Finite Group Databasebeta

← #15867762

Subgroups — #15867762

Order 3328.

1002 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718379C211yesmaximal, characteristic
2#13117238C211yesmaximal, characteristic
3C2×C4⋊C4×Dic13C211yesmaximal, characteristic
4#13115598C212yesmaximal, complemented
5#13115596C212yesmaximal, complemented
6#13115031C211yesmaximal, characteristic
7#13115029C211yesmaximal, characteristic
8#13115028C211yesmaximal, characteristic
9#106×Dic13C212yesmaximal, complemented
10#13114724C212yesmaximal, complemented
11C13×#1853C211yesmaximal, characteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic13C2211yescharacteristic
14C22×C4×Dic13—21no—
15C22×C4×Dic13C2211yescharacteristic
16C2×#88131—21no—
17C2×C13×#106C2212yes—
18C2×C13×C22⋊C8C2211yescharacteristic
19C13×#380C2211yescharacteristic
20#11718873C2211yescharacteristic
21C2×#88071—21no—
22C2×C22⋊#1055C2211yescharacteristic
23C2×#88057C2212yes—
24C2×#88054C2212yes—
25C2×C8×Dic13C2212yes—
26#11718637C2211yescharacteristic
27#11718637C2211yescharacteristic
28C4⋊C4×Dic13—21no—
29C4⋊C4×Dic13—21no—
30C4⋊C4×Dic13C2211yescharacteristic
31C4⋊C4×Dic13C2211yescharacteristic
32C4⋊C4×Dic13C2212yes—
33C4⋊C4×Dic13C2212yes—
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—24no—
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—24no—
59C2×C4⋊Dic13C2311yescharacteristic
60C2×C4⋊Dic13C2311yescharacteristic
61C2×C4⋊Dic13C2312yes—
62C2×#6101—21no—
63C2×#6101—21no—
64C2×#6101—21no—
65C2×#6101—21no—
66C2×C4×Dic13—21no—
67C2×C4×Dic13—21no—
68C2×C4×Dic13—22no—
69C2×C4×Dic13—22no—
70C2×C4×Dic13—44no—
71C2×C4×Dic13C2×C414yes—
72C2×C4×Dic13D411yescharacteristic
73C2×C4×Dic13D411yescharacteristic
74C2×C4×Dic13Q814yes—
75C2×C4×Dic13C2311yescharacteristic
76C2×C4×Dic13C2311yescharacteristic
77C22×#1055C2312yes—
78#88131—44no—
79C13×#106—24no—
80C13×C22⋊C8C2×C414yes—
81C13×#95—21no—
82#88071—21no—
83#88071—21no—
84#88071—21no—
85#88071—22no—
86#88071—22no—
87C22⋊#1055C2×C414yes—
88#88057—24no—
89#88054C2×C418yescomplemented
90C8×Dic13C2×C418yescomplemented
91#88048—24no—
92#88048—24no—
93#21359—131nomaximal, Sylow
94C24×C13C2×D411yescharacteristic
95C22×C4×C13—22no—
96C22×C4×C13—22no—
97C22×C4×C13—24no—
98C22×C4×C13—22no—
99C22×C4×C13—21no—
100C22×C4×C13—21no—
101C22×C4×C13—42no—
102C22×C4×C13—42no—
103C22×C4×C13C22×C412yes—
104C22×C4×C13C2×D411yescharacteristic
105C22×C4×C13C2×Q812yes—
106C22×C4×C13C2411yescharacteristic
107C22×Dic13—24no—
108C22×Dic13—22no—
109C22×Dic13—21no—
110C22×Dic13—21no—
111C22×Dic13—41no—
112C22×Dic13—41no—
113C22×Dic13—42no—
114C22×Dic13—42no—
115C22×Dic13C4⋊C414yes—
116C22×Dic13C4⋊C414yes—
117C22×Dic13SD1614yes—
118C22×Dic13SD1614yes—
119C22×Dic13C2×D411yescharacteristic
120C22×Dic13C2×D411yescharacteristic
121C22×Dic13C2×D411yescharacteristic
122C22×Dic13C2×D411yescharacteristic
123C2×C8×C13—22no—
124C2×C8×C13—22no—
125C2×C8×C13C22×C414yes—
126C13×C4⋊C4—24no—
127C13×C4⋊C4—22no—
128C13×C4⋊C4—22no—
129C13×C4⋊C4—22no—
130C13×C4⋊C4#4114yes—
131C13×C22⋊C4—42no—
132C22⋊Dic13—42no—
133C4⋊Dic13—24no—
134C4⋊Dic13—22no—
135C4⋊Dic13—22no—
136C4⋊Dic13—22no—
137C4⋊Dic13#4114yes—
138#6101—42no—
139#6101—42no—
140#6101—42no—
141#6101—42no—
142C4×Dic13—42no—
143C4×Dic13—42no—
144C4×Dic13—44no—
145C4×Dic13—44no—
146C4×Dic13C4⋊C414yes—
147C4×Dic13C4⋊C414yes—
148C4×Dic13C22×C412yes—
149C4×Dic13C22×C412yes—
150C4×Dic13C2×Q812yes—
151C4×Dic13C2×Q812yes—
152C2×#1055—22no—
153C2×#1055—22no—
