Finite Group Databasebeta

← #10822366

Subgroups — #10822366

Order 768.

1916 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#575C211yesmaximal, characteristic, complemented
2C2×#5822C212yesmaximal, complemented
3#84074C212yesmaximal, complemented
4#83762C211yesmaximal, characteristic, complemented
5#72121C218yesmaximal, complemented
6C2×C3⋊#338C211yesmaximal, characteristic, complemented
7#23434—31nomaximal, Sylow
8C22×C3×#41C2212yes—
9C22×#235C2211yescharacteristic
10C3×#575—24no—
11C3×#575—22no—
12C3×#575C2214yescomplemented
13C2×C3×#137—22no—
14C2×C3×AGL(1,ℤ/8)—22no—
15C2×C3×#135—22no—
16C22×C3×M16C2211yescharacteristic
17C2×#951—22no—
18#5822—24no—
19#5822—24no—
20#5822—24no—
21#5822C2218yescomplemented
22C2×#950—24no—
23C2×#949—22no—
24C2×#942—22no—
25C2×#932—22no—
26C22×#208C2212yes—
27C22×#208C2211yescharacteristic
28#5234—24no—
29#5234C2218yescomplemented
30#5207—24no—
31#5129C2214yescomplemented
32#5126—22no—
33#5110—22no—
34C3⋊#338C2214yescomplemented
35C2×#575—32no—
36C2×#575D311yescharacteristic, complemented
37#2879—32no—
38#2879—31no—
39#2017—38no—
40C2×#338—31no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41—22no—
46C2×C3×#41—22no—
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41D414yes—
52C2×C3×#41C2314yes—
53C22×C3×Q8D412yes—
54C22×C3×D4—22no—
55C22×C3×D4D412yes—
56C23×C3×C4—22no—
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C2×#235—21no—
60C2×#235—21no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235D412yes—
66C2×#235C2312yes—
67C22×D12D411yescharacteristic
68C22×C4×D3—21no—
69C22×Dic6D411yescharacteristic
70C3×#137—28no—
71C3×#137—28no—
72C3×AGL(1,ℤ/8)—28no—
73C3×AGL(1,ℤ/8)—28no—
74C3×#135—28no—
75C3×#135—28no—
76C2×C3×Q16—22no—
77C2×C3×SD16—24no—
78C2×C3×D8—22no—
79C2×C3×M16—22no—
80C2×C3×M16—22no—
81C2×C3×M16D412yes—
82C2×C3×M16C2312yes—
83C22×C3×C8D412yes—
84#951—28no—
85#951—28no—
86#950—216no—
87#950—216no—
88#949—28no—
89#949—28no—
90C2×#216—24no—
91#942—28no—
92#942—28no—
93C2×#215—24no—
94C2×#214—24no—
95#932—28no—
96#932—28no—
97C2×#213—24no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208—22no—
101C2×#208—22no—
102C2×#208—21no—
103C2×#208—22no—
104C2×#208—22no—
105C2×#208—21no—
106C2×#208D414yes—
107C2×#208C2314yes—
108C22×C3⋊C8D414yes—
109C22×C3⋊C8C2312yes—
110#830—216no—
111(C2×D3)⋊C8—28no—
112C22×#41—31no—
113C22×#41C2×D312yes—
114#575—24no—
115#575—22no—
116#575—38no—
117#575—64no—
118#575—64no—
119#575—64no—
120#575C2×D314yescomplemented
121C2×#137—22no—
122C2×#137—62no—
123C2×#137—62no—
124C2×AGL(1,ℤ/8)—22no—
125C2×AGL(1,ℤ/8)—62no—
126C2×AGL(1,ℤ/8)—62no—
127C2×#135—22no—
128C2×#135—64no—
129C22×M16—32no—
130C22×M16—31no—
131C22×M16C2×D311yescharacteristic
132#408—38no—
133#408—34no—
134#408—64no—
135#408—62no—
136#407—64no—
137#407—62no—
138#338—34no—
139C24×C3—22no—
140C24×C3#4111yescharacteristic
141C23×D3—21no—
142C3×#41—24no—
143C3×#41—24no—
144C3×#41—24no—
145C3×#41—24no—
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—24no—
149C3×#41—24no—
150C3×#41—42no—
151C3×#41—42no—
152C3×#41—42no—
153C3×#41—42no—
154C3×#41—42no—
155C3×#41—42no—
156C3×#41—42no—
157C3×#41—42no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D414yes—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4C2×D414yes—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4#4111yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—22no—
194C2×#67—22no—
195C2×#67—42no—
196C2×#67—42no—
197C22×Dic3—21no—
198#235—24no—
199#235—24no—
200#235—24no—
201#235—24no—
202#235—41no—
203#235—41no—
204#235—42no—
205#235—42no—
206#235—41no—
207#235—41no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—21no—
