Finite Group Databasebeta

← #10769248

Subgroups — #10769248

Order 768.

888 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86274C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic, complemented
3#79272C211yesmaximal, characteristic, complemented
4#74194C211yesmaximal, characteristic
5C3×#1518C211yesmaximal, characteristic, complemented
6#70782C211yesmaximal, characteristic
7#69546C211yesmaximal, characteristic, complemented
8#68331C214yesmaximal, complemented
9#68322C214yesmaximal
10#18311—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12#5729—21no—
13#5729—21no—
14#5689—21no—
15#5689—21no—
16#5634—21no—
17#5634C2212yes—
18#5623—21no—
19#5623C2212yes—
20#5578—22no—
21#5578—22no—
22D3×#117—21no—
23#5517—21no—
24C2×#872C2211yescharacteristic
25#5470C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C3×#431C2211yescharacteristic
28C3×#382C2211yescharacteristic
29#5079—21no—
30#5078—21no—
31#5044C2214yes—
32#5021—22no—
33#4987—21no—
34#4984—21no—
35#4925C2211yescharacteristic
36#4850—22no—
37#4722C2214yes—
38#4682—21no—
39#4681—21no—
40#4677C2212yes—
41#4673—21no—
42#4659C2212yes—
43#4656—21no—
44C3×#328C2214yes—
45#4469C2212yes—
46#4468C2212yes—
47#4452C2214yes—
48#3458—31no—
49#3230—31no—
50#3230—31no—
51#2628—31no—
52#1617—34no—
53#1614—34no—
54#1518—31no—
55#1518D311yescharacteristic, complemented
56#1458—31no—
57C22×C4×D3D411yescharacteristic
58C22×Dic6—21no—
59C3×C4⋊Q8D411yescharacteristic
60C3×C4⋊Q8D411yescharacteristic
61C3×#125—21no—
62C3×C22⋊Q8—21no—
63C3×C22⋊Q8C2312yes—
64C3×C4×Q8—21no—
65C3×C4×Q8D412yes—
66C3×#117—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C42C2311yescharacteristic
69#946—22no—
70#946—22no—
71#944—22no—
72#944—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4D411yescharacteristic
75C2×(C2×D3)⋊C4C2311yescharacteristic
76#926—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3C2312yes—
80#924—21no—
81#924C2312yes—
82C2×Dic3⋊C4D411yescharacteristic
83C2×Dic3⋊C4C2311yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85#897—22no—
86#897—22no—
87#896—22no—
88#896—22no—
89#896—22no—
90#896—22no—
91#892—21no—
92#892—21no—
93D3×C4⋊C4—21no—
94D3×C4⋊C4—21no—
95#890—21no—
96#890—21no—
97#889—22no—
98#889—22no—
99#889—22no—
100#889—22no—
101#888—21no—
102#888—21no—
103#888—22no—
104#888—22no—
105D3×C22⋊C4—22no—
106#878—22no—
107#878—22no—
108#877—22no—
109#872—22no—
110#872—21no—
111#872—21no—
112#872—21no—
113#872—21no—
114C42×D3—22no—
115#870—21no—
116#869—21no—
117#869—21no—
118#869—21no—
119#869—21no—
120#869D411yescharacteristic
121#869D411yescharacteristic
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6—21no—
125C4×Dic6D412yes—
126C3×C4⋊C8—21no—
127C3×C22⋊C8C2312yes—
128C3×C4×C8—21no—
129C3×#95—21no—
130C3×#95C2312yes—
131#835—24no—
132C3⋊#95—22no—
133C3⋊#95—22no—
134#830C2312yes—
135#818—24no—
136#816—24no—
137#807—24no—
138C4⋊(C3⋊C8)—21no—
139C4×C3⋊C8—21no—
140#544—32no—
141#544—61no—
142#544—61no—
143#544—61no—
144#544—61no—
145#544—61no—
146#543—32no—
147#543—61no—
148#533—31no—
149#533—61no—
150#533—62no—
151#533—62no—
152#533C2×D311yescharacteristic
153C2×C4⋊Q8—31no—
154C2×#117—31no—
155C2×#117—61no—
156#500—61no—
157#500—61no—
158#487—61no—
159#487—61no—
160#484—34no—
161#484—34no—
162#477—62no—
163#477—62no—
164#431—31no—
165#431C2×D311yescharacteristic
166#400—32no—
167#399—32no—
168#395—61no—
