Finite Group Databasebeta

← #10770319

Subgroups — #10770319

Order 768.

940 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
1C3×#3563C211yesmaximal, characteristic, complemented
2#83355C211yesmaximal, characteristic, complemented
3#83146C211yesmaximal, characteristic
4#83067C211yesmaximal, characteristic, complemented
5#71902C214yesmaximal, complemented
6#71888C214yesmaximal, complemented
7#71836C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic, complemented
9#71737C211yesmaximal, characteristic, complemented
10#18432—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×#874C2211yescharacteristic
27#5088—21no—
28#5085—21no—
29#5067—21no—
30#5058—21no—
31#5046C2214yes—
32#5046C2212yes—
33#5036C2212yes—
34#5027—22no—
35#5027—21no—
36#5015—21no—
37#4991—22no—
38#4991C2214yes—
39#4988—21no—
40#4988—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4538C2214yes—
46#4537C2212yes—
47#4528C2214yes—
48#3563D311yescharacteristic, complemented
49#3217—31no—
50#3208—31no—
51#2641—31no—
52#1599—34no—
53#1597—34no—
54#1506—31no—
55#1500—31no—
56#1470—31no—
57C22×D12—21no—
58C22×D12—21no—
59C22×D12C2312yes—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D411yescharacteristic
64C3×C4⋊Q8D411yescharacteristic
65C3×#126—22no—
66C3×#126—22no—
67C3×#126—22no—
68C3×#126—22no—
69C3×#126—21no—
70C3×#125—22no—
71C3×#125—22no—
72C3×#125—22no—
73C3×#125—22no—
74C3×#125—22no—
75C3×#125—21no—
76C3×#125—22no—
77C3×#125—22no—
78C3×#125—22no—
79C3×#125—21no—
80C3×#125—21no—
81C3×#125—21no—
82C3×#125—21no—
83C3×#125—21no—
84C3×#125D411yescharacteristic
85C3×#124D411yescharacteristic
86C3×#123—22no—
87C3×#123—21no—
88C3×#123—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—22no—
96C3×C22⋊Q8C2311yescharacteristic
97C3×C22⋊Q8C2312yes—
98C3×#121C2311yescharacteristic
99C3×C4×Q8—21no—
100C3×C4×Q8—22no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×Q8D411yescharacteristic
104C3×C4×Q8D412yes—
105C3×C4×D4—21no—
106C3×C4×D4—22no—
107C3×C4×D4D411yescharacteristic
108C3×#117—22no—
109C3×#117—21no—
110C3×#117—21no—
111C3×#117—22no—
112C3×#117—21no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4D411yescharacteristic
115C2×C3×C4⋊C4D411yescharacteristic
116C2×C3×C4⋊C4D412yes—
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C42C2311yescharacteristic
119C2×#215—22no—
120C2×#215—24no—
121C2×#213—22no—
122C2×(C2×D3)⋊C4—24no—
123C2×(C2×D3)⋊C4—24no—
124C2×C4×Dic3—21no—
125#874—21no—
126#874—21no—
127#874—21no—
128#874—21no—
129C3⋊#95—21no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#809—22no—
134#809—22no—
135#809—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×#127—31no—
154#529C2×D311yescharacteristic
155C2×#125C2×D311yescharacteristic
156#496—61no—
157#492—61no—
158#490—61no—
159#486—61no—
160#460—62no—
161#460—61no—
162#459—61no—
163#450—34no—
164#450—32no—
165#447—32no—
166#432—31no—
167#431—31no—
168#394—34no—
169#394—62no—
170#390—61no—
171#390—61no—
172#376—31no—
173#328—34no—
174#328—32no—
175#327—32no—
176#324—34no—
177C23×D3—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8C2×D411yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4C2×D411yescharacteristic
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—44no—
195C2×D12—41no—
196C2×D12—41no—
197C2×D12—41no—
198C2×D12—41no—
199C2×D12—82no—
200C2×D12—82no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
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—21no—
212C3×C4⋊C4—22no—
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—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4C2×D411yescharacteristic
227C3×C4⋊C4C2×D411yescharacteristic
228C3×C4⋊C4C2×D412yes—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—21no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—21no—
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247#215—82no—
248#215—84no—
249#213—82no—
250(C2×D3)⋊C4—84no—
251(C2×D3)⋊C4—84no—
252C4×Dic3—42no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C22×D4—32no—
256C22×D4—61no—
257C22×D4—61no—
258C2×SD16—64no—
259C2×SD16—62no—
260C2×D8—62no—
261C4⋊Q8—21no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8#6711yescharacteristic
265C4⋊Q8#6711yescharacteristic
266#127—61no—
267#127—61no—
268#127—61no—
269#127—61no—
270#126—22no—
271#126—22no—
272#126—22no—
273#126—22no—
274#126—21no—
275#125—22no—
276#125—22no—
277#125—22no—
278#125—22no—
279#125—21no—
280#125—22no—
281#125—22no—
282#125—22no—
283#125—22no—
284#125—21no—
