Finite Group Databasebeta

← #10770430

Subgroups — #10770430

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
1C3×#3563C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic, complemented
3#83124C211yesmaximal, characteristic
4#83043C211yesmaximal, characteristic, complemented
5#71899C214yesmaximal, complemented
6#71887C214yesmaximal
7#71836C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic
9#71737C211yesmaximal, characteristic, complemented
10#18443—31nomaximal, Sylow
11C3×#564—21no—
12C3×#563—21no—
13C3×#557—21no—
14C3×#554—21no—
15C3×#552—21no—
16C3×#552—21no—
17C3×#552—22no—
18C3×#544—21no—
19C3×#544C2212yes—
20C3×#541—21no—
21C3×#541—22no—
22C3×#539C2212yes—
23C3×#533C2211yescharacteristic
24C3×#529C2211yescharacteristic
25C2×C3×#125C2211yescharacteristic
26C2×#869C2211yescharacteristic
27#5079—21no—
28#5078—21no—
29#5064—21no—
30#5055—21no—
31#5044C2214yes—
32#5044C2212yes—
33#5034C2212yes—
34#5021—22no—
35#5020—21no—
36#5007—21no—
37#4986—22no—
38#4986C2214yes—
39#4973—21no—
40#4973—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4538C2214yes—
46#4537C2212yes—
47#4528C2214yes—
48#3563D311yescharacteristic
49#3239—31no—
50#3230—31no—
51#2642—31no—
52#1617—34no—
53#1615—34no—
54#1506—31no—
55#1500—31no—
56#1470—31no—
57C22×Dic6—21no—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8D411yescharacteristic
62C3×C4⋊Q8D411yescharacteristic
63C3×#126—22no—
64C3×#126—22no—
65C3×#126—22no—
66C3×#126—22no—
67C3×#126—21no—
68C3×#125—22no—
69C3×#125—22no—
70C3×#125—22no—
71C3×#125—22no—
72C3×#125—22no—
73C3×#125—21no—
74C3×#125—22no—
75C3×#125—22no—
76C3×#125—22no—
77C3×#125—21no—
78C3×#125—21no—
79C3×#125—21no—
80C3×#125—21no—
81C3×#125—21no—
82C3×#125D411yescharacteristic
83C3×#124D411yescharacteristic
84C3×#123—22no—
85C3×#123—21no—
86C3×#123—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8C2311yescharacteristic
95C3×C22⋊Q8C2312yes—
96C3×#121C2311yescharacteristic
97C3×C4×Q8—21no—
98C3×C4×Q8—22no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×Q8D411yescharacteristic
102C3×C4×Q8D412yes—
103C3×C4×D4—21no—
104C3×C4×D4—22no—
105C3×C4×D4D411yescharacteristic
106C3×#117—22no—
107C3×#117—21no—
108C3×#117—21no—
109C3×#117—22no—
110C3×#117—21no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4D411yescharacteristic
113C2×C3×C4⋊C4D411yescharacteristic
114C2×C3×C4⋊C4D412yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3C2312yes—
119C2×C4×Dic3—21no—
120#869—21no—
121#869—21no—
122#869—21no—
123#869—21no—
124#835—22no—
125#835—24no—
126#832—22no—
127C3⋊#95—21no—
128C3⋊#95—24no—
129C3⋊#95—24no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#808—22no—
134#807—22no—
135#807—24no—
136C4⋊(C3⋊C8)—21no—
137C4⋊(C3⋊C8)—21no—
138#803—21no—
139C4×C3⋊C8—21no—
140#564—21no—
141#563—21no—
142#557—21no—
143#554—21no—
144#552—21no—
145#552—21no—
146#552—22no—
147#544—21no—
148#544C2×D312yes—
149#541—21no—
150#541—22no—
151#539C2×D312yes—
152#533C2×D311yescharacteristic
153C2×C4⋊Q8—31no—
154#529C2×D311yescharacteristic
155C2×#125C2×D311yescharacteristic
156#501—61no—
157#500—61no—
158#489—61no—
159#487—61no—
160#484—34no—
161#484—32no—
162#481—32no—
163#477—62no—
164#475—61no—
165#474—61no—
166#432—31no—
167#431—31no—
168#400—34no—
169#400—62no—
170#391—61no—
171#391—61no—
172#376—31no—
173#328—34no—
174#328—32no—
175#327—32no—
176#324—34no—
177C2×C3×Q8—22no—
178C2×C3×Q8—21no—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8C2×D411yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4C2×D411yescharacteristic
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—22no—
191C22×Dic3—22no—
192C2×Dic6—44no—
193C2×Dic6—41no—
194C2×Dic6—41no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4C2×D411yescharacteristic
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D411yescharacteristic
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—21no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—21no—
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C4⋊Dic3—22no—
242C4⋊Dic3—22no—
243C4⋊Dic3—41no—
244C4⋊Dic3—41no—
245C4×Dic3—42no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C22×Q8—61no—
