Finite Group Databasebeta

← #15873471

Subgroups — #15873471

Order 3328.

858 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718380C211yesmaximal, characteristic
2#13117238C211yesmaximal, characteristic
3C2×#11718638C211yesmaximal, characteristic
4#13115603C212yesmaximal, complemented
5#13115601C212yesmaximal, complemented
6#13115031C211yesmaximal, characteristic
7#13115028C211yesmaximal, characteristic
8#13115024C211yesmaximal, characteristic
9#13115003C212yesmaximal, complemented
10#13114968C212yesmaximal, complemented
11C13×#1852C211yesmaximal, characteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic13C2211yescharacteristic
14C22×C4×Dic13C2211yescharacteristic
15C2×#88131—21no—
16C2×C13×#107C2212yes—
17C2×C13×C22⋊C8C2211yescharacteristic
18C13×#380C2211yescharacteristic
19#11718873C2211yescharacteristic
20C2×#88071—21no—
21C2×#88071—21no—
22C2×C22⋊#1055C2211yescharacteristic
23C2×#88057C2212yes—
24C2×#88055C2212yes—
25C2×#88054C2212yes—
26#11718638—21no—
27#11718638—21no—
28#11718638C2212yes—
29#11718638C2212yes—
30#11718637C2211yescharacteristic
31#11718637C2211yescharacteristic
32C4⋊C4×Dic13C2211yescharacteristic
33C4⋊C4×Dic13C2211yescharacteristic
34C2×#88048C2212yes—
35C2×#88047C2212yes—
36#11718603—21no—
37#11718380C418yescomplemented
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—21no—
65C2×#6101—21no—
66C2×C4×Dic13—21no—
67C2×C4×Dic13—21no—
68C2×C4×Dic13C2×C414yes—
69C2×C4×Dic13D411yescharacteristic
70C2×C4×Dic13D411yescharacteristic
71C2×C4×Dic13Q814yes—
72C2×C4×Dic13C2311yescharacteristic
73C2×C4×Dic13C2311yescharacteristic
74C22×#1055C2312yes—
75#88131—44no—
76C13×#107—42no—
77C13×C22⋊C8C2×C414yes—
78C13×#95—21no—
79#88071—21no—
80#88071—21no—
81#88071—21no—
82#88071—22no—
83#88071—22no—
84#88071—22no—
85#88071—22no—
86C22⋊#1055C2×C414yes—
87#88057—42no—
88#88055C2×C418yescomplemented
89#88054C2×C418yescomplemented
90#88048—42no—
91#88047—42no—
92#21986—131nomaximal, Sylow
93C24×C13C2×D411yescharacteristic
94C22×C4×C13—22no—
95C22×C4×C13—22no—
96C22×C4×C13—22no—
97C22×C4×C13—21no—
98C22×C4×C13—21no—
99C22×C4×C13—42no—
100C22×C4×C13—42no—
101C22×C4×C13—42no—
102C22×C4×C13C22×C412yes—
103C22×C4×C13C2×D411yescharacteristic
104C22×C4×C13C2×Q812yes—
105C22×C4×C13C2411yescharacteristic
106C22×Dic13—22no—
107C22×Dic13—22no—
108C22×Dic13—22no—
109C22×Dic13—21no—
110C22×Dic13—21no—
111C22×Dic13—41no—
112C22×Dic13—41no—
113C22×Dic13—42no—
114C22×Dic13—42no—
115C22×Dic13—42no—
116C22×Dic13C4⋊C414yes—
117C22×Dic13C4⋊C414yes—
118C22×Dic13C2×D411yescharacteristic
119C22×Dic13C2×D411yescharacteristic
120C22×Dic13C2×D411yescharacteristic
121C22×Dic13C2×D411yescharacteristic
122C2×C8×C13—22no—
123C2×C8×C13—22no—
124C2×C8×C13C22×C414yes—
125C13×C4⋊C4—22no—
126C13×C4⋊C4—22no—
127C13×C4⋊C4—22no—
128C13×C4⋊C4—22no—
129C13×C4⋊C4—42no—
130C13×C22⋊C4—42no—
131C22⋊Dic13—42no—
132C4⋊Dic13—22no—
133C4⋊Dic13—22no—
134C4⋊Dic13—22no—
135C4⋊Dic13—22no—
136C4⋊Dic13—42no—
137#6101—42no—
138#6101—42no—
139#6101—42no—
140#6101—42no—
141C4×Dic13—42no—
142C4×Dic13—42no—
143C4×Dic13C4⋊C414yes—
144C4×Dic13C4⋊C414yes—
145C4×Dic13C22×C412yes—
146C4×Dic13C22×C412yes—
147C4×Dic13C2×Q812yes—
148C4×Dic13C2×Q812yes—
149C2×#1055—22no—
150C2×#1055—22no—
151C2×#1055C22×C414yes—
152C2×#432—131no—
153#2285—131no—
154C2×#382—131no—
155#1949—132no—
156#1949—132no—