211C2×D12C2×D412yes—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—21no—
219C2×Dic6C2×D412yes—
220C3×Q16—28no—
221C3×SD16—216no—
222C3×D8—28no—
223C3×M16—24no—
224C3×M16—24no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8C2×D414yes—
228#216—216no—
229#215—216no—
230#214—216no—
231#213—216no—
232#208—24no—
233#208—24no—
234#208—24no—
235#208—24no—
236#208—42no—
237#208—41no—
238#208—41no—
239C2×C3⋊C8—24no—
240C2×C3⋊C8—24no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—24no—
243C2×C3⋊C8C2×D418yes—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—32no—
255C2×#41—32no—
256C2×#41—61no—
257C2×#41—61no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C2×#41#6714yes—
263C2×#41C22×D314yes—
264C22×Q8—31no—
265C22×Q8#6712yes—
266C22×D4—22no—
267C22×D4—31no—
268C22×D4—61no—
269C22×D4#6712yes—
270C23×C4—22no—
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273#137—28no—
274#137—28no—
275#137—68no—
276#137—68no—
277#137—68no—
278#137—68no—
279AGL(1,ℤ/8)—28no—
280AGL(1,ℤ/8)—28no—
281AGL(1,ℤ/8)—68no—
282AGL(1,ℤ/8)—68no—
283AGL(1,ℤ/8)—68no—
284AGL(1,ℤ/8)—68no—
285#135—28no—
286#135—28no—
287#135—616no—
288#135—616no—
289C2×Q16—22no—
290C2×Q16—64no—
291C2×SD16—24no—
292C2×SD16—64no—
293C2×SD16—64no—
294C2×D8—22no—
295C2×D8—64no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—34no—
299C2×M16—34no—
300C2×M16—62no—
301C2×M16—62no—
302C2×M16—62no—
303C2×M16—62no—
304C2×M16—62no—
305C2×M16—62no—
306C2×M16—61no—
307C2×M16—61no—
308C2×M16#6712yes—
309C2×M16C22×D312yes—
310C22×C8—34no—
311C22×C8—32no—
312C22×C8#6712yes—
313C22⋊C8—616no—
314C22⋊C8—68no—
315C23×C3—24no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—81no—
319C23×C3—84no—
320C23×C3#12111yescharacteristic
321C23×C3#12111yescharacteristic
322C23×C3#12112yes—
323C23×C3#12112yes—
324C23×C3C2×#4111yescharacteristic
325C22×D3—22no—
326C22×D3—42no—
327C22×D3—82no—
328C3×Q8—24no—
329C3×Q8—24no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—44no—
335C3×D4—44no—
336C3×D4—44no—
337C3×D4—44no—
338C3×D4—42no—
339C3×D4—42no—
340C2×C3×C4—24no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—44no—
344C2×C3×C4—44no—
345C2×C3×C4—44no—
346C2×C3×C4—44no—
347C2×C3×C4#12012yes—
348C2×C3×C4#12012yes—
349C2×C3×C4#12012yes—
350C2×C3×C4#12012yes—
351C2×C3×C4#12011yescharacteristic
352C2×C3×C4#12011yescharacteristic
353C2×C3×C4#12012yes—
354C2×C3×C4#12111yescharacteristic
355C2×C3×C4#12111yescharacteristic
356C2×C3×C4#12112yes—
357C2×C3×C4#12112yes—
358C2×C3×C4#12712yes—
359C2×C3×C4#12711yescharacteristic
360C2×C3×C4#12711yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D412yes—
363C2×C3×C4C22×D412yes—
364C2×C3×C4C22×D411yescharacteristic
365#67—44no—
366#67—42no—
367#67—42no—
368C2×Dic3—22no—
369C2×Dic3—42no—
370D12—24no—
371D12—41no—
372D12—41no—
373C4×D3—42no—
374C4×D3—42no—
375C4×D3—44no—
376Dic6—24no—
377Dic6—41no—
378Dic6—41no—
379C3×C8—24no—
380C3⋊C8—28no—
381C3⋊C8—42no—
382C24—22no—
383C24—61no—
384C24#23711yescharacteristic
385#41—24no—
386#41—24no—
387#41—24no—
388#41—24no—
389#41—24no—
390#41—24no—
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—42no—
396#41—42no—
397#41—42no—
398#41—42no—
399#41—42no—
400#41—42no—
401#41—64no—
402#41—64no—
403#41—64no—
404#41—64no—
405#41—122no—
406#41—122no—
407#41—121no—
408#41—121no—
409#41—121no—
410#41—121no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—62no—
419C2×Q8—62no—
420C2×Q8C2×#6714yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—32no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—61no—
438C2×D4—122no—
439C2×D4—122no—
440C2×D4C2×#6714yes—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—61no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4D3×D412yes—
456C22×C4D3×D412yes—
