Finite Group Databasebeta

← #27285258

Subgroups — #27285258

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic, complemented
2#20279661C211yesmaximal, characteristic
3#20275668C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#461C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133213—21no—
10C2×#13133191—22no—
11#16810971—22no—
12#16810971—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—22no—
15C2×C4×#88322C2211yescharacteristic
16#16810532—22no—
17#16810524—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—22no—
20#16810366—22no—
21D4⋊#88322—22no—
22C22×SD16×C53—21no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4×D4×C53C2211yescharacteristic
25C53×#461—22no—
26C53×#461—22no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#103×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C22×C22⋊D53—24no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133214—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214—21no—
39#13133213—42no—
40#13133213—42no—
41#13133191—44no—
42#13133191—44no—
43C2×C4⋊Dic53D412yes—
44C2×#11719921—24no—
45C2×C4×Dic53—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133137—28no—
49C4×#88322—21no—
50C4×#88322D412yescomplemented
51#13133124—21no—
52C22×Q8×C53—21no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C2×SD16×C53—24no—
56C2×SD16×C53—42no—
57C2×SD16×C53—42no—
58C22×C8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53D412yescomplemented
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73C4⋊C8×C53D412yescomplemented
74#103×C53—24no—
75C24×C53—21no—
76C23×D53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53D812yes—
89C22×C4×C53—21no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C2×C22⋊D53—48no—
99C2×C22⋊D53—48no—
100C2×C22⋊D53—84no—
101C22×Dic53—22no—
102C2×D212—22no—
103C2×D212—22no—
104C2×D212—44no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—42no—
111C2×D212—41no—
112C2×D212—42no—
113C2×D212—41no—
114C2×D212—41no—
115SD16×C53—48no—
116C2×C8×C53—22no—
117C2×C8×C53—41no—
118C2×C8×C53—41no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131#11719921—48no—
132C4⋊Dic53—22no—
133C4⋊Dic53SD1614yescomplemented
134C4×Dic53—41no—
135C4×Dic53—41no—
136C2×#88322—22no—
137C2×#88322—42no—
138C2×#88322—41no—
139C2×#88322—42no—
140C23×C53—22no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53#12011yescharacteristic
146C22×D53—42no—
147C22×D53—42no—
148C22×D53—42no—
149C22×D53—44no—
150C22×D53—44no—
151Q8×C53—42no—
152Q8×C53—42no—
153Q8×C53—81no—
154Q8×C53—81no—
155D4×C53—22no—
156D4×C53—22no—
157D4×C53—41no—
158D4×C53—41no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53—82no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12711yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53C2×SD1612yes—
176C2×C4×C53#13511yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C53AGL(1,ℤ/8)12yes—
180C22⋊D53—816no—
181C2×Dic53—24no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184D212—44no—
185D212—82no—
186D212—81no—
187D212—81no—
188D212—82no—
189D212—81no—
190D212—81no—
191D212—81no—
192D212—81no—
193C8×C53—42no—
194#88322—42no—
195#88322—81no—
196#10062—531nomaximal, Sylow
197C22×C53—44no—
198C22×C53—42no—
199C22×C53—82no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53#39211yescharacteristic
203C22×C53#44711yescharacteristic
204C22×C53#44911yescharacteristic
205C22×C53#45012yes—
206C22×C53#45512yes—
207C2×D53—84no—
208C2×D53—84no—
209C2×D53—82no—
210C2×D53—82no—
211C2×D53—82no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—82no—
215C4×C53—81no—
216C4×C53#44711yescharacteristic
217C4×C53#44911yescharacteristic
218C4×C53#46012yes—
219C4×C53#46611yescharacteristic
220C4×C53#46911yescharacteristic
221C4×C53#49211yescharacteristic
222C4×C53#49611yescharacteristic
223Dic53—44no—
224C2×#486—531no—
225C2×#461D5311yescharacteristic, complemented
226#2541—531no—
227#1988—531no—
228C2×#331—531no—
229C2×D4⋊C8—531no—
230#1279—531no—
231C2×C53—82no—
232C2×C53#163912yes—
233C2×C53#164212yes—
234C2×C53#199911yescharacteristic
235C2×C53#200411yescharacteristic
236C2×C53#201411yescharacteristic
237D53—162no—
238C22×SD16—21no—
239C2×C4⋊Q8C2×D5311yescharacteristic
240C2×#127—531no—
241C2×#127—1061no—
242C2×#121—1062no—
243C2×C4×D4C2×D5311yescharacteristic
244#486—1062no—
245#486—1062no—
246#461—22no—
247#461—22no—
248C2×C4⋊C8—531no—
249C2×C4⋊C8C2×D5311yescharacteristic
