Finite Group Databasebeta

← #27286392

Subgroups — #27286392

Order 13568.

744 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
1C2×#16810815C211yesmaximal, characteristic
2#20279633C211yesmaximal, characteristic, complemented
3#20275522C211yesmaximal, characteristic, complemented
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#460C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133193—21no—
10C2×C4×D212C2211yescharacteristic
11#16810815—22no—
12#16810815—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133138—21no—
15C2×#13133138—22no—
16C2×C4×#88322C2211yescharacteristic
17#16810545—22no—
18#16810525—21no—
19#16810521—21no—
20#16810493C2211yescharacteristic
21#16810480—21no—
22Q8⋊#88322—22no—
23#16810350—22no—
24C22×SD16×C53—21no—
25C2×#127×C53C2211yescharacteristic
26C2×C4×Q8×C53C2211yescharacteristic
27C53×#460—22no—
28C53×#460—22no—
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×#102×C53—21no—
31C22×D212D411yescharacteristic
32C22×C4×D53—21no—
33C2×#11719922—24no—
34C2×#11719922—42no—
35C2×#11719922—42no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133193—42no—
39#13133193—42no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×C4⋊Dic53D412yes—
48C2×#11719919—22no—
49C2×#11719921—22no—
50C2×#11719921—22no—
51C2×#11719921—21no—
52C4⋊#88322—21no—
53C4⋊#88322D412yes—
54#13133138—24no—
55#13133138—44no—
56C4×#88322—21no—
57C4×#88322—21no—
58#13133124—22no—
59C22×Q8×C53D411yescharacteristic
60C22×D4×C53—21no—
61C22×D4×C53—21no—
62C22×D4×C53—21no—
63C2×SD16×C53—24no—
64C2×SD16×C53—42no—
65C2×SD16×C53—42no—
66C22×C8×C53—21no—
67#127×C53—21no—
68#127×C53—21no—
69#127×C53—21no—
70#127×C53D412yes—
71C4×Q8×C53—21no—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53D411yescharacteristic
77C2×C42×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C4⋊C8×C53D412yes—
81#102×C53—24no—
82C24×C53—41no—
83C23×D53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53D812yescomplemented
90C2×D4×C53—22no—
91C2×D4×C53—22no—
92C2×D4×C53—22no—
93C2×D4×C53—42no—
94C2×D4×C53—42no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—41no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53—41no—
105C2×D4×C53—41no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×Dic53—21no—
112C22×Dic53—22no—
113C2×D212—21no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—21no—
117C2×D212—21no—
118C2×D212SD1612yescomplemented
119C2×C4×D53—41no—
120C2×C4×D53—41no—
121C2×C4×D53—41no—
122C2×C4×D53—41no—
123C2×C4×D53—41no—
124C2×C4×D53—41no—
125SD16×C53—48no—
126C2×C8×C53—22no—
127C2×C8×C53—41no—
128C2×C8×C53—41no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C42×C53—41no—
134C42×C53—41no—
135C42×C53C2×D411yescharacteristic
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139#11719922—48no—
140#11719921—44no—
141#11719921—41no—
142#11719921—41no—
143#11719921—82no—
144C4⋊Dic53—21no—
145C4⋊Dic53—21no—
146C4⋊Dic53—41no—
147C4⋊Dic53—41no—
148C4⋊Dic53—41no—
149C4⋊Dic53—41no—
150C4⋊Dic53—42no—
151#11719919—44no—
152C2×#88322—22no—
153C2×#88322—22no—
154C2×#88322—41no—
155C2×#88322—41no—
156C2×#88322—41no—
157C2×#88322—41no—
158C23×C53—42no—
159C23×C53—81no—
160C23×C53—81no—
161C23×C53—81no—
162C23×C53—81no—
163C23×C53—81no—
164C23×C53—81no—
165C23×C53#12011yescharacteristic
166C22×D53—22no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C22×D53—41no—
172C22×D53—41no—
173Q8×C53—22no—
174Q8×C53—22no—
175Q8×C53—41no—
176Q8×C53—41no—
177D4×C53—44no—
178D4×C53—42no—
179D4×C53—42no—
180D4×C53—42no—
181D4×C53—42no—
182D4×C53—82no—
183D4×C53—81no—
184D4×C53—81no—
185D4×C53—81no—
186D4×C53—81no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—22no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53#12011yescharacteristic
