Finite Group Databasebeta

← #27280280

Subgroups — #27280280

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810823C211yesmaximal, characteristic
2#20275521C211yesmaximal, characteristic
3#20274720C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C2×C8×Q8×C53C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10#16810823—22no—
11#16810823C414yescomplemented
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133141—21no—
15C2×#13133139—21no—
16C2×#13133139—21no—
17C2×#13133137—21no—
18#16810493C2211yescharacteristic
19#16810385—21no—
20#16810366—22no—
21C2×C4×Q8×C53C2211yescharacteristic
22C8×Q8×C53—22no—
23C8×Q8×C53—22no—
24C2×C4⋊C8×C53—21no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53—21no—
27C2×C4×C8×C53C2211yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214C2×C412yes—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322C2×C412yes—
47#13133141—42no—
48#13133139—42no—
49#13133139—42no—
50#13133137—24no—
51C22×Q8×C53D411yescharacteristic
52C22×C8×C53—21no—
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53C2×C412yes—
58C4×Q8×C53D412yes—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C42×C53—21no—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—24no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53D412yes—
66C4×C8×C53—24no—
67C4×C8×C53—21no—
68C4×C8×C53—21no—
69C23×D53—41no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53C22⋊C412yes—
74C2×Q8×C53D812yes—
75C2×Q8×C53SD1612yes—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—41no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—81no—
97C2×D212—81no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—22no—
102C2×C8×C53—21no—
103C2×C8×C53—21no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53SD1612yescomplemented
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53C22⋊C412yes—
111C42×C53—22no—
112C42×C53—22no—
113C42×C53C22×C411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117#11719924—48no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53—81no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C23×C53#12111yescharacteristic
130C22×D53—42no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53—81no—
134C22×D53—81no—
135C22×D53—81no—
136C22×D53—81no—
137Q8×C53—22no—
138Q8×C53—22no—
139Q8×C53—22no—
140Q8×C53#10214yescomplemented
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C2×C4×C53C2×SD1611yescharacteristic
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C53#13711yescharacteristic
163C2×Dic53—81no—
164C2×Dic53—81no—
165C2×Dic53—81no—
166D212—42no—
167D212—42no—
168D212—42no—
169D212—42no—
170D212—82no—
171D212—82no—
172D212—81no—
173D212—81no—
174D212—81no—
175D212—81no—
176C8×C53—22no—
177C8×C53—41no—
178C8×C53—42no—
179#88322—42no—
180#88322—81no—
181#9222—531nomaximal, Sylow
182C22×C53#38611yescharacteristic
183C22×C53#46011yescharacteristic
184C22×C53#46411yescharacteristic
185C22×C53#46511yescharacteristic
186C22×C53#46611yescharacteristic
187C22×C53#47111yescharacteristic
188C22×C53#47211yescharacteristic
189C2×D53—82no—
190C2×D53—82no—
191C2×D53—82no—
192C2×D53—161no—
193C2×D53—161no—
194C2×D53—161no—
195C2×D53—161no—
196C4×C53—24no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—42no—
200C4×C53C2×#10211yescharacteristic
201C4×C53#41611yescharacteristic
202C4×C53C4×SD1611yescharacteristic
203C4×C53#44011yescharacteristic
204C4×C53#44811yescharacteristic
205C4×C53#44911yescharacteristic
206C4×C53#46511yescharacteristic
207C4×C53#46611yescharacteristic
208Dic53—161no—
209C2×#460—531no—
210C2×C8×Q8D5311yescharacteristic, complemented
211#1957—531no—
212#1803—531no—
213C2×#331—531no—
214#1296—531no—
215#1277—531no—
216C2×C53#166611yescharacteristic
217C2×C53#166811yescharacteristic
218C2×C53#167811yescharacteristic
219C2×C53#168211yescharacteristic
220C2×C53#188011yescharacteristic
221C2×C53#189311yescharacteristic
222C2×C53#190711yescharacteristic
223D53—162no—
224C22×SD16—1061no—
225C2×#127—531no—
226C2×C4×Q8C2×D5311yescharacteristic
227#460—534no—
228#460—1062no—
229C8×Q8—22no—
230C8×Q8—22no—
231C2×C4⋊C8—21no—
232C2×C4⋊C8—531no—
233C2×C4⋊C8—531no—
234C2×C4⋊C8C2×D5311yescharacteristic
235C2×#103—1061no—
236C2×#102—1061no—
237C2×#102—1061no—
238C2×#102—1061no—
239C2×C4×C8—21no—
240C2×C4×C8C2×D5311yescharacteristic
241#384—531no—
242#340—1061no—
243#331—1062no—
244C53#922211yescharacteristic, complemented, Sylow
245C22×Q8D21211yescharacteristic
246C22×D4—1061no—
