Finite Group Databasebeta

← #27278949

Subgroups — #27278949

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284715C211yesmaximal, characteristic
2D4×#11719917C211yesmaximal, characteristic, complemented
3#20275633C211yesmaximal, characteristic, complemented
4#20273167C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C4×#102×C53C211yesmaximal, characteristic, complemented
8C22×#11719917—21no—
9#16811173C2211yescharacteristic
10#16810936—21no—
11#16810903C411yescharacteristic, complemented
12#16810903C411yescharacteristic, complemented
13#16810818C411yescharacteristic, complemented
14#16810818C411yescharacteristic, complemented
15#16810757—21no—
16C4×#11719922—21no—
17C4×#11719922—21no—
18#16810701—21no—
19#16810695—22no—
20D4×#88322—21no—
21#16810676C2211yescharacteristic
22#16810647C2211yescharacteristic
23C2×#13133145—21no—
24C2×#13133145—21no—
25#16810587—21no—
26C4×#11719917C2211yescharacteristic
27#16810352—22no—
28C2×C4×D4×C53C2211yescharacteristic
29C2×#102×C53—21no—
30C2×C4×C8×C53C2211yescharacteristic
31C4×C4⋊C4×C53C2211yescharacteristic
32C53×#340—21no—
33C2×#11719945—21no—
34C2×#11719922—21no—
35C2×#11719922—21no—
36C2×#11719917—22no—
37C2×#11719917—21no—
38C2×#11719917—42no—
39C2×#11719917—42no—
40C2×#11719917D411yescharacteristic
41C2×#11719917D411yescharacteristic
42C22×#88322—22no—
43#13133217C2×C411yescharacteristic
44#13133214C2×C411yescharacteristic
45#13133205—21no—
46#13133191—21no—
47#13133191—21no—
48C4×D212—21no—
49C4×D212—21no—
50#13133150—21no—
51#13133150—21no—
52C4⋊#88322—21no—
53C4⋊#88322C2×C411yescharacteristic
54C4⋊#88322C2×C411yescharacteristic
55C4⋊#88322D412yes—
56#13133145—42no—
57#13133145—42no—
58#13133138—21no—
59#13133138—21no—
60#13133137—21no—
61#13133137—21no—
62C22⋊#88322—22no—
63C22⋊#88322—22no—
64C22⋊#88322—21no—
65#13133128—21no—
66C4×#88322—21no—
67C4×#88322—21no—
68C22×D4×C53D411yescharacteristic
69C23×C4×C53—21no—
70C22×C8×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53C2×C411yescharacteristic
74C4×D4×C53C2×C411yescharacteristic
75C4×D4×C53D411yescharacteristic
76C4×D4×C53D411yescharacteristic
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53D411yescharacteristic
79C2×C22⋊C4×C53—21no—
80C2×C42×C53—21no—
81C2×C42×C53C2311yescharacteristic
82#102×C53—24no—
83C4×C8×C53—21no—
84C4×C8×C53—21no—
85#95×C53—21no—
86C24×C53—21no—
87C2×D4×C53—22no—
88C2×D4×C53—21no—
89C2×D4×C53—41no—
90C2×D4×C53C22⋊C411yescharacteristic
91C2×D4×C53C22⋊C411yescharacteristic
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C2×C22⋊D53—41no—
106#11719945—41no—
107#11719945—41no—
108#11719945—41no—
109#11719945—41no—
110C2×D212—21no—
111C2×D212—21no—
112C2×D212—21no—
113C2×C4×D53—41no—
114C2×Dic106—21no—
115C2×C8×C53—22no—
116C2×C8×C53—41no—
117C2×C8×C53—41no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53C22⋊C411yescharacteristic
122C4⋊C4×C53C22⋊C411yescharacteristic
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—21no—
127C42×C53—41no—
128C42×C53—41no—
129C42×C53C22×C411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53#4111yescharacteristic
133#11719922—42no—
134#11719922—42no—
135#11719921—41no—
136#11719921—41no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139#11719917—21no—
140#11719917—21no—
141#11719917—21no—
142#11719917—44no—
143#11719917—41no—
144#11719917—41no—
145#11719917D812yescomplemented
146C2×#88322—21no—
147C2×#88322—21no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—22no—
