Finite Group Databasebeta

← #10822211

Subgroups — #10822211

Order 768.

1980 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87268C212yesmaximal, complemented
2C2×#5748C211yesmaximal, characteristic, complemented
3C3×#2879C211yesmaximal, characteristic, complemented
4C2×#5139C212yesmaximal, complemented
5C2×#4557C211yesmaximal, characteristic, complemented
6#69993C218yesmaximal, complemented
7#23414—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×#235C2212yes—
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12#5833—24no—
13#5832—24no—
14#5830—22no—
15#5828—24no—
16#5825—22no—
17#5816—22no—
18#5805—22no—
19#5804—24no—
20#5791C2218yescomplemented
21C2×#926C2212yes—
22#5748—24no—
23#5748—22no—
24#5748C2214yes—
25C2×#909—22no—
26C2×#908—22no—
27C2×#904—22no—
28C3×#408—22no—
29C3×#408—22no—
30C3×#407C2214yescomplemented
31#5139—24no—
32#5139C2218yescomplemented
33C2×#825—24no—
34#4557C2214yescomplemented
35C2×#575—31no—
36#3449—32no—
37C2×#421—32no—
38#2879D311yescharacteristic, complemented
39#2010—38no—
40C2×H(3,ℤ/4)—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41—22no—
46C2×C3×#41—22no—
47C2×C3×#41—21no—
48C2×C3×#41—21no—
49C22×C3×Q8D411yescharacteristic
50C22×C3×D4D412yes—
51C22×C3×D4D411yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235D414yescomplemented
66C2×#235C2314yes—
67C22×D12D412yescomplemented
68C22×C4×D3—22no—
69C22×Dic6D412yescomplemented
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16—21no—
73C2×C3×M16—22no—
74C2×C3×M16—22no—
75C2×C3×M16D412yescomplemented
76C2×C3×M16C2312yes—
77C2×C3×M16C2312yes—
78C22×C3×C8—21no—
79C22×C3×C8D412yescomplemented
80#953—28no—
81C2×#217—22no—
82#947—24no—
83#946—28no—
84#944—24no—
85#940—24no—
86#939—216no—
87#938—28no—
88#937—28no—
89#936—24no—
90#935—28no—
91#930—28no—
92#929—28no—
93C4×#67—216no—
94#926—22no—
95#926—22no—
96#926D414yescomplemented
97#926C2314yes—
98C2×C4⋊Dic3—22no—
99#924—28no—
100C2×C4×Dic3D414yescomplemented
101#909—28no—
102#909—28no—
103#908—28no—
104#908—28no—
105C2×Dic12—22no—
106#904—28no—
107#904—28no—
108C2×D24—22no—
109C2×#204—24no—
110C3×C22⋊C8—28no—
111#825—216no—
112C22×#41—32no—
113C22×#41C2×D311yescharacteristic
114#575—34no—
115#575—64no—
116#575—62no—
117C2×#137—62no—
118C2×AGL(1,ℤ/8)—62no—
119C2×#135—62no—
120C22×M16C2×D311yescharacteristic
121C22×M16C2×D311yescharacteristic
122#537—64no—
123#536—62no—
124#536—62no—
125#535—62no—
126#535—62no—
127#534—64no—
128#532—64no—
129#525—38no—
130#525—64no—
131C2×#117—32no—
132#421—38no—
133#421—64no—
134C2×#104—64no—
135#408—22no—
136#408—22no—
137#407C2×D314yescomplemented
138H(3,ℤ/4)—34no—
139C24×C3C2×D412yes—
140C24×C3C2×D411yescharacteristic
141C23×D3—22no—
142C3×#41—41no—
143C3×#41—41no—
144C3×#41—42no—
145C3×#41—42no—
146C3×#41—42no—
147C3×#41—42no—
148C3×#41—42no—
149C3×#41—42no—
150C3×#41—41no—
151C3×#41—41no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4#4111yescharacteristic
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—216no—
182C2×#67—42no—
183C2×#67—42no—
184C2×#67—42no—
185C2×#67—42no—
186C22×Dic3—22no—
187C22×Dic3—24no—
188#235—24no—
189#235—24no—
190#235—24no—
191#235—24no—
192#235—24no—
193#235—24no—
194#235—24no—
195#235—24no—
196#235—42no—
197#235—42no—
198#235—42no—
199#235—42no—
200#235—42no—
201#235—42no—
202#235—42no—
203#235—42no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12C2×D414yes—
210C2×C4×D3—24no—
211C2×C4×D3—24no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6C2×D414yes—
222C3×M16—24no—
223C3×M16—24no—
224C3×M16—24no—
225C3×M16—41no—
226C3×M16—41no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—42no—
231C2×C3×C8C2×D414yes—
232#217—216no—
233#217—24no—
234#217—24no—
235(C2×D3)⋊C4—216no—
236C4⋊Dic3—24no—
237C4⋊Dic3—24no—
238Dic3⋊C4—216no—