247C22×D4—1061no—
248C22×D4—1061no—
249C2×SD16—1064no—
250C2×SD16—2122no—
251C2×SD16—2122no—
252C22×C8—21no—
253C22×C8—1061no—
254C22×C8—1061no—
255C22×C8C22⋊D5311yescharacteristic
256C22×C8C22⋊D5311yescharacteristic
257#127—532no—
258#127—1061no—
259#127—1061no—
260#127—1061no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C4×D5312yes—
264C4×Q8C22⋊D5312yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4D21211yescharacteristic
270C2×C42—21no—
271C2×C42C22×D5311yescharacteristic
272C4⋊C8—24no—
273C4⋊C8—21no—
274C4⋊C8—532no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278C4⋊C8C22⋊D5312yes—
279#103—2122no—
280#102—1064no—
281#102—2122no—
282#102—2122no—
283C4×C8—24no—
284C4×C8—21no—
285C4×C8—21no—
286C24—2121no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8#1171991312yes—
291C2×Q8D42412yes—
292C2×Q8#1171992112yes—
293C2×D4—1062no—
294C2×D4—1062no—
295C2×D4—1062no—
296C2×D4—4241no—
297C2×D4—4241no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2121no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2121no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—2121no—
312C22×C4C2×D21211yescharacteristic
313C22×C4#1171994511yescharacteristic
314C22×C4C2×C22⋊D5311yescharacteristic
315SD16—2128no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—1062no—
325C2×C8—2121no—
326C2×C8—2121no—
327C2×C8—2121no—
328C2×C8—2121no—
329C2×C8—2121no—
330C2×C8#1171992412yescomplemented
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—4241no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C4⋊C4—2121no—
338C4⋊C4—2121no—
339C4⋊C4#1171992112yes—
340C42—22no—
341C42—22no—
342C42C2×C4×D5311yescharacteristic
343C42#1171994511yescharacteristic
344C42C2×C22⋊D5311yescharacteristic
345C42C2×C22⋊D5311yescharacteristic
346C23—4241no—
347C23—4241no—
348C23—4241no—
349C23—4241no—
350C23—4241no—
351C23—4241no—
352C23—2122no—
353C23#1313319111yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—22no—
357Q8#1313313914yescomplemented
358D4—4242no—
359D4—4242no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—2122no—
365D4—2122no—
366D4—2122no—
367D4—2122no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—4241no—
377C2×C4—4241no—
378C2×C4—4241no—
379C2×C4C2×#1171992111yescharacteristic
380C2×C4C4×C22⋊D5311yescharacteristic
381C2×C4C4×C22⋊D5311yescharacteristic
382C2×C4#1313318611yescharacteristic
383C2×C4#1313319111yescharacteristic
384C2×C4#1313320011yescharacteristic
385C2×C4C2×D42411yescharacteristic
386C2×C4C2×#1171992411yescharacteristic
387C2×C4C2×#1171991311yescharacteristic
388C2×C4#1313323711yescharacteristic
389C2×C4#1313323911yescharacteristic
390C2×C4#1313323911yescharacteristic
391C2×C4#1313324211yescharacteristic
392C2×C4#1313324811yescharacteristic
393C8—22no—
394C8—41no—
395C8—42no—
396C8—4241no—
397C8—2122no—
398C22—8481no—
399C22—8481no—
400C22—8481no—
401C22—8481no—
402C22—4242no—
403C22—4242no—
404C22—4242no—
405C22#1681050211yescharacteristic
406C22#1681082311yescharacteristic
407C22#1681084511yescharacteristic
408C22#1681085211yescharacteristic
409C22#1681085911yescharacteristic
410C22#1681088611yescharacteristic
411C22#1681089311yescharacteristic
412C4—21no—
413C4—21no—
414C4—24no—
415C4—42no—
416C4—8481no—
417C4C2×#1313313911yescharacteristic
418C4#1681062011yescharacteristic
419C4C4×#1171992411yescharacteristic
420C4#1681072111yescharacteristic
421C4#1681076211yescharacteristic
422C4#1681076811yescharacteristic
423C4#1681085211yescharacteristic
424C4#1681085911yescharacteristic
425C2—8482no—
426C2#2027406311yescharacteristic
427C2#2027407511yescharacteristic
428C2#2027413111yescharacteristic
429C2#2027415911yescharacteristic
430C2#2027507011yescharacteristic
431C2#2027514311yescharacteristic
432C2#2027522311yescharacteristic
subgroupG/H
centerC23#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C8×Q8×C53C2
socleC23×C53#121
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterC2×C8×Q8D53
hypocenterC53#9222
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
3. #20274720C2semidirect
5. #20272599C2semidirect
6. #20272550C2semidirect
7. C2×C8×Q8×C53C2semidirect
11. #16810823C4semidirect
106. C2×C8×C53SD16semidirect
140. Q8×C53#102semidirect
210. C2×C8×Q8D53semidirect
244. C53#9222semidirect
330. C2×C8#11719924semidirect
357. Q8#13133139semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
#202755212
#202747202
C2×#168103662
#202725992
#202725502
C2×C8×Q8×C532
#922253