Finite Group Databasebeta

← #27287600

Subgroups — #27287600

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279628C211yesmaximal, characteristic
2#20274843C211yesmaximal, characteristic, complemented
3#20274716C211yesmaximal, characteristic, complemented
4#20274715C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133191—21no—
9C2×#13133188—21no—
10C2×C4×D212C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133139—21no—
13C2×#13133138—21no—
14#16810539—22no—
15#16810493C2211yescharacteristic
16#16810493C2211yescharacteristic
17#16810483—21no—
18#16810480—21no—
19#16810386—21no—
20#16810386—22no—
21#16810385—21no—
22#16810352—22no—
23C2×C4×D4×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×D4⋊C8—22no—
27C22×D212D411yescharacteristic
28C22×C22⋊D53—22no—
29C22×C4×D53—21no—
30C22×#88322—21no—
31#13133191—42no—
32#13133191—42no—
33#13133188—42no—
34#13133188—42no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53D411yescharacteristic
45C2×C4⋊Dic53D412yes—
46C2×C22⋊Dic53—22no—
47C2×#11719921—22no—
48C2×#11719921—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51#13133139—42no—
52#13133138—42no—
53#13133124—22no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53—21no—
56C22×C8×C53—21no—
57C22×C8×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4×C8×C53—21no—
68C4×C8×C53D412yescomplemented
69C24×C53—21no—
70C23×D53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53D812yes—
77C22×C4×C53—21no—
78C22×C4×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C2×C22⋊D53—44no—
88C2×C22⋊D53—44no—
89C2×C22⋊D53—82no—
90C22×Dic53—21no—
91C22×Dic53—22no—
92C22×Dic53—41no—
93C2×D212—21no—
94C2×D212—21no—
95C2×D212—21no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212SD1612yes—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C8×C53—22no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53#4111yescharacteristic
119C22⋊Dic53—44no—
120#11719921—44no—
121#11719921—41no—
122#11719921—41no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53—42no—
130C4⋊Dic53—42no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133C4⋊Dic53—41no—
134C4⋊Dic53—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C23×C53—22no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53#12011yescharacteristic
146C22×D53—22no—
147C22×D53—41no—
148C22×D53—41no—
149C22×D53—41no—
150C22×D53—41no—
151C22×D53—41no—
152C22×D53—41no—
153D4×C53—22no—
154D4×C53—22no—
155D4×C53—41no—
156D4×C53—41no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12311yescharacteristic
170C2×C4×C53#12311yescharacteristic
171C2×C4×C53#12411yescharacteristic
172C2×C4×C53C2×D811yescharacteristic
173C2×C4×C53C2×SD1611yescharacteristic
174C2×C4×C53#13511yescharacteristic
175C2×C4×C53#13511yescharacteristic
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C53#13711yescharacteristic
180C22⋊D53—88no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—42no—
184C2×Dic53—42no—
185C2×Dic53—41no—
186C2×Dic53—42no—
187C2×Dic53—81no—
188C2×Dic53—81no—
189C2×Dic53—81no—
190D212—22no—
191D212—22no—
192D212—41no—
193D212—41no—
194C4×D53—81no—
195C4×D53—81no—
196C4×D53—81no—
197C4×D53—81no—
198C8×C53—42no—
199C8×C53—81no—
200#88322—81no—
201#10431—531nomaximal, Sylow
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53#39411yescharacteristic
210C22×C53#44711yescharacteristic
211C22×C53#44911yescharacteristic
212C22×C53#45011yescharacteristic
213C22×C53#45211yescharacteristic
214C22×C53#45411yescharacteristic
215C22×C53#45511yescharacteristic
216C2×D53—42no—
217C2×D53—42no—
218C2×D53—42no—
219C2×D53—81no—
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—81no—
226C4×C53#46511yescharacteristic
227C4×C53#46611yescharacteristic
228C4×C53#46811yescharacteristic
229C4×C53#46911yescharacteristic
230C4×C53#48011yescharacteristic
231C4×C53#48111yescharacteristic
232C4×C53#48211yescharacteristic
233C4×C53#48211yescharacteristic
234Dic53—82no—
235Dic53—81no—
236Dic53—161no—
237#2537—531no—
238#1834—531no—
239#1802—531no—
240#1802—531no—
241C2×D4⋊C8—531no—
242C2×D4⋊C8D5311yescharacteristic, complemented
243#1296—531no—
244C2×C53—82no—
245C2×C53#162511yescharacteristic
246C2×C53#163511yescharacteristic
247C2×C53#163911yescharacteristic
248C2×C53#164611yescharacteristic
249C2×C53#203311yescharacteristic
250C2×C53#203411yescharacteristic
251C2×C53#204011yescharacteristic
252D53—82no—
253C2×#121—1061no—
254C2×#121—1061no—
255C2×C4×D4—531no—
256C2×C4×D4C2×D5311yescharacteristic
257C2×C4⋊C8—531no—
258C2×C4⋊C8C2×D5311yescharacteristic
259C2×#102—1061no—
260C2×#102—1061no—
261C2×C4×C8C2×D5311yescharacteristic
262#394—1062no—
263#384—531no—
264#384—531no—
265#380—1061no—
266#380—1061no—
267#340—1061no—
268#340—1062no—
269#340—1061no—
270D4⋊C8—22no—
271D4⋊C8—1062no—
