Finite Group Databasebeta

← #15873430

Subgroups — #15873430

Order 3328.

966 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
1C2×#11718379C211yesmaximal, characteristic
2#13117238C211yesmaximal, characteristic
3C2×#11718637C211yesmaximal, characteristic
4#13115598C212yesmaximal, complemented
5#13115596C212yesmaximal, complemented
6#13115031C211yesmaximal, characteristic
7#13115029C211yesmaximal, characteristic
8#13115028C211yesmaximal, characteristic
9#13114991C212yesmaximal, complemented
10#13114962C212yesmaximal, complemented
11C13×#1853C211yesmaximal, characteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic13C2211yescharacteristic
14C22×#6101—21no—
15C22×#6101—21no—
16C22×C4×Dic13C2211yescharacteristic
17C2×#88131—21no—
18C2×C13×#106C2212yes—
19C2×C13×C22⋊C8C2211yescharacteristic
20C13×#380C2211yescharacteristic
21#11718873C2211yescharacteristic
22C2×C22⋊#1055C2211yescharacteristic
23C2×#88057C2212yes—
24C2×#88054C2212yes—
25C2×C8×Dic13C2212yes—
26#11718637—21no—
27#11718637—21no—
28#11718637C2212yes—
29#11718637C2212yes—
30#11718637C2211yescharacteristic
31#11718637C2211yescharacteristic
32C4⋊C4×Dic13C2211yescharacteristic
33C4⋊C4×Dic13C2211yescharacteristic
34C2×#88048C2212yes—
35C2×#88048C2212yes—
36#11718603—21no—
37#11718379C418yescomplemented
38#11718311—21no—
39#11718310—21no—
40C13×#340C2212yes—
41#11718143C2212yes—
42#11718121C2212yes—
43#11718121C2212yes—
44C23×C4×C13—21no—
45C23×C4×C13C2311yescharacteristic
46C23×Dic13—21no—
47C23×Dic13D411yescharacteristic
48C23×Dic13D411yescharacteristic
49C22×C8×C13C2312yes—
50C2×C13×C4⋊C4—21no—
51C2×C13×C4⋊C4—42no—
52C2×C13×C4⋊C4C2311yescharacteristic
53C2×C13×C4⋊C4C2311yescharacteristic
54C2×C13×C4⋊C4C2312yes—
55C2×C13×C22⋊C4—21no—
56C2×C22⋊Dic13—21no—
57C2×C4⋊Dic13—21no—
58C2×C4⋊Dic13—42no—
59C2×C4⋊Dic13C2312yes—
60C2×C4⋊Dic13C2311yescharacteristic
61C2×C4⋊Dic13C2311yescharacteristic
62C2×#6101—21no—
63C2×#6101—21no—
64C2×#6101—22no—
65C2×#6101—22no—
66C2×#6101—22no—
67C2×#6101—22no—
68C2×#6101—21no—
69C2×#6101—21no—
70C2×#6101—44no—
71C2×#6101—44no—
72C2×C4×Dic13—21no—
73C2×C4×Dic13—21no—
74C2×C4×Dic13C2×C414yes—
75C2×C4×Dic13D411yescharacteristic
76C2×C4×Dic13D411yescharacteristic
77C2×C4×Dic13Q814yes—
78C2×C4×Dic13C2311yescharacteristic
79C2×C4×Dic13C2311yescharacteristic
80C22×#1055C2312yes—
81#88131—44no—
82C13×#106—42no—
83C13×C22⋊C8C2×C414yes—
84C13×#95—21no—
85#88071—21no—
86#88071—21no—
87#88071—21no—
88C22⋊#1055C2×C414yes—
89#88057—42no—
90#88054C2×C418yescomplemented
91C8×Dic13C2×C418yescomplemented
92#88048—42no—
93#88048—42no—
94#21981—131nomaximal, Sylow
95C24×C13C2×D411yescharacteristic
96C22×C4×C13—22no—
97C22×C4×C13—22no—
98C22×C4×C13—21no—
99C22×C4×C13—21no—
100C22×C4×C13—22no—
101C22×C4×C13—42no—
