Finite Group Databasebeta

← #10829953

Subgroups — #10829953

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic
3#82376C211yesmaximal, characteristic
4C2×C4×#214C211yesmaximal, characteristic
5#72057C212yesmaximal
6#72048C212yesmaximal
7#71785C211yesmaximal, characteristic, complemented
8#70485C211yesmaximal, characteristic, complemented
9#70472C211yesmaximal, characteristic, complemented
10#70284C212yesmaximal, complemented
11#70237C212yesmaximal, complemented
12#24187—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×Q8×Dic3C2212yes—
17C2×#944C2212yes—
18C22×#214C2211yescharacteristic
19C2×#924—21no—
20#5721—24no—
21#5721—24no—
22C2×#890—21no—
23C2×#889—22no—
24C2×#887—21no—
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×#461—24no—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5221C2212yes—
33C2×#832C2211yescharacteristic
34#5212—21no—
35C2×#816C2212yes—
36C4×#214—24no—
37#4972C2212yes—
38C2×#810C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4948—22no—
41#4935C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#557—31no—
47C2×#461D311yescharacteristic
48C2×C4×SD16—31no—
49#2538—31no—
50#2043—32no—
51#2013—32no—
52#1984—31no—
53#1956—31no—
54#1880—32no—
55#1865—32no—
56#1803—31no—
57C22×C3×Q8C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2311yescharacteristic
62C2×C3×SD16—24no—
63C2×C3×SD16—24no—
64C2×C3×SD16—42no—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yes—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#217—22no—
77Q8×Dic3—24no—
78Q8×Dic3—24no—
79#944—24no—
80#944—24no—
81C2×#214—22no—
82C2×#214—22no—
83C2×#214—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2311yescharacteristic
90#924—44no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4D412yes—
96C2×C4×Dic3—21no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101#890—28no—
102#889—216no—
103#887—28no—
104C4×Dic6—22no—
105C4×Dic6—22no—
106C4×Dic6—22no—
107C4×Dic6D414yes—
108C3×C4⋊C8—22no—
109C3×#103—24no—
110#835—24no—
111#835—42no—
112#832—22no—
113C3⋊#95—22no—
114#816—42no—
115#810—22no—
116#808—22no—
117C4×C3⋊C8—22no—
118C22×SD16—31no—
119C22×SD16C2×D312yes—
120#557—64no—
121#557—64no—
122C2×C4⋊Q8—32no—
123C2×C4⋊Q8C2×D311yescharacteristic
124C2×#125—62no—
125C2×#125—61no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—32no—
129C2×C4×Q8—31no—
130C2×C4×Q8—61no—
131C2×C4×D4C2×D311yescharacteristic
132#461—24no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—32no—
137C2×#103—31no—
138C2×#103—32no—
139C2×#103—32no—
140C2×#103C2×D312yes—
141C2×#102—31no—
142C2×C4×C8—31no—
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D414yes—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4#4112yes—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6SD1614yescomplemented
185C2×Dic6C2×D412yes—
186C3×SD16—48no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C22⋊C4—42no—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197#217—44no—
198#217—82no—
199#214—28no—
200C4⋊Dic3—28no—
201C4⋊Dic3—22no—
202C4⋊Dic3—21no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—42no—
206C4⋊Dic3—81no—
207C4⋊Dic3C2×D412yes—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210Dic3⋊C4—24no—
211Dic3⋊C4—24no—
212Dic3⋊C4—44no—
213C4×Dic3—28no—
214C4×Dic3—24no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8#4112yes—
223C22×Q8—31no—
224C22×Q8—31no—
225C22×Q8C22×D312yes—
226C22×D4C22×D311yescharacteristic
227C23×C4—21no—
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—42no—
231C2×SD16—62no—
232C2×SD16—62no—
233C2×SD16—62no—
234C22×C8—31no—
235C22×C8—31no—
236C22×C8C22×D312yes—
237C4⋊Q8—22no—
238C4⋊Q8—64no—
239C4⋊Q8—64no—
240C4⋊Q8#6714yes—
241#125—616no—
242#125—68no—
243C22⋊Q8—124no—
244C4×Q8—34no—
245C4×Q8—68no—
246C4×Q8—64no—
247C4×Q8—64no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—22no—
252C4×D4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C42—32no—
271C2×C42—62no—
272C2×C42—61no—
273C2×C42C22×D311yescharacteristic
274#106—62no—
275C4⋊C8—22no—
276#103—24no—
277#103—62no—
278#103—64no—
279#103—122no—
280#103—122no—
281#102—62no—
282C4×C8—62no—
283#95—62no—
