Finite Group Databasebeta

← #10770429

Subgroups — #10770429

Order 768.

832 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86142C211yesmaximal, characteristic
2C3×#3239C211yesmaximal, characteristic
3#83245C211yesmaximal, characteristic, complemented
4#83224C211yesmaximal, characteristic, complemented
5#71754C211yesmaximal, characteristic
6#71179C214yesmaximal, complemented
7#71170C214yesmaximal
8#70836C211yesmaximal, characteristic, complemented
9#70831C211yesmaximal, characteristic, complemented
10#18443—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×#529C2211yescharacteristic
13#5701—22no—
14C3⋊#564—21no—
15#5672—21no—
16#5650—22no—
17#5623C2212yes—
18#5602C2212yes—
19#5567—21no—
20#5543—21no—
21#5541—21no—
22#5541—21no—
23#5519—21no—
24#5518—21no—
25#5470C2211yescharacteristic
26C2×#870C2211yescharacteristic
27C3×#501—21no—
28C3×#489—21no—
29C3×#484C2212yes—
30C3×#481C2212yes—
31C3×#475—21no—
32C3×#474—21no—
33C3×#432C2211yescharacteristic
34#5229—21no—
35#5229—21no—
36#5093—21no—
37#5091—22no—
38#5078—21no—
39#4985—22no—
40#4985C2214yes—
41#4970C2214yes—
42#4932C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4537C2212yes—
46#4469C2214yes—
47#4467C2214yes—
48#3563—31no—
49#3239D311yescharacteristic
50#3230—31no—
51#2642—31no—
52#1617—34no—
53#1615—34no—
54#1506—31no—
55#1500—31no—
56#1470—31no—
57C22×C3×Q8—21no—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×#126—21no—
63C3×#125D411yescharacteristic
64C3×#124D411yescharacteristic
65C3×#123—21no—
66C3×C22⋊Q8C2311yescharacteristic
67C3×#121C2311yescharacteristic
68C3×C4×Q8D411yescharacteristic
69C3×C4×D4—21no—
70C3×C4×D4D411yescharacteristic
71C3×#117—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42C2311yescharacteristic
75#944—22no—
76#935—22no—
77#935—22no—
78D4×Dic3—22no—
79#926—21no—
80#926—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3D411yescharacteristic
84#924—21no—
85#924—21no—
86#924—21no—
87#924C2312yes—
88C2×Dic3⋊C4D412yes—
89C2×Dic3⋊C4C2312yes—
90C2×C4×Dic3—21no—
91#890—22no—
92#890—22no—
93#889—22no—
94#889—22no—
95#889—22no—
96#889—22no—
97#889—22no—
98#889—22no—
99#888—22no—
100#887—22no—
101#879—22no—
102#879—22no—
103#879—22no—
104#879—22no—
105#878—22no—
106#878—22no—
107#878—22no—
108#878—22no—
109#877—22no—
110#877—22no—
111#870—21no—
112#870—21no—
113#870—21no—
114#870—21no—
115#870—21no—
116#870—21no—
117#869—21no—
118#869D411yescharacteristic
119#869D411yescharacteristic
120C4×Dic6—21no—
121C4×Dic6—21no—
122C4×Dic6—21no—
123C4×Dic6D412yes—
124C3×#107—22no—
125C3×#106—22no—
126C3×C4⋊C8—21no—
127C3×#103—22no—
128C3×#102—22no—
129C3×C22⋊C8C2312yes—
130C3×#97—21no—
131C3⋊#95—21no—
132C3⋊#95—24no—
133C3⋊#95—24no—
134C3⋊#95C2312yes—
135#830C2312yes—
136#810—24no—
137#807—24no—
138C4⋊(C3⋊C8)—21no—
139C4×C3⋊C8—21no—
140#564—61no—
141#563—61no—
142#557—61no—
143#554—61no—
144#552—61no—
145#552—62no—
146#552—61no—
147#544—32no—
148#544—61no—
149#541—61no—
150#541—62no—
151#539—32no—
152#533—31no—
153C2×C4⋊Q8C2×D311yescharacteristic
154#529C2×D311yescharacteristic
155C2×#125—31no—
156#501—21no—
157#500—61no—
158#489—21no—
159#487—61no—
160#484—34no—
161#484C2×D312yes—
162#481C2×D312yes—
163#477—62no—
164#475—21no—
165#474—21no—
166#432C2×D311yescharacteristic
167#431—31no—
168#400—34no—
169#400—62no—
170#391—61no—
171#391—61no—
172#376—31no—
173#328—32no—
174#328—34no—
175#327—32no—
176#324—34no—
177C2×C3×Q8—44no—
178C2×C3×Q8—41no—
179C2×C3×Q8—41no—
180C2×C3×Q8C2×D411yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4C2×D411yescharacteristic
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3C2×D412yes—
191C22×Dic3C2×D412yes—
192C2×Dic6—21no—
193C2×Dic6—22no—
194C2×Dic6C2×D412yes—
195C2×C3×C8—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4C2×D411yescharacteristic
205C3×C4⋊C4C2×D411yescharacteristic
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—21no—
211C3×C42—21no—
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215C3×C42C2×D411yescharacteristic
216#217—22no—
217#217—22no—
218#217—22no—
219#217—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3C2×D412yes—
230Dic3⋊C4—22no—
231Dic3⋊C4—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236Dic3⋊C4—22no—
237Dic3⋊C4—22no—
238Dic3⋊C4—22no—
239Dic3⋊C4—22no—
240Dic3⋊C4—22no—
241Dic3⋊C4—22no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246C4×Dic3—42no—
