Finite Group Databasebeta

← #10770412

Subgroups — #10770412

Order 768.

888 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3559C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic, complemented
3#83124C211yesmaximal, characteristic, complemented
4#83043C211yesmaximal, characteristic, complemented
5#71887C214yesmaximal, complemented
6#71875C214yesmaximal, complemented
7#71812C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic, complemented
9#71737C211yesmaximal, characteristic, complemented
10#18441—31nomaximal, Sylow
11C3×#563—21no—
12C3×#562—21no—
13C3×#554—21no—
14C3×#553—21no—
15C3×#553—21no—
16C3×#547—22no—
17C3×#547—21no—
18C3×#544—21no—
19C3×#540—21no—
20C3×#539—22no—
21C3×#539C2212yes—
22C3×#537C2212yes—
23C3×#532C2211yescharacteristic
24C3×#529C2211yescharacteristic
25C2×C3×#125C2211yescharacteristic
26C2×#869C2211yescharacteristic
27#5080—21no—
28#5070—21no—
29#5064—21no—
30#5055—21no—
31#5044C2212yes—
32#5034C2214yes—
33#5034C2212yes—
34#5020—21no—
35#5008—22no—
36#5007—21no—
37#4986—22no—
38#4986C2214yes—
39#4973—21no—
40#4973—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4537C2212yes—
46#4537C2214yes—
47#4528C2214yes—
48#3559D311yescharacteristic
49#3239—31no—
50#3227—31no—
51#2642—31no—
52#1616—34no—
53#1615—34no—
54#1505—31no—
55#1500—31no—
56#1470—31no—
57C2×C3×#41—21no—
58C2×C3×#41—22no—
59C22×Dic6—21no—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8D411yescharacteristic
63C3×#127D411yescharacteristic
64C3×#126—22no—
65C3×#126—22no—
66C3×#126—21no—
67C3×#125—21no—
68C3×#125—22no—
69C3×#125—21no—
70C3×#125—21no—
71C3×#125—21no—
72C3×#125—21no—
73C3×#125D411yescharacteristic
74C3×#124—21no—
75C3×#124D411yescharacteristic
76C3×#123—22no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—21no—
82C3×#123—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8C2311yescharacteristic
90C3×#121—22no—
91C3×#121—21no—
92C3×#121—21no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—21no—
96C3×#121—22no—
97C3×#121C2311yescharacteristic
98C3×#121C2312yes—
99C3×C4×Q8D411yescharacteristic
100C3×C4×D4—22no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4—22no—
105C3×C4×D4—21no—
106C3×C4×D4D411yescharacteristic
107C3×C4×D4D412yes—
108C3×#117—21no—
109C3×#117—21no—
110C2×C3×C4⋊C4—22no—
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#832—22no—
126#832—24no—
127C3⋊#95—21no—
128C3⋊#95—24no—
129C3⋊#95—24no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#808—24no—
134#808—22no—
135#807—22no—
136C4⋊(C3⋊C8)—21no—
137C4⋊(C3⋊C8)—21no—
138#803—21no—
139C4×C3⋊C8—21no—
140#563—21no—
141#562—21no—
142#554—21no—
143#553—21no—
144#553—21no—
145#547—22no—
146#547—21no—
147#544—21no—
148#540—21no—
149#539—22no—
150#539C2×D312yes—
151#537C2×D312yes—
152#532C2×D311yescharacteristic
153C2×C4⋊Q8—31no—
154#529C2×D311yescharacteristic
155C2×#125C2×D311yescharacteristic
156#501—61no—
157#498—61no—
158#489—61no—
159#486—61no—
160#484—32no—
161#481—34no—
162#481—32no—
163#476—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—32no—
174#327—34no—
175#327—32no—
176#324—34no—
177C3×#41—82no—
178C3×#41—84no—
179C2×C3×Q8—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8C2×D411yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4C2×D411yescharacteristic
189C2×C3×D4C2×D412yes—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—22no—
201C22×Dic3—22no—
202C2×Dic6—44no—
203C2×Dic6—41no—
204C2×Dic6—41no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4C2×D411yescharacteristic
223C3×C4⋊C4C2×D412yes—
224C3×C4⋊C4C2×D411yescharacteristic
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—22no—
237C3×C42—22no—
238C3×C42—21no—
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C4⋊Dic3—22no—
244C4⋊Dic3—22no—
245C4⋊Dic3—41no—
246C4⋊Dic3—41no—
247C4×Dic3—42no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×#41—21no—
251C2×#41—22no—
252C22×Q8—61no—
253C4⋊Q8—21no—
254C4⋊Q8—22no—
255C4⋊Q8—61no—
256C4⋊Q8—61no—
257C4⋊Q8—61no—
258C4⋊Q8—61no—
259C4⋊Q8#6711yescharacteristic
260#127#6711yescharacteristic
261#126—22no—
262#126—22no—
263#126—21no—
264#125—21no—
265#125—22no—
266#125—21no—
267#125—21no—
268#125—21no—
269#125—21no—
270#125#6711yescharacteristic
271#124—21no—
272#124#6711yescharacteristic
273#123—22no—