154C2×#1055C22×C414yes—
155C2×#431—131no—
156#2285—131no—
157C2×C4×C4⋊C4—131no—
158#1948—132no—
159#1948—132no—
160#1853—131no—
161#1853—131no—
162#1853—131no—
163#1853D1311yescharacteristic
164C4×#106—132no—
165C4×#106—132no—
166C23×C13—21no—
167C23×C13—21no—
168C23×C13—21no—
169C23×C13—21no—
170C23×C13C2×C4⋊C412yes—
171C23×C13C2×SD1612yes—
172C23×C13C2×SD1612yes—
173C23×C13C22×D411yescharacteristic
174C2×C4×C13—21no—
175C2×C4×C13—21no—
176C2×C4×C13—24no—
177C2×C4×C13—24no—
178C2×C4×C13—24no—
179C2×C4×C13—48no—
180C2×C4×C13—42no—
181C2×C4×C13—42no—
182C2×C4×C13—41no—
183C2×C4×C13—41no—
184C2×C4×C13—41no—
185C2×C4×C13—41no—
186C2×C4×C13—41no—
187C2×C4×C13—41no—
188C2×C4×C13C2×C4⋊C412yes—
189C2×C4×C13C23×C411yescharacteristic
190C2×C4×C13C22×Q811yescharacteristic
191C2×C4×C13C2×#4112yes—
192C2×C4×C13C2×#4112yes—
193C2×Dic13—21no—
194C2×Dic13—21no—
195C2×Dic13—21no—
196C2×Dic13—21no—
197C2×Dic13—24no—
198C2×Dic13—44no—
199C2×Dic13—48no—
200C2×Dic13—42no—
201C2×Dic13—42no—
202C2×Dic13—41no—
203C2×Dic13—41no—
204C2×Dic13—41no—
205C2×Dic13—41no—
206C2×Dic13—41no—
207C2×Dic13—41no—
208C2×Dic13#106116yescomplemented
209C2×Dic13C2×C4⋊C412yes—
210C2×Dic13C2×C4⋊C412yes—
211C2×Dic13C2×C4⋊C412yes—
212C2×Dic13C2×C4⋊C412yes—
213C2×Dic13C2×SD1612yes—
214C2×Dic13C2×SD1612yes—
215C2×Dic13C2×SD1612yes—
216C2×Dic13C2×SD1612yes—
217C8×C13—24no—
218#1055—24no—
219C2×#117—261no—
220C22×C4⋊C4—131no—
221C22×C4⋊C4C2×D1311yescharacteristic
222C22×C42—131no—
223C22×C42—261no—
224#431—138no—
225C2×#106—132no—
226C2×#106—132no—
227C2×#106—132no—
228C2×#106C2×D1312yes—
229C2×C4⋊C8—132no—
230C2×C22⋊C8—131no—
231C2×C22⋊C8C2×D1311yescharacteristic
232C2×C4×C8—132no—
233#384—131no—
234#384—131no—
235#382—261no—
236#380—131no—
237#380—261no—
238#380C2×D1311yescharacteristic
239C4×C4⋊C4—132no—
240C4×C4⋊C4—132no—
241C4×C4⋊C4—131no—
242C4×C4⋊C4—131no—
243C4×C4⋊C4—261no—
244C4×C4⋊C4—261no—
245C4×C4⋊C4—261no—
246C2×#95—261no—
247#340—132no—
248#340—132no—
249#340—132no—
250#340C2×D1312yes—
251C22×C13—22no—
252C22×C13—22no—
253C22×C13—22no—
254C22×C13—22no—
255C22×C13—22no—
256C22×C13—22no—
257C22×C13C2×#10614yes—
258C22×C13C22×C4⋊C411yescharacteristic
259C22×C13#52511yescharacteristic
260C22×C13#52511yescharacteristic
261C22×C13C22×SD1611yescharacteristic
262C22×C13C22×SD1611yescharacteristic
263C22×C13#57511yescharacteristic
264C22×C13#57511yescharacteristic
265C4×C13—22no—
266C4×C13—44no—
267C4×C13—81no—
268C4×C13—81no—
269C4×C13C4×#4112yes—
270C4×C13#53312yes—
271Dic13—22no—
272Dic13—22no—
273Dic13—44no—
274Dic13—81no—
275Dic13—81no—
276Dic13C2×#10614yes—
277Dic13C2×#10614yes—
278C23×C4—21no—
279C23×C4—131no—
280C23×C4—131no—
281C23×C4—261no—
282C23×C4C22×D1311yescharacteristic
283C22×C8—132no—
284C22×C8C22×D1312yes—
285#117—524no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—132no—
289C2×C4⋊C4—131no—
290C2×C4⋊C4—131no—
291C2×C4⋊C4—264no—
292C2×C4⋊C4—261no—
293C2×C4⋊C4—261no—
294C2×C4⋊C4—261no—
295C2×C4⋊C4—261no—
296C2×C4⋊C4—261no—
297C2×C4⋊C4C22×D1311yescharacteristic
298C2×C4⋊C4C22×D1311yescharacteristic
299C2×C4⋊C4C22×D1312yes—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—261no—
302C2×C42—134no—
303C2×C42—131no—
304C2×C42—131no—
305C2×C42—134no—
306C2×C42—131no—
307C2×C42—131no—
308C2×C42—261no—
309C2×C42—261no—
310C2×C42—262no—
311C2×C42—262no—
312C2×C42—524no—
313#106—24no—
314#106—264no—
315#106—264no—
316#106—264no—
317C4⋊C8—138no—
318C22⋊C8—134no—
319C22⋊C8C4×D1314yes—
320C4×C8—138no—
