Finite Group Databasebeta

← #27280343

Subgroups — #27280343

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284723C211yesmaximal, characteristic
2#20275692C211yesmaximal, characteristic, complemented
3#20273213C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic
5#20272618C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C42×D4×C53C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810953—21no—
10#16810913C412yescomplemented
11#16810828C412yescomplemented
12#16810774—21no—
13C4×#11719923—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17#16810383—21no—
18#16810383—21no—
19#16810359—22no—
20D4⋊#88322—22no—
21C2×C4×D4×C53—21no—
22C2×C4×D4×C53C2211yescharacteristic
23C22×C42×C53—21no—
24C4×C4⋊C4×C53—21no—
25C4×C4⋊C4×C53C2211yescharacteristic
26C4×C22⋊C4×C53—21no—
27C4×C22⋊C4×C53—22no—
28C43×C53C2211yescharacteristic
29C2×#11719923—22no—
30C2×#11719917—21no—
31#13133217C2×C411yescharacteristic, complemented
32#13133217C2×C411yescharacteristic, complemented
33#13133209—21no—
34#13133196—21no—
35#13133196—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38#13133176—21no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322C2×C412yescomplemented
44#13133140—22no—
45#13133138—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C22×D4×C53D411yescharacteristic
51C23×C4×C53—22no—
52C23×C4×C53—21no—
53C4×D4×C53—28no—
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53C2×C412yes—
57C4×D4×C53D412yes—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C22⋊C4×C53—22no—
61C2×C22⋊C4×C53—21no—
62C2×C42×C53—22no—
63C2×C42×C53—21no—
64C2×C42×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53—42no—
67C2×C42×C53—42no—
68C2×C42×C53D411yescharacteristic
69C2×C42×C53D411yescharacteristic
70C2×C42×C53C2311yescharacteristic
71#95×C53—22no—
72#95×C53D412yes—
73C24×C53—21no—
74C2×D4×C53—24no—
75C2×D4×C53—21no—
76C2×D4×C53C22⋊C412yes—
77C22×C4×C53—21no—
78C22×C4×C53—28no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—44no—
85C22×C4×C53—44no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C2×Dic106—21no—
94C2×Dic106—21no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53C22⋊C412yes—
98C22⋊C4×C53—28no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C42×C53—24no—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53—21no—
105C42×C53—42no—
106C42×C53—44no—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53SD1612yescomplemented
110C42×C53C22×C411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53#4111yescharacteristic
113C42×C53#4111yescharacteristic
114C22⋊Dic53—41no—
115#11719923—44no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120#11719919—41no—
121#11719919—41no—
122C4×Dic53—41no—
123#11719917—41no—
124#11719917—41no—
125C2×#88322—22no—
126C2×#88322—41no—
127C2×#88322—41no—
128C23×C53—24no—
129C23×C53—22no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53#12011yescharacteristic
134D4×C53—24no—
135D4×C53—22no—
136D4×C53#10414yes—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—24no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—416no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53C2×C22⋊C411yescharacteristic
157C2×C4×C53C4×D411yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12311yescharacteristic
162C2×C4×C53C2×SD1611yescharacteristic
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53#13711yescharacteristic
166C2×Dic53—41no—
167C2×Dic53—41no—
168Dic106—42no—
169Dic106—81no—
170#88322—42no—
171#88322—81no—
172#88322—81no—
173#9231—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—24no—
177C22×C53—44no—
178C22×C53—44no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53#38611yescharacteristic
182C22×C53#45011yescharacteristic
183C22×C53#45211yescharacteristic
184C4×C53—24no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—42no—
189C4×C53—44no—
190C4×C53—42no—
191C4×C53C2×#10412yes—
192C4×C53C4×SD1611yescharacteristic
193C4×C53#44011yescharacteristic
194C4×C53#46111yescharacteristic
195C4×C53#46311yescharacteristic
196Dic53—81no—
197Dic53—81no—
198#3101—531no—
199C42×D4D5311yescharacteristic
200#1994—531no—
201#1503—531no—
202#1486—531no—
203#1303—531no—
204#1276—531no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53—44no—
