Finite Group Databasebeta

← #10848864

Subgroups — #10848864

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×#5534C211yesmaximal, characteristic, complemented
2#82286C211yesmaximal, characteristic, complemented
3C2×#4712C211yesmaximal, characteristic, complemented
4C2×#4694C211yesmaximal, characteristic, complemented
5C3×#1750C211yesmaximal, characteristic, complemented
6#69962C212yesmaximal, complemented
7#69961C212yesmaximal, complemented
8#69828C211yesmaximal, characteristic, complemented
9#69773C211yesmaximal, characteristic, complemented
10#68686C212yesmaximal, complemented
11#68673C212yesmaximal, complemented
12#25614—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×#930C2212yes—
16C2×#924C2211yescharacteristic
17C22×D24C2212yes—
18C22×#204C2212yes—
19#5534—24no—
20#5534—24no—
21C2×#898—21no—
22C2×#896—21no—
23C2×#885—21no—
24C2×#883—21no—
25C2×#875C2211yescharacteristic
26C2×#874C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C22⋊C8C2212yes—
31C2×C3×#97C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33C2×#821C2212yes—
34C2×#821C2212yes—
35C2×#821C2212yes—
36C2×#816C2212yes—
37#4991—21no—
38#4991—21no—
39#4960—21no—
40#4960—21no—
41#4939C2212yes—
42#4936C2211yescharacteristic
43#4712—24no—
44#4694—24no—
45C3×#336C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#542—31no—
49C2×#496—31no—
50C2×#486—31no—
51#2619—31no—
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1958—31no—
56#1880—32no—
57#1879—32no—
58#1750D311yescharacteristic, complemented
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C22×#67—22no—
62C22×D12—21no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6—21no—
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×#117—22no—
72C3×#117—22no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C22⋊C4D411yescharacteristic
76C2×C3×C22⋊C4D411yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×#217—21no—
80#930—22no—
81#930—22no—
82#930—42no—
83#930—42no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95#924—22no—
96#924—22no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×C4×Dic3—21no—
100C2×D24—24no—
101C2×D24—24no—
102C2×D24—42no—
103C2×#204—24no—
104C2×#204—24no—
105C2×#204—42no—
106#898—28no—
107#896—28no—
108#885—28no—
109#883—28no—
110#875—22no—
111#875D414yescomplemented
112#874—22no—
113#874—22no—
114C4×D12—22no—
115C4×D12—22no—
116#869—22no—
117#869D414yescomplemented
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×C22⋊C8—24no—
121C3×#97—22no—
122C3×C4×C8—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125#821—24no—
126#821—24no—
127#821—42no—
128#816—42no—
129C22×SD16—32no—
130C22×D8—32no—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8—31no—
134C2×#127—31no—
135C2×#126—61no—
136C2×#124—31no—
137C2×#124—61no—
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C2×#121—31no—
142C2×#121—32no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117C2×D311yescharacteristic
146#496—64no—
147#486—64no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×C22⋊C8C2×D312yes—
153C2×#97C2×D311yescharacteristic
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#336C2×D312yes—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C23×D3—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×#67—24no—
178C2×#67—44no—
179C2×#67—82no—
180C2×#67—82no—
181C2×#67—82no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—21no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—42no—
196C2×D12—42no—
197C2×D12—82no—
198C2×D12—82no—
199C2×D12C2×D412yes—
200C2×C4×D3—24no—
201C2×C4×D3—82no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—21no—
205C2×Dic6—24no—
206C2×Dic6—82no—
207C2×Dic6C2×D412yes—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8#4114yes—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222#217—24no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—44no—
228(C2×D3)⋊C4—82no—
229(C2×D3)⋊C4—82no—
230C4⋊Dic3—24no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235C4⋊Dic3C2×D412yes—
236C4⋊Dic3C2×D412yes—
237Dic3⋊C4—24no—
238Dic3⋊C4—24no—
239C4×Dic3—24no—
240D24—48no—
241#204—48no—
242C22×Q8—31no—
243C22×Q8—61no—
244C22×D4—31no—
245C22×D4—32no—
246C22×D4—61no—
247C22×D4—62no—
248C22×D4—61no—
249C22×D4—61no—
250C23×C4—61no—
251C23×C4C22×D311yescharacteristic
252C2×SD16—64no—
253C2×SD16—64no—
254C2×SD16—122no—
255C2×D8—64no—
256C2×D8—64no—
257C2×D8—122no—
258C22×C8C22×D312yes—
259C22×C8C22×D312yes—
260C4⋊Q8—34no—
261C4⋊Q8—62no—
262#127—62no—
263#127—62no—
264#126—68no—
265#124—34no—
266#124—68no—
267#124—62no—
268#123—68no—
269C22⋊Q8—68no—
270C22⋊Q8—62no—
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—22no—
