Finite Group Databasebeta

← #10822212

Subgroups — #10822212

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
1C2×#5748C211yesmaximal, characteristic, complemented
2#85735C212yesmaximal, complemented
3#83762C211yesmaximal, characteristic, complemented
4C2×#5005C212yesmaximal, complemented
5C2×C3×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6#68703C218yesmaximal, complemented
7#23414—31nomaximal, Sylow
8C22×#235C2211yescharacteristic
9C22×#235C2212yes—
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2212yes—
12C22×#208C2211yescharacteristic
13#5748—24no—
14#5748—22no—
15#5748C2214yes—
16C2×#909—22no—
17C2×#908—22no—
18C2×#904—22no—
19#5526—22no—
20#5526—22no—
21#5525—22no—
22#5525—22no—
23#5524—24no—
24#5524C2218yescomplemented
25#5510—24no—
26#5493—24no—
27#5477—24no—
28#5129—22no—
29#5126C2214yescomplemented
30#5110—22no—
31#5005—24no—
32#5005C2218yescomplemented
33C2×#805—24no—
34C3×H(3,ℤ/4)C2214yescomplemented
35C2×#575—31no—
36#3449—32no—
37C2×#421—32no—
38#2879—31no—
39#2010—38no—
40C2×H(3,ℤ/4)D311yescharacteristic, complemented
41C23×C3×C4C2311yescharacteristic
42C22×#67—22no—
43C22×#67D412yes—
44C2×#235—21no—
45C2×#235—22no—
46C2×#235—22no—
47C2×#235—22no—
48C2×#235—22no—
49C2×#235—22no—
50C2×#235—22no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235—21no—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235D414yescomplemented
63C2×#235C2314yes—
64C22×D12D411yescharacteristic
65C22×D12D412yescomplemented
66C22×C4×D3—22no—
67C22×C4×D3C2312yes—
68C22×Dic6D411yescharacteristic
69C22×Dic6D412yescomplemented
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16D412yescomplemented
73C2×C3×M16C2312yes—
74C22×C3×C8D412yescomplemented
75C3×#117—22no—
76C3×#117—22no—
77C3×#117D414yescomplemented
78C3×#117C2314yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42D414yescomplemented
82#930—216no—
83C2×#208—21no—
84C2×#208—22no—
85C2×#208—22no—
86C2×#208C2312yes—
87C22×C3⋊C8—21no—
88#909—28no—
89#909—28no—
90#908—28no—
91#908—28no—
92C2×Dic12—22no—
93#904—28no—
94#904—28no—
95C2×D24—22no—
96C2×#204—24no—
97#897—28no—
98#897—28no—
99#895—28no—
100#895—28no—
101#886—28no—
102#886—28no—
103#882—28no—
104#882—28no—
105#875—24no—
106#875—24no—
107#874—24no—
108C4×D12—216no—
109#869—24no—
110(C2×D3)⋊C8—28no—
111#805—216no—
112C22×#41—31no—
113C22×#41—32no—
114#575—34no—
115#575—64no—
116#575—62no—
117C2×#137—62no—
118C2×AGL(1,ℤ/8)—62no—
119C2×#135—62no—
120C22×M16—31no—
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×#117C2×D312yes—
132#421—38no—
133#421—64no—
134C2×#104—64no—
135#408—62no—
136#408—62no—
137#407—34no—
138H(3,ℤ/4)C2×D314yescomplemented
139C24×C3C2×D411yescharacteristic
140C23×D3—22no—
141C23×D3C2×D412yes—
142C22×C3×C4—24no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4#4111yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C22×C3×C4C2411yescharacteristic
153C2×#67—24no—
154C2×#67—24no—
155C2×#67—24no—
156C2×#67—22no—
157C2×#67—22no—
158C2×#67—24no—
159C2×#67—42no—
160C2×#67—42no—
161C2×#67—42no—
162C2×#67—42no—
163C22×Dic3—22no—
164C22×Dic3C2×D412yes—
165#235—24no—
166#235—24no—
167#235—24no—
168#235—24no—
169#235—24no—
170#235—24no—
171#235—24no—
172#235—24no—
173#235—41no—
174#235—41no—
175#235—42no—
176#235—42no—
177#235—42no—
178#235—42no—
179#235—42no—
180#235—42no—
181#235—42no—
182#235—42no—
183#235—42no—
184#235—42no—
185#235—42no—
186#235—42no—
187#235—41no—
188#235—41no—
189#235—42no—
190#235—42no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—216no—
201C2×D12C2×D414yes—
202C2×C4×D3—24no—
203C2×C4×D3—22no—
204C2×C4×D3—24no—
205C2×C4×D3—24no—
206C2×C4×D3—24no—
207C2×C4×D3—22no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6C2×D414yes—
222C3×M16—24no—