321#95—21no—
322#95—262no—
323#95—262no—
324#95—261no—
325#95—261no—
326#95—261no—
327C2×C13—24no—
328C2×C13#229611yescharacteristic
329C2×C13C22×#10611yescharacteristic
330C2×C13#291111yescharacteristic
331C2×C13#310111yescharacteristic
332C2×C13#310111yescharacteristic
333C2×C13#310311yescharacteristic
334C2×C13#310311yescharacteristic
335C24D4×D1311yescharacteristic
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—134no—
345C22×C4—134no—
346C22×C4—134no—
347C22×C4—131no—
348C22×C4—131no—
349C22×C4—131no—
350C22×C4—131no—
351C22×C4—134no—
352C22×C4—264no—
353C22×C4—262no—
354C22×C4—261no—
355C22×C4—261no—
356C22×C4—521no—
357C22×C4—521no—
358C22×C4—522no—
359C22×C4—522no—
360C22×C4C2×C4×D1312yes—
361C22×C4D4×D1311yescharacteristic
362C22×C4Q8×D1312yes—
363C22×C4C23×D1311yescharacteristic
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—134no—
367C2×C8—262no—
368C2×C8—262no—
369C2×C8C2×C4×D1314yes—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—134no—
375C4⋊C4—264no—
376C4⋊C4—262no—
377C4⋊C4—262no—
378C4⋊C4—262no—
379C4⋊C4—522no—
380C4⋊C4—522no—
381C4⋊C4—522no—
382C4⋊C4—522no—
383C4⋊C4#612914yes—
384C22⋊C4—42no—
385C22⋊C4—522no—
386C42—134no—
387C42—134no—
388C42—132no—
389C42—132no—
390C42—132no—
391C42—132no—
392C42—524no—
393C42—524no—
394C42—522no—
395C42—522no—
396C13#2135911yescharacteristic, complemented, Sylow
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23C4⋊C4×D1312yes—
402C23SD16×D1312yes—
403C23SD16×D1312yes—
404C23C2×D4×D1311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—48no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—132no—
420C2×C4—132no—
421C2×C4—132no—
422C2×C4—132no—
423C2×C4—132no—
424C2×C4—132no—
425C2×C4—1316no—
426C2×C4—132no—
427C2×C4—132no—
428C2×C4—261no—
429C2×C4—261no—
430C2×C4—261no—
431C2×C4—261no—
432C2×C4—264no—
433C2×C4—524no—
434C2×C4—528no—
435C2×C4—522no—
436C2×C4—522no—
437C2×C4—521no—
438C2×C4—521no—
439C2×C4—521no—
440C2×C4—521no—
441C2×C4—521no—
442C2×C4—521no—
443C2×C4C4⋊C4×D1312yes—
444C2×C4C22×C4×D1311yescharacteristic
445C2×C4C2×#612912yes—
446C2×C4C2×Q8×D1311yescharacteristic
447C2×C4#41×D1312yes—
448C8—24no—
449C8—264no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22D13×#10614yes—
457C22C2×C4⋊C4×D1311yescharacteristic
458C22#1171929111yescharacteristic
459C22#1171936111yescharacteristic
460C22C2×SD16×D1311yescharacteristic
461C22C2×SD16×D1311yescharacteristic
462C22#1171947811yescharacteristic
463C22#1171948211yescharacteristic
464C4—22no—
465C4—44no—
466C4—81no—
467C4—81no—
468C4—134no—
469C4—134no—
470C4—262no—
471C4—262no—
472C4—524no—
473C4—1041no—
474C4—1041no—
475C4C4×#612912yes—
476C4#1171932712yes—
477C2—24no—
478C2#1311731511yescharacteristic
479C2C2×D13×#10611yescharacteristic
480C2#1312342311yescharacteristic
481C2#1312483911yescharacteristic
482C2#1312484111yescharacteristic
483C2#1312486211yescharacteristic
484C2#1312486411yescharacteristic
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#21359
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. #106×Dic13C2semidirect
10. #13114724C2semidirect
37. #11718379C4semidirect
89. #88054C2×C4semidirect
90. C8×Dic13C2×C4semidirect
208. C2×Dic13#106semidirect
396. C13#21359semidirect

Maximal subgroups

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

subgroupindex
C2×#117183792
#131172382
C2×C4⋊C4×Dic132
#131155982
#131155962
#131150312
#131150292
#131150282
#106×Dic132
#131147242
C13×#18532
#2135913