102C22×C4×C13—42no—
103C22×C4×C13—42no—
104C22×C4×C13C22×C412yes—
105C22×C4×C13C2×D411yescharacteristic
106C22×C4×C13C2×Q812yes—
107C22×C4×C13C2411yescharacteristic
108C22×Dic13—22no—
109C22×Dic13—22no—
110C22×Dic13—21no—
111C22×Dic13—21no—
112C22×Dic13—22no—
113C22×Dic13—42no—
114C22×Dic13—42no—
115C22×Dic13—42no—
116C22×Dic13C4⋊C414yes—
117C22×Dic13C4⋊C414yes—
118C22×Dic13SD1614yes—
119C22×Dic13SD1614yes—
120C22×Dic13C2×D411yescharacteristic
121C22×Dic13C2×D411yescharacteristic
122C22×Dic13C2×D411yescharacteristic
123C22×Dic13C2×D411yescharacteristic
124C2×C8×C13—22no—
125C2×C8×C13—22no—
126C2×C8×C13C22×C414yes—
127C13×C4⋊C4—22no—
128C13×C4⋊C4—22no—
129C13×C4⋊C4—22no—
130C13×C4⋊C4—22no—
131C13×C4⋊C4—42no—
132C13×C22⋊C4—42no—
133C22⋊Dic13—42no—
134C4⋊Dic13—22no—
135C4⋊Dic13—22no—
136C4⋊Dic13—22no—
137C4⋊Dic13—22no—
138C4⋊Dic13—42no—
139#6101—44no—
140#6101—42no—
141#6101—42no—
142#6101—44no—
143#6101—44no—
144#6101—42no—
145#6101—44no—
146#6101—42no—
147C4×Dic13—42no—
148C4×Dic13—42no—
149C4×Dic13C4⋊C414yes—
150C4×Dic13C4⋊C414yes—
151C4×Dic13C22×C412yes—
152C4×Dic13C22×C412yes—
153C4×Dic13C2×Q812yes—
154C4×Dic13C2×Q812yes—
155C2×#1055—22no—
156C2×#1055—22no—
157C2×#1055C22×C414yes—
158C2×#431—131no—
159#2285—131no—
160C2×#384—131no—
161#1948—132no—
162#1948—132no—
163#1853—131no—
164#1853—131no—
165#1853—131no—
166#1853D1311yescharacteristic
167#1844—132no—
168#1834—132no—
169C23×C13—21no—
170C23×C13—21no—
171C23×C13—21no—
172C23×C13—21no—
173C23×C13C2×C4⋊C412yes—
174C23×C13C2×SD1612yes—
175C23×C13C2×SD1612yes—
176C23×C13C22×D411yescharacteristic
177C2×C4×C13—21no—
178C2×C4×C13—21no—
179C2×C4×C13—24no—
180C2×C4×C13—24no—
181C2×C4×C13—42no—
182C2×C4×C13—41no—
183C2×C4×C13—41no—
184C2×C4×C13—41no—
185C2×C4×C13—41no—
186C2×C4×C13—42no—
187C2×C4×C13—42no—
188C2×C4×C13—41no—
189C2×C4×C13—41no—
190C2×C4×C13—84no—
191C2×C4×C13C2×C4⋊C412yes—
192C2×C4×C13C23×C411yescharacteristic
193C2×C4×C13C22×Q811yescharacteristic
194C2×C4×C13H(5,𝔽2)12yes—
195C2×C4×C132−1+412yes—
196C2×Dic13—21no—
197C2×Dic13—21no—
198C2×Dic13—28no—
199C2×Dic13—42no—
200C2×Dic13—41no—
201C2×Dic13—41no—
202C2×Dic13—41no—
203C2×Dic13—41no—
204C2×Dic13—42no—
205C2×Dic13—42no—
206C2×Dic13—41no—
207C2×Dic13—41no—
208C2×Dic13—84no—
209C2×Dic13#106116yescomplemented
210C2×Dic13C2×C4⋊C412yes—
211C2×Dic13C2×C4⋊C412yes—
212C2×Dic13C2×C4⋊C412yes—
213C2×Dic13C2×C4⋊C412yes—
214C2×Dic13C2×SD1612yes—
215C2×Dic13C2×SD1612yes—
216C2×Dic13C2×SD1612yes—
217C2×Dic13C2×SD1612yes—
218C2×Dic13AGL(1,ℤ/8)12yes—
219C2×Dic13#13712yes—
220C8×C13—24no—
221#1055—24no—
222C2×#117—261no—
223C22×C4⋊C4—131no—
224C22×C4⋊C4—261no—
225C22×C4⋊C4—261no—
226C22×C4⋊C4C2×D1311yescharacteristic
227C22×C42—131no—
228#431—138no—
229C2×#106—132no—