223C3×M16—24no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8C2×D414yes—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C42—24no—
232C3×C42C2×D418yes—
233(C2×D3)⋊C4—216no—
234(C2×D3)⋊C4—216no—
235C4⋊Dic3—216no—
236#208—24no—
237#208—41no—
238#208—41no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—42no—
241Dic12—28no—
242D24—28no—
243#204—216no—
244C2×#41—34no—
245C2×#41—34no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—61no—
250C2×#41—61no—
251C2×#41—62no—
252C2×#41—62no—
253C2×#41—62no—
254C2×#41—61no—
255C2×#41—61no—
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—31no—
265C22×Q8—32no—
266C22×D4—32no—
267C22×D4—32no—
268C22×D4—31no—
269C22×D4—62no—
270C23×C4—32no—
271C23×C4—62no—
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—32no—
285C2×M16—61no—
286C2×M16—62no—
287C2×M16—62no—
288C2×M16D1212yescomplemented
289C2×M16C22×D312yes—
290C22×C8—61no—
291C22×C8D1212yescomplemented
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—22no—
307#117—22no—
308#117D1214yescomplemented
309#117C22×D314yes—
310C2×C4⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C42D1214yescomplemented
313#104—616no—
314C22⋊C8—68no—
315C23×C3—81no—
316C23×C3#12012yes—
317C23×C3#12012yes—
318C23×C3#12111yescharacteristic
319C23×C3#12111yescharacteristic
320C23×C3C22×D411yescharacteristic
321C22×D3—24no—
322C22×D3—24no—
323C22×D3—22no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—82no—
327C22×D3—84no—
328C2×C3×C4—28no—
329C2×C3×C4—84no—
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12012yes—
333C2×C3×C4#12012yes—
334C2×C3×C4#12012yes—
335C2×C3×C4#12112yes—
336C2×C3×C4#12112yes—
337C2×C3×C4#12112yes—
338C2×C3×C4#12112yes—
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12712yes—
342C2×C3×C4#12711yescharacteristic
343C2×C3×C4#12711yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D412yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C2×#4111yescharacteristic
348#67—44no—
349#67—44no—
350#67—44no—
351#67—44no—
352#67—42no—
353#67—44no—
354#67—42no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360D12—24no—
361D12—24no—
362D12—42no—
363D12—41no—
364D12—42no—
365D12—42no—
366D12—41no—
367D12—816no—
368C4×D3—42no—
369C4×D3—44no—
370C4×D3—44no—
371C4×D3—42no—
372C4×D3—44no—
373C4×D3—44no—
374C4×D3—44no—
375Dic6—24no—
376Dic6—24no—
377Dic6—42no—
378Dic6—42no—
379Dic6—41no—
380Dic6—42no—
381Dic6—41no—
382C3×C8—24no—
383C3⋊C8—42no—
384C24—32no—
385C24—62no—
386C24D3×D411yescharacteristic
387#41—64no—
388#41—64no—
389#41—64no—
390#41—64no—
391#41—64no—
392#41—64no—
393#41—64no—
394#41—64no—
395#41—122no—
396#41—121no—
397#41—121no—
398#41—122no—
399#41—122no—
400#41—122no—
401#41—122no—
402#41—121no—
403#41—121no—
404#41—122no—
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—34no—
414C2×Q8—62no—
415C2×Q8—62no—
416C2×Q8—62no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×Q8—62no—
422C2×Q8—62no—
423C2×D4—34no—
424C2×D4—616no—
425C2×D4—64no—
426C2×D4—64no—
427C2×D4—64no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—64no—
431C2×D4—62no—
432C2×D4—62no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—122no—
441C2×D4—122no—
442C2×D4—122no—
443C2×D4—122no—
444C22×C4—24no—
445C22×C4—32no—
446C22×C4—62no—
447C22×C4—64no—
448C22×C4—62no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—62no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4C2×D1211yescharacteristic
458C22×C4C2×D1211yescharacteristic
459C22×C4C2×D1212yes—
460C22×C4C2×D1212yes—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D412yes—
463C22×C4D3×D412yes—
464C22×C4#23911yescharacteristic
465C22×C4#23911yescharacteristic
466C22×C4C23×D311yescharacteristic
467Q16—68no—
468SD16—616no—
469D8—68no—
470M16—24no—
471M16—24no—
472M16—64no—
473M16—121no—
