Finite Group Databasebeta

← #10796779

Subgroups — #10796779

Order 768.

1694 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×#5526C211yesmaximal, characteristic
2C2×#5516C211yesmaximal, characteristic, complemented
3C22×#824C211yesmaximal, characteristic, complemented
4#83768C211yesmaximal, characteristic
5#82213C211yesmaximal, characteristic
6C3×#1733C211yesmaximal, characteristic, complemented
7#71578C211yesmaximal, characteristic, complemented
8#68766C214yesmaximal, complemented
9#68765C214yesmaximal, complemented
10#20954—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C22×#208C2211yescharacteristic
18#5526—24no—
19#5526C418yescomplemented
20#5516—22no—
21#5516—22no—
22#5516—22no—
23#5516C2214yes—
24C2×#897—22no—
25C2×#887—22no—
26C2×#886—22no—
27C2×#877—22no—
28C2×#875C2212yes—
29C2×#869C2212yes—
30C2×C4×Dic6—22no—
31C2×#824—24no—
32C2×#824C2214yes—
33#5133—22no—
34#5131C2212yes—
35#5120C2212yes—
36#5113—22no—
37#5004C2212yes—
38#5002—22no—
39#5001C2212yes—
40#4966—22no—
41C3×#339C2214yes—
42#4523C2214yes—
43C2×#536—31no—
44C2×#519—31no—
45#2896—31no—
46#2896—31no—
47C22×#101—31no—
48#1888—34no—
49#1886—34no—
50#1733—31no—
51#1733D311yescharacteristic, complemented
52C23×C3×C4C2311yescharacteristic
53C22×#67—21no—
54C23×Dic3D412yes—
55C2×#235—22no—
56C2×#235—21no—
57C2×#235—21no—
58C2×#235—21no—
59C2×#235—21no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235C2×C414yes—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×Dic6—21no—
66C22×Dic6—21no—
67C22×Dic6—21no—
68C22×Dic6—21no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6C2×C414yes—
72C22×Dic6D412yes—
73C22×Dic6D414yes—
74C22×Dic6C2311yescharacteristic
75C22×Dic6C2312yes—
76C2×C3×M16—21no—
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16C2312yes—
80C22×C3×C8—21no—
81C3×#117—21no—
82C3×#117—21no—
83C3×#117—22no—
84C3×#117C2×C414yes—
85C3×#117D412yes—
86C3×#117C2312yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D412yes—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2312yes—
93C2×#217—21no—
94C2×(C2×D3)⋊C4—24no—
95#926—21no—
96#926—21no—
97#926D412yes—
98#926C2312yes—
99C2×C4⋊Dic3—24no—
100C2×C4⋊Dic3D412yes—
101C2×Dic3⋊C4—24no—
102C2×C4×Dic3—24no—
103C2×C4×Dic3—21no—
104C2×#208—21no—
105C2×#208—22no—
106C2×#208—22no—
107C2×#208C2312yes—
108C22×C3⋊C8—21no—
109#897—216no—
110#887—28no—
111#887—28no—
112#886—216no—
113#877—28no—
114#877—28no—
115#875—24no—
116#875C2×C418yescomplemented
117#869—24no—
118#869C2×C418yescomplemented
119C4×Dic6—28no—
120C4×Dic6—28no—
121#824—28no—
122#821—24no—
123#816—24no—
124#810—24no—
125#809—24no—
126C22×#41—31no—
127C23×Q8—31no—
128C22×M16—31no—
129C22×M16C2×D311yescharacteristic
130#536—38no—
131#536—64no—
132C2×C4⋊Q8—32no—
133C2×#124—32no—
134C2×#123—62no—
135C2×C22⋊Q8—62no—
136#519—34no—
137#519—62no—
138#519—62no—
139#519—62no—
140C2×C4×Q8—62no—
141C2×C4×Q8—62no—
142C2×#117—31no—
143C2×#117—62no—
144C2×#117C2×D311yescharacteristic
145C2×#117C2×D311yescharacteristic
146#419—32no—
147#419—32no—
148#418—32no—
149#418—32no—
150#417—62no—
151#417—62no—
152#416—62no—
153#416—62no—
154C2×#101—34no—
155C2×#101—64no—
156#339—34no—
157#339C2×D314yes—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4C22×C412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4#4112yes—
170C22×C3×C4#4112yes—
171C22×C3×C4C2411yescharacteristic
172C2×#67—24no—
173C2×#67—41no—
174C2×#67—41no—
175C2×#67—42no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—24no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3C22⋊C418yes—
182C22×Dic3C2×D414yes—
183#235—24no—
184#235—24no—
185#235—24no—
186#235—24no—
187#235—42no—
188#235—42no—
189#235—42no—
190#235—42no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12C22×C412yes—
197C2×C4×D3—24no—
198C2×C4×D3—41no—
199C2×C4×D3—41no—
200C2×C4×D3—42no—
201C2×Dic6—24no—
202C2×Dic6—24no—
203C2×Dic6—24no—
