Finite Group Databasebeta

← #15834347

Subgroups — #15834347

Order 3328.

656 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123851C211yesmaximal, characteristic, complemented
2C4×#88204C211yesmaximal, characteristic
3#13113731C212yesmaximal, complemented
4#13113672C212yesmaximal, complemented
5#13113670C212yesmaximal, complemented
6C8×#6104C212yesmaximal, complemented
7#13113597C211yesmaximal, characteristic, complemented
8#13113270C211yesmaximal, characteristic, complemented
9#13113268C211yesmaximal, characteristic, complemented
10M16×#1055C211yesmaximal, characteristic, complemented
11C13×#2970C211yesmaximal, characteristic, complemented
12C4×C13×#41C2211yescharacteristic
13C2×#88204C411yescharacteristic
14C4×#6127C2211yescharacteristic
15C13×#446C2212yes—
16C13×#435—22no—
17C8×D4×C13C2212yes—
18C13×#432—21no—
19C13×#432C2211yescharacteristic
20C4×C13×M16C2211yescharacteristic
21C2×C13×#97—21no—
22#11718894C411yescharacteristic
23#11718788—22no—
24C4×#6107C411yescharacteristic
25C4×#6107C2211yescharacteristic
26C4×#6106C411yescharacteristic
27C4×#6106C2211yescharacteristic
28C4×#6105C411yescharacteristic
29C4×#6105C2211yescharacteristic
30C4×#6104C411yescharacteristic
31C4×#6104C2211yescharacteristic
32#11718663C411yescharacteristic
33#11718659C411yescharacteristic
34#11718587C2211yescharacteristic
35#11718586C2211yescharacteristic
36C2×C4×#1055C2211yescharacteristic
37#11718366—21no—
38D13×#97—21no—
39C8×D52C2212yes—
40#11718340C2212yes—
41Q8⋊#1055C2212yes—
42D4⋊#1055C2212yes—
43#11718076C2212yes—
44#11718075C2212yes—
45#11718072C2212yes—
46#11718071C2212yes—
47#11718047C2212yes—
48C8×#1055C2212yes—
49C2×C13×#41C2×C411yescharacteristic
50C2×#6127C2×C411yescharacteristic
51C2×C13×M16D412yes—
52C22×C8×C13—22no—
53C4×Q8×C13C2×C411yescharacteristic
54C4×Q8×C13C2311yescharacteristic
55C4×D4×C13—21no—
56C4×D4×C13—21no—
57C4×D4×C13C2×C411yescharacteristic
58C4×D4×C13C2311yescharacteristic
59C13×#117—21no—
60C13×#117C2×C411yescharacteristic
61C2×C42×C13—21no—
62C2×C42×C13C2311yescharacteristic
63#88204—24no—
64C2×#6107C2×C411yescharacteristic
65C2×#6106C2×C411yescharacteristic
66C2×#6105C2×C411yescharacteristic
67C2×#6104C2×C411yescharacteristic
68C4×C22⋊D13—21no—
69C4×C22⋊D13—21no—
70#88180C2×C411yescharacteristic
71C22×#1055C2×C411yescharacteristic
72C22×#1055C2×C411yescharacteristic
73C2×C8×D13—22no—
74C4×D52C2×C411yescharacteristic
75C4×D52C2311yescharacteristic
76#88126—21no—
77C42×D13—21no—
78C4×Dic26C2×C411yescharacteristic
79C4×Dic26C2311yescharacteristic
80C13×C4⋊C8—22no—
81C13×C4⋊C8C2312yes—
82C13×C22⋊C8—21no—
83C13×C22⋊C8—22no—
84C13×#97—22no—
85C13×#97—21no—
86C13×#97D411yescharacteristic
87C13×#97D411yescharacteristic
88C4×C8×C13C2312yes—
89#88075C2×C411yescharacteristic
90#88075C2×C411yescharacteristic
91#88072C2×C411yescharacteristic
92#88072C2×C411yescharacteristic
93C22⋊#1055—21no—
94C22⋊#1055—21no—
95#88060—22no—
96#88055—22no—
97#88054—22no—
