Finite Group Databasebeta

← #10828750

Subgroups — #10828750

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×#5507C211yesmaximal, characteristic
2C22×C3×#107C211yesmaximal, characteristic, complemented
3#82196C211yesmaximal, characteristic, complemented
4C2×#4967C211yesmaximal, characteristic, complemented
5#71975C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#71553C211yesmaximal, characteristic, complemented
8#70638C212yesmaximal, complemented
9#70634C212yesmaximal, complemented
10#70632C212yesmaximal, complemented
11#70617C212yesmaximal, complemented
12#24088—31nomaximal, Sylow
13C22×#235—21no—
14C23×C3×C8C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C2×#929—21no—
19C2×#929—21no—
20C2×C4×#67C2211yescharacteristic
21C2×#896—21no—
22#5507—24no—
23#5507—24no—
24C2×#895C2211yescharacteristic
25C2×#894—21no—
26C2×#893C2211yescharacteristic
27C2×D3×C4⋊C4C2211yescharacteristic
28C2×#888C2211yescharacteristic
29C2×C3×#107—24no—
30C2×C3×#107C2212yes—
31C2×C3×#107C2212yes—
32#5212C2211yescharacteristic
33#5211—21no—
34C2×#830C2211yescharacteristic
35C2×#821C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37#4992—21no—
38#4987C2212yes—
39#4983—21no—
40#4972C2212yes—
41#4967—24no—
42C2×#809C2212yes—
43C2×#807C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#547—31no—
49C2×#480—31no—
50C22×#107D311yescharacteristic, complemented
51#2486—31no—
52#2057—32no—
53#2033—32no—
54#1893—31no—
55#1869—31no—
56#1862—32no—
57#1852—32no—
58#1823—31no—
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67D411yescharacteristic
64C2×#235—28no—
65C2×#235—44no—
66C2×#235—44no—
67C22×D12—21no—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×C4×D3—21no—
71C22×C4×D3C2311yescharacteristic
72C22×Dic6—21no—
73C22×C3×C8—22no—
74C22×C3×C8—24no—
75C22×C3×C8C2312yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D414yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×#217—21no—
86C2×#217D411yescharacteristic
87#930—22no—
88#930D414yes—
89#929—28no—
90#929—44no—
91C4×#67—22no—
92C4×#67—22no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106#896—28no—
107#895—22no—
108#895—22no—
109#894—28no—
110#893—22no—
111#893—22no—
112D3×C4⋊C4—22no—
113D3×C4⋊C4—22no—
114#888—22no—
115#888—22no—
116C3×#107—24no—
117C3×#107—24no—
118C3⋊#95—21no—
119C3⋊#95—22no—
120C3⋊#95—22no—
121#830—22no—
122#821—24no—
123(C2×D3)⋊C8—24no—
124#809—24no—
125#807—24no—
126C22×#41—61no—
127C23×C8C2×D311yescharacteristic
128#547—64no—
129#547—64no—
130C2×C4⋊Q8—31no—
131C2×#123—61no—
132C2×#123—61no—
133C2×#123—61no—
134C2×C22⋊Q8—61no—
135C2×#121—31no—
136C2×#121—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C4⋊C4C2×D311yescharacteristic
142#480—64no—
143C2×#107—24no—
144C2×#107—32no—
145C2×#107C2×D312yes—
146C2×#107C2×D312yes—
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×C3C2×D411yescharacteristic
162C23×D3—21no—
163C23×D3C2×D411yescharacteristic
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—22no—
180C2×#67—22no—
181C2×#67—24no—
182C2×#67—42no—
183C2×#67—42no—
184C2×#67D814yes—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190#235—432no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—24no—
194C2×D12—21no—
195C2×D12—42no—
196C2×D12—42no—
197C2×D12C2×D412yes—
198C2×C4×D3—24no—
199C2×C4×D3—24no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×Dic6—24no—
209C2×Dic6—42no—
210C2×Dic6—42no—
211C2×C3×C8—28no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8#4114yescomplemented
215C3×C4⋊C4—24no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223#217—22no—
224#217—42no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—44no—
230C4⋊Dic3—21no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4⋊Dic3C2×D412yes—
234Dic3⋊C4—24no—
235Dic3⋊C4—24no—
236Dic3⋊C4—22no—
237Dic3⋊C4—44no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×#41—68no—
244C2×#41—124no—
245C2×#41—124no—
246C22×Q8—61no—
247C22×D4—31no—
248C22×D4—31no—
249C22×D4—61no—
250C22×D4—61no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—31no—
254C23×C4—61no—
255C23×C4—61no—
256C23×C4C22×D311yescharacteristic
257C22×C8—22no—
258C22×C8—24no—
259C22×C8—32no—
260C22×C8C22×D312yes—
261C4⋊Q8—62no—
262C4⋊Q8—62no—
263#123—68no—
264#123—68no—
265#123—124no—
266C22⋊Q8—68no—
267#121—34no—
268#121—62no—
269#121—62no—
270#121—62no—
271C4×D4—62no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—62no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4#6714yes—
289C2×C4⋊C4C22×D312yes—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—61no—
299C2×C42—31no—
300#107—24no—
301#107—24no—
302#107—64no—
303#102—64no—
304#102—64no—
305C22⋊C8—62no—
306C22⋊C8—64no—
307#95—62no—
308#95—62no—