285#125—21no—
286#125—21no—
287#125—21no—
288#125—21no—
289#125#6711yescharacteristic
290#124#6711yescharacteristic
291#123—22no—
292#123—21no—
293#123—22no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—21no—
300C22⋊Q8—22no—
301C22⋊Q8C22×D311yescharacteristic
302C22⋊Q8C22×D312yes—
303#121C22×D311yescharacteristic
304C4×Q8—21no—
305C4×Q8—22no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8#6711yescharacteristic
309C4×Q8#6712yes—
310C4×D4—21no—
311C4×D4—22no—
312C4×D4#6711yescharacteristic
313#117—22no—
314#117—21no—
315#117—21no—
316#117—22no—
317#117—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4#6711yescharacteristic
320C2×C4⋊C4#6712yes—
321C2×C4⋊C4#6711yescharacteristic
322C2×C4⋊C4C22×D312yes—
323C2×C22⋊C4—64no—
324C2×C22⋊C4—64no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C4⋊C8—61no—
328C4⋊C8—61no—
329#102—62no—
330#102—62no—
331#102—64no—
332C22⋊C8—32no—
333C22⋊C8—32no—
334#97—61no—
335C4×C8—61no—
336#95—32no—
337#95—61no—
338C23×C3—21no—
339C23×C3C22×D411yescharacteristic
340C22×D3—44no—
341C22×D3—44no—
342C22×D3—42no—
343C22×D3—44no—
344C3×Q8—41no—
345C3×Q8—42no—
346C3×Q8—42no—
347C3×Q8—82no—
348C3×D4—41no—
349C3×D4—42no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—82no—
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12011yescharacteristic
366C2×C3×C4#12011yescharacteristic
367C2×C3×C4#12011yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373D12—42no—
374D12—42no—
375D12—44no—
376D12—82no—
377D12—82no—
378D12—82no—
379D12—82no—
380C3⋊C8—42no—
381C3⋊C8—42no—
382C24—62no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8C2×#6712yes—
386C2×Q8C2×#6711yescharacteristic
387C2×D4—21no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—124no—
391C2×D4—121no—
392C2×D4—121no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4—242no—
396C2×D4—242no—
397C2×D4C2×#6711yescharacteristic
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—62no—
401C22×C4C2×#6712yes—
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6712yes—
405C22×C4C23×D311yescharacteristic
406SD16—242no—
407SD16—244no—
408D8—242no—
409C2×C8—62no—
410C2×C8—62no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4C2×#6711yescharacteristic
437C4⋊C4C2×#6711yescharacteristic
438C4⋊C4C2×#6712yes—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—244no—
449C22⋊C4—244no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—21no—
455C42—122no—
456C42C2×#6711yescharacteristic
457C42C2×#6711yescharacteristic
458C42C2×#6711yescharacteristic
459C42C2×#6711yescharacteristic
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3C2×#12011yescharacteristic
466C22×C3#57712yes—
467C2×D3—88no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—41no—
483C3×C4#45412yes—
484Dic3—82no—
485C23—21no—
486C23—124no—
487C23—124no—
488C23—124no—
489C23—122no—
490C23C22×#6711yescharacteristic
491Q8—41no—
492Q8—42no—
493Q8—42no—
494Q8—82no—
495D4—41no—
496D4—42no—
497D4—122no—
498D4—122no—
499D4—124no—
500D4—242no—
501D4—242no—
502D4—242no—
503D4—242no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—82no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4#95311yescharacteristic
522C2×C4#95311yescharacteristic
523C2×C4#95311yescharacteristic
524C2×C4#95311yescharacteristic
525C2×C4C22×#6711yescharacteristic
526C2×C4C22×#6711yescharacteristic
527C8—122no—
528C8—122no—
529C2×C3—22no—
530C2×C3—81no—
531C2×C3#301011yescharacteristic
532C2×C3C2×#45411yescharacteristic
533C2×C3#302711yescharacteristic
534D3—162no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—248no—
541C22—242no—
542C22—242no—
543C22—242no—
544C22#583712yes—
545C22C2×#95311yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—242no—
559C4#524512yes—
560C3#1843211yescharacteristic, complemented, Sylow
561C2—22no—
562C2—81no—
563C2—482no—
564C2#8411311yescharacteristic
565C2C2×#524511yescharacteristic
566C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3563C2
socleC22×C3C2×#120
prime radical#83146C2
hypercenter#3563D3
hypocenterC3#18432
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
1. C3×#3563C2semidirect
2. #83355C2semidirect
4. #83067C2semidirect
5. #71902C2semidirect
6. #71888C2semidirect
7. #71836C2semidirect
8. #71754C2semidirect
9. #71737C2semidirect
48. #3563D3semidirect
560. C3#18432semidirect

Maximal subgroups

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

subgroupindex
C3×#35632
#833552
#831462
#830672
#719022
#718882
#718362
#717542
#717372
#184323