yes—
222C2×Dic6#4112yes—
223C3×SD16—48no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4C2×D412yes—
231C3×C22⋊C4—24no—
232C3×C42C2×D412yes—
233C3×C42C2×D412yes—
234#217—24no—
235#216—48no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—44no—
239C4⋊Dic3—21no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3C2×D412yes—
243Dic3⋊C4—24no—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—44no—
249C4×Dic3—24no—
250C4×Dic3—24no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—22no—
256C2×#41—34no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C22×Q8—31no—
263C22×Q8C22×D312yes—
264C22×D4—61no—
265C22×D4C22×D311yescharacteristic
266C23×C4—21no—
267C23×C4—31no—
268C23×C4—61no—
269C23×C4—61no—
270C23×C4—61no—
271C2×Q16—64no—
272C2×Q16—64no—
273C2×Q16—64no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C22×C8—32no—
278C22×C8C22×D312yes—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C22⋊Q8—64no—
282C22⋊Q8—64no—
283C4×Q8—34no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×Q8—62no—
287C4×D4—22no—
288C4×D4—68no—
289C4×D4—62no—
290C4×D4—68no—
291C4×D4—62no—
292C4×D4#6714yes—
293#117—68no—
294#117—62no—
295#117—62no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—68no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—62no—
315C2×C42—31no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319C4⋊C8—22no—
320C4⋊C8—62no—
321#103—24no—
322#103—64no—
323#103—64no—
324#102—64no—
325C22⋊C8—64no—
326#95—62no—
327#95—62no—
328#95—61no—
329#95—61no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3C22×D411yescharacteristic
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—22no—
339C22×D3—22no—
340C22×D3#12112yes—
341C3×Q8—42no—
342C3×Q8—42no—
343C3×Q8—42no—
344C3×Q8—42no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4#12114yes—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—84no—
357C2×C3×C4#12012yes—
358C2×C3×C4#12012yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365#67—216no—
366C2×Dic3—216no—
367C2×Dic3—22no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—84no—
382C2×Dic3#12112yes—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—84no—
388Dic6—22no—
389Dic6—22no—
390Dic6—22no—
391Dic6#12114yes—
392C3×C8—42no—
393C3⋊C8—42no—
394C24—21no—
395C24—31no—
396#41—68no—
397#41—68no—
398#41—68no—
399#41—68no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—32no—
407C2×Q8—32no—
408C2×Q8—32no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—64no—
418C2×D4D3×D412yes—
419C2×D4#23712yes—
420C2×D4C2×#6712yes—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—31no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—82no—
442C22×C4—242no—
443C22×C4—242no—
444C22×C4D3×D411yescharacteristic
445C22×C4D3×D411yescharacteristic
446C22×C4C23×D311yescharacteristic
447Q16—128no—
448SD16—48no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C4⋊C4—21no—
456C4⋊C4—32no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—64no—
460C4⋊C4—64no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—64no—
464C4⋊C4—61no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—124no—
468C4⋊C4C2×#6712yes—
469C22⋊C4—24no—
470C22⋊C4—64no—
471C22⋊C4—62no—
472C22⋊C4—62no—
473C22⋊C4—124no—
474C42—64no—
475C42—64no—
476C42—61no—
477C42—61no—
478C42D3×D412yes—
479C42C2×#6712yes—
480C22×C3—24no—
481C22×C3—44no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3C2×#12011yescharacteristic
486C22×C3C2×#12111yescharacteristic
487C22×C3#52511yescharacteristic
488C22×C3#57712yes—
489C22×C3#57812yes—
490C2×D3—22no—
491C2×D3—24no—
492C2×D3—42no—
493C2×D3—41no—
494C2×D3—41no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—82no—
499C3×C4C2×#12111yescharacteristic
500C3×C4C2×#12111yescharacteristic
501C3×C4#52511yescharacteristic
502C3×C4#52511yescharacteristic
503Dic3—24no—
504Dic3—22no—
505Dic3—42no—
506Dic3—41no—
507Dic3—41no—
508Dic3—82no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—32no—
513C23—42no—
514C23—42no—
515C23—62no—
516C23—62no—
517C23—61no—
518C23—61no—
519C23C2×D3×D411yescharacteristic
520Q8—34no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—62no—
526Q8—62no—
527Q8—62no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4—616no—
532D4#93814yes—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—32no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—616no—
542C2×C4—62no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—61no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—61no—
554C2×C4—62no—
555C2×C4—61no—
556C2×C4—84no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—244no—
562C2×C4—244no—
563C2×C4#93712yes—
564C2×C4#93712yes—
565C2×C4C2×D3×D411yescharacteristic
566C2×C4C2×D3×D411yescharacteristic
567C2×C4C2×#23711yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4C22×#6711yescharacteristic
571C8—42no—
572C8—122no—
573C2×C3—44no—
574C2×C3#284711yescharacteristic
575C2×C3#300911yescharacteristic
576C2×C3#301111yescharacteristic
577C2×C3#304612yes—
578C2×C3#312012yes—
579D3—42no—
580C22—24no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—62no—
586C22—64no—
587C22—121no—
588C22—122no—
589C22—121no—
590C22D3×#12111yescharacteristic
591C22#560611yescharacteristic
592C22#576212yes—
593C22#577912yes—
594C22C2×#93711yescharacteristic
595C4—22no—
596C4—22no—
597C4—42no—
598C4—62no—
599C4—64no—
600C4—82no—
601C4—121no—
602C4—121no—
603C4—122no—
604C4—242no—
605C4D3×#12111yescharacteristic
606C4#560611yescharacteristic
607C4C2×#93811yescharacteristic
608C4#582811yescharacteristic
609C3#3291611yescharacteristic, complemented, Sylow
610C2—44no—
611C2—122no—
612C2#8041212yes—
613C2#8043112yes—
614C2#8272311yescharacteristic
615C2#8385911yescharacteristic
616C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenterC2×#461D3
hypocenterC3#32916
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. C2×C3×#461C2semidirect
4. #83261C2semidirect
5. C2×#5090C2semidirect
6. #72047C2semidirect
7. #71605C2semidirect
8. #70486C2semidirect
9. #70476C2semidirect
10. #70281C2semidirect
11. #70268C2semidirect
51. C2×#461D3semidirect
609. C3#32916semidirect

Maximal subgroups

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

subgroupindex
C2×#55462
C2×C3×#4612
C2×#51732
#832612
C2×#50902
#720472
#716052
#704862
#704762
#702812
#702682
#329163