Finite Group Databasebeta

← #27367012

Subgroups — #27367012

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285109C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3C2×#16810918C211yesmaximal, characteristic, complemented
4#20274297C212yesmaximal, complemented
5#20274255C212yesmaximal, complemented
6#20274173C212yesmaximal, complemented
7#20274131C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273397C211yesmaximal, characteristic, complemented
10#20273393C211yesmaximal, characteristic
11Q8×M16×C53C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16810983—22no—
15#16810974C2212yes—
16#16810970C2212yes—
17#16810941—21no—
18#16810918—21no—
19#16810918—21no—
20#16810918C2212yes—
21#16810918C2212yes—
22#16810908C2211yescharacteristic
23#16810908C2211yescharacteristic
24#16810823C2211yescharacteristic
25#16810823C2211yescharacteristic
26#16810762—21no—
27C4×#11719924—21no—
28C4×#11719924—21no—
29#16810676C2211yescharacteristic
30C2×#13133150—21no—
31C2×C4⋊#88322C2211yescharacteristic
32#16810625—21no—
33#16810625—21no—
34C2×#13133141—21no—
35#16810376C2212yes—
36#16810374C2212yes—
37#16810371C2212yes—
38#16810371C2212yes—
39#16810370C2212yes—
40#16810366C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#446—22no—
43C53×#446C2212yes—
44C8×Q8×C53C2212yes—
45C53×#423—21no—
46C53×#423C2211yescharacteristic
47C4×M16×C53—21no—
48C4×M16×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51C2×#11719924—21no—
52C2×#11719924—21no—
53C22×#88322—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217—21no—
57#13133217C2311yescharacteristic
58#13133217C2312yes—
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67#13133150—22no—
68#13133150—22no—
69#13133148—21no—
70#13133148—21no—
71C4⋊#88322—21no—
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75#13133142—22no—
76#13133142—22no—
77#13133141—21no—
78#13133141—21no—
79#13133141—22no—
80#13133141—22no—
81#13133139—22no—
82#13133139—22no—
83#13133137—21no—
84#13133137—21no—
85C22⋊#88322—21no—
86C4×#88322—21no—
87C22×Q8×C53D411yescharacteristic
88C2×M16×C53—21no—
89C2×M16×C53D411yescharacteristic
90C2×M16×C53D411yescharacteristic
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53D411yescharacteristic
94C4×Q8×C53D411yescharacteristic
95C4×Q8×C53C2311yescharacteristic
96C4×Q8×C53C2311yescharacteristic
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C42×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53—22no—
102C4⋊C8×C53—22no—
103C4⋊C8×C53—21no—
104C4⋊C8×C53C2312yes—
105#97×C53—22no—
106#97×C53D412yes—
107C4×C8×C53—22no—
108C4×C8×C53C2312yes—
109C2×Q8×C53—21no—
110C2×Q8×C53—22no—
111C2×Q8×C53—41no—
112C2×Q8×C53C2×D411yescharacteristic
113C2×Q8×C53C2×D411yescharacteristic
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×C22⋊D53—41no—
120C22×Dic53—41no—
121#11719945—42no—
122#11719945—42no—
123C2×D212—21no—
124C2×D212—21no—
125C2×D212—21no—
126C2×C4×D53—41no—
127C2×Dic106—21no—
128C2×Dic106—22no—
129C2×Dic106—41no—
130C2×Dic106—82no—
131M16×C53—22no—
132M16×C53—22no—
133M16×C53—42no—
134C2×C8×C53—22no—
135C2×C8×C53C2×D412yes—
136C2×C8×C53C2×D412yes—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C42×C53—21no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152#11719924—42no—
153#11719924—42no—
154#11719921—41no—
155#11719921—41no—