169#393—61no—
170#389—61no—
171#382C2×D311yescharacteristic
172#380—61no—
173#328—34no—
174#328C2×D314yes—
175#324—32no—
176#323—32no—
177C23×D3C2×D411yescharacteristic
178C2×C3×Q8—21no—
179C2×C3×Q8C2×D412yes—
180C22×C3×C4—22no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—22no—
185C22×Dic3C2×D411yescharacteristic
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C2×C4×D3—21no—
189C2×C4×D3—21no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3Q1614yes—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—21no—
197C2×Dic6—41no—
198C2×Dic6—41no—
199C2×Dic6—82no—
200C2×Dic6C2×D412yes—
201C2×C3×C8—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—22no—
210C3×C42—22no—
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215#217—22no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—21no—
220(C2×D3)⋊C4—21no—
221(C2×D3)⋊C4—21no—
222(C2×D3)⋊C4—82no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3—41no—
231C4⋊Dic3—41no—
232C4⋊Dic3—82no—
233C4⋊Dic3C2×D412yes—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236Dic3⋊C4—22no—
237Dic3⋊C4—22no—
238Dic3⋊C4—22no—
239Dic3⋊C4—22no—
240Dic3⋊C4—22no—
241Dic3⋊C4—21no—
242Dic3⋊C4—21no—
243Dic3⋊C4—22no—
244Dic3⋊C4—21no—
245Dic3⋊C4—21no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—21no—
249C4×Dic3—22no—
250C4×Dic3—21no—
251C2×C3⋊C8—22no—
252C22×Q8—61no—
253C23×C4—31no—
254C4⋊Q8—31no—
255C4⋊Q8—31no—
256C4⋊Q8—62no—
257C4⋊Q8—62no—
258C4⋊Q8—61no—
259C4⋊Q8—61no—
260C4⋊Q8—62no—
261C4⋊Q8—62no—
262C4⋊Q8—61no—
263C4⋊Q8—61no—
264C4⋊Q8—61no—
265C4⋊Q8—61no—
266C4⋊Q8D1211yescharacteristic
267C4⋊Q8D1211yescharacteristic
268#125—21no—
269#125—62no—
270#125—62no—
271#125—62no—
272#125—61no—
273#125—61no—
274#125—62no—
275#125—61no—
276C22⋊Q8—21no—
277C22⋊Q8—32no—
278C22⋊Q8—62no—
279C22⋊Q8—62no—
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282C22⋊Q8—62no—
283C22⋊Q8—62no—
284C22⋊Q8—62no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C22⋊Q8—61no—
288C22⋊Q8—62no—
289C22⋊Q8C22×D312yes—
290C4×Q8—21no—
291C4×Q8—32no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—61no—
295C4×Q8D1212yes—
296#117—21no—
297#117—62no—
298#117—62no—
299#117—61no—
300#117—61no—
301#117—61no—
302#117—61no—
303#117—61no—
304#117—61no—
305#117—61no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—62no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C42—31no—
319C2×C42—62no—
320C2×C42C22×D311yescharacteristic
321#107—64no—
322#107—64no—
323C4⋊C8—21no—
324C4⋊C8—61no—
325#103—64no—
326#103—64no—
327C22⋊C8—32no—
328C22⋊C8C22×D312yes—
329C4×C8—21no—
330C4×C8—61no—
331#95—21no—
332#95—62no—
333#95—62no—
334#95C22×D312yes—
335C23×C3C22×D411yescharacteristic
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3C2×Q1612yes—
341C3×Q8—42no—
342C3×Q8—81no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4#12011yescharacteristic
351C2×C3×C4#12011yescharacteristic
352C2×C3×C4#12011yescharacteristic
353C2×C3×C4#12011yescharacteristic
354C2×C3×C4C2×Q1612yes—
355C2×C3×C4#13712yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D411yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3C2×Q1612yes—
370C4×D3—24no—
371C4×D3—24no—
372C4×D3—82no—
373C4×D3—82no—
374Dic6—42no—
375Dic6—81no—
376Dic6—82no—
377Dic6—82no—
378Dic6—82no—
379Dic6—82no—
380C3×C8—42no—
381C3⋊C8—42no—
382C24—31no—
383C2×Q8—21no—
384C2×Q8—32no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×Q8—61no—