247C2×C3⋊C8—22no—
248C22×Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—31no—
254C4⋊Q8—31no—
255C4⋊Q8—61no—
256C4⋊Q8—62no—
257C4⋊Q8—62no—
258#126—21no—
259#126—62no—
260#126—62no—
261#126—62no—
262#126—62no—
263#125—61no—
264#125—62no—
265#125—62no—
266#125—62no—
267#125—62no—
268#125—62no—
269#125—62no—
270#125—62no—
271#125—62no—
272#125—61no—
273#125—61no—
274#125—61no—
275#125—61no—
276#125—61no—
277#125#6711yescharacteristic
278#124#6711yescharacteristic
279#123—21no—
280#123—62no—
281#123—62no—
282C22⋊Q8—32no—
283C22⋊Q8—62no—
284C22⋊Q8—62no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C22⋊Q8—61no—
288C22⋊Q8—61no—
289C22⋊Q8—61no—
290C22⋊Q8C22×D311yescharacteristic
291#121C22×D311yescharacteristic
292C4×Q8—32no—
293C4×Q8—61no—
294C4×Q8—61no—
295C4×Q8—62no—
296C4×Q8—61no—
297C4×Q8#6711yescharacteristic
298C4×D4—21no—
299C4×D4—62no—
300C4×D4#6711yescharacteristic
301#117—21no—
302#117—62no—
303#117—62no—
304#117—61no—
305#117—61no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D312yes—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—22no—
316#107—64no—
317#106—22no—
318C4⋊C8—21no—
319C4⋊C8—61no—
320#103—22no—
321#103—64no—
322#102—22no—
323C22⋊C8—32no—
324C22⋊C8C22×D312yes—
325#97—21no—
326C4×C8—61no—
327#95—32no—
328#95—64no—
329#95—64no—
330#95—61no—
331C23×C3—21no—
332C23×C3C22×D411yescharacteristic
333C3×Q8—41no—
334C3×Q8—82no—
335C3×Q8—82no—
336C3×D4—41no—
337C3×D4—42no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—44no—
346C2×C3×C4—42no—
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4#12011yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—82no—
363Dic6—42no—
364Dic6—42no—
365Dic6—82no—
366C3×C8—42no—
367C3⋊C8—42no—
368C2×Q8—32no—
369C2×Q8—44no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—62no—
373C2×Q8—61no—
374C2×Q8C2×#6711yescharacteristic
375C2×D4—21no—
376C2×D4C2×#6711yescharacteristic
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D311yescharacteristic
386C2×C8—22no—
387C2×C8—62no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—32no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4—62no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4C2×#6711yescharacteristic
419C4⋊C4C2×#6711yescharacteristic
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C42—21no—
430C42—62no—
431C42—62no—
432C42—62no—
433C42—62no—
434C42—122no—
435C42D3×D411yescharacteristic
436C42D3×D411yescharacteristic
437C42C2×#6711yescharacteristic
438C42C2×#6711yescharacteristic
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3C2×#12011yescharacteristic
445C22×C3#57812yes—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—82no—
455C3×C4#45412yes—
456Dic3—42no—
457Dic3—42no—
458Dic3—42no—
459Dic3—42no—
460Dic3—82no—
461C23—21no—
462C23C2×D3×D411yescharacteristic
463Q8—41no—
464Q8—82no—
465Q8—82no—
466Q8—122no—
467Q8—122no—
468Q8—242no—
469D4—41no—
470D4—42no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—64no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—242no—
490C2×C4#93711yescharacteristic
491C2×C4#93711yescharacteristic
492C2×C4#93711yescharacteristic
493C2×C4#93711yescharacteristic
494C2×C4C2×D3×D411yescharacteristic
495C2×C4C22×#6711yescharacteristic
496C8—42no—
497C8—122no—
498C2×C3—22no—
499C2×C3—81no—
500C2×C3#301111yescharacteristic
501C2×C3C2×#45411yescharacteristic
502C2×C3#302811yescharacteristic
503C22—21no—
504C22—21no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22#577912yes—
509C22C2×#93711yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—82no—
519C4—122no—
520C4—122no—
521C4—122no—
522C4—122no—
523C4—242no—
524C4#520112yes—
525C3#1844311yescharacteristic, complemented, Sylow
526C2—22no—
527C2—81no—
528C2#8341711yescharacteristic
529C2C2×#520111yescharacteristic
530C2#8398311yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3239C2
socleC22×C3C2×#120
prime radicalC3×#121C23
hypercenter#3239D3
hypocenterC3#18443
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3239

Complements

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

NH ≅ G/Ntype
3. #83245C2semidirect
4. #83224C2semidirect
6. #71179C2semidirect
8. #70836C2semidirect
9. #70831C2semidirect
525. C3#18443semidirect

Maximal subgroups

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

subgroupindex
#861422
C3×#32392
#832452
#832242
#717542
#711792
#711702
#708362
#708312
#184433