Finite Group Databasebeta

← #10822564

Subgroups — #10822564

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#544C211yesmaximal, characteristic, complemented
2#78956C212yesmaximal
3#76482C211yesmaximal, characteristic
4C3×#2028C211yesmaximal, characteristic, complemented
5#71903C211yesmaximal, characteristic, complemented
6#70333C212yesmaximal, complemented
7#69673C211yesmaximal, characteristic, complemented
8#68995C211yesmaximal, characteristic, complemented
9D3×#343C211yesmaximal, characteristic, complemented
10#68763C212yesmaximal, complemented
11#68753C212yesmaximal, complemented
12#23462—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C3×#544C2211yescharacteristic
15C2×D3×M16C2212yes—
16#5729—22no—
17D3×C4⋊Q8—22no—
18#5689—22no—
19D3×#125—22no—
20D3×C22⋊Q8—22no—
21#5623—22no—
22D3×#117C2212yes—
23#5519C2212yes—
24C2×D3×C4⋊C4C2211yescharacteristic
25#5500C2211yescharacteristic
26C2×#872C2211yescharacteristic
27C3×#419C2212yes—
28C3×#391C2211yescharacteristic
29#5229C2211yescharacteristic
30#5225C2212yes—
31#5131C2212yes—
32#5001C2212yes—
33#4988C2211yescharacteristic
34#4939C2211yescharacteristic
35#4804—22no—
36#4804—22no—
37#4803—22no—
38#4803—22no—
39#4667—22no—
40#4665—22no—
41C3×#343C2211yescharacteristic
42C3×#339C2212yes—
43#4555C2212yes—
44#4554C2212yes—
45#4532C2212yes—
46#4530C2211yescharacteristic
47#4523C2212yes—
48C2×#544—31no—
49#2896—32no—
50#2311—31no—
51#2028—32no—
52#2028—31no—
53#2028D311yescharacteristic, complemented
54#2027—31no—
55#2027—32no—
56#2027—31no—
57C2×#343—31no—
58#1733—32no—
59C22×C3×Q8C2311yescharacteristic
60C2×#239—22no—
61C2×#239—22no—
62C2×#239—22no—
63C2×#239—24no—
64C2×#239Q814yescomplemented
65C2×D3×Q8—22no—
66C22×D12—21no—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×C4×D3D412yes—
70C22×C4×D3C2311yescharacteristic
71C2×C3×M16C2312yes—
72C3×C4⋊Q8—22no—
73C3×#125—22no—
74C3×C22⋊Q8—22no—
75C3×#117C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78#946—22no—
79#944—24no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83#926C2312yes—
84C2×C4⋊Dic3C2311yescharacteristic
85#924—22no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4D412yes—
88C2×C4×Dic3C2311yescharacteristic
89C2×#208C2312yes—
90D3×M16—24no—
91D3×M16—24no—
92C2×ΓL(1,𝔽25)—22no—
93C2×D3×C8—22no—
94#897—22no—
95#896—24no—
96#892D412yes—
97#892D412yes—
98D3×C4⋊C4—22no—
99D3×C4⋊C4—24no—
100D3×C4⋊C4—22no—
101D3×C4⋊C4—22no—
102D3×C4⋊C4D412yes—
103D3×C4⋊C4D412yes—
104#890—22no—
105#890—22no—
106#889—24no—
107#889—24no—
108#888—24no—
109#888—22no—
110#888—22no—
111D3×C22⋊C4—22no—
112#878—24no—
113#877—22no—
114#872—22no—
115#872—24no—
116C42×D3—22no—
117#870—22no—
118#869—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#95—22no—
122#835—22no—
123#835—22no—
124C3⋊#95—22no—
125#821—22no—
126#821—22no—
127#809—22no—
128#809—22no—
129C22×#41—31no—
130C22×M16—32no—
131#544—32no—
132#544—31no—
133#544—62no—
134#544—62no—
135#544—62no—
136#544C2×D311yescharacteristic
137C2×C4⋊Q8—62no—
138C2×#125—62no—
139C2×C22⋊Q8—62no—
140C2×#117—31no—
141C2×#117—32no—
142C22×C4⋊C4—31no—
143#419—32no—
144#419C2×D312yes—
145#418—32no—
146#418—32no—
147#417—62no—
148#417—62no—
149#416—62no—
150#416—62no—
151#391—31no—
152#391C2×D311yescharacteristic
153#390—31no—
154#390—31no—
155#386—62no—
156#380—62no—
157#343—32no—
158#343—31no—
159#343C2×D311yescharacteristic
160#339—32no—
161#339—32no—
162#339—32no—
163#339C2×D312yes—
164C23×D3—21no—
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8C2×Q812yescomplemented
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—21no—