208C2×C53#187711yescharacteristic
209C2×C53#189611yescharacteristic
210C2×C53#190611yescharacteristic
211#533—531no—
212C2×C4×D4—21no—
213C2×C4×D4C2×D5311yescharacteristic
214C22×C42—21no—
215#483—1061no—
216#476—532no—
217#461—532no—
218#450—1061no—
219C4×SD16—1062no—
220#431—531no—
221#431—531no—
222#423—531no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4C2×D5311yescharacteristic
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C43C2×D5311yescharacteristic
228#339—1061no—
229#339—1061no—
230#328—1062no—
231D4⋊C8—1062no—
232C53#923111yescharacteristic, complemented, Sylow
233C22×D4C22⋊D5311yescharacteristic
234C23×C4—22no—
235C23×C4—21no—
236C2×SD16—1062no—
237C2×M16—1061no—
238C4⋊Q8—531no—
239C4⋊Q8—531no—
240#125—1061no—
241C22⋊Q8—1061no—
242C22⋊Q8—1061no—
243C4×Q8—1061no—
244C4×Q8—1061no—
245C4×D4—28no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4C4×D5312yes—
249C4×D4D21212yes—
250#117—1061no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22⋊D5311yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C22⋊D5311yescharacteristic
262C2×C42C22⋊D5311yescharacteristic
263C2×C42C22×D5311yescharacteristic
264#106—1062no—
265C4⋊C8—532no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269#103—1062no—
270#102—1062no—
271C22⋊C8—1061no—
272C22⋊C8—1061no—
273C4×C8—1061no—
274C4×C8—1061no—
275#95—22no—
276#95C22⋊D5312yes—
277C24—21no—
278C2×Q8—1061no—
279C2×Q8—1061no—
280C2×D4—24no—
281C2×D4—21no—
282C2×D4#1171992112yes—
283C22×C4—21no—
284C22×C4—28no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C2×C22⋊D5311yescharacteristic
297C22×C4C2×C22⋊D5311yescharacteristic
298C22×C4C2×C22⋊D5311yescharacteristic
299SD16—2124no—
300M16—2121no—
301M16—2121no—
302C2×C8—1062no—
303C2×C8—2121no—
304C2×C8—2121no—
305C4⋊C4—24no—
306C4⋊C4—21no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—2121no—
312C4⋊C4—2121no—
313C4⋊C4#1171992112yes—
314C22⋊C4—28no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—2121no—
318C42—24no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—44no—
323C42—42no—
324C42—41no—
325C42—41no—
326C42—2121no—
327C42#1171992312yescomplemented
328C42C2×C4×D5311yescharacteristic
329C42C2×D21211yescharacteristic
330C42#1171994511yescharacteristic
331C42#1171994511yescharacteristic
332C23—24no—
333C23—22no—
334C23—42no—
335C23—41no—
336C23—41no—
337C23#1313318611yescharacteristic
338Q8—4241no—
339Q8—2122no—
340D4—24no—
341D4—22no—
342D4#1313314514yes—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—416no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4C2×#1171992111yescharacteristic
365C2×C4C4×C22⋊D5311yescharacteristic
366C2×C4C4×C22⋊D5311yescharacteristic
367C2×C4#1313319111yescharacteristic
368C2×C4#1313319111yescharacteristic
369C2×C4#1313320011yescharacteristic
370C2×C4C2×#1171992311yescharacteristic
371C2×C4#1313323711yescharacteristic
372C2×C4#1313324311yescharacteristic
373C2×C4#1313324711yescharacteristic
374C8—4241no—
375C8—4241no—
376C8—2122no—
377C22—21no—
378C22—21no—
379C22—24no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22#1681050211yescharacteristic
385C22#1681077411yescharacteristic
386C22#1681078511yescharacteristic
387C4—24no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—44no—
392C4—42no—
393C4—42no—
394C4—4241no—
395C4—4241no—
396C4C2×#1313314512yes—
397C4C4×#1171992311yescharacteristic
398C4#1681072311yescharacteristic
399C4#1681082811yescharacteristic
400C4#1681083811yescharacteristic
401C2—21no—
402C2—21no—
403C2—44no—
404C2#2027504911yescharacteristic
405C2#2027515811yescharacteristic
406C2#2027521611yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC42×D4×C53C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenterC42×D4D53
hypocenterC53#9231
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #20275692C2semidirect
3. #20273213C2semidirect
5. #20272618C2semidirect
6. #20272548C2semidirect
10. #16810913C4semidirect
11. #16810828C4semidirect
31. #13133217C2×C4semidirect
32. #13133217C2×C4semidirect
43. C4⋊#88322C2×C4semidirect
109. C42×C53SD16semidirect
232. C53#9231semidirect
327. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202847232
#202756922
#202732132
#202731402
#202726182
#202725482
C42×D4×C532
#923153