196C2×C4×C53#12011yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12711yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C2×SD1611yescharacteristic
203C2×C4×C53#13511yescharacteristic
204C2×C4×C53#13511yescharacteristic
205C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
206C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
207C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
208C2×C4×C53#13711yescharacteristic
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—42no—
212C2×Dic53—41no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215D212—22no—
216D212—22no—
217D212—41no—
218D212—41no—
219C4×D53—81no—
220C4×D53—81no—
221C4×D53—81no—
222C4×D53—81no—
223C8×C53—42no—
224#88322—42no—
225#88322—42no—
226#10229—531nomaximal, Sylow
227C22×C53—82no—
228C22×C53—82no—
229C22×C53—82no—
230C22×C53—161no—
231C22×C53—161no—
232C22×C53—161no—
233C22×C53—161no—
234C22×C53#39211yescharacteristic
235C22×C53#44711yescharacteristic
236C22×C53#44811yescharacteristic
237C22×C53#44911yescharacteristic
238C22×C53#44911yescharacteristic
239C22×C53#45311yescharacteristic
240C22×C53#45411yescharacteristic
241C2×D53—42no—
242C2×D53—42no—
243C2×D53—42no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—81no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—42no—
251C4×C53—81no—
252C4×C53#44711yescharacteristic
253C4×C53#44911yescharacteristic
254C4×C53#45011yescharacteristic
255C4×C53#45211yescharacteristic
256C4×C53#46511yescharacteristic
257C4×C53#46611yescharacteristic
258C4×C53#46811yescharacteristic
259C4×C53#46911yescharacteristic
260Dic53—82no—
261Dic53—81no—
262C2×#460D5311yescharacteristic, complemented
263C2×#459—531no—
264#2538—531no—
265#1957—531no—
266C2×Q8⋊C8—531no—
267C2×D4⋊C8—531no—
268#1296—531no—
269C2×C53—162no—
270C2×C53#162111yescharacteristic
271C2×C53#162211yescharacteristic
272C2×C53#162711yescharacteristic
273C2×C53#162811yescharacteristic
274C2×C53#199911yescharacteristic
275C2×C53#200011yescharacteristic
276C2×C53#200611yescharacteristic
277D53—82no—
278C22×SD16—21no—
279C22×D8—1061no—
280C2×#127C2×D5311yescharacteristic
281C2×C22⋊Q8—1061no—
282C2×C4×Q8C2×D5311yescharacteristic
283C2×C4×D4—531no—
284#460—22no—
285#460—22no—
286#459—1062no—
287#459—1062no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8C2×D5311yescharacteristic
290C2×#102—21no—
291C2×#102—1061no—
292C2×#102—1062no—
293C2×C4×C8—531no—
294#396—1062no—
295#390—1061no—
296#389—1061no—
297#384—531no—
298#380—1061no—
299Q8⋊C8—1062no—
300D4⋊C8—1062no—
301C53#1022911yescharacteristic, complemented, Sylow
302C22×Q8C22⋊D5311yescharacteristic
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—531no—
307C23×C4—1061no—
308C2×SD16—24no—
309C2×SD16—42no—
310C2×SD16—42no—
311C2×D8—1064no—
312C2×D8—2122no—
313C2×D8—2122no—
314C22×C8—21no—
315C22×C8—1061no—
316C22×C8—1061no—
317#127—21no—
318#127—21no—
319#127—21no—
320#127C22⋊D5312yes—
321C22⋊Q8—2122no—
322C22⋊Q8—2122no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4—1061no—
330C4×D4—1061no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—531no—
333C2×C4⋊C4—532no—
334C2×C4⋊C4—1062no—
335C2×C4⋊C4—1061no—
336C2×C4⋊C4—1061no—
337C2×C4⋊C4C22⋊D5311yescharacteristic
338C2×C22⋊C4—1062no—
339C2×C22⋊C4—1062no—
340C2×C22⋊C4—1061no—
341C2×C42—21no—
342C2×C42C22×D5311yescharacteristic
343C4⋊C8—21no—
344C4⋊C8—532no—
345C4⋊C8—1061no—
346C4⋊C8C22⋊D5312yes—
347#102—24no—
348#102—1064no—
349#102—2124no—
350C4×C8—1061no—
351C4×C8—1061no—
352#95—1062no—
353C24—41no—
354C24—1061no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8#1171992212yescomplemented
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—532no—