284C23×C3—22no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—24no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—84no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4C2×SD1612yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C2×C3×C42−1+411yescharacteristic
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3C2×SD1614yes—
334Dic6—28no—
335Dic6—22no—
336Dic6—22no—
337Dic6—41no—
338Dic6—41no—
339C3×C8—42no—
340C3⋊C8—22no—
341C3⋊C8—41no—
342C24—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—34no—
349C2×Q8—32no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8C2×#6714yes—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4#23712yes—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4D3×D411yescharacteristic
375C22×C4D3×D411yescharacteristic
376C22×C4C23×D311yescharacteristic
377SD16—48no—
378SD16—68no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—32no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—61no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—32no—
393C4⋊C4—68no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C4⋊C4—64no—
397C4⋊C4—62no—
398C4⋊C4—64no—
399C4⋊C4—61no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—124no—
403C4⋊C4—122no—
404C4⋊C4—241no—
405C22⋊C4—42no—
406C22⋊C4—124no—
407C22⋊C4—242no—
408C42—68no—
409C42—64no—
410C42—62no—
411C42—62no—
412C42D3×D412yes—
413C42C2×#6712yes—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—81no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54811yescharacteristic
423C22×C3#55011yescharacteristic
424C22×C3C22×SD1611yescharacteristic
425C22×C3C2×#13711yescharacteristic
426C22×C3#57711yescharacteristic
427C22×C3#57811yescharacteristic
428C3×C4—22no—
429C3×C4—44no—
430C3×C4—81no—
431C3×C4#44814yes—
432C3×C4#52511yescharacteristic
433C3×C4#53711yescharacteristic
434C3×C4#54611yescharacteristic
435C3×C4#54711yescharacteristic
436Dic3—24no—
437Dic3—42no—
438Dic3—42no—
439Dic3—42no—
440Dic3—82no—
441C23—22no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23C2×D3×D411yescharacteristic
449Q8—24no—
450Q8—24no—
451Q8—42no—
452Q8—42no—
453Q8—68no—
454Q8—62no—
455Q8—62no—
456Q8—121no—
457Q8—121no—
458D4—22no—
459D4—22no—
460D4—41no—
461D4—41no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—34no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—84no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4D3×SD1612yes—
487C2×C4#91512yes—
488C2×C4#93712yes—
489C2×C4#93712yes—
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×#23711yescharacteristic
492C2×C4#100411yescharacteristic
493C2×C4D3×#4111yescharacteristic
494C2×C4#101011yescharacteristic
495C2×C4C22×#6711yescharacteristic
496C8—42no—
497C8—62no—
498C8—121no—
499C2×C3—82no—
500C2×C3#240711yescharacteristic
501C2×C3C2×#44811yescharacteristic
502C2×C3#301311yescharacteristic
503C2×C3#335911yescharacteristic
504C2×C3#337211yescharacteristic
505C2×C3#339211yescharacteristic
506C2×C3#339811yescharacteristic
507C22—42no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22#559811yescharacteristic
515C22#559911yescharacteristic
516C22C2×D3×SD1611yescharacteristic
517C22C2×#91511yescharacteristic
518C22#577811yescharacteristic
519C22#577911yescharacteristic
520C22C2×#93711yescharacteristic
521C4—22no—
522C4—44no—
523C4—64no—
524C4—81no—
525C4—122no—
526C4—122no—
527C4—122no—
528C4—242no—
529C4#517014yes—
530C4#560611yescharacteristic
531C4#560811yescharacteristic
532C4#580611yescharacteristic
533C4#582411yescharacteristic
534C3#2418711yescharacteristic, complemented, Sylow
535C2—82no—
536C2#8041411yescharacteristic
537C2#8041711yescharacteristic
538C2#8042011yescharacteristic
539C2#8042211yescharacteristic
540C2#8270111yescharacteristic
541C2C2×#517011yescharacteristic
542C2#8398411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#461D3
hypocenterC3#24187
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#5721C2semidirect
7. #71785C2semidirect
8. #70485C2semidirect
9. #70472C2semidirect
10. #70284C2semidirect
11. #70237C2semidirect
184. C2×Dic6SD16semidirect
534. C3#24187semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×C3×#4612
#823762
C2×C4×#2142
#720572
#720482
#717852
#704852
#704722
#702842
#702372
#241873