Finite Group Databasebeta

← #10828746

Subgroups — #10828746

Order 768.

1354 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×#5033C211yesmaximal, characteristic, complemented
3#82373C211yesmaximal, characteristic, complemented
4C22×#807C211yesmaximal, characteristic, complemented
5#71782C212yesmaximal, complemented
6#71681C211yesmaximal, characteristic, complemented
7#71679C212yesmaximal, complemented
8#71610C211yesmaximal, characteristic, complemented
9#71606C212yesmaximal, complemented
10#71555C212yesmaximal, complemented
11#71553C211yesmaximal, characteristic, complemented
12#24088—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#547—24no—
15C3×#547—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#123—21no—
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C2×#935—21no—
27C22×C4⋊Dic3C2211yescharacteristic
28C23×C3⋊C8C2211yescharacteristic
29C2×#832C2212yes—
30C2×#832C2212yes—
31#5212C2211yescharacteristic
32#5212C2212yes—
33C2×#830C2211yescharacteristic
34C2×#830C2212yes—
35#5033—24no—
36#4981—21no—
37C2×#807—24no—
38C2×#807C2212yes—
39C2×#807C2212yes—
40C2×#807C2212yes—
41#4953—21no—
42#4943—21no—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#547D311yescharacteristic
49C2×#480—31no—
50C22×#107—31no—
51#2486—31no—
52#2057—32no—
53#2033—32no—
54#1893—31no—
55#1869—31no—
56#1862—32no—
57#1852—32no—
58#1823—31no—
59C2×C3×#41—28no—
60C2×C3×#41—44no—
61C2×C3×#41—44no—
62C22×C3×Q8—21no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4D411yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×Dic3—21no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×#123—28no—
76C3×#123—28no—
77C3×C22⋊Q8—28no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121D414yes—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D414yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104C2×#217—22no—
105#935—44no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—22no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2312yes—
110C2×C4⋊Dic3C2312yes—
111C2×Dic3⋊C4—22no—
112C22×C3⋊C8—22no—
113C22×C3⋊C8—24no—
114C22×C3⋊C8C2312yes—
115C22×C3⋊C8C2312yes—
116#832—24no—
117#832—24no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120C3⋊#95—21no—
121#830—22no—
122#830—24no—
123#807—24no—
124#807—24no—
125#807—24no—
126C22×#41—21no—
127C23×C8—31no—
128#547—24no—
129#547—24no—
130C2×C4⋊Q8C2×D311yescharacteristic
131C2×#123—21no—
132C2×#123—21no—
133C2×#123—61no—
134C2×C22⋊Q8—21no—
135C2×#121C2×D311yescharacteristic
136C2×#121C2×D311yescharacteristic
137C2×C4×D4C2×D311yescharacteristic
138C2×C4×D4C2×D311yescharacteristic
139C22×C4⋊C4—31no—
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C4⋊C4C2×D311yescharacteristic
142#480—64no—
143C2×#107—32no—
144C2×#107—32no—
145C2×#107—32no—
146C2×#107—64no—
147C2×#102—32no—
148C2×#102—32no—
149C2×C22⋊C8—31no—
150C2×C22⋊C8—32no—
151#398—61no—
152#397—61no—
153#389—32no—
154#381—61no—
155#380—31no—
156#380—32no—
157#340—32no—
158#340—32no—
159#340—32no—
160#336—32no—
161C24×C3—21no—
162C24×C3C2×D411yescharacteristic
163C24×C3C2×D411yescharacteristic
164C3×#41—432no—
165C2×C3×Q8—24no—
166C2×C3×Q8—42no—
167C2×C3×Q8—42no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4D814yes—
180C2×C3×D4C2×D412yes—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—42no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4C2×D412yes—
219C3×C4⋊C4C2×D412yes—
220C3×C4⋊C4C2×D412yes—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—22no—
226C3×C42—21no—
227C3×C42—21no—
228#217—42no—
229#217—44no—
230C4⋊Dic3—22no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3—22no—
234C4⋊Dic3—42no—
235Dic3⋊C4—44no—
236C2×C3⋊C8—28no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8#4114yescomplemented
243C2×#41—28no—
244C2×#41—44no—
245C2×#41—44no—
246C22×Q8—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4#6711yescharacteristic
250C22×D4C22×D311yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—61no—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C22×C8—32no—
258C22×C8—32no—
259C22×C8—64no—
260C22×C8—62no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263#123—28no—
264#123—28no—
265#123—124no—
266C22⋊Q8—28no—
267#121—22no—
268#121—22no—
269#121—22no—