157#1853—131no—
158#1853—131no—
159#1852—131no—
160#1852D1311yescharacteristic
161#1847—132no—
162#1836—132no—
163C23×C13—21no—
164C23×C13—21no—
165C23×C13—21no—
166C23×C13—21no—
167C23×C13C2×C4⋊C412yes—
168C23×C13AGL(1,ℤ/8)12yes—
169C23×C13#13712yes—
170C23×C13C22×D411yescharacteristic
171C2×C4×C13—21no—
172C2×C4×C13—21no—
173C2×C4×C13—24no—
174C2×C4×C13—24no—
175C2×C4×C13—42no—
176C2×C4×C13—41no—
177C2×C4×C13—41no—
178C2×C4×C13—41no—
179C2×C4×C13—41no—
180C2×C4×C13—42no—
181C2×C4×C13—41no—
182C2×C4×C13—41no—
183C2×C4×C13—42no—
184C2×C4×C13—84no—
185C2×C4×C13C2×C4⋊C412yes—
186C2×C4×C13C23×C411yescharacteristic
187C2×C4×C13C22×Q811yescharacteristic
188C2×C4×C13H(5,𝔽2)12yes—
189C2×C4×C132−1+412yes—
190C2×Dic13—21no—
191C2×Dic13—21no—
192C2×Dic13—21no—
193C2×Dic13—21no—
194C2×Dic13—21no—
195C2×Dic13—21no—
196C2×Dic13—21no—
197C2×Dic13—21no—
198C2×Dic13—28no—
199C2×Dic13—44no—
200C2×Dic13—42no—
201C2×Dic13—41no—
202C2×Dic13—41no—
203C2×Dic13—41no—
204C2×Dic13—41no—
205C2×Dic13—42no—
206C2×Dic13—41no—
207C2×Dic13—41no—
208C2×Dic13—42no—
209C2×Dic13—84no—
210C2×Dic13C2×C4⋊C412yes—
211C2×Dic13C2×C4⋊C412yes—
212C2×Dic13C2×C4⋊C412yes—
213C2×Dic13C2×C4⋊C412yes—
214C8×C13—24no—
215#1055—24no—
216C2×#117—261no—
217C22×C4⋊C4—131no—
218C22×C4⋊C4C2×D1311yescharacteristic
219C22×C42—131no—
220#432—138no—
221C2×#107—132no—
222C2×#107C2×D1312yes—
223C2×#106—132no—
224C2×#106—132no—
225C2×C4⋊C8—132no—
226C2×C22⋊C8—131no—
227C2×C22⋊C8C2×D1311yescharacteristic
228C2×#97—132no—
229#384—131no—
230#384—131no—
231#382—132no—
232#382—132no—
233#382—261no—
234#382—261no—
235#382—261no—
236#380—131no—
237#380—261no—
238#380C2×D1311yescharacteristic
239C4×C4⋊C4—131no—
240C4×C4⋊C4—131no—
241C4×C4⋊C4—261no—
242C2×#95—261no—
243C2×#95—261no—
244#340—132no—
245#340—132no—
246#340—132no—
247#340C2×D1312yes—
248C22×C13—22no—
249C22×C13—22no—
250C22×C13—22no—
251C22×C13—22no—
252C22×C13—22no—
253C22×C13—22no—
254C22×C13#42814yes—
255C22×C13C22×C4⋊C411yescharacteristic
256C22×C13#53511yescharacteristic
257C22×C13#53611yescharacteristic
258C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C13C2×#13711yescharacteristic
260C22×C13#57511yescharacteristic
261C22×C13#57511yescharacteristic
262C4×C13—22no—
263C4×C13—81no—
264C4×C13—82no—
265C4×C13—81no—
266C4×C13#52012yes—
267C4×C13#54412yes—
268Dic13—22no—
269Dic13—22no—
270Dic13—22no—
271Dic13—22no—
272Dic13—81no—
273Dic13—82no—
274Dic13—81no—
275C23×C4—21no—
276C23×C4—131no—
277C23×C4—131no—
278C23×C4—261no—
279C23×C4C22×D1311yescharacteristic
280C22×C8—132no—
281C22×C8C22×D1312yes—
282#117—524no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—132no—
286C2×C4⋊C4—131no—
287C2×C4⋊C4—131no—
288C2×C4⋊C4—261no—
289C2×C4⋊C4—261no—
290C2×C4⋊C4—261no—
291C2×C4⋊C4—261no—
292C2×C4⋊C4—261no—
293C2×C4⋊C4—522no—
294C2×C4⋊C4C22×D1312yes—
295C2×C4⋊C4C22×D1311yescharacteristic
296C2×C4⋊C4C22×D1311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—261no—
299C2×C42—134no—
300C2×C42—131no—
301C2×C42—131no—
302C2×C42—134no—
303C2×C42—131no—
304C2×C42—131no—
305C2×C42—261no—
306C2×C42—261no—
307#107—42no—
308#107—522no—
309#106—522no—
310#106—522no—
311C4⋊C8—138no—
312C22⋊C8—134no—
313C22⋊C8C4×D1314yes—
314#97—138no—
315#95—21no—
316#95—262no—
317#95—262no—
318#95—262no—
319#95—262no—
320#95—261no—
321#95—261no—
322#95—261no—
323C2×C13—24no—