274#123—22no—
275#123—22no—
276#123—22no—
277#123—22no—
278#123—21no—
279#123—22no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8C22×D311yescharacteristic
287#121—21no—
288#121—22no—
289#121—21no—
290#121—22no—
291#121—22no—
292#121—21no—
293#121—22no—
294#121C22×D311yescharacteristic
295#121C22×D312yes—
296C4×Q8#6711yescharacteristic
297C4×D4—22no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—22no—
302C4×D4—21no—
303C4×D4#6711yescharacteristic
304C4×D4#6712yes—
305#117—21no—
306#117—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—61no—
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—62no—
319#106—64no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322#103—62no—
323#102—62no—
324#102—64no—
325C22⋊C8—32no—
326C22⋊C8—32no—
327#97—61no—
328C4×C8—61no—
329#95—32no—
330#95—61no—
331#95—64no—
332#95—64no—
333C23×C3—21no—
334C23×C3—22no—
335C23×C3C22×D411yescharacteristic
336C3×Q8—41no—
337C3×Q8—81no—
338C3×Q8—82no—
339C3×D4—42no—
340C3×D4—41no—
341C3×D4—44no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—81no—
345C3×D4—82no—
346C3×D4—82no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—82no—
360C2×C3×C4—82no—
361C2×C3×C4—82no—
362C2×C3×C4—82no—
363C2×C3×C4#12011yescharacteristic
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12011yescharacteristic
366C2×C3×C4#12011yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×Dic3—42no—
370C2×Dic3—44no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374Dic6—82no—
375Dic6—82no—
376C3⋊C8—42no—
377C3⋊C8—42no—
378#41—82no—
379#41—84no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—124no—
383C2×Q8—121no—
384C2×Q8—121no—
385C2×Q8C2×#6711yescharacteristic
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×#6711yescharacteristic
393C2×D4C2×#6712yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4C2×#6712yes—
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6712yes—
405C22×C4C23×D311yescharacteristic
406C2×C8—62no—
407C2×C8—62no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—121no—
428C4⋊C4—121no—
429C4⋊C4C2×#6712yes—
430C4⋊C4C2×#6711yescharacteristic
431C4⋊C4C2×#6711yescharacteristic
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C42—22no—
445C42—21no—
446C42—122no—
447C42C2×#6711yescharacteristic
448C42C2×#6711yescharacteristic
449C42C2×#6711yescharacteristic
450C42C2×#6711yescharacteristic
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—41no—
459C22×C3C2×#12011yescharacteristic
460C22×C3#57712yes—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—42no—
471C3×C4—41no—
472C3×C4#45412yes—
473Dic3—82no—
474Dic3—82no—
475C23—21no—
476C23—22no—
477C23C22×#6711yescharacteristic
478Q8—41no—
479Q8—81no—
480Q8—82no—
481Q8—242no—
482Q8—242no—
483D4—42no—
484D4—41no—
485D4—44no—
486D4—42no—
487D4—42no—
488D4—81no—
489D4—82no—
490D4—82no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—122no—
508C2×C4—124no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4#95311yescharacteristic
513C2×C4#95311yescharacteristic
514C2×C4#95311yescharacteristic
515C2×C4#95311yescharacteristic
516C2×C4C22×#6711yescharacteristic
517C2×C4C22×#6711yescharacteristic
518C8—122no—
519C8—122no—
520C2×C3—22no—
521C2×C3—81no—
522C2×C3—82no—
523C2×C3#300911yescharacteristic
524C2×C3C2×#45411yescharacteristic
525C2×C3#302811yescharacteristic
526C22—21no—
527C22—21no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22#583612yes—
535C22C2×#95311yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—41no—
545C4—42no—
546C4—41no—
547C4—242no—
548C4—242no—
549C4#524512yes—
550C3#1844111yescharacteristic, complemented, Sylow
551C2—22no—
552C2—81no—
553C2—82no—
554C2#8410511yescharacteristic
555C2C2×#524511yescharacteristic
556C2#8412411yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3559C2
socleC22×C3C2×#120
hypercenter#3559D3
hypocenterC3#18441
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
2. #83271C2semidirect
3. #83124C2semidirect
4. #83043C2semidirect
5. #71887C2semidirect
6. #71875C2semidirect
7. #71812C2semidirect
8. #71754C2semidirect
9. #71737C2semidirect
550. C3#18441semidirect

Maximal subgroups

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

subgroupindex
C3×#35592
#832712
#831242
#830432
#718872
#718752
#718122
#717542
#717372
#184413