270#121#6714yes—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4#6711yescharacteristic
288C2×C4⋊C4#6714yes—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4#6711yescharacteristic
298C2×C22⋊C4#6711yescharacteristic
299C2×C42C22×D311yescharacteristic
300#107—64no—
301#107—64no—
302#107—64no—
303#102—64no—
304#102—64no—
305C22⋊C8—64no—
306C22⋊C8—62no—
307#95—62no—
308#95—61no—
309#95—62no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—42no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3#12012yes—
322C23×C3C2×D812yes—
323C23×C3C2×D812yes—
324C23×C3AGL(1,ℤ/8)12yes—
325C23×C3C22×D411yescharacteristic
326C3×Q8—48no—
327C3×D4—28no—
328C3×D4—24no—
329C3×D4—48no—
330C3×D4—48no—
331C3×D4—41no—
332C3×D4—41no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—48no—
347C2×C3×C4—48no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—48no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4#12012yes—
359C2×C3×C4C2×D812yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4112yes—
363C2×C3×C42−1+412yes—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—84no—
371C3⋊C8—24no—
372C3⋊C8—42no—
373C24—21no—
374C24C2×#6711yescharacteristic
375C24C2×#6711yescharacteristic
376#41—432no—
377C2×Q8—24no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4#21314yes—
392C2×D4C2×#6712yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6712yes—
421C22×C4C2×#6712yes—
422C22×C4C23×D311yescharacteristic
423C2×C8—34no—
424C2×C8—68no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—62no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—124no—
442C4⋊C4—122no—
443C4⋊C4C2×#6712yes—
444C4⋊C4C2×#6712yes—
445C4⋊C4C2×#6712yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—122no—
452C22⋊C4—124no—
453C42—21no—
454C42—21no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—24no—
460C22×C3—22no—
461C22×C3—42no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3#44714yes—
470C22×C3C2×#12011yescharacteristic
471C22×C3#54712yes—
472C22×C3C22×D811yescharacteristic
473C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
474C22×C3#57812yes—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—41no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—41no—
482C3×C4#52511yescharacteristic
483C3×C4#53611yescharacteristic
484C3×C4#54712yes—
485Dic3—82no—
486Dic3—82no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—21no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23C2×#21312yes—
499C23C2×#21312yes—
500C23#93212yes—
501C23#95312yes—
502C23C22×#6711yescharacteristic
503Q8—48no—
504D4—28no—
505D4—24no—
506D4—48no—
507D4—48no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—244no—
542C2×C4C2×#21312yes—
543C2×C4#95312yes—
544C2×C4D3×#4112yes—
545C2×C4#101012yes—
546C2×C4C22×#6711yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—64no—
549C8—122no—
550C2×C3—24no—
551C2×C3—42no—
552C2×C3—82no—
553C2×C3#240611yescharacteristic
554C2×C3C2×#44711yescharacteristic
555C2×C3#301111yescharacteristic
556C2×C3#333412yes—
557C2×C3#334312yes—
558C22—24no—
559C22—24no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—22no—
564C22—42no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—82no—
571C22—82no—
572C22#521714yes—
573C22C22×#21311yescharacteristic
574C22C2×#93211yescharacteristic
575C22#582412yes—
576C22#583812yes—
577C22C2×#95311yescharacteristic
578C4—22no—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—41no—
585C4—242no—
586C4—242no—
587C4#582412yes—
588C4#582511yescharacteristic
589C4#582811yescharacteristic
590C3#2408811yescharacteristic, complemented, Sylow
591C2—24no—
592C2—42no—
593C2—82no—
594C2#8351612yes—
595C2#8351812yes—
596C2C2×#521711yescharacteristic
597C2#8409011yescharacteristic
598C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3#24088
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#5033C2semidirect
3. #82373C2semidirect
4. C22×#807C2semidirect
5. #71782C2semidirect
6. #71681C2semidirect
7. #71679C2semidirect
8. #71610C2semidirect
9. #71606C2semidirect
10. #71555C2semidirect
11. #71553C2semidirect
242. C2×C3⋊C8#41semidirect
590. C3#24088semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×#50332
#823732
C22×#8072
#717822
#716812
#716792
#716102
#716062
#715552
#715532
#240883