98#88050C2×C411yescharacteristic
99#88050C2×C411yescharacteristic
100#88049C2×C411yescharacteristic
101#88049C2×C411yescharacteristic
102#88048C2×C411yescharacteristic
103#88048C2×C411yescharacteristic
104#88047C2×C411yescharacteristic
105#88047C2×C411yescharacteristic
106C4⋊#1055—21no—
107C4⋊#1055C2312yes—
108#88043—21no—
109C4×#1055C2×C411yescharacteristic
110C4×#1055C2×C411yescharacteristic
111C4×#1055C2312yes—
112C4×#1055C2311yescharacteristic
113C4×#1055C2311yescharacteristic
114#18035—131nomaximal, Sylow
115C13×#41—22no—
116C13×#41—41no—
117C2×Q8×C13C22×C411yescharacteristic
118C2×D4×C13—21no—
119C2×D4×C13C22×C411yescharacteristic
120C22×C4×C13—21no—
121C22×C4×C13—21no—
122C22×C4×C13—21no—
123C22×C4×C13C22×C411yescharacteristic
124C22×C4×C13C2×D411yescharacteristic
125C22×C4×C13#4111yescharacteristic
126C2×C22⋊D13—21no—
127#6127—22no—
128#6127—41no—
129C2×D52C22×C411yescharacteristic
130C2×C4×D13—21no—
131C2×C4×D13—21no—
132C2×C4×D13—21no—
133C2×Dic26C22×C411yescharacteristic
134C13×M16—24no—
135C2×C8×C13—22no—
136C2×C8×C13—44no—
137C2×C8×C13C2×D412yes—
138C2×C8×C13C2×D412yes—
139C13×C4⋊C4—21no—
140C13×C4⋊C4—21no—
141C13×C4⋊C4C22×C411yescharacteristic
142C13×C4⋊C4C22×C411yescharacteristic
143C13×C22⋊C4—21no—
144C13×C22⋊C4—21no—
145C13×C22⋊C4—21no—
146C42×C13—21no—
147C42×C13—21no—
148C42×C13—21no—
149C42×C13C22×C411yescharacteristic
150C42×C13C2×D411yescharacteristic
151C42×C13#4111yescharacteristic
152C42×C13C2411yescharacteristic
153C22⋊Dic13—21no—
154#6107—22no—
155#6106—22no—
156#6105—22no—
157#6104M1614yescomplemented
158#6103—21no—
159#6103—21no—
160C4⋊Dic13C22×C411yescharacteristic
161C4⋊Dic13C22×C411yescharacteristic
162#6101—21no—
163#6101—21no—
164C4×Dic13—21no—
165C4×Dic13—21no—
166C4×Dic13—21no—
167C2×#1055—22no—
168C2×#1055—22no—
169C2×#1055—22no—
170C2×#1055—41no—
171C2×#1055C22×C411yescharacteristic
172C2×#1055C22×C411yescharacteristic
173C2×#1055C22×C411yescharacteristic
174C2×#1055C22×C411yescharacteristic
175C8×D13—44no—
176#2970—131no—
177#2970D1311yescharacteristic, complemented
178C4×#135—131no—
179#1591—132no—
180#1582—132no—
181#1582—132no—
182C8×D8—132no—
183#1568—131no—
184#1488—131no—
185#1488—131no—
186C8×M16—131no—
187C23×C13—21no—
188C23×C13C4×D411yescharacteristic
189C22×D13—21no—
190Q8×C13—21no—
191Q8×C13—21no—
192D4×C13—21no—
193D4×C13—42no—
194D4×C13C2×M1612yes—
195C2×C4×C13—21no—
196C2×C4×C13—21no—
197C2×C4×C13—22no—
198C2×C4×C13—21no—
199C2×C4×C13—21no—
200C2×C4×C13—21no—
201C2×C4×C13—41no—
202C2×C4×C13—42no—
203C2×C4×C13—42no—
204C2×C4×C13—81no—
205C2×C4×C13C4×D411yescharacteristic
206C2×C4×C13C4×D411yescharacteristic
207C2×C4×C13C4×D411yescharacteristic
208C2×C4×C13C23×C411yescharacteristic
209C2×C4×C13C22×D411yescharacteristic
210C2×C4×C13C2×#4111yescharacteristic
211C22⋊D13—42no—
212C2×Dic13—21no—
213C2×Dic13—21no—
214C2×Dic13—21no—
215D52—21no—
216D52C2×M1612yes—
217C4×D13—42no—
218C4×D13—42no—
219C4×D13—81no—