204C2×Dic6—24no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C2×Dic6—24no—
208C2×Dic6—24no—
209C2×Dic6—24no—
210C2×Dic6—24no—
211C2×Dic6—24no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6C22⋊C418yes—
222C2×Dic6C22×C412yes—
223C2×Dic6C22×C414yes—
224C2×Dic6C2×D414yes—
225C3×M16—22no—
226C3×M16—22no—
227C3×M16—42no—
228C2×C3×C8—22no—
229C2×C3×C8—42no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4C2×D414yes—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42C22×C414yes—
240#217—22no—
241#217—22no—
242(C2×D3)⋊C4—216no—
243C4⋊Dic3—22no—
244C4⋊Dic3—216no—
245C4⋊Dic3C2×D414yes—
246Dic3⋊C4—216no—
247C4×Dic3—216no—
248C4×Dic3—22no—
249C4×Dic3—22no—
250#208—22no—
251#208—22no—
252#208—42no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—42no—
255C2×#41—34no—
256C2×#41—62no—
257C2×#41—61no—
258C2×#41—61no—
259C2×#41—62no—
260C2×#41—61no—
261C2×#41—61no—
262C2×#41—62no—
263C22×Q8—34no—
264C22×Q8—32no—
265C22×Q8—34no—
266C22×Q8—32no—
267C22×Q8—31no—
268C22×Q8—61no—
269C22×Q8—61no—
270C22×Q8—62no—
271C22×Q8—61no—
272C22×Q8—61no—
273C22×Q8—62no—
274C22×D4—31no—
275C22×D4—61no—
276C23×C4—32no—
277C23×C4—61no—
278C23×C4C22×D311yescharacteristic
279C2×M16—21no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—32no—
283C2×M16—61no—
284C2×M16—62no—
285C2×M16—62no—
286C2×M16C22×D312yes—
287C22×C8—21no—
288C22×C8—61no—
289C4⋊Q8—38no—
290C4⋊Q8—64no—
291#124—38no—
292#124—64no—
293#123—616no—
294C22⋊Q8—616no—
295C4×Q8—68no—
296C4×Q8—68no—
297C4×Q8—68no—
298C4×Q8—68no—
299#117—21no—
300#117—21no—
301#117—22no—
302#117—32no—
303#117—32no—
304#117—68no—
305#117—68no—
306#117—61no—
307#117—61no—
308#117C4×D314yes—
309#117D1212yes—
310#117C22×D312yes—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—64no—
314C2×C4⋊C4—64no—
315C2×C4⋊C4D1212yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—64no—
319C2×C22⋊C4—61no—
320C2×C42—21no—
321C2×C42—64no—
322C2×C42—61no—
323C2×C42C22×D312yes—
324#103—64no—
325#103—64no—
326#102—64no—
327#102—64no—
328#101—68no—
329C23×C3—42no—
330C23×C3C2×C22⋊C412yes—
331C23×C3#12011yescharacteristic
332C23×C3#12011yescharacteristic
333C23×C3#12312yes—
334C23×C3C22×D411yescharacteristic
335C22×D3—22no—
336C22×D3—44no—
337C22×D3—41no—
338C22×D3—41no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—44no—
342C2×C3×C4—44no—
343C2×C3×C4C2×C22⋊C412yes—
344C2×C3×C4C4×D412yes—
345C2×C3×C4C4×D412yes—
346C2×C3×C4C4×D412yes—
347C2×C3×C4C4×D412yes—
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4#12112yes—
351C2×C3×C4#12112yes—
352C2×C3×C4#12112yes—
353C2×C3×C4#12112yes—
354C2×C3×C4#12312yes—
355C2×C3×C4C23×C411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361#67—44no—
362#67—44no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—28no—
366C2×Dic3—28no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—44no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3C2×C22⋊C418yes—
373D12—22no—
374D12—22no—
375D12—41no—
376D12—41no—
377C4×D3—44no—
378C4×D3—44no—
379Dic6—24no—
380Dic6—24no—
381Dic6—24no—
382Dic6—24no—
383Dic6—28no—
384Dic6—28no—
385Dic6—22no—
386Dic6—22no—
387Dic6—41no—
388Dic6—41no—
389C3×C8—42no—
390C3⋊C8—42no—
391C24—61no—
392C24D3×D411yescharacteristic
393#41—64no—
394#41—64no—
395#41—64no—
396#41—64no—
397#41—122no—
398#41—122no—
399#41—122no—
400#41—122no—
401C2×Q8—38no—
402C2×Q8—34no—
403C2×Q8—34no—
404C2×Q8—32no—
405C2×Q8—64no—
406C2×Q8—64no—
407C2×Q8—62no—
408C2×Q8—62no—
409C2×Q8—64no—
410C2×Q8—64no—
411C2×Q8—64no—
412C2×Q8—64no—
413C2×Q8—64no—
414C2×Q8—64no—
415C2×Q8—64no—
416C2×Q8—62no—
417C2×Q8—62no—
418C2×Q8—62no—
419C2×Q8—61no—
420C2×Q8—62no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×D4—32no—
426C2×D4—64no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—121no—
433C2×D4—121no—
434C2×D4—122no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—38no—
438C22×C4—34no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—64no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—64no—