176C22×Dic3C2×D412yes—
177C22×Dic3C2×D411yescharacteristic
178#239—28no—
179#239—28no—
180#239—44no—
181#239—44no—
182D3×Q8—84no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—24no—
186C2×D12—21no—
187C2×D12—41no—
188C2×D12—41no—
189C2×D12—42no—
190C2×D12C2×Q812yescomplemented
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—24no—
195C2×C4×D3—22no—
196C2×C4×D3—42no—
197C2×C4×D3—42no—
198C2×C4×D3—82no—
199C2×C4×D3C2×D412yes—
200C2×C4×D3C2×D412yes—
201C2×C4×D3C2×Q812yes—
202C2×C4×D3#4114yes—
203C2×C4×D3#4112yes—
204C2×Dic6—24no—
205C2×Dic6—22no—
206C3×M16—22no—
207C3×M16—22no—
208C2×C3×C8—22no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—22no—
215C3×C42—22no—
216C3×C42—22no—
217#217—22no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—42no—
221(C2×D3)⋊C4—82no—
222(C2×D3)⋊C4—82no—
223C4⋊Dic3—24no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3C2×D412yes—
227C4⋊Dic3C2×D412yes—
228Dic3⋊C4—24no—
229Dic3⋊C4—24no—
230Dic3⋊C4—24no—
231Dic3⋊C4—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—24no—
234Dic3⋊C4—42no—
235C4×Dic3—22no—
236C4×Dic3—22no—
237C4×Dic3—22no—
238C4×Dic3—22no—
239#208—22no—
240#208—22no—
241C2×C3⋊C8—22no—
242ΓL(1,𝔽25)—44no—
243D3×C8—44no—
244C2×#41—34no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—64no—
249C22×Q8—62no—
250C22×Q8C22×D311yescharacteristic
251C22×D4—31no—
252C22×D4—61no—
253C23×C4—31no—
254C23×C4—32no—
255C23×C4—61no—
256C2×M16—32no—
257C2×M16—62no—
258C2×M16—64no—
259C2×M16—64no—
260C2×M16C22×D312yes—
261C22×C8—62no—
262C4⋊Q8—22no—
263C4⋊Q8—62no—
264C4⋊Q8—62no—
265C4⋊Q8—64no—
266C4⋊Q8—64no—
267C4⋊Q8—62no—
268#125—22no—
269#125—62no—
270#125—62no—
271#125—64no—
272#125—64no—
273#125—62no—
274C22⋊Q8—22no—
275C22⋊Q8—64no—
276C22⋊Q8—64no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—62no—
280#117—32no—
281#117—32no—
282#117—32no—
283#117—62no—
284#117—64no—
285#117—62no—
286#117C22×D312yes—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—62no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—62no—
301C2×C42—31no—
302C2×C42—62no—
303C2×C42C22×D311yescharacteristic
304#103—22no—
305#103—22no—
306#103—62no—
307#103—62no—
308#102—62no—
309#102—62no—
310#102—62no—
311#102—62no—
312#95—22no—
313#95—62no—
314C23×C3C22×D411yescharacteristic
315C22×D3—22no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3#12011yescharacteristic
323C22×D3#12011yescharacteristic
324C22×D3C22⋊Q812yes—
325C22×D3#12412yes—
326C3×Q8—24no—
327C3×Q8—42no—
328C3×Q8—82no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4C22×D412yes—
340C2×C3×C4C22×Q811yescharacteristic
341C2×C3×C4C2×#4112yes—
342C2×C3×C4C2×#4111yescharacteristic
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3#12012yes—
351C2×Dic3C22⋊Q812yes—
352C2×Dic3#12412yes—
353D12—24no—
354D12—44no—
355D12—44no—
356D12—42no—
357C4×D3—44no—
358C4×D3—44no—
359C4×D3—44no—
360C4×D3—48no—
361C4×D3—82no—
362C4×D3C22⋊Q814yes—
363C4×D3C22⋊Q814yes—
364C4×D3#12414yes—
365C4×D3#12414yes—
366Dic6—82no—
367Dic6—84no—
368C3×C8—42no—
369C3⋊C8—42no—
370C24—31no—
371C24—61no—
372#41—68no—
373#41—68no—
374#41—124no—
375#41—124no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—64no—
381C2×Q8—62no—
382C2×Q8—244no—
383C2×Q8D3×Q812yescomplemented
384C2×D4—32no—
385C2×D4—64no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4—122no—
390C2×D4—121no—
391C2×D4—121no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—31no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—32no—
398C22×C4—34no—
399C22×C4—32no—
400C22×C4—32no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—242no—