220Dic26—21no—
221Dic26—21no—
222C8×C13—22no—
223C8×C13—42no—
224C8×C13#13514yescomplemented
225#1055—21no—
226#1055—21no—
227#1055—21no—
228#1055—42no—
229#1055—42no—
230#1055C2×M1612yes—
231C2×#135—131no—
232C4×#41—131no—
233C4×#41C2×D1311yescharacteristic
234#446—132no—
235#446C2×D1312yes—
236C4×Q16—131no—
237C4×Q16—131no—
238C4×SD16—131no—
239C4×SD16—131no—
240C4×SD16—131no—
241C4×SD16—131no—
242C4×D8—131no—
243C4×D8—131no—
244#435—22no—
245#435—262no—
246C8×D4—132no—
247C8×D4C2×D1312yes—
248#432—21no—
249#432—131no—
250#432—261no—
251#432C2×D1311yescharacteristic
252#431—131no—
253#427—131no—
254#416—131no—
255#416—131no—
256C4×M16C2×D1311yescharacteristic
257C2×#97—21no—
258C2×#97—261no—
259C2×C4×C8—131no—
260#335—132no—
261#334—132no—
262Q8⋊C8—132no—
263Q8⋊C8—132no—
264D4⋊C8—132no—
265D4⋊C8—132no—
266#322—132no—
267C82—132no—
268C22×C13—21no—
269C22×C13—22no—
270C22×C13—21no—
271C22×C13—21no—
272C22×C13—41no—
273C22×C13—41no—
274C22×C13C2×C4×D411yescharacteristic
275C2×D13—22no—
276C2×D13—41no—
277C2×D13—41no—
278C4×C13—21no—
279C4×C13—21no—
280C4×C13—21no—
281C4×C13—21no—
282C4×C13—41no—
283C4×C13—41no—
284C4×C13—41no—
285C4×C13C22×M1611yescharacteristic
286C4×C13#56811yescharacteristic
287C4×C13C2×#13512yes—
288Dic13—41no—
289Dic13—41no—
290Dic13—41no—
291C2×#41—131no—
292C2×#41C4×D1311yescharacteristic
293#135—264no—
294C2×Q16—131no—
295C2×SD16—131no—
296C2×SD16—131no—
297C2×D8—131no—
298C2×M16C22⋊D1312yes—
299C22×C8—22no—
300C22×C8—131no—
301C22×C8—131no—
302C22×C8—262no—
303C4×Q8—131no—
304C4×Q8—131no—
305C4×Q8C4×D1311yescharacteristic
306C4×Q8C22×D1311yescharacteristic
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—131no—
310C4×D4—131no—
311C4×D4—261no—
312C4×D4—261no—
313C4×D4C4×D1311yescharacteristic
314C4×D4C22×D1311yescharacteristic
315#117—21no—
316#117—131no—
317#117—261no—
318#117C4×D1311yescharacteristic
319C2×C42—21no—
320C2×C42—261no—
321C2×C42C22×D1311yescharacteristic
322#107—131no—
323#107—131no—
324#106—131no—
325#106—131no—
326C4⋊C8—22no—
327C4⋊C8—132no—
328C4⋊C8—262no—
329C4⋊C8—261no—
330C4⋊C8C22×D1312yes—
331#103—131no—
332#103—131no—
333#103—131no—
334#103—131no—
335#102—131no—
336#102—131no—
337#102—131no—
338#102—131no—
339C22⋊C8—21no—
340C22⋊C8—22no—
341C22⋊C8—261no—
342C22⋊C8—261no—
343C22⋊C8—262no—
344#97—22no—
345#97—21no—
346#97—262no—
347#97—261no—
348#97C22⋊D1311yescharacteristic
349#97C22⋊D1311yescharacteristic
350C4×C8—131no—
351C4×C8—131no—
352C4×C8—131no—
353C4×C8—131no—
354C4×C8—132no—
355C4×C8C22×D1312yes—
356C2×C13—41no—
357C2×C13—42no—
358C2×C13D4×M1611yescharacteristic
359C2×C13C4×#13511yescharacteristic
360C2×C13#295711yescharacteristic
361D13—42no—
362#41—22no—
363#41—41no—
364#41—262no—
365#41—521no—
366C2×Q8—131no—
367C2×Q8C2×C4×D1311yescharacteristic
368C2×D4—21no—
369C2×D4—131no—
370C2×D4—261no—
371C2×D4C2×C4×D1311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—261no—
376C22×C4—261no—