324C2×C13#231711yescharacteristic
325C2×C13C2×#42811yescharacteristic
326C2×C13#291111yescharacteristic
327C2×C13#321311yescharacteristic
328C2×C13#321511yescharacteristic
329C2×C13#323411yescharacteristic
330C2×C13#323611yescharacteristic
331C24D4×D1311yescharacteristic
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—134no—
341C22×C4—131no—
342C22×C4—131no—
343C22×C4—131no—
344C22×C4—131no—
345C22×C4—134no—
346C22×C4—262no—
347C22×C4—262no—
348C22×C4—262no—
349C22×C4—261no—
350C22×C4—261no—
351C22×C4—521no—
352C22×C4—521no—
353C22×C4—522no—
354C22×C4—522no—
355C22×C4—522no—
356C22×C4C2×C4×D1312yes—
357C22×C4D4×D1311yescharacteristic
358C22×C4Q8×D1312yes—
359C22×C4C23×D1311yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—134no—
363C2×C8—262no—
364C2×C8—262no—
365C2×C8C2×C4×D1314yes—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—262no—
372C4⋊C4—262no—
373C4⋊C4—262no—
374C4⋊C4—262no—
375C4⋊C4—522no—
376C4⋊C4—522no—
377C4⋊C4—522no—
378C4⋊C4—522no—
379C4⋊C4—522no—
380C22⋊C4—42no—
381C22⋊C4—522no—
382C42—134no—
383C42—134no—
384C42—132no—
385C42—132no—
386C42—132no—
387C42—132no—
388C42—522no—
389C42—522no—
390C13#2198611yescharacteristic, complemented, Sylow
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23C4⋊C4×D1312yes—
396C23#8816512yes—
397C23#8817212yes—
398C23C2×D4×D1311yescharacteristic
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—84no—
413C2×C4—132no—
414C2×C4—132no—
415C2×C4—132no—
416C2×C4—132no—
417C2×C4—261no—
418C2×C4—261no—
419C2×C4—261no—
420C2×C4—261no—
421C2×C4—261no—
422C2×C4—261no—
423C2×C4—261no—
424C2×C4—261no—
425C2×C4—268no—
426C2×C4—524no—
427C2×C4—521no—
428C2×C4—521no—
429C2×C4—522no—
430C2×C4—521no—
431C2×C4—521no—
432C2×C4—522no—
433C2×C4—522no—
434C2×C4—521no—
435C2×C4—521no—
436C2×C4—1044no—
437C2×C4C4⋊C4×D1312yes—
438C2×C4C22×C4×D1311yescharacteristic
439C2×C4#8825112yes—
440C2×C4C2×Q8×D1311yescharacteristic
441C2×C4#8825712yes—
442C8—24no—
443C8—264no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22#1171854214yes—
451C22C2×C4⋊C4×D1311yescharacteristic
452C22#1171929711yescharacteristic
453C22#1171935311yescharacteristic
454C22C2×#8816511yescharacteristic
455C22C2×#8817211yescharacteristic
456C22#1171947811yescharacteristic
457C22#1171948211yescharacteristic
458C4—22no—
459C4—81no—
460C4—81no—
461C4—82no—
462C4—262no—
463C4—262no—
464C4—262no—
465C4—262no—
466C4—1041no—
467C4—1041no—
468C4—1042no—
469C4#1171925112yes—
470C4#1171932812yes—
471C2—24no—
472C2#1311745511yescharacteristic
473C2C2×#1171854211yescharacteristic
474C2#1312342311yescharacteristic
475C2#1312582011yescharacteristic
476C2#1312583811yescharacteristic
477C2#1312602311yescharacteristic
478C2#1312604111yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1852C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1852D13
hypocenterC13#21986
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #13115603C2semidirect
5. #13115601C2semidirect
9. #13115003C2semidirect
10. #13114968C2semidirect
37. #11718380C4semidirect
88. #88055C2×C4semidirect
89. #88054C2×C4semidirect
390. C13#21986semidirect

Maximal subgroups

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

subgroupindex
C2×#117183802
#131172382
C2×#117186382
#131156032
#131156012
#131150312
#131150282
#131150242
#131150032
#131149682
C13×#18522
#2198613