Finite Group Databasebeta

← #16807236

Subgroups — #16807236

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131615C211yesmaximal, characteristic, complemented
2#13131599C211yesmaximal, characteristic, complemented
3#13131247C212yesmaximal, complemented
4#13131246C212yesmaximal, complemented
5#13131244C212yesmaximal, complemented
6#13131241C212yesmaximal, complemented
7C2×#11718220C211yesmaximal, characteristic, complemented
8#13131164C211yesmaximal, characteristic, complemented
9D13⋊#360C211yesmaximal, characteristic, complemented
10#13131055C211yesmaximal, characteristic, complemented
11#13127969C211yesmaximal, characteristic
12#11719631—21no—
13#11719629—21no—
14D13×#135C2211yescharacteristic
15D13×#135C2211yescharacteristic
16#11719166—21no—
17#11719164—21no—
18#11719112C2212yes—
19#11719110C2212yescomplemented
20#11719107—22no—
21SD16×#261—22no—
22#11719104C2212yes—
23#11719102C2212yescomplemented
24C2×#88103C2211yescharacteristic
25C2×#88101C2211yescharacteristic
26#11719089C2211yescharacteristic
27D13⋊M32C2211yescharacteristic
28C13×#508C411yescharacteristic, complemented
29#11718855C411yescharacteristic, complemented
30#11718855C411yescharacteristic, complemented
31#11718579C411yescharacteristic, complemented
32#11718579C2211yescharacteristic
33#11718576C411yescharacteristic, complemented
34#11718576C2211yescharacteristic
35#11718575C411yescharacteristic, complemented
36#11718575C2211yescharacteristic
37#11718572C411yescharacteristic, complemented
38#11718572C2211yescharacteristic
39#11718565C411yescharacteristic, complemented
40C2×#88038C2211yescharacteristic
41#11718282C2212yescomplemented
42#11718279C2212yes—
43#11718278C2212yescomplemented
44#11718275C2212yes—
45#11718220C2212yescomplemented
46#11718220C2212yescomplemented
47#11718218C2212yescomplemented
48#11718217C2212yescomplemented
49#41×D13—21no—
50#41×D13—21no—
51Q8×#261—22no—
52#88240—22no—
53D4×#261—22no—
54D4×#261—42no—
55#88238—22no—
56#88238—42no—
57#88237—41no—
58#88237D411yescharacteristic
59C2×C4⋊#261—21no—
60C2×C4⋊#261—41no—
61D13⋊M16—41no—
62D13⋊M16D411yescharacteristic
63C2×#6118—21no—
64C2×#6118—41no—
65C13×#135C2×C411yescharacteristic
66C13×#135C2×C411yescharacteristic, complemented
67#88204—21no—
68#88204—21no—
69#88173C2×C411yescharacteristic
70#88173C2311yescharacteristic
71Q16×D13C2×C411yescharacteristic, complemented
72Q16×D13C2311yescharacteristic
73#88170—21no—
74#88170—21no—
75SD16×D13—21no—
76SD16×D13—21no—
77#88166C2×C411yescharacteristic
78#88166C2311yescharacteristic
79D8×D13C2×C411yescharacteristic, complemented
80D8×D13C2311yescharacteristic
81#88158C2×C411yescharacteristic
82#88158C2×C411yescharacteristic, complemented
83C2×C8×D13C2311yescharacteristic
84#88118—22no—
85#88117—22no—
86#88116—22no—
87#88115—22no—
88#88106—21no—
89#88103—21no—
90#88103C2312yes—
91#88101—21no—
92#88101C2312yes—
93#88100C2312yes—
94C8×#261D412yes—
95#88098C2312yes—
96D13⋊C16—21no—
97C13×Q32C2×C411yescharacteristic, complemented
98C13×SD32C2×C411yescharacteristic, complemented
99C13×SD32C2×C411yescharacteristic, complemented
100C13×D16C2×C411yescharacteristic, complemented
101C2×C13×C16C2×C411yescharacteristic, complemented
102#88069C2×C411yescharacteristic, complemented
103#88069C2×C411yescharacteristic, complemented
104#88068C2×C411yescharacteristic, complemented
105#88068C2×C411yescharacteristic, complemented
106#88067C2×C411yescharacteristic, complemented
107#88067C2×C411yescharacteristic, complemented