272C53#1043111yescharacteristic, complemented, Sylow
273C22×D4—531no—
274C22×D4—1062no—
275C22×D4D21211yescharacteristic
276C23×C4—21no—
277C23×C4—1061no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—1061no—
281#121—2122no—
282#121—2122no—
283#121—2122no—
284#121—2122no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—1061no—
290C4×D4—1061no—
291C4×D4—1061no—
292C4×D4—1061no—
293C2×C4⋊C4—531no—
294C2×C4⋊C4—532no—
295C2×C4⋊C4—1061no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4—1061no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4D21211yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—1062no—
303C2×C22⋊C4—1062no—
304C2×C22⋊C4—1061no—
305C2×C42C22×D5311yescharacteristic
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—1061no—
309C4⋊C8—1061no—
310#102—2122no—
311#102—2122no—
312C4×C8—21no—
313C4×C8C22⋊D5312yescomplemented
314#95—1062no—
315C24—21no—
316C24—1061no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—532no—
323C2×D4—1061no—
324C2×D4—1061no—
325C2×D4—1061no—
326C2×D4—1061no—
327C2×D4—1061no—
328C2×D4—4242no—
329C2×D4—2124no—
330C2×D4—2124no—
331C2×D4D42412yes—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—1061no—
340C22×C4—1062no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2121no—
345C22×C4—2121no—
346C22×C4—2121no—
347C22×C4—2121no—
348C22×C4C2×D21211yescharacteristic
349C22×C4D4×D5311yescharacteristic
350C22×C4D4×D5311yescharacteristic
351C2×C8—22no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—2121no—
358C2×C8—2121no—
359C2×C8—2121no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—1061no—
363C4⋊C4—1061no—
364C4⋊C4—2122no—
365C4⋊C4—2122no—
366C4⋊C4—2121no—
367C4⋊C4—2121no—
368C4⋊C4—2121no—
369C4⋊C4—2121no—
370C4⋊C4—2121no—
371C4⋊C4—2121no—
372C4⋊C4—2121no—
373C4⋊C4—2121no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—2124no—
377C22⋊C4—2124no—
378C22⋊C4—2121no—
379C22⋊C4—2121no—
380C42#1171994511yescharacteristic
381C42#1171994711yescharacteristic
382C42#1171994711yescharacteristic
383C42C2×C22⋊D5311yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—1062no—
392C23—2121no—
393C23—2121no—
394C23—2121no—
395C23—2121no—
396C23—2121no—
397C23—2121no—
398C23#1313318711yescharacteristic
399D4—22no—
400D4—22no—
401D4—41no—
402D4—41no—
403D4—1062no—
404D4—1062no—
405D4—4248no—
406D4—2121no—
407D4—2121no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—4241no—
418C2×C4—4241no—
419C2×C4—4241no—
420C2×C4—4241no—
421C2×C4—4241no—
422C2×C4—4241no—
423C2×C4—4241no—
424C2×C4—2121no—
425C2×C4—2122no—
426C2×C4—2122no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2122no—
430C2×C4#1313318811yescharacteristic
431C2×C4#1313318911yescharacteristic
432C2×C4#1313319111yescharacteristic
433C2×C4#1313319911yescharacteristic
434C2×C4#1313320111yescharacteristic
435C2×C4#1313320311yescharacteristic
436C2×C4C2×D42411yescharacteristic
437C2×C4SD16×D5311yescharacteristic
438C2×C4#1313323811yescharacteristic
439C2×C4#1313323911yescharacteristic
440C2×C4#1313324111yescharacteristic
441C2×C4#1313324111yescharacteristic
442C2×C4#1313324511yescharacteristic
443C2×C4#1313325011yescharacteristic
444C8—42no—
445C8—81no—
446C8—4241no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—4241no—
455C22—4241no—
456C22—4241no—
457C22—4241no—
458C22—2122no—
459C22—2122no—
460C22—2122no—
461C22#1681053911yescharacteristic
462C22#1681075911yescharacteristic
463C22#1681076911yescharacteristic
464C22#1681077511yescharacteristic
465C22#1681078711yescharacteristic
466C22#1681079711yescharacteristic
467C22#1681080211yescharacteristic
468C4—21no—
469C4—21no—
470C4—81no—
471C4—8481no—
472C4—4241no—
473C4—4242no—
474C4#1681084911yescharacteristic
475C4#1681085911yescharacteristic
476C4#1681086911yescharacteristic
477C4#1681087311yescharacteristic
478C4#1681093711yescharacteristic
479C4#1681094011yescharacteristic
480C4#1681094611yescharacteristic
481C4#1681094811yescharacteristic
482C2—82no—
483C2—4242no—
484C2#2027380611yescharacteristic
485C2#2027386911yescharacteristic
486C2#2027389611yescharacteristic
487C2#2027394611yescharacteristic
488C2#2027589811yescharacteristic
489C2#2027590311yescharacteristic
490C2#2027594311yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×D4⋊C8C2
socleC23×C53#120
prime radical#20279628C2
hypercenterC2×D4⋊C8D53
hypocenterC53#10431
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20274843C2semidirect
3. #20274716C2semidirect
4. #20274715C2semidirect
5. C2×#16810352C2semidirect
6. #20272599C2semidirect
7. C2×C53×D4⋊C8C2semidirect
68. C4×C8×C53D4semidirect
242. C2×D4⋊C8D53semidirect
272. C53#10431semidirect
313. C4×C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202796282
#202748432
#202747162
#202747152
C2×#168103522
#202725992
C2×C53×D4⋊C82
#1043153