249C4⋊Q8—21no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—61no—
253C4⋊Q8—61no—
254C4⋊Q8—61no—
255C4⋊Q8—61no—
256C4⋊Q8#6711yescharacteristic
257C4⋊Q8#6711yescharacteristic
258#126—22no—
259#126—22no—
260#126—22no—
261#126—22no—
262#126—21no—
263#125—22no—
264#125—22no—
265#125—22no—
266#125—22no—
267#125—21no—
268#125—22no—
269#125—22no—
270#125—22no—
271#125—22no—
272#125—21no—
273#125—21no—
274#125—21no—
275#125—21no—
276#125—21no—
277#125#6711yescharacteristic
278#124#6711yescharacteristic
279#123—22no—
280#123—21no—
281#123—22no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—21no—
288C22⋊Q8—22no—
289C22⋊Q8C22×D311yescharacteristic
290C22⋊Q8C22×D312yes—
291#121C22×D311yescharacteristic
292C4×Q8—21no—
293C4×Q8—22no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8#6711yescharacteristic
297C4×Q8#6712yes—
298C4×D4—21no—
299C4×D4—22no—
300C4×D4#6711yescharacteristic
301#117—22no—
302#117—21no—
303#117—21no—
304#117—22no—
305#117—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4#6711yescharacteristic
310C2×C4⋊C4#6712yes—
311C2×C4⋊C4#6711yescharacteristic
312C2×C4⋊C4C22×D312yes—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#107—64no—
317#106—62no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320#103—62no—
321#103—64no—
322#102—62no—
323C22⋊C8—32no—
324C22⋊C8—32no—
325#97—61no—
326C4×C8—61no—
327#95—32no—
328#95—61no—
329#95—64no—
330#95—64no—
331C23×C3—21no—
332C23×C3C22×D411yescharacteristic
333C3×Q8—41no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8—82no—
337C3×D4—41no—
338C3×D4—42no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—82no—
353C2×C3×C4#12011yescharacteristic
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×Dic3—42no—
360C2×Dic3—44no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364Dic6—82no—
365Dic6—82no—
366C3⋊C8—42no—
367C3⋊C8—42no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—124no—
371C2×Q8—121no—
372C2×Q8—121no—
373C2×Q8C2×#6712yes—
374C2×Q8C2×#6711yescharacteristic
375C2×D4—21no—
376C2×D4C2×#6711yescharacteristic
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4C2×#6712yes—
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6712yes—
385C22×C4C23×D311yescharacteristic
386C2×C8—62no—
387C2×C8—62no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—121no—
416C4⋊C4—121no—
417C4⋊C4C2×#6711yescharacteristic
418C4⋊C4C2×#6711yescharacteristic
419C4⋊C4C2×#6712yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—21no—
434C42—122no—
435C42C2×#6711yescharacteristic
436C42C2×#6711yescharacteristic
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—42no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—41no—
458C3×C4#45412yes—
459Dic3—82no—
460Dic3—82no—
461C23—21no—
462C23C22×#6711yescharacteristic
463Q8—41no—
464Q8—42no—
465Q8—42no—
466Q8—82no—
467Q8—242no—
468Q8—242no—
469D4—41no—
470D4—42no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—122no—
486C2×C4—124no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4#95311yescharacteristic
491C2×C4#95311yescharacteristic
492C2×C4#95311yescharacteristic
493C2×C4#95311yescharacteristic
494C2×C4C22×#6711yescharacteristic
495C2×C4C22×#6711yescharacteristic
496C8—122no—
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#583812yes—
509C22C2×#95311yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—42no—
521C4—41no—
522C4—242no—
523C4—242no—
524C4#524512yes—
525C3#1844311yescharacteristic, complemented, Sylow
526C2—22no—
527C2—81no—
528C2C2×#524511yescharacteristic
529C2#8411711yescharacteristic
530C2#8412411yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3563C2
socleC22×C3C2×#120
prime radicalC3×#121C23
hypercenter#3563D3
hypocenterC3#18443
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3563

Complements

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

NH ≅ G/Ntype
2. #83324C2semidirect
4. #83043C2semidirect
5. #71899C2semidirect
7. #71836C2semidirect
9. #71737C2semidirect
525. C3#18443semidirect

Maximal subgroups

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

subgroupindex
C3×#35632
#833242
#831242
#830432
#718992
#718872
#718362
#717542
#717372
#184433