239C4×Dic3—24no—
240C4×Dic3C2×D418yes—
241Dic12—28no—
242D24—28no—
243#204—216no—
244C2×#41—21no—
245C2×#41—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—21no—
251C2×#41—21no—
252C2×#41—34no—
253C2×#41—34no—
254C2×#41—62no—
255C2×#41—62no—
256C2×#41—62no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C2×#41—62no—
263C2×#41—62no—
264C22×Q8—32no—
265C22×Q8D1211yescharacteristic
266C22×D4—32no—
267C22×D4—62no—
268C22×D4D1212yes—
269C22×D4D1211yescharacteristic
270C23×C4—62no—
271C23×C4C22×D312yes—
272C23×C4C22×D311yescharacteristic
273#137—68no—
274#137—68no—
275AGL(1,ℤ/8)—68no—
276AGL(1,ℤ/8)—68no—
277#135—68no—
278#135—68no—
279C2×Q16—62no—
280C2×SD16—64no—
281C2×D8—62no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—21no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16#6712yescomplemented
288C2×M16C22×D312yes—
289C2×M16C22×D312yes—
290C22×C8—21no—
291C22×C8#6712yescomplemented
292C4⋊Q8—64no—
293#127—64no—
294#124—64no—
295#124—64no—
296#123—68no—
297#123—68no—
298C22⋊Q8—68no—
299C22⋊Q8—68no—
300#121—68no—
301#121—68no—
302#121—616no—
303#120—68no—
304#120—68no—
305C4×D4—616no—
306#117—34no—
307#117—34no—
308#117—62no—
309#117—62no—
310C2×C4⋊C4—62no—
311C2×C22⋊C4—62no—
312C2×C42—34no—
313#104—616no—
314C22⋊C8—28no—
315C23×C3—24no—
316C23×C3—22no—
317C23×C3—81no—
318C23×C3—82no—
319C23×C3#12012yes—
320C23×C3#12012yes—
321C23×C3#12111yescharacteristic
322C23×C3#12111yescharacteristic
323C23×C3C22×D411yescharacteristic
324C22×D3—24no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—84no—
328C3×Q8—41no—
329C3×Q8—42no—
330C3×Q8—41no—
331C3×D4—42no—
332C3×D4—44no—
333C3×D4—42no—
334C3×D4—41no—
335C3×D4—42no—
336C3×D4—41no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4#12012yes—
343C2×C3×C4#12012yes—
344C2×C3×C4#12011yescharacteristic
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12012yes—
347C2×C3×C4#12112yes—
348C2×C3×C4#12112yes—
349C2×C3×C4#12112yes—
350C2×C3×C4#12112yes—
351C2×C3×C4#12111yescharacteristic
352C2×C3×C4#12111yescharacteristic
353C2×C3×C4#12712yes—
354C2×C3×C4#12711yescharacteristic
355C2×C3×C4#12711yescharacteristic
356C2×C3×C4C22×D412yes—
357C2×C3×C4C22×D412yes—
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360#67—44no—
361#67—44no—
362#67—44no—
363#67—44no—
364#67—816no—
365C2×Dic3—28no—
366C2×Dic3—24no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—84no—
370D12—24no—
371D12—24no—
372D12—42no—
373D12—42no—
374C4×D3—44no—
375C4×D3—44no—
376C4×D3—44no—
377C4×D3—44no—
378Dic6—24no—
379Dic6—24no—
380Dic6—42no—
381Dic6—42no—
382C3×C8—24no—
383C3×C8—42no—
384C24—62no—
385C24C2×D1212yes—
386C24C2×D1211yescharacteristic
387#41—42no—
388#41—42no—
389#41—42no—
390#41—42no—
391#41—41no—
392#41—41no—
393#41—42no—
394#41—42no—
395#41—41no—
396#41—41no—
397#41—64no—
398#41—64no—
399#41—64no—
400#41—64no—
401#41—64no—
402#41—64no—
403#41—64no—
404#41—64no—
405#41—122no—
406#41—122no—
407#41—122no—
408#41—122no—
409#41—122no—
410#41—122no—
411#41—122no—
412#41—122no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—34no—
418C2×Q8—62no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×Q8—62no—
422C2×Q8—62no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—34no—
432C2×D4—64no—
433C2×D4—64no—
434C2×D4—616no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—62no—
439C2×D4—62no—
440C2×D4—122no—
441C2×D4—122no—
442C2×D4—122no—
443C2×D4—122no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—62no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4C2×D1212yes—
457C22×C4C2×D1211yescharacteristic
458C22×C4#23511yescharacteristic
459C22×C4#23511yescharacteristic
460C22×C4D3×D412yes—
461C22×C4D3×D412yes—
462C22×C4D3×D412yes—
463C22×C4D3×D412yes—
464C22×C4C2×#6711yescharacteristic
465C22×C4C2×#6711yescharacteristic