151C2×#88322—44no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—42no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—81no—
158C23×C53#12111yescharacteristic
159C22×D53—41no—
160D4×C53—21no—
161D4×C53—21no—
162D4×C53—21no—
163D4×C53—42no—
164D4×C53#10412yescomplemented
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—81no—
182C2×C4×C53—81no—
183C2×C4×C53C2×C22⋊C411yescharacteristic
184C2×C4×C53C4×D411yescharacteristic
185C2×C4×C53C4×D411yescharacteristic
186C2×C4×C53#12011yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12311yescharacteristic
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53#13511yescharacteristic
191C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
193C22⋊D53—81no—
194C22⋊D53—81no—
195C2×Dic53—41no—
196D212—42no—
197D212—41no—
198D212—41no—
199D212—41no—
200D212—41no—
201C4×D53—81no—
202C4×D53—81no—
203Dic106—41no—
204Dic106—41no—
205C8×C53—42no—
206#88322—22no—
207#88322—41no—
208#88322—41no—
209#88322—42no—
210#88322—41no—
211#8991—531nomaximal, Sylow
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—22no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53#38611yescharacteristic
223C22×C53#45911yescharacteristic
224C22×C53#46311yescharacteristic
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—81no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—82no—
234C4×C53C2×#10411yescharacteristic
235C4×C53#42111yescharacteristic
236C4×C53C4×D811yescharacteristic
237C4×C53#44211yescharacteristic
238C4×C53#44711yescharacteristic
239C4×C53#44911yescharacteristic
240Dic53—81no—
241#3100—531no—
242D4×M16—531no—
243#1980—531no—
244C4×#102D5311yescharacteristic, complemented
245#1492—531no—
246#1303—531no—
247#1278—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—42no—
251C2×C53—81no—
252C2×C53#187111yescharacteristic
253C2×C53#187911yescharacteristic
254C2×C53#188111yescharacteristic
255D53—161no—
256C22×M16—1061no—
257#532—531no—
258C2×C4×D4C2×D5311yescharacteristic
259#480—1061no—
260#474—531no—
261#474—531no—
262#459—531no—
263#459—531no—
264#447—1061no—
265C4×D8—1061no—
266C4×D8—1061no—
267#436—1061no—
268#435—1062no—
269C8×D4—1061no—
270#431—531no—
271#423—531no—
272C2×#104—1061no—
273C2×#104—1061no—
274C2×#102—21no—
275#407—1061no—
276C4×M16—531no—
277C2×C4×C8C2×D5311yescharacteristic
278C4×C4⋊C4C2×D5311yescharacteristic
279#340—21no—
280D4⋊C8—1062no—
281C53#899111yescharacteristic, complemented, Sylow
282C2×#41—1061no—
283C22×D4C22⋊D5311yescharacteristic
284C23×C4—21no—
285C2×D8—1061no—
286C2×D8—1061no—
287C2×M16—531no—
288C2×M16—531no—
289C2×M16—1062no—
290C2×M16—1061no—
291C2×M16—2122no—
292C2×M16—2122no—
293C22×C8—21no—
294C22×C8—1062no—
295C4⋊Q8—531no—
296#127—531no—
297#124—1061no—
298#121—1061no—
299#121—1061no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—1061no—
303C4×D4—1061no—
304C4×D4C4×D5311yescharacteristic
305C4×D4C4×D5311yescharacteristic
306C4×D4D21211yescharacteristic
307C4×D4D21211yescharacteristic
308C2×C4⋊C4—21no—
309C2×C4⋊C4C22⋊D5311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C42—21no—
312C2×C42C22×D5311yescharacteristic
313#107—1061no—
314#107—1061no—
315C4⋊C8—531no—
316C4⋊C8—531no—
317C4⋊C8—532no—
318C4⋊C8—1061no—
319#104—2122no—
320#104—2122no—
321#102—24no—
322#102—1061no—
323#102—1061no—
324#102—1061no—
325#102—1061no—
326C22⋊C8—1062no—
327C22⋊C8—1062no—
328C22⋊C8—1061no—
