Finite Group Databasebeta

← #10828478

Subgroups — #10828478

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5622C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic
3C2×#4972C211yesmaximal, characteristic
4C2×#4971C211yesmaximal, characteristic
5#71692C211yesmaximal, characteristic
6#70588C212yesmaximal, complemented
7#70565C212yesmaximal, complemented
8#70284C212yesmaximal, complemented
9#70237C212yesmaximal, complemented
10#69930C211yesmaximal, characteristic
11#69910C211yesmaximal, characteristic
12#24063—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C2×#944C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C22×Dic3⋊C4—21no—
20C22×C4×Dic3C2211yescharacteristic
21#5622—24no—
22#5622—24no—
23C2×#889—21no—
24C2×#888C2211yescharacteristic
25C2×#887—21no—
26C2×#887C2211yescharacteristic
27C2×#878—21no—
28C2×#877—21no—
29C3×#483—24no—
30C2×C3×#106C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×#835C2212yes—
34C2×C3⋊#95—21no—
35C2×#830C2211yescharacteristic
36C2×#816C2212yes—
37#4972—21no—
38#4972—21no—
39#4972C2212yes—
40#4972C2212yes—
41#4971—24no—
42C2×#810C2212yes—
43C2×#808C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#533—31no—
49C2×#483—31no—
50C2×#483D311yescharacteristic
51C2×#389—31no—
52#2058—32no—
53#2043—32no—
54#1872—31no—
55#1872—31no—
56#1865—32no—
57#1865—32no—
58#1817—31no—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—21no—
63C23×Dic3D411yescharacteristic
64C23×Dic3D411yescharacteristic
65C22×Dic6—21no—
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—21no—
77C2×#217—21no—
78Q8×Dic3—22no—
79Q8×Dic3—22no—
80#944—22no—
81#944—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87#924—22no—
88#924D414yes—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—44no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106#889—28no—
107#888—22no—
108#888—22no—
109#887—28no—
110#887—22no—
111#887—22no—
112#878—28no—
113#877—28no—
114C3×#106—24no—
115C3×#103—24no—
116C3×C22⋊C8—22no—
117#835—24no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120#830—22no—
121#816—24no—
122#810—24no—
123#808—24no—
124#533—64no—
125#533—64no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—31no—
128C2×#125—61no—
129C2×C22⋊Q8—31no—
130C2×C22⋊Q8—61no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×C4×Q8—31no—
133C2×C4×Q8—31no—
134C2×C4×Q8—61no—
135C2×#117—61no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—61no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C42—31no—
140#483—24no—
141#483—64no—
142C2×#106—32no—
143C2×#106C2×D312yes—
144C2×#103—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×#103C2×D312yes—
148C2×C22⋊C8—31no—
149C2×C22⋊C8C2×D311yescharacteristic
150#389—32no—
151#389—32no—
152#389—61no—
153#389—61no—
154C2×#95—61no—
155#340—32no—
156#340—32no—
157#336—32no—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—24no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—41no—
185C22×Dic3—42no—
186C22×Dic3—42no—
187C22×Dic3SD1614yes—
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—24no—
195C2×Dic6—21no—
196C2×Dic6—42no—
197C2×Dic6—42no—
198C2×Dic6C2×D412yes—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4C2×D412yes—
208C3×C22⋊C4—24no—
209#217—24no—
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3C2×D412yes—
215C4⋊Dic3#4114yes—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4—24no—
221Dic3⋊C4—44no—
222Dic3⋊C4—44no—
223C4×Dic3—24no—
224C4×Dic3—24no—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230C4×Dic3—21no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C22×Q8—31no—
235C22×Q8—61no—
236C22×Q8C22×D311yescharacteristic
237C23×C4—21no—
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—61no—
241C23×C4C22×D311yescharacteristic
242C22×C8—32no—
243C22×C8C22×D312yes—
244C4⋊Q8—62no—
245C4⋊Q8—62no—
246C4⋊Q8—62no—
247C4⋊Q8—62no—
248#125—68no—
249C22⋊Q8—22no—
250C22⋊Q8—34no—
251C22⋊Q8—68no—
252C22⋊Q8—62no—
253C22⋊Q8#6714yes—
254C4×Q8—62no—
255C4×Q8—68no—
256C4×Q8—62no—
257C4×Q8—62no—
258C4×Q8—62no—
259#117—68no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—124no—
277C2×C4⋊C4C22×D312yes—
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—61no—
281C2×C42—31no—
282C2×C42—31no—
283C2×C42—31no—
284C2×C42—31no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42—62no—
288C2×C42—62no—
289#106—24no—
290#106—64no—
291#103—24no—