377C22×C4—261no—
378C22×C4C2×C4×D1311yescharacteristic
379C22×C4#612711yescharacteristic
380C22×C4C2×C22⋊D1311yescharacteristic
381Q16—262no—
382SD16—262no—
383SD16—262no—
384D8—134no—
385M16—24no—
386C2×C8—22no—
387C2×C8—44no—
388C2×C8—131no—
389C2×C8—131no—
390C2×C8—131no—
391C2×C8—131no—
392C2×C8—262no—
393C2×C8—262no—
394C2×C8—262no—
395C2×C8—521no—
396C2×C8—524no—
397C2×C8C2×C22⋊D1312yes—
398C2×C8C2×C22⋊D1312yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—131no—
402C4⋊C4—131no—
403C4⋊C4—261no—
404C4⋊C4—261no—
405C4⋊C4C2×C4×D1311yescharacteristic
406C4⋊C4C2×C4×D1311yescharacteristic
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—261no—
411C22⋊C4—261no—
412C22⋊C4—261no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—261no—
417C42—261no—
418C42—261no—
419C42C2×C4×D1311yescharacteristic
420C42#612711yescharacteristic
421C42C2×C22⋊D1311yescharacteristic
422C42C23×D1311yescharacteristic
423C13#1803511yescharacteristic, complemented, Sylow
424C23—21no—
425C23—261no—
426C23C4×C22⋊D1311yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—261no—
430Q8—261no—
431D4—21no—
432D4—42no—
433D4—132no—
434D4—261no—
435D4—522no—
436D4M16×D1312yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—261no—
448C2×C4—261no—
449C2×C4—261no—
450C2×C4—522no—
451C2×C4—522no—
452C2×C4—1041no—
453C2×C4C4×C22⋊D1311yescharacteristic
454C2×C4C4×C22⋊D1311yescharacteristic
455C2×C4C4×C22⋊D1311yescharacteristic
456C2×C4C22×C4×D1311yescharacteristic
457C2×C4C2×#612711yescharacteristic
458C2×C4C22×C22⋊D1311yescharacteristic
459C8—22no—
460C8—42no—
461C8—132no—
462C8—261no—
463C8—261no—
464C8—261no—
465C8—522no—
466C8—522no—
467C8#8820414yescomplemented
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—262no—
475C22—521no—
476C22—521no—
477C22C2×C4×C22⋊D1311yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—521no—
486C4—521no—
487C4—521no—
488C4C2×M16×D1311yescharacteristic
489C4#1171945011yescharacteristic
490C4C2×#8820412yes—
491C2—41no—
492C2—42no—
493C2—522no—
494C2M16×C22⋊D1311yescharacteristic
495C2C4×#8820411yescharacteristic
496C2#1312375111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC42C23×D13
FittingC13×#2970C2
socleC22×C13C2×C4×D4
prime radicalC2×#88204C4
cosocleC42×C13C24
hypercenter#2970D13
hypocenterC13#18035
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
1. #13123851C2semidirect
3. #13113731C2semidirect
4. #13113672C2semidirect
5. #13113670C2semidirect
6. C8×#6104C2semidirect
7. #13113597C2semidirect
8. #13113270C2semidirect
9. #13113268C2semidirect
10. M16×#1055C2semidirect
11. C13×#2970C2semidirect
157. #6104M16semidirect
177. #2970D13semidirect
224. C8×C13#135semidirect
423. C13#18035semidirect
467. C8#88204semidirect

Maximal subgroups

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

subgroupindex
#131238512
C4×#882042
#131137312
#131136722
#131136702
C8×#61042
#131135972
#131132702
#131132682
M16×#10552
C13×#29702
#1803513