Finite Group Databasebeta

← #27278853

Subgroups — #27278853

Order 13568.

764 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810818C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic, complemented
3#20275520C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5#20272598C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7D4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10#16810818—22no—
11#16810818C414yescomplemented
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15C2×#13133137—22no—
16C2×#13133137—21no—
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19D4⋊#88322—22no—
20C2×C4×D4×C53—21no—
21C2×C4×D4×C53C2211yescharacteristic
22C22×C4⋊C4×C53—21no—
23C53×#389—21no—
24C53×#385—22no—
25C53×#384C2211yescharacteristic
26C53×#380—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719922—24no—
32C2×#11719922—42no—
33C2×#11719922—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214C2×C412yescomplemented
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322C2×C412yescomplemented
45#13133137—24no—
46#13133137—44no—
47#13133137—42no—
48#13133137—42no—
49C4×#88322—21no—
50C4×#88322—21no—
51C22×D4×C53D411yescharacteristic
52C23×C4×C53—21no—
53C23×C4×C53—22no—
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53—42no—
57C4×D4×C53—42no—
58C4×D4×C53C2×C412yes—
59C4×D4×C53D412yes—
60C2×C4⋊C4×C53—22no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—42no—
64C2×C4⋊C4×C53—42no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C4⋊C4×C53D412yes—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—21no—
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#95×C53—22no—
74#95×C53—21no—
75C24×C53—21no—
76C23×D53—41no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53C22⋊C412yes—
81C2×D4×C53D812yes—
82C2×D4×C53SD1612yes—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—44no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—44no—
95C22×C4×C53—82no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—42no—
103C2×D212—42no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C2×D212—41no—
112C2×D212—41no—
113C2×D212—41no—
114C2×D212—41no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—44no—
120C4⋊C4×C53—42no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53C22⋊C412yes—
126C4⋊C4×C53D812yescomplemented
127C22⋊C4×C53—22no—
128C22⋊C4×C53—22no—
129C22⋊C4×C53—44no—
130C42×C53—41no—
131C42×C53—41no—
132C42×C53C22×C411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C42×C53#4111yescharacteristic
136#11719922—48no—
137C2×#88322—22no—
138C2×#88322—22no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C23×C53—22no—
147C23×C53—22no—
148C23×C53—22no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53#12011yescharacteristic
154C22×D53—42no—
155C22×D53—81no—
156C22×D53—81no—
157C22×D53—81no—
158C22×D53—81no—
159C22×D53—81no—
160C22×D53—81no—
161D4×C53—22no—
162D4×C53—22no—
163D4×C53—22no—
164D4×C53#10214yescomplemented
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—88no—
184C2×C4×C53C2×C22⋊C411yescharacteristic
185C2×C4×C53C4×D411yescharacteristic
186C2×C4×C53C4×D411yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12311yescharacteristic
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53C2×SD1611yescharacteristic
193C2×C4×C53#13511yescharacteristic
194C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
195C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C53#13711yescharacteristic
198D212—44no—
199D212—42no—
200D212—42no—
201D212—42no—
202D212—42no—
203D212—82no—
204D212—81no—
205D212—81no—
206D212—81no—
207D212—81no—
208#88322—42no—
209#88322—42no—
210#88322—81no—
211#8977—531nomaximal, Sylow
212C22×C53—24no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—42no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—42no—
219C22×C53#38611yescharacteristic
220C22×C53#44711yescharacteristic
221C22×C53#44711yescharacteristic
222C22×C53#44811yescharacteristic
223C22×C53#44911yescharacteristic
224C22×C53#45311yescharacteristic
225C22×C53#45411yescharacteristic
226C2×D53—82no—
227C2×D53—82no—
228C2×D53—82no—
229C2×D53—161no—
230C2×D53—161no—
231C2×D53—161no—
232C2×D53—161no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—82no—
238C4×C53—81no—
239C4×C53C2×#10211yescharacteristic
240C4×C53#41711yescharacteristic
241C4×C53C4×D811yescharacteristic
242C4×C53#44211yescharacteristic
243C4×C53#45911yescharacteristic
244C4×C53#46311yescharacteristic
245C4×C53#48011yescharacteristic
246C4×C53#48111yescharacteristic
247C2×#459—531no—
248D4×C4⋊C4D5311yescharacteristic, complemented
249#1982—531no—
250#1957—531no—
251C2×D4⋊C8—531no—
252#1296—531no—
253#1279—531no—
254C2×C53—44no—
255C2×C53#161911yescharacteristic
256C2×C53#162211yescharacteristic
257C2×C53#162511yescharacteristic
258C2×C53#162711yescharacteristic
259C2×C53#187411yescharacteristic
260C2×C53#187911yescharacteristic
261C2×C53#188311yescharacteristic
262D53—162no—
263C22×D8—1061no—
264C2×#127—531no—
265C2×C4×D4—21no—
266C2×C4×D4C2×D5311yescharacteristic
267C22×C4⋊C4—21no—
268#459—534no—
269#459—1062no—
270C2×C4⋊C8—531no—
271C2×C4⋊C8—531no—