466C22×C4C23×D311yescharacteristic
467Q16—68no—
468SD16—616no—
469D8—68no—
470M16—24no—
471M16—24no—
472M16—24no—
473M16—41no—
474M16—41no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—42no—
479C2×C8C2×#6714yes—
480C4⋊C4—616no—
481C4⋊C4—64no—
482C4⋊C4—64no—
483C22⋊C4—616no—
484C22⋊C4—616no—
485C22⋊C4—64no—
486C22⋊C4—64no—
487C42—38no—
488C42—64no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—48no—
495C22×C3—44no—
496C22×C3—42no—
497C22×C3#39211yescharacteristic
498C22×C3#39212yes—
499C22×C3#39211yescharacteristic
500C22×C3C2×#12012yes—
501C22×C3C2×#12111yescharacteristic
502C2×D3—48no—
503C2×D3—44no—
504C2×D3—84no—
505C2×D3—84no—
506C3×C4—42no—
507C3×C4#39211yescharacteristic
508C3×C4#39212yes—
509C3×C4#39211yescharacteristic
510C3×C4C2×#12011yescharacteristic
511C3×C4C2×#12112yes—
512C3×C4C2×#12711yescharacteristic
513Dic3—48no—
514Dic3—44no—
515C23—24no—
516C23—22no—
517C23—64no—
518C23—81no—
519C23—82no—
520C23—122no—
521C23—122no—
522C23—244no—
523C23#88212yes—
524C23#88212yes—
525C23#93011yescharacteristic
526C23#93011yescharacteristic
527C23C22×D1211yescharacteristic
528Q8—41no—
529Q8—42no—
530Q8—41no—
531Q8—64no—
532Q8—64no—
533Q8—122no—
534Q8—122no—
535D4—42no—
536D4—44no—
537D4—42no—
538D4—41no—
539D4—42no—
540D4—41no—
541D4—64no—
542D4—64no—
543D4—124no—
544D4—124no—
545D4—124no—
546D4—124no—
547D4—122no—
548D4—122no—
549D4—2416no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—68no—
556C2×C4—64no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—124no—
560C2×C4—124no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4—244no—
564C2×C4#88212yes—
565C2×C4#88212yes—
566C2×C4#88412yes—
567C2×C4#88412yes—
568C2×C4#88412yes—
569C2×C4#88412yes—
570C2×C4#93011yescharacteristic
571C2×C4#93011yescharacteristic
572C2×C4#93711yescharacteristic
573C2×C4#93711yescharacteristic
574C2×C4#93712yes—
575C2×C4#94012yes—
576C2×C4#94011yescharacteristic
577C2×C4#94011yescharacteristic
578C2×C4C2×#23511yescharacteristic
579C2×C4C2×D3×D412yes—
580C2×C4C2×D3×D412yes—
581C2×C4C22×#6711yescharacteristic
582C8—24no—
583C8—42no—
584C2×C3—22no—
585C2×C3—21no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—82no—
590C2×C3C2×#39211yescharacteristic
591D3—84no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—48no—
598C22—44no—
599C22—42no—
600C22—128no—
601C22—124no—
602C22—244no—
603C22—244no—
604C22#496011yescharacteristic
605C22#496012yes—
606C22#496011yescharacteristic
607C22C2×#88212yes—
608C22C2×#93011yescharacteristic
609C4—42no—
610C4—128no—
611C4—124no—
612C4#496011yescharacteristic
613C4#496012yes—
614C4#496011yescharacteristic
615C4C2×#88412yes—
616C4C2×#93711yescharacteristic
617C4C2×#94011yescharacteristic
618C3#2341411yescharacteristic, complemented, Sylow
619C2—22no—
620C2—21no—
621C2—21no—
622C2—22no—
623C2—21no—
624C2—82no—
625C2—244no—
626C2C2×#496011yescharacteristic
subgroupG/H
centerC4C2×#940
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2879C2
socleC2×C3C2×#392
hypercenter#2879D3
hypocenterC3#23414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
1. #87268C2semidirect
2. C2×#5748C2semidirect
3. C3×#2879C2semidirect
4. C2×#5139C2semidirect
5. C2×#4557C2semidirect
6. #69993C2semidirect
20. #5791C22semidirect
30. C3×#407C22semidirect
32. #5139C22semidirect
34. #4557C22semidirect
38. #2879D3semidirect
65. C2×#235D4semidirect
67. C22×D12D4semidirect
69. C22×Dic6D4semidirect
75. C2×C3×M16D4semidirect
79. C22×C3×C8D4semidirect
96. #926D4semidirect
100. C2×C4×Dic3D4semidirect
137. #407C2×D3semidirect
287. C2×M16#67semidirect
291. C22×C8#67semidirect
618. C3#23414semidirect

Maximal subgroups

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

subgroupindex
#872682
C2×#57482
C3×#28792
C2×#51392
C2×#45572
#699932
#234143