230C2×#106—132no—
231C2×#106—132no—
232C2×#106C2×D1312yes—
233C2×C4⋊C8—132no—
234C2×C22⋊C8—131no—
235C2×C22⋊C8C2×D1311yescharacteristic
236C2×C4×C8—132no—
237#384—131no—
238#384—132no—
239#384—131no—
240#384—132no—
241#384—261no—
242#384—261no—
243#382—261no—
244#380—131no—
245#380—261no—
246#380C2×D1311yescharacteristic
247C4×C4⋊C4—131no—
248C4×C4⋊C4—131no—
249C4×C4⋊C4—261no—
250#340—132no—
251#340—132no—
252#340—132no—
253#340C2×D1312yes—
254C22×C13—22no—
255C22×C13—22no—
256C22×C13—22no—
257C22×C13—22no—
258C22×C13—22no—
259C22×C13—22no—
260C22×C13C2×#10614yes—
261C22×C13C22×C4⋊C411yescharacteristic
262C22×C13#53511yescharacteristic
263C22×C13#53611yescharacteristic
264C22×C13C22×SD1611yescharacteristic
265C22×C13C22×SD1611yescharacteristic
266C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C13C2×#13711yescharacteristic
268C4×C13—22no—
269C4×C13—81no—
270C4×C13—82no—
271C4×C13—81no—
272C4×C13#52012yes—
273C4×C13#54412yes—
274Dic13—22no—
275Dic13—81no—
276Dic13—82no—
277Dic13—81no—
278Dic13C2×#10614yes—
279Dic13C2×#10614yes—
280Dic13#42814yes—
281C23×C4—21no—
282C23×C4—131no—
283C23×C4—131no—
284C23×C4—261no—
285C23×C4C22×D1311yescharacteristic
286C22×C8—132no—
287C22×C8C22×D1312yes—
288#117—524no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—132no—
292C2×C4⋊C4—131no—
293C2×C4⋊C4—131no—
294C2×C4⋊C4—262no—
295C2×C4⋊C4—262no—
296C2×C4⋊C4—262no—
297C2×C4⋊C4—262no—
298C2×C4⋊C4—261no—
299C2×C4⋊C4—261no—
300C2×C4⋊C4—261no—
301C2×C4⋊C4—261no—
302C2×C4⋊C4—261no—
303C2×C4⋊C4—522no—
304C2×C4⋊C4—524no—
305C2×C4⋊C4—524no—
306C2×C4⋊C4C22×D1312yes—
307C2×C4⋊C4C22×D1311yescharacteristic
308C2×C4⋊C4C22×D1311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—261no—
311C2×C42—134no—
312C2×C42—131no—
313C2×C42—131no—
314C2×C42—134no—
315C2×C42—131no—
316C2×C42—131no—
317C2×C42—261no—
318C2×C42—261no—
319#106—42no—
320#106—522no—
321#106—522no—
322#106—522no—
323C4⋊C8—138no—
324C22⋊C8—134no—
325C22⋊C8C4×D1314yes—
326C4×C8—138no—
327#95—21no—
328#95—261no—
329#95—261no—
330#95—261no—
331C2×C13—24no—
332C2×C13#231711yescharacteristic
333C2×C13C22×#10611yescharacteristic
334C2×C13C2×#42811yescharacteristic
335C2×C13#320611yescharacteristic
336C2×C13#320811yescharacteristic
337C2×C13#322711yescharacteristic
338C2×C13#322911yescharacteristic
339C24D4×D1311yescharacteristic
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—134no—
349C22×C4—134no—
350C22×C4—134no—
351C22×C4—131no—
352C22×C4—131no—
353C22×C4—131no—
354C22×C4—134no—
355C22×C4—131no—
356C22×C4—262no—
357C22×C4—262no—
358C22×C4—261no—
359C22×C4—261no—
360C22×C4—262no—
361C22×C4—522no—
362C22×C4—522no—
363C22×C4—522no—
364C22×C4C2×C4×D1312yes—
365C22×C4D4×D1311yescharacteristic
366C22×C4Q8×D1312yes—
367C22×C4C23×D1311yescharacteristic
