Finite Group Databasebeta

← #10848869

Subgroups — #10848869

Order 768.

1216 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5733C211yesmaximal, characteristic, complemented
2C2×C3×#486C211yesmaximal, characteristic, complemented
3#83189C211yesmaximal, characteristic, complemented
4C2×#5074C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70213C212yesmaximal, complemented
7#70208C212yesmaximal, complemented
8#70205C212yesmaximal, complemented
9#70204C212yesmaximal, complemented
10#69773C211yesmaximal, characteristic, complemented
11#69753C211yesmaximal, characteristic, complemented
12#25614—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×#947—21no—
16C2×#946—21no—
17C2×#938C2212yes—
18C22×#214C2212yes—
19C22×#213C2212yes—
20#5733—24no—
21#5733—24no—
22C2×#898—21no—
23C2×#897C2211yescharacteristic
24C2×#895C2211yescharacteristic
25C2×#894—21no—
26C2×#893C2211yescharacteristic
27C2×#887C2211yescharacteristic
28C2×#875C2211yescharacteristic
29C2×#872C2211yescharacteristic
30C3×#486—24no—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C4×C8C2211yescharacteristic
34#5223—21no—
35#5223—21no—
36#5222—21no—
37#5222—21no—
38#5221C2212yes—
39C2×#821C2212yes—
40C2×(C2×D3)⋊C8C2212yes—
41C2×#816C2212yes—
42#5074—24no—
43#4936C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#542—31no—
49C2×#496—31no—
50C2×#486D311yescharacteristic, complemented
51#2619—31no—
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1958—31no—
56#1880—32no—
57#1879—32no—
58#1750—31no—
59C22×C3×Q8—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2312yes—
63C22×#67—22no—
64C22×D12—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yescomplemented
72C3×#127—22no—
73C3×#127—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81#947—28no—
82#946—28no—
83#938—42no—
84#938—42no—
85C2×#214—24no—
86C2×#214—24no—
87C2×#214—42no—
88C2×#213—24no—
89C2×#213—24no—
90C2×#213—42no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2312yes—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4D411yescharacteristic
102C2×Dic3⋊C4D411yescharacteristic
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106#898—28no—
107#897—22no—
108#897—22no—
109#895—22no—
110#895—22no—
111#894—28no—
112#893—22no—
113#893—22no—
114#887—22no—
115#887—22no—
116#875—22no—
117#875D414yescomplemented
118#872—22no—
119#872—22no—
120C3×#102—24no—
121C3×#102—24no—
122C3×C4×C8—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125#821—42no—
126(C2×D3)⋊C8—24no—
127#816—42no—
128#803—22no—
129C22×SD16—32no—
130C22×D8—32no—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8C2×D311yescharacteristic
134C2×#127C2×D311yescharacteristic
135C2×#126—61no—
136C2×#124—31no—
137C2×#124—61no—
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C2×#121—32no—
142C2×#121—31no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—31no—
146#496—64no—
147#486—24no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102C2×D312yes—
151C2×#102C2×D312yes—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×C4×C8C2×D311yescharacteristic
155#394—61no—
156#394—61no—
157#392—61no—
158#392—61no—
159#390—32no—
160#382—31no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—22no—
165C23×D3—21no—
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—24no—
168C2×C3×Q8—82no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—82no—
178C2×C3×D4—82no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C2×#67—44no—
185C2×#67—82no—
186C2×#67—82no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—21no—
195C2×D12—24no—
196C2×D12—82no—
197C2×D12C2×D412yes—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—24no—
203C2×C4×D3—82no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—21no—
207C2×Dic6C2×D412yes—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4C2×D412yes—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218#217—44no—
219#217—82no—
220#217—82no—
221#214—48no—
222#213—48no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—24no—
230C4⋊Dic3—24no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233Dic3⋊C4—24no—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236C4×Dic3—24no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8#4114yes—
242C22×Q8—21no—
243C22×Q8—31no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—31no—
247C22×D4—61no—
248C22×D4—62no—
249C22×D4C22×D312yes—
250C23×C4—31no—
251C23×C4—61no—
252C2×SD16—64no—
253C2×SD16—64no—
254C2×SD16—122no—
255C2×D8—64no—
256C2×D8—64no—
257C2×D8—122no—
258C22×C8—32no—
259C22×C8C22×D312yes—
260C4⋊Q8—22no—
261C4⋊Q8#6714yescomplemented
262#127—22no—
263#127—22no—
264#126—68no—
265#124—34no—
266#124—68no—
267#124—62no—
268#123—68no—
269C22⋊Q8—62no—
270C22⋊Q8—68no—
271C22⋊Q8—62no—
272#121—62no—
273#121—62no—
274#121—122no—
275#121—122no—
276C4×Q8—62no—