292#103—64no—
293#103—64no—
294#103—64no—
295C22⋊C8—22no—
296C22⋊C8—62no—
297#95—62no—
298#95—62no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3#12012yes—
304C23×C3C2×SD1612yes—
305C23×C3#13712yes—
306C23×C3C22×D411yescharacteristic
307C3×Q8—24no—
308C3×Q8—42no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—84no—
323C2×C3×C4#12012yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4112yes—
327C2×C3×C42−1+412yes—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—28no—
335C2×Dic3—21no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—44no—
342C2×Dic3—48no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3C2×SD1612yes—
350C2×Dic3C2×SD1612yes—
351C2×Dic3#13712yes—
352Dic6—24no—
353Dic6—48no—
354Dic6—42no—
355C3×C8—42no—
356C3⋊C8—42no—
357C24D3×D411yescharacteristic
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—32no—
362C2×Q8—64no—
363C2×Q8—61no—
364C2×Q8—62no—
365C2×Q8—62no—
366C2×Q8—122no—
367C2×Q8—122no—
368C2×Q8C2×#6712yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—34no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—64no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—121no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4D3×D411yescharacteristic
394C22×C4D3×D412yes—
395C22×C4C2×#6712yes—
396C22×C4C23×D311yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—32no—
408C4⋊C4—34no—
409C4⋊C4—42no—
410C4⋊C4—64no—
411C4⋊C4—64no—
412C4⋊C4—64no—
413C4⋊C4—64no—
414C4⋊C4—64no—
415C4⋊C4—64no—
416C4⋊C4—64no—
417C4⋊C4—62no—
418C4⋊C4—61no—
419C4⋊C4—124no—
420C4⋊C4—124no—
421C4⋊C4C2×#6712yes—
422C22⋊C4—24no—
423C22⋊C4—64no—
424C42—64no—
425C42—64no—
426C42—62no—
427C42—62no—
428C42—61no—
429C42—61no—
430C42—61no—
431C42—61no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3#44814yes—
439C22×C3C2×#12011yescharacteristic
440C22×C3#52511yescharacteristic
441C22×C3#53611yescharacteristic
442C22×C3C22×SD1611yescharacteristic
443C22×C3C2×#13711yescharacteristic
444C22×C3C2×#13711yescharacteristic
445C22×C3#57511yescharacteristic
446C3×C4—22no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4#54712yes—
451C3×C4#54712yes—
452Dic3—22no—
453Dic3—22no—
454Dic3—22no—
455Dic3—44no—
456Dic3—42no—
457Dic3—42no—
458Dic3—41no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23D3×SD1612yes—
464C23#91512yes—
465C23#93712yes—
466C23C2×D3×D411yescharacteristic
467Q8—24no—
468Q8—42no—
469Q8—64no—
470Q8—128no—
471Q8—122no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—68no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—64no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—84no—
502C2×C4—124no—
503C2×C4—128no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4—122no—
509C2×C4—121no—
510C2×C4#93712yes—
511C2×C4C2×D3×D411yescharacteristic
512C2×C4D3×#4112yes—
513C2×C4#101012yes—
514C2×C4C22×#6711yescharacteristic
515C8—42no—
516C8—122no—
517C2×C3—24no—
518C2×C3#240611yescharacteristic
519C2×C3C2×#44811yescharacteristic
520C2×C3#300711yescharacteristic
521C2×C3#310111yescharacteristic
522C2×C3#311311yescharacteristic
523C2×C3#322911yescharacteristic
524C2×C3#323611yescharacteristic
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22#517014yes—
532C22#564811yescharacteristic
533C22#565611yescharacteristic
534C22C2×D3×SD1611yescharacteristic
535C22C2×#91511yescharacteristic
536C22D3×#13711yescharacteristic
537C22#577711yescharacteristic
538C22C2×#93711yescharacteristic
539C4—22no—
540C4—42no—
541C4—42no—
542C4—62no—
543C4—62no—
544C4—62no—
545C4—82no—
546C4—124no—
547C4—121no—
548C4—122no—
549C4—122no—
550C4#565512yes—
551C4#582412yes—
552C3#2406311yescharacteristic, complemented, Sylow
553C2—24no—
554C2#8141211yescharacteristic
555C2#8141411yescharacteristic
556C2#8141811yescharacteristic
557C2#8142011yescharacteristic
558C2#8284111yescharacteristic
559C2C2×#517011yescharacteristic
560C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#483C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#483D3
hypocenterC3#24063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
6. #70588C2semidirect
7. #70565C2semidirect
8. #70284C2semidirect
9. #70237C2semidirect
552. C3#24063semidirect

Maximal subgroups

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

subgroupindex
C2×#56222
C2×C3×#4832
C2×#49722
C2×#49712
#716922
#705882
#705652
#702842
#702372
#699302
#699102
#240633