410C22×C4D3×D412yes—
411C22×C4C23×D311yescharacteristic
412M16—22no—
413M16—22no—
414M16—62no—
415M16—62no—
416M16—124no—
417C2×C8—22no—
418C2×C8—62no—
419C2×C8—124no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—32no—
424C4⋊C4—32no—
425C4⋊C4—64no—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—64no—
429C4⋊C4—62no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—62no—
434C4⋊C4—122no—
435C4⋊C4D3×D412yes—
436C4⋊C4D3×D412yes—
437C22⋊C4—22no—
438C22⋊C4—62no—
439C22⋊C4—62no—
440C22⋊C4—62no—
441C22⋊C4—122no—
442C22⋊C4—242no—
443C22⋊C4—242no—
444C42—22no—
445C42—22no—
446C42—62no—
447C42—62no—
448C42—62no—
449C42—62no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3C2×#12011yescharacteristic
453C22×C3C2×C22⋊Q811yescharacteristic
454C22×C3C2×#12411yescharacteristic
455C2×D3—21no—
456C2×D3—21no—
457C2×D3—21no—
458C2×D3—21no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—44no—
462C2×D3—42no—
463C2×D3—82no—
464C2×D3—81no—
465C2×D3—81no—
466C2×D3#39312yes—
467C2×D3#39312yes—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—81no—
473C3×C4C2×C22⋊Q812yes—
474C3×C4C2×#12412yes—
475Dic3—42no—
476Dic3—42no—
477Dic3—42no—
478Dic3—81no—
479Dic3#39314yes—
480C23—31no—
481C23—31no—
482C23—32no—
483C23—32no—
484C23—62no—
485C23—121no—
486C23—121no—
487C23—122no—
488C23—122no—
489C23—121no—
490C23—121no—
491C23C2×D3×D411yescharacteristic
492Q8—24no—
493Q8—42no—
494Q8—82no—
495Q8—242no—
496Q8—244no—
497D4—64no—
498D4—124no—
499D4—124no—
500D4—122no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—34no—
509C2×C4—34no—
510C2×C4—34no—
511C2×C4—34no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—128no—
522C2×C4—124no—
523C2×C4—124no—
524C2×C4—124no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—242no—
530C2×C4C2×D3×D412yes—
531C2×C4C2×D3×Q811yescharacteristic
532C2×C4D3×#4111yescharacteristic
533C2×C4D3×#4112yes—
534C8—42no—
535C8—122no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3C2×#39311yescharacteristic
539C2×C3#301911yescharacteristic
540C2×C3#302011yescharacteristic
541D3—22no—
542D3—22no—
543D3—82no—
544C22—21no—
545C22—21no—
546C22—32no—
547C22—32no—
548C22—61no—
549C22—61no—
550C22—61no—
551C22—61no—
552C22—122no—
553C22—122no—
554C22—124no—
555C22—122no—
556C22—242no—
557C22—241no—
558C22—241no—
559C22D3×#12011yescharacteristic
560C22D3×C22⋊Q811yescharacteristic
561C22D3×#12411yescharacteristic
562C4—34no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—122no—
569C4—122no—
570C4—122no—
571C4—241no—
572C4D3×C22⋊Q812yes—
573C4D3×#12412yes—
574C3#2346211yescharacteristic, complemented, Sylow
575C2—21no—
576C2—21no—
577C2—62no—
578C2—62no—
579C2—242no—
580C2D3×#39311yescharacteristic
581C2#8029211yescharacteristic
582C2#8029911yescharacteristic
subgroupG/H
centerC22D3×#120
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2028C2
socleC22×C3C2×#120
prime radicalC22×#239C22
hypercenter#2028D3
hypocenterC3#23462
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
1. D3×#544C2semidirect
4. C3×#2028C2semidirect
5. #71903C2semidirect
6. #70333C2semidirect
7. #69673C2semidirect
8. #68995C2semidirect
9. D3×#343C2semidirect
10. #68763C2semidirect
11. #68753C2semidirect
53. #2028D3semidirect
64. C2×#239Q8semidirect
170. C2×C3×Q8C2×Q8semidirect
190. C2×D12C2×Q8semidirect
383. C2×Q8D3×Q8semidirect
574. C3#23462semidirect

Maximal subgroups

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

subgroupindex
D3×#5442
#789562
#764822
C3×#20282
#719032
#703332
#696732
#689952
D3×#3432
#687632
#687532
#234623