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—134no—
371C2×C8—262no—
372C2×C8—262no—
373C2×C8C2×C4×D1314yes—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—262no—
380C4⋊C4—262no—
381C4⋊C4—262no—
382C4⋊C4—262no—
383C4⋊C4—522no—
384C4⋊C4—524no—
385C4⋊C4—524no—
386C4⋊C4—524no—
387C4⋊C4—522no—
388C4⋊C4—522no—
389C4⋊C4—522no—
390C4⋊C4—522no—
391C4⋊C4—524no—
392C22⋊C4—42no—
393C22⋊C4—522no—
394C42—134no—
395C42—134no—
396C42—132no—
397C42—132no—
398C42—132no—
399C42—132no—
400C42—522no—
401C42—522no—
402C13#2198111yescharacteristic, complemented, Sylow
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23C4⋊C4×D1312yes—
408C23SD16×D1312yes—
409C23SD16×D1312yes—
410C23C2×D4×D1311yescharacteristic
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—84no—
425C2×C4—132no—
426C2×C4—132no—
427C2×C4—132no—
428C2×C4—132no—
429C2×C4—132no—
430C2×C4—132no—
431C2×C4—132no—
432C2×C4—132no—
433C2×C4—132no—
434C2×C4—1316no—
435C2×C4—132no—
436C2×C4—268no—
437C2×C4—261no—
438C2×C4—261no—
439C2×C4—521no—
440C2×C4—521no—
441C2×C4—522no—
442C2×C4—522no—
443C2×C4—521no—
444C2×C4—521no—
445C2×C4—522no—
446C2×C4—521no—
447C2×C4—521no—
448C2×C4—1044no—
449C2×C4C4⋊C4×D1312yes—
450C2×C4C22×C4×D1311yescharacteristic
451C2×C4#8825112yes—
452C2×C4C2×Q8×D1311yescharacteristic
453C2×C4#8825712yes—
454C8—24no—
455C8—264no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22D13×#10614yes—
463C22C2×C4⋊C4×D1311yescharacteristic
464C22#1171929711yescharacteristic
465C22#1171935311yescharacteristic
466C22C2×SD16×D1311yescharacteristic
467C22C2×SD16×D1311yescharacteristic
468C22D13×AGL(1,ℤ/8)11yescharacteristic
469C22D13×#13711yescharacteristic
470C4—22no—
471C4—81no—
472C4—81no—
473C4—82no—
474C4—134no—
475C4—134no—
476C4—134no—
477C4—262no—
478C4—1041no—
479C4—1041no—
480C4—1042no—
481C4#1171925112yes—
482C4#1171932812yes—
483C2—24no—
484C2#1311745511yescharacteristic
485C2C2×D13×#10611yescharacteristic
486C2D13×#42811yescharacteristic
487C2#1312575911yescharacteristic
488C2#1312577511yescharacteristic
489C2#1312596211yescharacteristic
490C2#1312597811yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1853C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1853D13
hypocenterC13#21981
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #13115598C2semidirect
5. #13115596C2semidirect
9. #13114991C2semidirect
10. #13114962C2semidirect
37. #11718379C4semidirect
90. #88054C2×C4semidirect
91. C8×Dic13C2×C4semidirect
209. C2×Dic13#106semidirect
402. C13#21981semidirect

Maximal subgroups

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

subgroupindex
C2×#117183792
#131172382
C2×#117186372
#131155982
#131155962
#131150312
#131150292
#131150282
#131149912
#131149622
C13×#18532
#2198113