108#88066C2×C411yescharacteristic, complemented
109#88066C2×C411yescharacteristic, complemented
110C2×#6090C2×C411yescharacteristic, complemented
111Dic104C2×C411yescharacteristic, complemented
112#88040C2×C411yescharacteristic, complemented
113#88040C2×C411yescharacteristic, complemented
114D208C2×C411yescharacteristic, complemented
115#88038C2×C411yescharacteristic, complemented
116#88038C2×C411yescharacteristic, complemented
117#88038C2311yescharacteristic
118#88038C2311yescharacteristic
119#33469—131nomaximal, Sylow
120C22×#261—42no—
121C22×#261—41no—
122C13×#41—21no—
123C13×#41—21no—
124#6131—21no—
125#6131—21no—
126Q8×D13—21no—
127Q8×D13—21no—
128#6129—21no—
129#6129—21no—
130#6129—41no—
131#6129—41no—
132D4×D13—21no—
133D4×D13—21no—
134D4×D13—41no—
135D4×D13—41no—
136#6127—21no—
137#6127—21no—
138#6127—41no—
139#6127—41no—
140C2×C4×D13—41no—
141C2×C4×D13—41no—
142C2×C4×D13C2×D411yescharacteristic
143C22⋊#261—21no—
144C22⋊#261—42no—
145#6122—21no—
146#6122—42no—
147C2×#1057—42no—
148C2×#1057—41no—
149C4⋊#261—21no—
150C4⋊#261—22no—
151C4⋊#261—42no—
152C4⋊#261—41no—
153C4×#261—42no—
154C4×#261C2×D412yes—
155#6118—21no—
156#6118—22no—
157#6118—42no—
158#6118—41no—
159D13⋊C8—42no—
160D13⋊C8C2×D412yes—
161C13×Q16C22×C411yescharacteristic
162C13×Q16C22×C411yescharacteristic
163C13×SD16—21no—
164C13×SD16—21no—
165C13×D8C22×C411yescharacteristic
166C13×D8C22×C411yescharacteristic
167C2×C8×C13C22×C411yescharacteristic
168#6107—21no—
169#6107—21no—
170#6106—21no—
171#6106—21no—
172#6105—21no—
173#6105—21no—
174#6104—21no—
175#6104—21no—
176C2×#1055#4111yescharacteristic
177Dic52C22×C411yescharacteristic
178Dic52C22×C411yescharacteristic
179D104C22×C411yescharacteristic
180D104C22×C411yescharacteristic
181#6095—21no—
182#6095—21no—
183C8×D13C22×C411yescharacteristic
184C8×D13C2×D411yescharacteristic
185C8×D13#4111yescharacteristic
186C8×D13C2411yescharacteristic
187#6092—22no—
188C13×C16C22×C411yescharacteristic
189C13×C16C22×C411yescharacteristic
190#6090C22×C411yescharacteristic
191#6090C22×C411yescharacteristic
192#3414—131no—
193#2957—131no—
194#2952—131no—
195AGL(1,ℤ/16)—132no—
196AGL(1,ℤ/16)—132no—
197#2180—132no—
198#2180—132no—
199C2×#365—131no—
200#2148—131no—
201C2×#360—131no—
202#2110—131no—
203C22×D13—41no—
204C22×D13—41no—
205C22×D13AGL(1,ℤ/8)11yescharacteristic
206C2×#261—22no—
207C2×#261—42no—
208C2×#261—162no—
209C2×#261—161no—
210Q8×C13—21no—
211Q8×C13—21no—
212D4×C13—21no—
213D4×C13—21no—
214D4×C13—41no—
215D4×C13—41no—
216C2×C4×C13—41no—
217C2×C4×C13—41no—
218C2×C4×C13C4×D411yescharacteristic
219C22⋊D13—41no—
220C22⋊D13—41no—
221C22⋊D13—41no—
222C22⋊D13—41no—
223C22⋊D13—41no—
224C22⋊D13—41no—
225C2×Dic13—41no—
226C2×Dic13—41no—
227C2×Dic13AGL(1,ℤ/8)11yescharacteristic
228D52—21no—
229D52—21no—
230D52—41no—
231D52—41no—
232C4×D13—41no—
233C4×D13—41no—
234C4×D13—41no—
235C4×D13—41no—
236C4×D13—41no—
237C4×D13—41no—
238C4×D13C4×D411yescharacteristic
239C4×D13AGL(1,ℤ/8)11yescharacteristic
240C4×D13AGL(1,ℤ/8)11yescharacteristic
241C4×D13C22×D411yescharacteristic
242Dic26—21no—
243Dic26—21no—
244Dic26—41no—
245Dic26—41no—
246#1057—22no—
247#1057—42no—
248C8×C13C4×D411yescharacteristic
249C8×C13C23×C411yescharacteristic
250#1055C4×D411yescharacteristic
251#1055C2×#4111yescharacteristic
252C2×#135—131no—
253C2×#135—131no—
254#568—261no—
255#520—261no—
256#510—131no—
257#510—131no—
258#510—131no—
259#510—131no—
260#509—131no—
261#509—131no—