329#97—1061no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—1061no—
333C4×C8—1061no—
334#95—21no—
335C24—21no—
336#41—2121no—
337#41—2121no—
338#41—2121no—
339#41—2121no—
340C2×Q8—1061no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—1061no—
347C2×D4—2121no—
348C2×D4#1171992111yescharacteristic
349C2×D4#1171992111yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—2121no—
361C22×C4D4×D5311yescharacteristic
362C22×C4#1171994711yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364D8—2122no—
365D8—2122no—
366M16—532no—
367M16—1061no—
368M16—1061no—
369M16—1061no—
370M16—2124no—
371M16—2121no—
372M16—2121no—
373C2×C8—22no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—1061no—
377C2×C8—1061no—
378C2×C8—1061no—
379C2×C8—1061no—
380C2×C8—1062no—
381C2×C8—2124no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—1061no—
386C4⋊C4—1061no—
387C4⋊C4#1171992111yescharacteristic
388C4⋊C4#1171992111yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42—41no—
396C42—41no—
397C42C2×C4×D5311yescharacteristic
398C42C2×D21211yescharacteristic
399C42D4×D5311yescharacteristic
400C42#1171994711yescharacteristic
401C23—21no—
402C23—21no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—81no—
407C23—2121no—
408C23#1313318911yescharacteristic
409Q8—2121no—
410Q8—2121no—
411D4—21no—
412D4—21no—
413D4—21no—
414D4—42no—
415D4—4241no—
416D4—4241no—
417D4—2122no—
418D4—2121no—
419D4—2121no—
420D4—2121no—
421D4—2121no—
422D4#1313314512yescomplemented
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—4241no—
442C2×C4—4241no—
443C2×C4—2121no—
444C2×C4C2×#1171992111yescharacteristic
445C2×C4#1313317711yescharacteristic
446C2×C4#1313317711yescharacteristic
447C2×C4#1313318711yescharacteristic
448C2×C4#1313318911yescharacteristic
449C2×C4#1313320111yescharacteristic
450C2×C4D8×D5311yescharacteristic
451C2×C4#1313323811yescharacteristic
452C2×C4#1313324111yescharacteristic
453C2×C4#1313324411yescharacteristic
454C8—42no—
455C8—1062no—
456C8—2122no—
457C8—2121no—
458C8—2121no—
459C8—2121no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—82no—
469C22—82no—
470C22—4241no—
471C22—4241no—
472C22—4241no—
473C22#1681050011yescharacteristic
474C22#1681081611yescharacteristic
475C22#1681084011yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—4241no—
483C4C2×#1313314511yescharacteristic
484C4#1681064111yescharacteristic
485C4#1681070311yescharacteristic
486C4#1681073211yescharacteristic
487C4#1681075811yescharacteristic
488C4#1681076911yescharacteristic
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—8481no—
494C2#2027501511yescharacteristic
495C2#2027506111yescharacteristic
496C2#2027507511yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×#102×C53C2
socleC22×C53#386
prime radical#20284715C2
cosocleC2×C42×C53C23
hypercenterC4×#102D53
hypocenterC53#8991
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. D4×#11719917C2semidirect
3. #20275633C2semidirect
4. #20273167C2semidirect
5. #20272618C2semidirect
6. #20272552C2semidirect
7. C4×#102×C53C2semidirect
11. #16810903C4semidirect
12. #16810903C4semidirect
13. #16810818C4semidirect
14. #16810818C4semidirect
145. #11719917D8semidirect
164. D4×C53#104semidirect
244. C4×#102D53semidirect
281. C53#8991semidirect
422. D4#13133145semidirect

Maximal subgroups

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

subgroupindex
#202847152
D4×#117199172
#202756332
#202731672
#202726182
#202725522
C4×#102×C532
#899153