388C2×Q8—121no—
389C2×Q8—121no—
390C2×Q8—242no—
391C2×Q8C2×D1212yes—
392C22×C4—22no—
393C22×C4—31no—
394C22×C4—34no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4D3×D411yescharacteristic
404C22×C4D3×D411yescharacteristic
405C22×C4C23×D311yescharacteristic
406C2×C8—22no—
407C2×C8—62no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—32no—
414C4⋊C4—42no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—62no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—62no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—62no—
431C4⋊C4—61no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4—121no—
435C4⋊C4—121no—
436C4⋊C4—242no—
437C4⋊C4C2×D1212yes—
438C22⋊C4—22no—
439C22⋊C4—62no—
440C22⋊C4—62no—
441C22⋊C4—62no—
442C22⋊C4—61no—
443C22⋊C4—61no—
444C22⋊C4—61no—
445C22⋊C4—61no—
446C22⋊C4—242no—
447C42—22no—
448C42—62no—
449C42—62no—
450C42—61no—
451C42—61no—
452C42—62no—
453C42C2×D1211yescharacteristic
454C42C2×D1211yescharacteristic
455C42D3×D411yescharacteristic
456C42D3×D411yescharacteristic
457C22×C3—21no—
458C22×C3—21no—
459C22×C3C2×#12011yescharacteristic
460C22×C3C22×Q1611yescharacteristic
461C22×C3C2×#13711yescharacteristic
462C2×D3—22no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—42no—
466C2×D3—81no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—82no—
474C3×C4#45112yes—
475C3×C4#45112yes—
476Dic3—22no—
477Dic3—42no—
478Dic3—41no—
479Dic3—42no—
480Dic3—41no—
481Dic3—82no—
482C23—32no—
483C23—61no—
484C23—61no—
485C23—121no—
486C23—121no—
487C23C2×D3×D411yescharacteristic
488Q8—42no—
489Q8—81no—
490Q8—122no—
491Q8—241no—
492Q8—242no—
493Q8—242no—
494Q8—242no—
495Q8—242no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—32no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—62no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—242no—
518C2×C4—242no—
519C2×C4#88211yescharacteristic
520C2×C4#88211yescharacteristic
521C2×C4#88211yescharacteristic
522C2×C4#88211yescharacteristic
523C2×C4#91512yes—
524C2×C4D3×Q1612yes—
525C2×C4C22×D1211yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C8—42no—
528C8—122no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3C2×#45111yescharacteristic
532C2×C3#302111yescharacteristic
533C2×C3#302811yescharacteristic
534D3—42no—
535C22—21no—
536C22—21no—
537C22—62no—
538C22—122no—
539C22—121no—
540C22—121no—
541C22—241no—
542C22C2×#88211yescharacteristic
543C22C2×#91511yescharacteristic
544C22C2×D3×Q1611yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—42no—
550C4—42no—
551C4—62no—
552C4—82no—
553C4—121no—
554C4—122no—
555C4—121no—
556C4—122no—
557C4—242no—
558C4#480912yes—
559C4#480912yes—
560C3#1831111yescharacteristic, complemented, Sylow
561C2—21no—
562C2—21no—
563C2—122no—
564C2#7869211yescharacteristic
565C2C2×#480911yescharacteristic
566C2#7899411yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1518C2
socleC22×C3C2×#120
prime radicalC23×D3C2×D4
hypercenter#1518D3
hypocenterC3#18311
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #83324C2semidirect
3. #79272C2semidirect
5. C3×#1518C2semidirect
7. #69546C2semidirect
8. #68331C2semidirect
55. #1518D3semidirect
560. C3#18311semidirect

Maximal subgroups

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

subgroupindex
#862742
#833242
#792722
#741942
C3×#15182
#707822
#695462
#683312
#683222
#183113