262#509—131no—
263#509—131no—
264#508—131no—
265#508—131no—
266#508—131no—
267#508#26111yescharacteristic, complemented
268C2×M32—131no—
269C2×M32—131no—
270#444—132no—
271#444—132no—
272#443—262no—
273#442—132no—
274#441—132no—
275C4×SD16—262no—
276C2×#108—131no—
277C2×#106—131no—
278#421—261no—
279#417—261no—
280#405—131no—
281#365—132no—
282#365—132no—
283#362—132no—
284#361—132no—
285#360—132no—
286#360—132no—
287#347—132no—
288#347—132no—
289C22×C13—41no—
290C22×C13—41no—
291C22×C13#44211yescharacteristic
292C2×D13—41no—
293C2×D13—41no—
294C2×D13—41no—
295C2×D13—81no—
296C2×D13—81no—
297C2×D13—161no—
298C2×D13#44211yescharacteristic
299C2×D13C2×AGL(1,ℤ/8)11yescharacteristic
300#261—42no—
301#261—82no—
302C4×C13—41no—
303C4×C13—41no—
304C4×C13#44211yescharacteristic
305C4×C13C2×C4×D411yescharacteristic
306Dic13—41no—
307Dic13—41no—
308Dic13#44211yescharacteristic
309Dic13C2×AGL(1,ℤ/8)11yescharacteristic
310C2×#41—261no—
311C2×#41—261no—
312#135—131no—
313#135—131no—
314#135—131no—
315#135—131no—
316#135—131no—
317#135—131no—
318#135—261no—
319#135—261no—
320#135—261no—
321#135—261no—
322#135C2×#26111yescharacteristic
323#135C2×#26111yescharacteristic, complemented
324C2×Q16—131no—
325C2×Q16—131no—
326C2×SD16—261no—
327C2×SD16—261no—
328C2×D8—131no—
329C2×D8—131no—
330#131—262no—
331#131—262no—
332#131—522no—
333C2×M16—131no—
334C2×M16—261no—
335C2×M16—521no—
336C2×M16—521no—
337C22×C8—131no—
338C4×Q8—262no—
339C4×D4—262no—
340C4×D4—522no—
341#117—131no—
342#117—521no—
343C2×C4⋊C4—261no—
344C2×C4⋊C4—521no—
345Q32—131no—
346Q32—131no—
347Q32—131no—
348Q32C2×#26111yescharacteristic, complemented
349SD32—131no—
350SD32—131no—
351SD32—131no—
352SD32—131no—
353SD32—131no—
354SD32—131no—
355SD32C2×#26111yescharacteristic, complemented
356SD32C2×#26111yescharacteristic, complemented
357D16—131no—
358D16—131no—
359D16—131no—
360D16C2×#26111yescharacteristic, complemented
361M32—131no—
362M32—131no—
363M32—131no—
364M32—131no—
365M32—132no—
366M32—261no—
367C2×C16—131no—
368C2×C16—261no—
369C2×C16C2×#26111yescharacteristic, complemented
370#108—132no—
371#108—261no—
372#106—132no—
373#106—261no—
374#104—262no—
375#104—262no—
376#103—262no—
377#102—262no—
378#97—132no—
379C4×C8—132no—
380C2×C13—21no—
381C2×C13—81no—
382C2×C13—81no—
383C2×C13C2×#44211yescharacteristic
384D13—21no—
385D13—21no—
386D13—81no—
387D13—81no—
388#41—21no—
389#41—21no—
390#41—261no—
391#41—261no—
392#41—261no—
393#41—261no—
394#41—261no—
395#41—261no—
396#41—521no—
397#41—521no—
398#41—521no—
399#41—521no—
400C2×Q8—261no—
401C2×Q8—261no—
402C2×D4—261no—
403C2×D4—261no—
404C2×D4—521no—
405C2×D4—521no—
406C22×C4—131no—
407C22×C4—521no—
408C22×C4—521no—
409C22×C4—522no—
410C22×C4—521no—
411Q16—131no—
412Q16—131no—
413Q16—261no—
414Q16—261no—
415Q16C22×#26111yescharacteristic
416Q16C22×#26111yescharacteristic
417SD16—21no—
418SD16—21no—
419SD16—261no—
420SD16—261no—
421SD16—261no—
422SD16—261no—
423SD16—261no—
424SD16—261no—
425D8—131no—
426D8—131no—
427D8—261no—
428D8—261no—
429D8C22×#26111yescharacteristic
430D8C22×#26111yescharacteristic
431M16—261no—
432M16—261no—
433M16—262no—
434M16—522no—
435M16—522no—
436M16—521no—
437C2×C8—131no—
438C2×C8—131no—
439C2×C8—131no—
440C2×C8—131no—
441C2×C8—131no—
442C2×C8—132no—
443C2×C8—522no—
444C2×C8—522no—
445C2×C8—521no—
446C2×C8C22×#26111yescharacteristic
447C4⋊C4—261no—
448C4⋊C4—262no—
449C4⋊C4—521no—
450C4⋊C4—522no—
451C22⋊C4—261no—
452C22⋊C4—522no—
453C42—132no—