281#117—22no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—32no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4D1211yescharacteristic
289C2×C4⋊C4D1211yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4D1211yescharacteristic
299C2×C22⋊C4D1211yescharacteristic
300C2×C42—61no—
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#103—122no—
304#102—64no—
305#102—64no—
306#102—122no—
307C22⋊C8—24no—
308#97—22no—
309C4×C8—22no—
310#95—61no—
311#95—61no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3C22×D411yescharacteristic
318C22×D3—22no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4#12012yes—
342C2×C3×C4#12012yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4112yes—
346C2×C3×C4H(5,𝔽2)12yes—
347#67—84no—
348#67—88no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359D12—24no—
360D12—42no—
361D12—42no—
362D12—42no—
363D12—42no—
364D12—42no—
365D12—84no—
366D12—84no—
367C4×D3—84no—
368Dic6—24no—
369Dic6—42no—
370Dic6—84no—
371C3×C8—24no—
372C3×C8—42no—
373C24—61no—
374C24—62no—
375C24C2×D1211yescharacteristic
376C2×Q8—32no—
377C2×Q8—64no—
378C2×Q8—61no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—242no—
382C2×D4—32no—
383C2×D4—64no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—62no—
393C2×D4—124no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4—242no—
397C2×D4—242no—
398C2×D4—242no—
399C2×D4—242no—
400C2×D4—242no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—64no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—242no—
411C22×C4C2×D1211yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4C2×D1211yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C2×D1211yescharacteristic
416C22×C4C23×D311yescharacteristic
417SD16—128no—
418D8—128no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8#23514yes—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—32no—
428C4⋊C4—32no—
429C4⋊C4—64no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—64no—
439C22⋊C4—64no—
440C22⋊C4—64no—
441C22⋊C4—64no—
442C22⋊C4—64no—
443C22⋊C4—124no—
444C22⋊C4—242no—
445C22⋊C4—242no—
446C42—21no—
447C42—21no—
448C42—64no—
449C42D3×D412yes—
450C42D3×D412yes—
451C22×C3—42no—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#53211yescharacteristic
458C22×C3#53511yescharacteristic
459C22×C3#57612yes—
460C22×C3#57712yes—
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—42no—
464C2×D3—82no—
465C2×D3—82no—
466C2×D3—82no—
467C2×D3—82no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4#54612yes—
479C3×C4#55012yes—
480Dic3—42no—
481Dic3—42no—
482Dic3—42no—
483Dic3—82no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—21no—
489C23—62no—
490C23—121no—
491C23—122no—
492C23—122no—
493C23—121no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23C22×D1211yescharacteristic
502Q8—64no—
503Q8—122no—
504Q8—244no—
505D4—64no—
506D4—122no—
507D4—122no—
508D4—122no—
509D4—122no—
510D4—122no—
511D4—244no—
512D4—244no—
513D4—244no—
514D4—248no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—244no—
537C2×C4#88212yes—
538C2×C4#88212yes—
539C2×C4C2×#23512yes—
540C2×C4C2×D3×D411yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4#100912yes—
543C8—24no—
544C8—42no—
545C2×C3—81no—
546C2×C3#243111yescharacteristic
547C2×C3#300811yescharacteristic
548C2×C3#301011yescharacteristic
549C2×C3#317912yes—
550C2×C3#322412yes—
551D3—82no—
552D3—162no—
553C22—42no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—122no—
559C22—122no—
560C22—122no—
561C22—242no—
562C22—242no—
563C22—242no—
564C22—242no—
565C22—242no—
566C22—242no—
567C22—242no—
568C22—242no—
569C22—242no—
570C22#547711yescharacteristic
571C22C2×#88211yescharacteristic
572C22#552511yescharacteristic
573C22#575112yes—
574C22#575212yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—122no—
581C4—122no—
582C4—122no—
583C4—242no—
584C4#555012yes—
585C4#555312yes—
586C3#2561411yescharacteristic, complemented, Sylow
587C2—81no—
588C2—242no—
589C2—482no—
590C2#7358611yescharacteristic
591C2#7857812yes—
592C2#7867011yescharacteristic
593C2#7867811yescharacteristic
594C2#7939212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1750C2
socleC23×C3C22×D4
hypercenter#1750D3
hypocenterC3#25614
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
1. C2×#5534C2semidirect
2. #82286C2semidirect
3. C2×#4712C2semidirect
4. C2×#4694C2semidirect
5. C3×#1750C2semidirect
6. #69962C2semidirect
7. #69961C2semidirect
8. #69828C2semidirect
9. #69773C2semidirect
10. #68686C2semidirect
11. #68673C2semidirect
58. #1750D3semidirect
111. #875D4semidirect
117. #869D4semidirect
586. C3#25614semidirect

Maximal subgroups

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

subgroupindex
C2×#55342
#822862
C2×#47122
C2×#46942
C3×#17502
#699622
#699612
#698282
#697732
#686862
#686732
#256143