272C2×#102—1061no—
273C2×#102—1062no—
274C2×#102—1061no—
275C2×#102—1061no—
276C2×C4×C8—531no—
277#389—21no—
278#385—22no—
279#384C2×D5311yescharacteristic
280#380—21no—
281C4×C4⋊C4C2×D5311yescharacteristic
282D4⋊C8—1062no—
283C53#897711yescharacteristic, complemented, Sylow
284C22×D4—1061no—
285C22×D4—1061no—
286C22×D4—1061no—
287C22×D4C22⋊D5311yescharacteristic
288C23×C4—21no—
289C23×C4—22no—
290C2×D8—1064no—
291C2×D8—2122no—
292C2×D8—2122no—
293C22×C8—1061no—
294C22×C8—1061no—
295C22×C8—1061no—
296#127—532no—
297#127—1061no—
298#127—1061no—
299#127—1061no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—42no—
303C4×D4—42no—
304C4×D4C4×D5312yes—
305C4×D4D21212yes—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4C22⋊D5311yescharacteristic
312C2×C4⋊C4C22⋊D5311yescharacteristic
313C2×C4⋊C4C22⋊D5311yescharacteristic
314C2×C4⋊C4C22⋊D5312yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C42—21no—
318C2×C42C22×D5311yescharacteristic
319C4⋊C8—532no—
320C4⋊C8—1061no—
321C4⋊C8—1061no—
322C4⋊C8—1061no—
323#102—1064no—
324#102—2122no—
325#102—2124no—
326#102—2122no—
327C4×C8—1061no—
328C4×C8—1061no—
329#95—22no—
330#95—21no—
331C24—21no—
332C24—2121no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—1062no—
337C2×D4—1062no—
338C2×D4—1062no—
339C2×D4—2122no—
340C2×D4—2122no—
341C2×D4—2121no—
342C2×D4—2121no—
343C2×D4—2121no—
344C2×D4—2121no—
345C2×D4—2121no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4—2121no—
349C2×D4—2121no—
350C2×D4—2121no—
351C2×D4—2121no—
352C2×D4#1171992112yes—
353C2×D4#1171992212yes—
354C2×D4#1171992412yes—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—44no—
367C22×C4—82no—
368C22×C4C2×C22⋊D5311yescharacteristic
369C22×C4C2×C22⋊D5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371D8—2128no—
372C2×C8—1062no—
373C2×C8—1062no—
374C2×C8—2121no—
375C2×C8—2121no—
376C2×C8—2121no—
377C2×C8—2121no—
378C2×C8—2121no—
379C2×C8—2121no—
380C2×C8—2121no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4#1171992112yes—
392C4⋊C4#1171992212yescomplemented
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—44no—
396C42—41no—
397C42—41no—
398C42C2×C4×D5311yescharacteristic
399C42C2×D21211yescharacteristic
400C42#1171994511yescharacteristic
401C42#1171994511yescharacteristic
402C23—22no—
403C23—22no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—4241no—
410C23—4241no—
411C23—4241no—
412C23—4241no—
413C23—4241no—
414C23—4241no—
415C23—2122no—
416C23#1313318611yescharacteristic
417D4—22no—
418D4—22no—
419D4—22no—
420D4—4242no—
421D4—4241no—
422D4—4241no—
423D4—4241no—
424D4—4241no—
425D4—2124no—
426D4—2122no—
427D4—2122no—
428D4—2122no—
429D4—2122no—
430D4#1313313714yescomplemented
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—88no—
450C2×C4C2×#1171992111yescharacteristic
451C2×C4C4×C22⋊D5311yescharacteristic
452C2×C4C4×C22⋊D5311yescharacteristic
453C2×C4#1313319111yescharacteristic
454C2×C4#1313319111yescharacteristic
455C2×C4#1313320011yescharacteristic
456C2×C4C2×#1171992211yescharacteristic
457C2×C4C2×#1171992211yescharacteristic
458C2×C4C2×#1171992411yescharacteristic
459C2×C4#1313323711yescharacteristic
460C2×C4#1313324211yescharacteristic
461C2×C4#1313324311yescharacteristic
462C2×C4#1313324311yescharacteristic
463C2×C4#1313324811yescharacteristic
464C8—4241no—
465C8—2122no—
466C8—2122no—
467C22—24no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—8481no—
475C22—8481no—
476C22—8481no—
477C22—8481no—
478C22—4242no—
479C22—4242no—
480C22—4242no—
481C22#1681050211yescharacteristic
482C22#1681075711yescharacteristic
483C22#1681075711yescharacteristic
484C22#1681076211yescharacteristic
485C22#1681076811yescharacteristic
486C22#1681079111yescharacteristic
487C22#1681079611yescharacteristic
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—82no—
493C4—81no—
494C4C2×#1313313711yescharacteristic
495C4#1681062211yescharacteristic
496C4C4×#1171992211yescharacteristic
497C4#1681073311yescharacteristic
498C4#1681081811yescharacteristic
499C4#1681083811yescharacteristic
500C4#1681093611yescharacteristic
501C4#1681094111yescharacteristic
502C2—44no—
503C2—8482no—
504C2#2027376911yescharacteristic
505C2#2027378711yescharacteristic
506C2#2027380411yescharacteristic
507C2#2027381811yescharacteristic
508C2#2027503211yescharacteristic
509C2#2027506311yescharacteristic
510C2#2027509111yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingD4×C4⋊C4×C53C2
socleC23×C53#120
prime radicalC2×#16810818C2
cosocleC2×C42×C53C23
hypercenterD4×C4⋊C4D53
hypocenterC53#8977
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20275642C2semidirect
3. #20275520C2semidirect
4. C2×D4⋊#88322C2semidirect
5. #20272598C2semidirect
6. #20272556C2semidirect
7. D4×C4⋊C4×C53C2semidirect
11. #16810818C4semidirect
40. #13133214C2×C4semidirect
44. C4⋊#88322C2×C4semidirect
126. C4⋊C4×C53D8semidirect
164. D4×C53#102semidirect
248. D4×C4⋊C4D53semidirect
283. C53#8977semidirect
392. C4⋊C4#11719922semidirect
430. D4#13133137semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
#202756422
#202755202
C2×D4⋊#883222
#202725982
#202725562
D4×C4⋊C4×C532
#897753