309#95—61no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3#12012yes—
313C23×C3C2×D812yes—
314C23×C3AGL(1,ℤ/8)12yes—
315C23×C3C22×D411yescharacteristic
316C22×D3—21no—
317C22×D3—22no—
318C22×D3—21no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—42no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—42no—
325C22×D3C2×D812yes—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—84no—
341C2×C3×C4#12012yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4112yes—
345C2×C3×C42−1+412yes—
346#67—28no—
347#67—48no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3C2×D812yes—
361D12—24no—
362D12—48no—
363D12—41no—
364D12—41no—
365C4×D3—48no—
366C4×D3—48no—
367C4×D3—41no—
368C4×D3—41no—
369C4×D3—42no—
370Dic6—48no—
371C3×C8—24no—
372C3⋊C8—42no—
373C24—31no—
374C24—61no—
375C24D3×D411yescharacteristic
376#41—1232no—
377C2×Q8—64no—
378C2×Q8—122no—
379C2×Q8—122no—
380C2×D4—34no—
381C2×D4—32no—
382C2×D4—64no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—64no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4—122no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D412yes—
421C22×C4C2×#6712yes—
422C22×C4C23×D311yescharacteristic
423C2×C8—28no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8#23514yescomplemented
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—32no—
436C4⋊C4—42no—
437C4⋊C4—64no—
438C4⋊C4—64no—
439C4⋊C4—62no—
440C4⋊C4—61no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—124no—
444C4⋊C4C2×#6712yes—
445C4⋊C4C2×#6712yes—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—62no—
449C22⋊C4—62no—
450C22⋊C4—64no—
451C22⋊C4—124no—
452C22⋊C4—122no—
453C42—61no—
454C42—61no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3#44714yes—
460C22×C3C2×#12011yescharacteristic
461C22×C3#54712yes—
462C22×C3C22×D811yescharacteristic
463C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
464C22×C3#57812yes—
465C2×D3—24no—
466C2×D3—22no—
467C2×D3—42no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—82no—
474C2×D3—82no—
475C3×C4—22no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—82no—
479C3×C4#52511yescharacteristic
480C3×C4#53611yescharacteristic
481C3×C4#54712yes—
482Dic3—22no—
483Dic3—41no—
484Dic3—42no—
485Dic3—41no—
486Dic3—82no—
487C23—22no—
488C23—22no—
489C23—32no—
490C23—61no—
491C23—62no—
492C23—62no—
493C23—61no—
494C23—62no—
495C23—121no—
496C23—121no—
497C23—122no—
498C23—122no—
499C23D3×D812yes—
500C23#91112yes—
501C23#93712yes—
502C23C2×D3×D411yescharacteristic
503Q8—128no—
504D4—68no—
505D4—64no—
506D4—128no—
507D4—128no—
508D4—121no—
509D4—121no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—32no—
517C2×C4—48no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—62no—
532C2×C4—84no—
533C2×C4—128no—
534C2×C4—128no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—122no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4#93712yes—
544C2×C4C2×#23512yes—
545C2×C4C2×D3×D411yescharacteristic
546C2×C4#100712yes—
547C2×C4C22×#6711yescharacteristic
548C8—24no—
549C8—122no—
550C2×C3—24no—
551C2×C3#240611yescharacteristic
552C2×C3C2×#44711yescharacteristic
553C2×C3#301111yescharacteristic
554C2×C3#333412yes—
555C2×C3#334312yes—
556D3—42no—
557D3—82no—
558C22—22no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—64no—
563C22—62no—
564C22—121no—
565C22—122no—
566C22—121no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—242no—
571C22—242no—
572C22#515614yes—
573C22#550712yes—
574C22C2×D3×D811yescharacteristic
575C22C2×#91111yescharacteristic
576C22#576612yes—
577C22C2×#93711yescharacteristic
578C4—22no—
579C4—42no—
580C4—42no—
581C4—62no—
582C4—82no—
583C4—121no—
584C4—122no—
585C4—121no—
586C4—242no—
587C4#550712yes—
588C4#579111yescharacteristic
589C4#581611yescharacteristic
590C3#2408811yescharacteristic, complemented, Sylow
591C2—24no—
592C2—122no—
593C2—242no—
594C2#7915412yes—
595C2#7915812yes—
596C2#8219211yescharacteristic
597C2C2×#515611yescharacteristic
598C2#8388411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#107C2
socleC23×C3C22×D4
prime radicalC2×#5507C2
hypercenterC22×#107D3
hypocenterC3#24088
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C22×C3×#107C2semidirect
3. #82196C2semidirect
4. C2×#4967C2semidirect
5. #71975C2semidirect
6. #71973C2semidirect
7. #71553C2semidirect
8. #70638C2semidirect
9. #70634C2semidirect
10. #70632C2semidirect
11. #70617C2semidirect
50. C22×#107D3semidirect
214. C2×C3×C8#41semidirect
429. C2×C8#235semidirect
590. C3#24088semidirect

Maximal subgroups

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

subgroupindex
C2×#55072
C22×C3×#1072
#821962
C2×#49672
#719752
#719732
#715532
#706382
#706342
#706322
#706172
#240883