250C2×#103—21no—
251C2×#102—1062no—
252C2×C4×C8—531no—
253#392—1062no—
254#390—1061no—
255C4×C4⋊C4—531no—
256#340—1062no—
257#331—1062no—
258D4⋊C8—1062no—
259C53#1006211yescharacteristic, complemented, Sylow
260C22×Q8—21no—
261C22×D4—532no—
262C22×D4—1064no—
263C22×D4—1061no—
264C22×D4—1061no—
265C22×D4C22⋊D5311yescharacteristic
266C23×C4—21no—
267C2×SD16—24no—
268C2×SD16—42no—
269C2×SD16—42no—
270C22×C8—21no—
271C22×C8—1061no—
272C22×C8—1061no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8C22⋊D5312yescomplemented
277#127—1061no—
278#127—1061no—
279#127—1061no—
280#127—1061no—
281#127—2122no—
282#127—2122no—
283#121—2124no—
284#121—2124no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—532no—
292C2×C4⋊C4C22⋊D5311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—1064no—
295C2×C42—1061no—
296C2×C42C22×D5311yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—1061no—
299C4⋊C8—1061no—
300C4⋊C8C22⋊D5312yescomplemented
301#103—24no—
302#102—1068no—
303C4×C8—532no—
304C4×C8—1061no—
305#95—1061no—
306C24—21no—
307C24—1062no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—1062no—
320C2×D4—1062no—
321C2×D4—4244no—
322C2×D4—2124no—
323C2×D4—2128no—
324C2×D4—2128no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C2×D4—2121no—
329C2×D4—2121no—
330C2×D4—2122no—
331C2×D4—2121no—
332C2×D4—2121no—
333C2×D4—2121no—
334C2×D4—2122no—
335C2×D4#1171992212yes—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—1062no—
343C22×C4D4×D5311yescharacteristic
344C22×C4D4×D5311yescharacteristic
345C22×C4C2×C22⋊D5311yescharacteristic
346SD16—48no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—1062no—
351C2×C8—2121no—
352C2×C8—2122no—
353C2×C8—2122no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—534no—
361C4⋊C4—1062no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—2128no—
365C42—2121no—
366C42—2121no—
367C42D4×D5311yescharacteristic
368C42#1171994711yescharacteristic
369C42C2×C22⋊D5311yescharacteristic
370C42C2×C22⋊D5311yescharacteristic
371C23—22no—
372C23—41no—
373C23—42no—
374C23—41no—
375C23—42no—
376C23—2122no—
377C23—2122no—
378C23—2124no—
379C23—2122no—
380C23—2124no—
381C23#1313318511yescharacteristic
382Q8—42no—
383Q8—42no—
384Q8—81no—
385Q8—81no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390D4—4242no—
391D4—42416no—
392D4—4241no—
393D4—4241no—
394D4—4241no—
395D4—4242no—
396D4—4241no—
397D4—4241no—
398D4—4241no—
399D4—2124no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—1064no—
410C2×C4—2122no—
411C2×C4—2122no—
412C2×C4#1313318511yescharacteristic
413C2×C4#1313318611yescharacteristic
414C2×C4#1313318811yescharacteristic
415C2×C4#1313318911yescharacteristic
416C2×C4#1313318911yescharacteristic
417C2×C4#1313321311yescharacteristic
418C2×C4C2×#1171992211yescharacteristic
419C2×C4SD16×D5312yes—
420C2×C4#1313323711yescharacteristic
421C2×C4#1313324211yescharacteristic
422C2×C4#1313324311yescharacteristic
423C2×C4#1313324412yes—
424C8—42no—
425C8—4241no—
426C8—2122no—
427C22—44no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22—4242no—
433C22—4242no—
434C22—4244no—
435C22—4244no—
436C22—4242no—
437C22#1681053111yescharacteristic
438C22#1681075611yescharacteristic
439C22#1681076611yescharacteristic
440C22#1681077212yes—
441C22#1681079912yes—
442C4—21no—
443C4—21no—
444C4—81no—
445C4—82no—
446C4—2124no—
447C4#1681075711yescharacteristic
448C4#1681076811yescharacteristic
449C4#1681082112yes—
450C4#1681085711yescharacteristic
451C4#1681087411yescharacteristic
452C4#1681099411yescharacteristic
453C4#1681100911yescharacteristic
454C2—82no—
455C2—8482no—
456C2#2027389212yes—
457C2#2027391312yes—
458C2#2027571111yescharacteristic
459C2#2027574111yescharacteristic
460C2#2027580111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#461C2
socleC23×C53#120
prime radical#20279661C2
hypercenterC2×#461D53
hypocenterC53#10062
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#16810971C2semidirect
3. #20275668C2semidirect
4. C2×#16810366C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272554C2semidirect
7. C2×C53×#461C2semidirect
50. C4×#88322D4semidirect
62. C4⋊Q8×C53D4semidirect
73. C4⋊C8×C53D4semidirect
133. C4⋊Dic53SD16semidirect
225. C2×#461D53semidirect
259. C53#10062semidirect
276. C4⋊Q8C22⋊D53semidirect
300. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
#202796612
#202756682
C2×#168103662
C2×D4⋊#883222
#202725542
C2×C53×#4612
#1006253