156C4⋊Dic53—22no—
157C4⋊Dic53—21no—
158C4⋊Dic53—22no—
159C4⋊Dic53—21no—
160C4⋊Dic53—41no—
161C4⋊Dic53—41no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322—22no—
165C2×#88322—42no—
166C23×C53#12111yescharacteristic
167C22×D53—41no—
168Q8×C53—21no—
169Q8×C53—21no—
170Q8×C53—21no—
171Q8×C53—42no—
172Q8×C53#13712yes—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—81no—
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#13512yes—
186C2×C4×C53#13712yes—
187C2×C4×C53C22×D411yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C2×#4111yescharacteristic
190C22⋊D53—82no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—82no—
194D212—41no—
195D212—41no—
196D212—41no—
197D212—41no—
198D212—81no—
199C4×D53—82no—
200Dic106—41no—
201Dic106—41no—
202Dic106—82no—
203Dic106—82no—
204C8×C53—22no—
205C8×C53—22no—
206C8×C53—42no—
207#88322—41no—
208#88322—42no—
209#88322—41no—
210#19176—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53#46412yes—
214C22×C53C2×#12111yescharacteristic
215C2×D53—81no—
216C2×D53—81no—
217C2×D53—81no—
218C4×C53—21no—
219C4×C53—22no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—41no—
223C4×C53—41no—
224C4×C53—41no—
225C4×C53#46412yes—
226C4×C53C2×#13511yescharacteristic
227C4×C53C2×#13711yescharacteristic
228C4×C53C2×#13711yescharacteristic
229C4×C53#57511yescharacteristic
230Dic53—81no—
231Dic53—82no—
232#3141—531no—
233#3103—531no—
234C2×#477—531no—
235Q8×M16D5311yescharacteristic, complemented
236#1704—532no—
237#1698—532no—
238#1684—532no—
239#1678—532no—
240#1565—531no—
241#1548—531no—
242#1547—531no—
243C2×C53—22no—
244C2×C53C2×#46411yescharacteristic
245C2×C53#305611yescharacteristic
246C2×C53#306811yescharacteristic
247D53—161no—
248#532—531no—
249C2×C4⋊Q8—531no—
250C2×C4×Q8C2×D5311yescharacteristic
251#489—1062no—
252#487—532no—
253#486—532no—
254#481—1061no—
255#477—532no—
256#477—532no—
257#477—1061no—
258#477—1061no—
259#475—531no—
260#475—531no—
261#460—531no—
262#460—531no—
263#448—1061no—
264#446—22no—
265#446C2×D5312yes—
266C8×Q8C2×D5312yes—
267C4×SD16—1061no—
268C4×SD16—1061no—
269#431—531no—
270C2×#107—1061no—
271#423—21no—
272#423C2×D5311yescharacteristic
273C2×C4⋊C8—531no—
274#417—1061no—
275#417—1061no—
276C2×#103—1061no—
277C4×M16—21no—
278C4×M16C2×D5311yescharacteristic
279#335—532no—
280#334—532no—
281#333—532no—
282#333—532no—
283#332—532no—
284#331—532no—
285C53#1917611yescharacteristic, complemented, Sylow
286C2×#41—1061no—
287C22×Q8—1061no—
288C22×Q8D21211yescharacteristic
289C2×SD16—1061no—
290C2×SD16—1061no—
291C2×M16—21no—
292C2×M16C22⋊D5311yescharacteristic
293C2×M16C22⋊D5311yescharacteristic
294C22×C8—1061no—
295C4⋊Q8—532no—
296C4⋊Q8—531no—
297C4⋊Q8—1061no—
298C4⋊Q8—1061no—
299C4⋊Q8—1061no—
300#127—531no—
301#124—1061no—
302#121—1061no—
303#121—1061no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8D21211yescharacteristic
307C4×Q8D21211yescharacteristic
308C4×Q8C22×D5311yescharacteristic
309C4×Q8C22×D5311yescharacteristic
310C4×D4—1061no—
311C4×D4—1061no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—1061no—
314C2×C4⋊C4—1061no—
315C2×C4⋊C4D21211yescharacteristic
316C2×C42—21no—
317C2×C42C22×D5311yescharacteristic
318#107—1062no—
319#107—1062no—
320#106—1061no—
321#106—1061no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—21no—
325C4⋊C8—532no—