454C42—522no—
455C16—131no—
456C16—131no—
457C16—262no—
458C16C22×#26111yescharacteristic
459C16C22×#26111yescharacteristic
460C13#3346911yescharacteristic, complemented, Sylow
461C23—131no—
462C23—521no—
463C23—521no—
464Q8—21no—
465Q8—21no—
466Q8—261no—
467Q8—261no—
468Q8—521no—
469Q8—521no—
470D4—21no—
471D4—21no—
472D4—41no—
473D4—41no—
474D4—261no—
475D4—261no—
476D4—521no—
477D4—521no—
478D4—521no—
479D4—521no—
480D4—521no—
481D4—521no—
482D4—521no—
483D4—521no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—131no—
487C2×C4—131no—
488C2×C4—131no—
489C2×C4—131no—
490C2×C4—131no—
491C2×C4—262no—
492C2×C4—521no—
493C2×C4—521no—
494C2×C4—521no—
495C2×C4—521no—
496C2×C4—521no—
497C2×C4—521no—
498C2×C4—521no—
499C2×C4—521no—
500C2×C4—522no—
501C2×C4—2082no—
502C2×C4—2081no—
503C2×C4D4×#26111yescharacteristic
504C8—131no—
505C8—131no—
506C8—262no—
507C8—522no—
508C8D4×#26111yescharacteristic
509C8C23×#26111yescharacteristic
510C22—41no—
511C22—41no—
512C22—131no—
513C22—131no—
514C22—521no—
515C22—521no—
516C22—521no—
517C22—1041no—
518C22—1041no—
519C22—2081no—
520C22#1171910211yescharacteristic
521C4—41no—
522C4—41no—
523C4—131no—
524C4—131no—
525C4—522no—
526C4—521no—
527C4—521no—
528C4—1042no—
529C4#1171910211yescharacteristic
530C4C2×D4×#26111yescharacteristic
531C2—21no—
532C2—81no—
533C2—81no—
534C2—261no—
535C2—261no—
536C2—1041no—
537C2—1041no—
538C2C2×#1171910211yescharacteristic
subgroupG/H
centerC2C2×#11719102
commutatorC8×C13C23×C4
FrattiniC8C23×#261
FittingC13×#508C4
socleC2×C13C2×#442
prime radical#13127969C2
cosocleC8×D13C24
hypercenter#508#261
hypocenterC13#33469
LCSG ▷ C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #13131615C2semidirect
2. #13131599C2semidirect
3. #13131247C2semidirect
4. #13131246C2semidirect
5. #13131244C2semidirect
6. #13131241C2semidirect
7. C2×#11718220C2semidirect
8. #13131164C2semidirect
9. D13⋊#360C2semidirect
10. #13131055C2semidirect
19. #11719110C22semidirect
23. #11719102C22semidirect
28. C13×#508C4semidirect
29. #11718855C4semidirect
30. #11718855C4semidirect
31. #11718579C4semidirect
33. #11718576C4semidirect
35. #11718575C4semidirect
37. #11718572C4semidirect
39. #11718565C4semidirect
41. #11718282C22semidirect
43. #11718278C22semidirect
45. #11718220C22semidirect
46. #11718220C22semidirect
47. #11718218C22semidirect
48. #11718217C22semidirect
66. C13×#135C2×C4semidirect
71. Q16×D13C2×C4semidirect
79. D8×D13C2×C4semidirect
82. #88158C2×C4semidirect
97. C13×Q32C2×C4semidirect
98. C13×SD32C2×C4semidirect
99. C13×SD32C2×C4semidirect
100. C13×D16C2×C4semidirect
101. C2×C13×C16C2×C4semidirect
102. #88069C2×C4semidirect
103. #88069C2×C4semidirect
104. #88068C2×C4semidirect
105. #88068C2×C4semidirect
106. #88067C2×C4semidirect
107. #88067C2×C4semidirect
108. #88066C2×C4semidirect
109. #88066C2×C4semidirect
110. C2×#6090C2×C4semidirect
111. Dic104C2×C4semidirect
112. #88040C2×C4semidirect
113. #88040C2×C4semidirect
114. D208C2×C4semidirect
115. #88038C2×C4semidirect
116. #88038C2×C4semidirect
267. #508#261semidirect
323. #135C2×#261semidirect
348. Q32C2×#261semidirect
355. SD32C2×#261semidirect
356. SD32C2×#261semidirect
360. D16C2×#261semidirect
369. C2×C16C2×#261semidirect
460. C13#33469semidirect

Maximal subgroups

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

subgroupindex
#131316152
#131315992
#131312472
#131312462
#131312442
#131312412
C2×#117182202
#131311642
D13⋊#3602
#131310552
#131279692
#3346913