457C22×C4#23711yescharacteristic
458C22×C4C2×#6712yes—
459C22×C4C2×#6712yes—
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C23×D311yescharacteristic
463Q16—28no—
464Q16—616no—
465SD16—216no—
466SD16—616no—
467SD16—616no—
468D8—28no—
469D8—616no—
470M16—24no—
471M16—24no—
472M16—64no—
473M16—64no—
474M16—64no—
475M16—64no—
476M16—121no—
477M16—122no—
478M16—121no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—38no—
482C2×C8—64no—
483C2×C8—64no—
484C2×C8—64no—
485C2×C8—62no—
486C2×C8C2×#6714yes—
487C22×C3—41no—
488C22×C3—42no—
489C22×C3—41no—
490C22×C3—42no—
491C22×C3—41no—
492C22×C3—48no—
493C22×C3—44no—
494C22×C3—84no—
495C22×C3—84no—
496C22×C3#39212yes—
497C22×C3#39211yescharacteristic
498C22×C3#39211yescharacteristic
499C22×C3C2×#12111yescharacteristic
500C22×C3C2×#12112yes—
501C2×D3—44no—
502C2×D3—42no—
503C2×D3—84no—
504C3×C4—44no—
505C3×C4#39211yescharacteristic
506C3×C4#39211yescharacteristic
507C3×C4#39212yes—
508C3×C4C2×#12012yes—
509C3×C4C2×#12011yescharacteristic
510C3×C4C2×#12711yescharacteristic
511Dic3—42no—
512C23—24no—
513C23—42no—
514C23—42no—
515C23—62no—
516C23—81no—
517C23—84no—
518C23—122no—
519C23—242no—
520C23#93811yescharacteristic
521C23#93811yescharacteristic
522C23#93912yes—
523C23#93912yes—
524C23C2×#23711yescharacteristic
525Q8—24no—
526Q8—24no—
527Q8—42no—
528Q8—42no—
529Q8—64no—
530Q8—121no—
531Q8—121no—
532D4—24no—
533D4—24no—
534D4—44no—
535D4—44no—
536D4—44no—
537D4—44no—
538D4—42no—
539D4—42no—
540D4—64no—
541D4—122no—
542D4—122no—
543D4—124no—
544D4—121no—
545D4—121no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—62no—
554C2×C4—122no—
555C2×C4—124no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4#93712yes—
559C2×C4#93712yes—
560C2×C4#93712yes—
561C2×C4#93712yes—
562C2×C4#93811yescharacteristic
563C2×C4#93811yescharacteristic
564C2×C4#93912yes—
565C2×C4#93912yes—
566C2×C4#94011yescharacteristic
567C2×C4#94012yes—
568C2×C4#94011yescharacteristic
569C2×C4#95311yescharacteristic
570C2×C4#95311yescharacteristic
571C2×C4#95312yes—
572C2×C4C2×D3×D411yescharacteristic
573C2×C4C2×D3×D412yes—
574C2×C4C22×#6712yes—
575C2×C4C22×#6711yescharacteristic
576C8—24no—
577C8—68no—
578C8—122no—
579C2×C3—21no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—84no—
585C2×C3C2×#39211yescharacteristic
586D3—82no—
587C22—42no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—48no—
593C22—44no—
594C22—84no—
595C22—84no—
596C22—124no—
597C22—122no—
598C22—244no—
599C22#522311yescharacteristic
600C22#522312yes—
601C22#522311yescharacteristic
602C22C2×#93811yescharacteristic
603C22C2×#93912yes—
604C4—44no—
605C4—122no—
606C4#522311yescharacteristic
607C4#522312yes—
608C4#522311yescharacteristic
609C4C2×#93712yes—
610C4C2×#94011yescharacteristic
611C4C2×#95311yescharacteristic
612C3#2343411yescharacteristic, complemented, Sylow
613C2—22no—
614C2—21no—
615C2—22no—
616C2—21no—
617C2—21no—
618C2—84no—
619C2—242no—
620C2C2×#522311yescharacteristic
subgroupG/H
centerC4C2×#940
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#575C2
socleC2×C3C2×#392
hypercenterC2×#575D3
hypocenterC3#23434
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#575

Complements

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

NH ≅ G/Ntype
1. C2×C3×#575C2semidirect
2. C2×#5822C2semidirect
3. #84074C2semidirect
4. #83762C2semidirect
5. #72121C2semidirect
6. C2×C3⋊#338C2semidirect
12. C3×#575C22semidirect
21. #5822C22semidirect
29. #5234C22semidirect
31. #5129C22semidirect
34. C3⋊#338C22semidirect
36. C2×#575D3semidirect
120. #575C2×D3semidirect
612. C3#23434semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5752
C2×#58222
#840742
#837622
#721212
C2×C3⋊#3382
#234343