Finite Group Databasebeta

← #10770328

Subgroups — #10770328

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3563C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic, complemented
3#83154C211yesmaximal, characteristic
4#83044C211yesmaximal, characteristic, complemented
5#71904C214yesmaximal, complemented
6#71889C214yesmaximal
7#71836C211yesmaximal, characteristic, complemented
8#71754C211yesmaximal, characteristic
9#71737C211yesmaximal, characteristic, complemented
10#18433—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#5093—21no—
28#5092—21no—
29#5069—21no—
30#5060—21no—
31#5050C2214yes—
32#5050C2212yes—
33#5040C2212yes—
34#5031—22no—
35#5031—21no—
36#5019—21no—
37#4981—22no—
38#4979C2214yes—
39#4977—21no—
40#4975—21no—
41#4932C2211yescharacteristic
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4538C2212yes—
45#4538C2214yes—
46#4537C2212yes—
47#4528C2214yes—
48#3563D311yescharacteristic
49#3218—31no—
50#3209—31no—
51#2660—31no—
52#1605—34no—
53#1603—34no—
54#1506—31no—
55#1500—31no—
56#1470—31no—
57C22×Dic6C2312yes—
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×#216—22no—
118C2×#216—24no—
119C2×#214—22no—
120C2×C4⋊Dic3—21no—
121C2×C4⋊Dic3—21no—
122C2×Dic3⋊C4—24no—
123C2×C4×Dic3—21no—
124#869—21no—
125#869—21no—
126#869—21no—
127#869—21no—
128C3⋊#95—21no—
129C3⋊#95—24no—
130C3⋊#95C2312yes—
131#830C2312yes—
132#830C2312yes—
133#810—22no—
134#810—22no—
135#810—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#497—61no—
157#494—61no—
158#491—61no—
159#487—61no—
160#462—62no—
161#462—61no—
162#461—61no—
163#451—34no—
164#451—32no—
165#448—32no—
166#432—31no—
167#431—31no—
168#398—62no—
169#395—34no—
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—22no—
193C2×Dic6—22no—
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
241#216—82no—
242#216—84no—
243#214—82no—
244C4⋊Dic3—41no—
245C4⋊Dic3—41no—
246C4⋊Dic3—41no—
247C4⋊Dic3—41no—
248Dic3⋊C4—84no—
249C4×Dic3—24no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C22×Q8—32no—
253C2×Q16—64no—
254C2×Q16—62no—
255C2×SD16—62no—
256C4⋊Q8—21no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—61no—
260C4⋊Q8—61no—
261C4⋊Q8—61no—
262C4⋊Q8—61no—
263C4⋊Q8#6711yescharacteristic
264C4⋊Q8#6711yescharacteristic
265#126—22no—
266#126—22no—
267#126—22no—
268#126—22no—
269#126—21no—
270#125—22no—
271#125—22no—
272#125—22no—
273#125—22no—
274#125—21no—
275#125—22no—
276#125—22no—
277#125—22no—
278#125—22no—
279#125—21no—
280#125—21no—
281#125—21no—
282#125—21no—
283#125—21no—
284#125#6711yescharacteristic
285#124#6711yescharacteristic
286#123—22no—
287#123—21no—
288#123—22no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—21no—
295C22⋊Q8—22no—
296C22⋊Q8C22×D311yescharacteristic
297C22⋊Q8C22×D312yes—
298#121C22×D311yescharacteristic
299C4×Q8—21no—
300C4×Q8—22no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8#6711yescharacteristic
304C4×Q8#6712yes—
305C4×D4—21no—
306C4×D4—22no—
307C4×D4#6711yescharacteristic
308#117—22no—
309#117—21no—
310#117—21no—
311#117—22no—
312#117—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—64no—
317C2×C4⋊C4#6711yescharacteristic
318C2×C4⋊C4#6712yes—
319C2×C4⋊C4#6711yescharacteristic
320C2×C4⋊C4C22×D312yes—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323C4⋊C8—61no—
324C4⋊C8—61no—
325#103—62no—
326#103—62no—
327#103—64no—
328C22⋊C8—32no—
329C22⋊C8—32no—
330#97—61no—
331C4×C8—61no—
332#95—32no—
333#95—61no—
334#95—64no—
335C23×C3—21no—
336C23×C3C22×D411yescharacteristic
337C3×Q8—41no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—82no—
341C3×D4—41no—
342C3×D4—42no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—82no—
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4#12011yescharacteristic
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×Dic3—24no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—44no—
367C2×Dic3—42no—
368Dic6—42no—
369Dic6—42no—
370Dic6—44no—
371C3⋊C8—42no—
372C3⋊C8—42no—
373C2×Q8—22no—
374C2×Q8—21no—
375C2×Q8—62no—
376C2×Q8—62no—
377C2×Q8—242no—
378C2×Q8C2×#6712yes—
379C2×Q8C2×#6711yescharacteristic
380C2×D4—21no—
381C2×D4C2×#6711yescharacteristic
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4C2×#6712yes—
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C2×#6712yes—
390C22×C4C23×D311yescharacteristic
391Q16—242no—
392Q16—244no—
393SD16—242no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—121no—
422C4⋊C4—121no—
423C4⋊C4—121no—
424C4⋊C4—121no—
425C4⋊C4—244no—
426C4⋊C4C2×#6711yescharacteristic
427C4⋊C4C2×#6711yescharacteristic
428C4⋊C4C2×#6712yes—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—21no—
443C42—64no—
444C42C2×#6711yescharacteristic
445C42C2×#6711yescharacteristic
446C42C2×#6711yescharacteristic
447C42C2×#6711yescharacteristic
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3C2×#12011yescharacteristic
454C22×C3#57812yes—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—42no—
461C3×C4—41no—
462C3×C4—41no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—41no—
467C3×C4#45612yes—
468Dic3—44no—
469Dic3—82no—
470C23—21no—
471C23C22×#6711yescharacteristic
472Q8—41no—
473Q8—42no—
474Q8—42no—
475Q8—82no—
476Q8—122no—
477Q8—122no—
478Q8—124no—
479D4—41no—
480D4—42no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—64no—
495C2×C4—82no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—124no—
499C2×C4—122no—
500C2×C4#95311yescharacteristic
501C2×C4#95311yescharacteristic
502C2×C4#95311yescharacteristic
503C2×C4#95311yescharacteristic
504C2×C4C22×#6711yescharacteristic
505C2×C4C22×#6711yescharacteristic
506C8—122no—
507C8—122no—
508C2×C3—22no—
509C2×C3—81no—
510C2×C3#301111yescharacteristic
511C2×C3C2×#45611yescharacteristic
512C2×C3#302711yescharacteristic
513C22—21no—
514C22—21no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22#583812yes—
519C22C2×#95311yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—124no—
533C4—242no—
534C4#524612yes—
535C3#1843311yescharacteristic, complemented, Sylow
536C2—22no—
537C2—81no—
538C2#8411711yescharacteristic
539C2C2×#524611yescharacteristic
540C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3563C2
socleC22×C3C2×#120
prime radicalC3×#121C23
hypercenter#3563D3
hypocenterC3#18433
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. #83370C2semidirect
4. #83044C2semidirect
5. #71904C2semidirect
7. #71836C2semidirect
9. #71737C2semidirect
535. C3#18433semidirect

Maximal subgroups

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

subgroupindex
C3×#35632
#833702
#831542
#830442
#719042
#718892
#718362
#717542
#717372
#184333