326C4⋊C8—531no—
327C4⋊C8—531no—
328C4⋊C8—1061no—
329C4⋊C8C22×D5312yes—
330#103—1062no—
331#103—1062no—
332#103—1061no—
333#103—1061no—
334#103—1062no—
335#103—1062no—
336#102—1062no—
337#102—1062no—
338#102—1061no—
339#102—1061no—
340C22⋊C8—1061no—
341#97—22no—
342#97C22⋊D5312yes—
343C4×C8—22no—
344C4×C8—1061no—
345C4×C8C22×D5312yes—
346#41—2122no—
347#41—2122no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—41no—
351C2×Q8—1061no—
352C2×Q8—1062no—
353C2×Q8—4242no—
354C2×Q8—2121no—
355C2×Q8C2×D21211yescharacteristic
356C2×Q8C2×D21211yescharacteristic
357C2×D4—1061no—
358C2×D4—1061no—
359C2×D4—1061no—
360C2×D4—2121no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—2121no—
364C22×C4—2121no—
365C22×C4C2×D21211yescharacteristic
366C22×C4#1171994511yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368SD16—2122no—
369SD16—2122no—
370M16—22no—
371M16—22no—
372M16—42no—
373C2×C8—22no—
374C2×C8—1061no—
375C2×C8—1061no—
376C2×C8—1062no—
377C2×C8—2122no—
378C2×C8C2×C22⋊D5312yes—
379C2×C8C2×C22⋊D5312yes—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—1062no—
386C4⋊C4—1061no—
387C4⋊C4—1062no—
388C4⋊C4—1061no—
389C4⋊C4—2121no—
390C4⋊C4—2121no—
391C4⋊C4C2×D21211yescharacteristic
392C4⋊C4C2×D21211yescharacteristic
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42C2×D21211yescharacteristic
400C42#1171994511yescharacteristic
401C42C2×C22⋊D5311yescharacteristic
402C42C23×D5311yescharacteristic
403C23—2121no—
404C23#1313319111yescharacteristic
405Q8—21no—
406Q8—21no—
407Q8—21no—
408Q8—42no—
409Q8—4242no—
410Q8—4242no—
411Q8—2121no—
412Q8—2121no—
413Q8#1313325012yes—
414D4—4241no—
415D4—4242no—
416D4—2121no—
417D4—2121no—
418D4—2121no—
419D4—2121no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4—4242no—
430C2×C4—4242no—
431C2×C4—2121no—
432C2×C4—2122no—
433C2×C4#1313319111yescharacteristic
434C2×C4#1313319111yescharacteristic
435C2×C4#1313319111yescharacteristic
436C2×C4#1313323712yes—
437C2×C4#1313324812yes—
438C2×C4C22×C22⋊D5311yescharacteristic
439C2×C4C22×D21211yescharacteristic
440C2×C4C2×#1171994511yescharacteristic
441C8—22no—
442C8—22no—
443C8—42no—
444C8—2121no—
445C8—2122no—
446C8—2121no—
447C22—21no—
448C22—21no—
449C22—4241no—
450C22—4241no—
451C22—4241no—
452C22#1681084512yes—
453C22C2×#1313319111yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—4242no—
462C4—4241no—
463C4#1681084512yes—
464C4C2×#1313323711yescharacteristic
465C4C2×#1313324811yescharacteristic
466C4C2×#1313325011yescharacteristic
467C4#1681142311yescharacteristic
468C2—22no—
469C2—8481no—
470C2C2×#1681084511yescharacteristic
471C2#2028434511yescharacteristic
472C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×M16×C53C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterQ8×M16D53
hypocenterC53#19176
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16810918C2semidirect
4. #20274297C2semidirect
5. #20274255C2semidirect
6. #20274173C2semidirect
7. #20274131C2semidirect
8. #20273467C2semidirect
9. #20273397C2semidirect
11. Q8×M16×C53C2semidirect
235. Q8×M16D53semidirect
285. C53#19176semidirect

Maximal subgroups

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

subgroupindex
#202851092
#202847462
C2×#168109182
#202742972
#202742552
#202741732
#202741312
#202734672
#202733972
#202733932
Q8×M16×C532
#1917653