Finite Group Databasebeta

← #10770421

Subgroups — #10770421

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3563C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic, complemented
3#83123C211yesmaximal, characteristic
4#83044C211yesmaximal, characteristic, complemented
5#71898C214yesmaximal, complemented
6#71886C214yesmaximal
7#71836C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic
9#71737C211yesmaximal, characteristic, complemented
10#18442—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#5080—21no—
28#5079—21no—
29#5064—21no—
30#5055—21no—
31#5043C2214yes—
32#5043C2212yes—
33#5033C2212yes—
34#5021—21no—
35#5020—22no—
36#5008—21no—
37#4981C2214yes—
38#4979—22no—
39#4977—21no—
40#4975—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4538C2214yes—
46#4537C2212yes—
47#4528C2214yes—
48#3563D311yescharacteristic
49#3238—31no—
50#3229—31no—
51#2660—31no—
52#1611—34no—
53#1609—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×Q8D412yes—
102C3×C4×Q8D411yescharacteristic
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×Dic3⋊C4—24no—
120C2×C4×Dic3—21no—
121#869—21no—
122#869—21no—
123#869—21no—
124#869—21no—
125#835—24no—
126#835—22no—
127#832—22no—
128C3⋊#95—21no—
129C3⋊#95—24no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#808—22no—
134#808—24no—
135#807—22no—
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#498—61no—
158#489—61no—
159#487—61no—
160#483—34no—
161#483—32no—
162#480—32no—
163#477—61no—
164#476—61no—
165#475—62no—
166#432—31no—
167#431—31no—
168#398—34no—
169#395—62no—
170#384—61no—
171#382—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—41no—
193C2×Dic6—41no—
194C2×Dic6—82no—
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×D411yescharacteristic
222C3×C4⋊C4C2×D412yes—
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—
245Dic3⋊C4—84no—
246C4×Dic3—24no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C22×Q8—61no—
250C4⋊Q8—21no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—61no—
254C4⋊Q8—61no—
255C4⋊Q8—61no—
256C4⋊Q8—61no—
257C4⋊Q8#6711yescharacteristic
258C4⋊Q8#6711yescharacteristic
259#126—22no—
260#126—22no—
261#126—22no—
262#126—22no—
263#126—21no—
264#125—22no—
265#125—22no—
266#125—22no—
267#125—22no—
268#125—21no—
269#125—22no—
270#125—22no—
271#125—22no—
272#125—22no—
273#125—21no—
274#125—21no—
275#125—21no—
276#125—21no—
277#125—21no—
278#125#6711yescharacteristic
279#124#6711yescharacteristic
280#123—22no—
281#123—21no—
282#123—22no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—21no—
289C22⋊Q8—22no—
290C22⋊Q8C22×D311yescharacteristic
291C22⋊Q8C22×D312yes—
292#121C22×D311yescharacteristic
293C4×Q8—21no—
294C4×Q8—22no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8#6711yescharacteristic
298C4×Q8#6712yes—
299C4×D4—21no—
300C4×D4—22no—
301C4×D4#6711yescharacteristic
302#117—22no—
303#117—21no—
304#117—21no—
305#117—22no—
306#117—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—64no—
311C2×C4⋊C4#6711yescharacteristic
312C2×C4⋊C4#6712yes—
313C2×C4⋊C4#6711yescharacteristic
314C2×C4⋊C4C22×D312yes—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#106—64no—
319#106—62no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322#103—64no—
323#103—62no—
324#102—62no—
325C22⋊C8—32no—
326C22⋊C8—32no—
327#97—61no—
328C4×C8—61no—
329#95—32no—
330#95—61no—
331#95—64no—
332C23×C3—21no—
333C23×C3C22×D411yescharacteristic
334C3×Q8—41no—
335C3×Q8—42no—
336C3×Q8—42no—
337C3×Q8—82no—
338C3×D4—41no—
339C3×D4—42no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—82no—
354C2×C3×C4#12011yescharacteristic
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×Dic3—24no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—44no—
365Dic6—82no—
366Dic6—82no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—121no—
372C2×Q8—121no—
373C2×Q8—242no—
374C2×Q8C2×#6712yes—
375C2×Q8C2×#6711yescharacteristic
376C2×D4—21no—
377C2×D4C2×#6711yescharacteristic
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4C2×#6712yes—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6712yes—
386C22×C4C23×D311yescharacteristic
387C2×C8—62no—
388C2×C8—62no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—121no—
417C4⋊C4—121no—
418C4⋊C4—244no—
419C4⋊C4C2×#6711yescharacteristic
420C4⋊C4C2×#6712yes—
421C4⋊C4C2×#6711yescharacteristic
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—21no—
436C42—64no—
437C42C2×#6711yescharacteristic
438C42C2×#6711yescharacteristic
439C42C2×#6711yescharacteristic
440C42C2×#6711yescharacteristic
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3C2×#12011yescharacteristic
447C22×C3#57712yes—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—42no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—41no—
460C3×C4#45612yes—
461Dic3—44no—
462Dic3—82no—
463C23—21no—
464C23C22×#6711yescharacteristic
465Q8—41no—
466Q8—42no—
467Q8—42no—
468Q8—82no—
469Q8—242no—
470Q8—242no—
471D4—41no—
472D4—42no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—64no—
487C2×C4—82no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—124no—
491C2×C4—122no—
492C2×C4#95311yescharacteristic
493C2×C4#95311yescharacteristic
494C2×C4#95311yescharacteristic
495C2×C4#95311yescharacteristic
496C2×C4C22×#6711yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—122no—
499C8—122no—
500C2×C3—22no—
501C2×C3—81no—
502C2×C3#301011yescharacteristic
503C2×C3C2×#45611yescharacteristic
504C2×C3#302811yescharacteristic
505C22—21no—
506C22—21no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22#583712yes—
511C22C2×#95311yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—41no—
524C4—124no—
525C4—242no—
526C4#524612yes—
527C3#1844211yescharacteristic, complemented, Sylow
528C2—22no—
529C2—81no—
530C2#8411311yescharacteristic
531C2C2×#524611yescharacteristic
532C2#8412411yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3563C2
socleC22×C3C2×#120
prime radicalC3×#121C23
hypercenter#3563D3
hypocenterC3#18442
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. #83323C2semidirect
4. #83044C2semidirect
5. #71898C2semidirect
7. #71836C2semidirect
9. #71737C2semidirect
527. C3#18442semidirect

Maximal subgroups

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

subgroupindex
C3×#35632
#833232
#831232
#830442
#718982
#718862
#718362
#717542
#717372
#184423