474M16—121no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—62no—
478C2×C8—122no—
479C2×C8C2×D1214yes—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—616no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—616no—
486C22⋊C4—616no—
487C42—24no—
488C42C2×D1218yes—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3#39211yescharacteristic
495C22×C3#39212yes—
496C22×C3#39211yescharacteristic
497C22×C3C2×#12012yes—
498C22×C3C2×#12111yescharacteristic
499C2×D3—48no—
500C2×D3—48no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—42no—
504C2×D3—84no—
505C2×D3—84no—
506C3×C4—48no—
507C3×C4#39211yescharacteristic
508C3×C4#39212yes—
509C3×C4#39211yescharacteristic
510C3×C4C2×#12011yescharacteristic
511C3×C4C2×#12112yes—
512C3×C4C2×#12711yescharacteristic
513Dic3—44no—
514Dic3—42no—
515C23—64no—
516C23—64no—
517C23—62no—
518C23—81no—
519C23—122no—
520C23—122no—
521C23—242no—
522C23—244no—
523C23#88212yes—
524C23#88212yes—
525C23#89511yescharacteristic
526C23#89511yescharacteristic
527C23C2×D3×D411yescharacteristic
528Q8—64no—
529Q8—64no—
530Q8—121no—
531Q8—122no—
532Q8—121no—
533Q8—122no—
534Q8—122no—
535D4—64no—
536D4—64no—
537D4—122no—
538D4—124no—
539D4—122no—
540D4—124no—
541D4—124no—
542D4—124no—
543D4—124no—
544D4—121no—
545D4—122no—
546D4—122no—
547D4—122no—
548D4—121no—
549D4—2416no—
550C2×C4—28no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—84no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—124no—
558C2×C4—124no—
559C2×C4—122no—
560C2×C4—124no—
561C2×C4—122no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4#87412yes—
565C2×C4#87411yescharacteristic
566C2×C4#87411yescharacteristic
567C2×C4#88212yes—
568C2×C4#88212yes—
569C2×C4#88211yescharacteristic
570C2×C4#88211yescharacteristic
571C2×C4#88212yes—
572C2×C4#89511yescharacteristic
573C2×C4#89511yescharacteristic
574C2×C4#89512yes—
575C2×C4#89512yes—
576C2×C4#89512yes—
577C2×C4#89512yes—
578C2×C4C22×D1211yescharacteristic
579C2×C4C22×D1212yes—
580C2×C4C2×D3×D412yes—
581C2×C4C2×#23911yescharacteristic
582C8—24no—
583C8—122no—
584C2×C3—22no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3C2×#39211yescharacteristic
590D3—82no—
591D3—84no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—128no—
598C22—128no—
599C22—124no—
600C22—124no—
601C22—122no—
602C22—244no—
603C22—244no—
604C22#467111yescharacteristic
605C22#467112yes—
606C22#467111yescharacteristic
607C22C2×#88212yes—
608C22C2×#89511yescharacteristic
609C4—48no—
610C4—122no—
611C4—124no—
612C4#467111yescharacteristic
613C4#467112yes—
614C4#467111yescharacteristic
615C4C2×#87411yescharacteristic
616C4C2×#88211yescharacteristic
617C4C2×#89512yes—
618C3#2341411yescharacteristic, complemented, Sylow
619C2—22no—
620C2—21no—
621C2—21no—
622C2—22no—
623C2—21no—
624C2—242no—
625C2—244no—
626C2C2×#467111yescharacteristic
subgroupG/H
centerC4C2×#874
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×H(3,ℤ/4)C2
socleC2×C3C2×#392
hypercenterC2×H(3,ℤ/4)D3
hypocenterC3#23414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#5748C2semidirect
2. #85735C2semidirect
3. #83762C2semidirect
4. C2×#5005C2semidirect
5. C2×C3×H(3,ℤ/4)C2semidirect
6. #68703C2semidirect
24. #5524C22semidirect
29. #5126C22semidirect
32. #5005C22semidirect
34. C3×H(3,ℤ/4)C22semidirect
40. C2×H(3,ℤ/4)D3semidirect
62. C2×#235D4semidirect
65. C22×D12D4semidirect
69. C22×Dic6D4semidirect
72. C2×C3×M16D4semidirect
74. C22×C3×C8D4semidirect
77. C3×#117D4semidirect
81. C2×C3×C42D4semidirect
138. H(3,ℤ/4)C2×D3semidirect
288. C2×M16D12semidirect
291. C22×C8D12semidirect
308. #117D12semidirect
312. C2×C42D12semidirect
618. C3#23414semidirect

Maximal subgroups

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

subgroupindex
C2×#57482
#857352
#837622
C2×#50052
C2×C3×H(3,ℤ/4)2
#687032
#234143