445C22×C4—121no—
446C22×C4—121no—
447C22×C4—122no—
448C22×C4C2×C4×D312yes—
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4#23512yes—
452C22×C4D3×D411yescharacteristic
453C22×C4D3×D412yes—
454C22×C4D3×D411yescharacteristic
455C22×C4D3×D411yescharacteristic
456C22×C4#23912yes—
457C22×C4C23×D311yescharacteristic
458M16—22no—
459M16—22no—
460M16—42no—
461M16—62no—
462M16—62no—
463M16—122no—
464C2×C8—22no—
465C2×C8—42no—
466C2×C8—62no—
467C2×C8—122no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—34no—
471C4⋊C4—616no—
472C4⋊C4—616no—
473C4⋊C4—62no—
474C4⋊C4C2×D1214yes—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—616no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—616no—
485C42—62no—
486C42—62no—
487C42C2×C4×D314yes—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3#38612yes—
494C22×C3#38612yes—
495C22×C3C22×C22⋊C411yescharacteristic
496C22×C3C2×#12011yescharacteristic
497C22×C3C2×#12311yescharacteristic
498C2×D3—44no—
499C2×D3—42no—
500C2×D3—82no—
501C2×D3—82no—
502C3×C4—44no—
503C3×C4—44no—
504C3×C4#38612yes—
505C3×C4#38612yes—
506C3×C4C2×C4×D411yescharacteristic
507C3×C4C2×C4×D411yescharacteristic
508C3×C4C2×#12111yescharacteristic
509C3×C4C2×#12111yescharacteristic
510Dic3—28no—
511Dic3—44no—
512Dic3—42no—
513C23—42no—
514C23—62no—
515C23—124no—
516C23—121no—
517C23—121no—
518C23D3×C22⋊C412yes—
519C23#88211yescharacteristic
520C23#88211yescharacteristic
521C23#89412yes—
522C23C2×D3×D411yescharacteristic
523Q8—68no—
524Q8—68no—
525Q8—64no—
526Q8—64no—
527Q8—64no—
528Q8—64no—
529Q8—62no—
530Q8—62no—
531Q8—121no—
532Q8—121no—
533D4—62no—
534D4—62no—
535D4—124no—
536D4—124no—
537D4—121no—
538D4—121no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—38no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—68no—
545C2×C4—68no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—124no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—124no—
554C2×C4—124no—
555C2×C4C4×D1212yes—
556C2×C4C4×D1212yes—
557C2×C4D3×C22⋊C412yes—
558C2×C4#88211yescharacteristic
559C2×C4#88211yescharacteristic
560C2×C4#88412yes—
561C2×C4#88412yes—
562C2×C4#89312yes—
563C2×C4#89312yes—
564C2×C4#89412yes—
565C2×C4#89512yes—
566C2×C4#89512yes—
567C2×C4C22×C4×D311yescharacteristic
568C2×C4C22×D1211yescharacteristic
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×#23911yescharacteristic
573C8—42no—
574C8—122no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—22no—
578C2×C3—22no—
579C2×C3C2×#38611yescharacteristic
580C2×C3#302012yes—
581D3—82no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—124no—
588C22—122no—
589C22—242no—
590C22—242no—
591C22#466512yes—
592C22#466512yes—
593C22C2×D3×C22⋊C411yescharacteristic
594C22C2×#88211yescharacteristic
595C22C2×#89411yescharacteristic
596C4—44no—
597C4—44no—
598C4—68no—
599C4—124no—
600C4—122no—
601C4#466512yes—
602C4#466512yes—
603C4C2×C4×D1211yescharacteristic
604C4C2×#88411yescharacteristic
605C4C2×#89311yescharacteristic
606C4C2×#89511yescharacteristic
607C3#2095411yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—22no—
611C2—22no—
612C2—242no—
613C2C2×#466511yescharacteristic
614C2#7899612yes—
subgroupG/H
centerC22C2×#882
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1733C2
socleC22×C3C2×#120
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#1733D3
hypocenterC3#20954
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
2. C2×#5516C2semidirect
3. C22×#824C2semidirect
6. C3×#1733C2semidirect
7. #71578C2semidirect
8. #68766C2semidirect
9. #68765C2semidirect
19. #5526C4semidirect
51. #1733D3semidirect
116. #875C2×C4semidirect
118. #869C2×C4semidirect
607. C3#20954semidirect

Maximal subgroups

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

subgroupindex
C2×#55262
C2×#55162
C22×#8242
#837682
#822132
C3×#17332
#715782
#687662
#687652
#209543