277C4×Q8—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—62no—
281#117—62no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#103—122no—
304#102—24no—
305#102—24no—
306#102—122no—
307C22⋊C8—64no—
308#97—62no—
309C4×C8—22no—
310#95—61no—
311#95—61no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3C22×D411yescharacteristic
320C22×D3—21no—
321C22×D3—22no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—42no—
329C22×D3—42no—
330C3×Q8—84no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—84no—
336C3×D4—84no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4#12012yes—
344C2×C3×C4#12012yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348C2×C3×C4H(5,𝔽2)12yes—
349#67—88no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363D12—24no—
364D12—42no—
365D12—84no—
366C4×D3—42no—
367C4×D3—42no—
368C4×D3—84no—
369Dic6—24no—
370Dic6—42no—
371C3×C8—42no—
372C3⋊C8—24no—
373C24—22no—
374C24—31no—
375C24—61no—
376C2×Q8—24no—
377C2×Q8—32no—
378C2×Q8—61no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—82no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—32no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—64no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—61no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—124no—
398C2×D4—242no—
399C2×D4—242no—
400C2×D4—242no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—64no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—242no—
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4C23×D311yescharacteristic
417SD16—128no—
418D8—128no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—34no—
423C2×C8—62no—
424C2×C8—62no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—64no—
429C4⋊C4—64no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4C2×#6712yes—
435C4⋊C4C2×#6712yes—
436C22⋊C4—64no—
437C22⋊C4—64no—
438C22⋊C4—64no—
439C22⋊C4—62no—
440C22⋊C4—62no—
441C22⋊C4—64no—
442C22⋊C4—64no—
443C22⋊C4—124no—
444C22⋊C4—242no—
445C22⋊C4—242no—
446C42—64no—
447C42—61no—
448C42—61no—
449C42D3×D412yes—
450C42C2×#6712yes—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3C2×#12011yescharacteristic
459C22×C3#53211yescharacteristic
460C22×C3#53511yescharacteristic
461C22×C3#57612yes—
462C22×C3#57712yes—
463C2×D3—42no—
464C2×D3—41no—
465C2×D3—42no—
466C2×D3—42no—
467C2×D3—42no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—42no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4#54612yes—
478C3×C4#55012yes—
479Dic3—42no—
480Dic3—41no—
481Dic3—41no—
482Dic3—41no—
483Dic3—82no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—61no—
492C23—61no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—61no—
497C23—121no—
498C23—121no—
499C23—122no—
500C23—122no—
501C23C2×D3×D411yescharacteristic
502Q8—64no—
503Q8—84no—
504Q8—122no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—64no—
510D4—84no—
511D4—84no—
512D4—122no—
513D4—244no—
514D4—248no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—61no—
526C2×C4—62no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—244no—
537C2×C4#93712yes—
538C2×C4#93712yes—
539C2×C4C2×D3×D411yescharacteristic
540C2×C4D3×#4112yes—
541C2×C4#100912yes—
542C2×C4C22×#6711yescharacteristic
543C8—42no—
544C8—64no—
545C2×C3—162no—
546C2×C3#243111yescharacteristic
547C2×C3#300811yescharacteristic
548C2×C3#301011yescharacteristic
549C2×C3#317912yes—
550C2×C3#322412yes—
551D3—81no—
552D3—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—121no—
561C22—122no—
562C22—122no—
563C22—121no—
564C22—121no—
565C22—122no—
566C22—122no—
567C22—122no—
568C22—242no—
569C22—242no—
570C22#566811yescharacteristic
571C22#567411yescharacteristic
572C22#575712yes—
573C22#576212yes—
574C22C2×#93711yescharacteristic
575C4—22no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—121no—
580C4—122no—
581C4—121no—
582C4—121no—
583C4—242no—
584C4#567612yes—
585C4#583412yes—
586C3#2561411yescharacteristic, complemented, Sylow
587C2—162no—
588C2—241no—
589C2—242no—
590C2#8155812yes—
591C2#8157012yes—
592C2#8295811yescharacteristic
593C2#8383311yescharacteristic
594C2#8386011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#486C2
socleC23×C3C22×D4
hypercenterC2×#486D3
hypocenterC3#25614
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. C2×#5733C2semidirect
2. C2×C3×#486C2semidirect
3. #83189C2semidirect
4. C2×#5074C2semidirect
5. #71788C2semidirect
6. #70213C2semidirect
7. #70208C2semidirect
8. #70205C2semidirect
9. #70204C2semidirect
10. #69773C2semidirect
11. #69753C2semidirect
50. C2×#486D3semidirect
71. C3×C4⋊Q8D4semidirect
117. #875D4semidirect
261. C4⋊Q8#67semidirect
586. C3#25614semidirect

Maximal subgroups

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

subgroupindex
C2×#57332
C2×C3×#4862
#831892
C2×#50742
#717882
#702132
#702082
#702052
#702042
#697732
#697532
#256143