378C2×D4—1061no—
379C2×D4—1061no—
380C2×D4—1061no—
381C2×D4—1061no—
382C2×D4—1061no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—1061no—
386C22×C4—1062no—
387C22×C4—2121no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4—2121no—
393C22×C4D4×D5311yescharacteristic
394C22×C4D4×D5311yescharacteristic
395C22×C4C2×C22⋊D5311yescharacteristic
396SD16—48no—
397D8—2128no—
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—1062no—
402C2×C8—1062no—
403C2×C8—2121no—
404C2×C8—2121no—
405C2×C8—2121no—
406C2×C8—2121no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4—2122no—
414C4⋊C4—2124no—
415C4⋊C4—2121no—
416C4⋊C4—2121no—
417C4⋊C4—2121no—
418C4⋊C4—2121no—
419C22⋊C4—4242no—
420C22⋊C4—2124no—
421C22⋊C4—2121no—
422C22⋊C4—2121no—
423C42—41no—
424C42—41no—
425C42D4×D5311yescharacteristic
426C42#1171994711yescharacteristic
427C42C2×C22⋊D5311yescharacteristic
428C42C2×C22⋊D5311yescharacteristic
429C23—42no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—1062no—
437C23—2121no—
438C23—2121no—
439C23—2121no—
440C23—2121no—
441C23—2121no—
442C23—2121no—
443C23#1313318511yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—82no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—1062no—
459D4—1062no—
460D4—2121no—
461D4—2121no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—4241no—
471C2×C4—4241no—
472C2×C4—4241no—
473C2×C4—4241no—
474C2×C4—2122no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2122no—
478C2×C4—2121no—
479C2×C4—2122no—
480C2×C4#1313318511yescharacteristic
481C2×C4#1313318611yescharacteristic
482C2×C4#1313318811yescharacteristic
483C2×C4#1313318911yescharacteristic
484C2×C4#1313318911yescharacteristic
485C2×C4#1313321311yescharacteristic
486C2×C4C2×#1171992211yescharacteristic
487C2×C4SD16×D5311yescharacteristic
488C2×C4#1313323711yescharacteristic
489C2×C4#1313323811yescharacteristic
490C2×C4#1313324211yescharacteristic
491C2×C4#1313324311yescharacteristic
492C2×C4#1313324411yescharacteristic
493C2×C4#1313324611yescharacteristic
494C8—42no—
495C8—2122no—
496C8—2122no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—4241no—
505C22—4241no—
506C22—4241no—
507C22—4241no—
508C22—2122no—
509C22—2122no—
510C22—2122no—
511C22#1681053111yescharacteristic
512C22#1681075611yescharacteristic
513C22#1681076011yescharacteristic
514C22#1681076611yescharacteristic
515C22#1681076711yescharacteristic
516C22#1681078911yescharacteristic
517C22#1681079411yescharacteristic
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—4242no—
523C4—4241no—
524C4#1681075711yescharacteristic
525C4#1681076811yescharacteristic
526C4#1681077211yescharacteristic
527C4#1681078211yescharacteristic
528C4#1681084911yescharacteristic
529C4#1681085711yescharacteristic
530C4#1681086611yescharacteristic
531C4#1681087411yescharacteristic
532C2—162no—
533C2—4242no—
534C2#2027377811yescharacteristic
535C2#2027378611yescharacteristic
536C2#2027381611yescharacteristic
537C2#2027382411yescharacteristic
538C2#2027571111yescharacteristic
539C2#2027571511yescharacteristic
540C2#2027575511yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#460C2
socleC23×C53#120
prime radicalC2×#16810815C2
hypercenterC2×#460D53
hypocenterC53#10229
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #20279633C2semidirect
3. #20275522C2semidirect
4. C2×Q8⋊#88322C2semidirect
5. C2×#16810350C2semidirect
6. #20272599C2semidirect
7. C2×C53×#460C2semidirect
89. C2×Q8×C53D8semidirect
118. C2×D212SD16semidirect
262. C2×#460D53semidirect
301. C53#10229semidirect
360. C2×Q8#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168108152
#202796332
#202755222
C2×Q8⋊#883222
C2×#168103502
#202725992
C2×C53×#4602
#1022953