Finite Group Databasebeta

← #15901333

Subgroups — #15901333

Order 3328.

984 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718695C211yesmaximal, characteristic
2C22×#88075C211yesmaximal, characteristic, complemented
3#13117711C211yesmaximal, characteristic, complemented
4#13116217C212yesmaximal, complemented
5#13116072C212yesmaximal, complemented
6#13115769C212yesmaximal, complemented
7#13115467C212yesmaximal, complemented
8#13115265C211yesmaximal, characteristic
9#13115016C211yesmaximal, characteristic, complemented
10#13114771C211yesmaximal, characteristic, complemented
11C13×#2436C211yesmaximal, characteristic
12C23×Q8×C13C2211yescharacteristic
13C2×C13×#124C2212yes—
14C2×C13×#121C2211yescharacteristic
15C22×C4⋊Dic13C2211yescharacteristic
16C23×#1055C2211yescharacteristic
17C13×#393—22no—
18C13×#391C2212yes—
19C13×#386—21no—
20C13×#386—21no—
21C4×C13×C22⋊C4C2211yescharacteristic
22C2×#88075—24no—
23C2×#88075C2212yes—
24C2×#88075C2212yes—
25C2×#88075C2212yes—
26C2×#88075C2212yes—
27C2×#88072C2211yescharacteristic
28C2×#88072C2211yescharacteristic
29C2×#88072C2212yes—
30#11718873C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32#11718695—24no—
33C2×#88048C2211yescharacteristic
34C2×#88048C2211yescharacteristic
35#11718603—21no—
36C22⋊C4×Dic13—21no—
37C4×C4⋊Dic13C2212yes—
38#11718589C2212yes—
39C2×C4⋊#1055C2212yes—
40#11718315—22no—
41#11718121C2212yes—
42#11718116C2212yes—
43C22×Q8×C13—21no—
44C22×Q8×C13—24no—
45C22×Q8×C13—22no—
46C22×Q8×C13—22no—
47C22×Q8×C13—42no—
48C22×Q8×C13C2312yes—
49C22×D4×C13—21no—
50C22×D4×C13C2311yescharacteristic
51C23×C4×C13—21no—
52C23×C4×C13C2311yescharacteristic
53C23×Dic13—21no—
54C13×#124—24no—
55C13×#124—42no—
56C13×#121—22no—
57C13×#121—22no—
58C2×C13×C4⋊C4C2311yescharacteristic
59C2×C13×C22⋊C4—22no—
60C2×C13×C22⋊C4—22no—
61C2×C13×C22⋊C4—21no—
62C2×C13×C22⋊C4D411yescharacteristic
63C2×C13×C22⋊C4D411yescharacteristic
64C2×C42×C13C2312yes—
65C2×C22⋊Dic13—21no—
66C2×C4⋊Dic13—21no—
67C2×C4⋊Dic13—22no—
68C2×C4⋊Dic13D414yes—
69C2×C4⋊Dic13C2312yes—
70C2×C4⋊Dic13C2311yescharacteristic
71C2×C4⋊Dic13C2311yescharacteristic
72C2×#6101—22no—
73C2×C4×Dic13—22no—
74C22×#1055—22no—
75C22×#1055—42no—
76C22×#1055C2312yes—
77C22×#1055C2311yescharacteristic
78C22×#1055C2311yescharacteristic
79C13×#95—22no—
80C13×#95—22no—
81C13×#95D412yes—
82#88075—24no—
83#88075—24no—
84#88075—24no—
85#88075—42no—
86#88072—22no—
87#88072—22no—
88#88072—42no—
89#88071—21no—
90#88071—22no—
91#88071—22no—
92C22⋊#1055—22no—
93#88048—22no—
94#88048—22no—
95C4⋊#1055—42no—
96#24483—131nomaximal, Sylow
97C24×C13—21no—
98C24×C13C2×D411yescharacteristic
99C2×Q8×C13—22no—
100C2×Q8×C13—28no—
101C2×Q8×C13—22no—
102C2×Q8×C13—22no—
103C2×Q8×C13—22no—
104C2×Q8×C13—22no—
105C2×Q8×C13—44no—
106C2×Q8×C13—48no—
107C2×Q8×C13—44no—
108C2×Q8×C13—42no—
109C2×Q8×C13#4114yes—
110C2×D4×C13—21no—
111C2×D4×C13—21no—
112C2×D4×C13—22no—
113C2×D4×C13—22no—
114C2×D4×C13—42no—
115C2×D4×C13—42no—
116C2×D4×C13—42no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—21no—
121C22×C4×C13—44no—
122C22×C4×C13—42no—
123C22×C4×C13C2×D412yes—
124C22×C4×C13C2×D412yes—
125C22×C4×C13C2×D412yes—
126C22×C4×C13C2×D411yescharacteristic
127C22×C4×C13#4112yes—
128C22×C4×C13C2411yescharacteristic
129C22×Dic13—21no—
130C22×Dic13—21no—
131C22×Dic13—24no—
132C22×Dic13—22no—
133C22×Dic13—82no—
134C13×C4⋊C4—21no—
135C13×C4⋊C4—21no—
136C13×C22⋊C4—22no—
137C13×C22⋊C4—22no—
138C13×C22⋊C4—44no—
139C13×C22⋊C4—44no—
140C13×C22⋊C4—42no—
141C42×C13—22no—
142C42×C13—22no—
143C22⋊Dic13—42no—
144C4⋊Dic13—24no—
145C4⋊Dic13—21no—
146C4⋊Dic13—21no—
147C4⋊Dic13—22no—
148C4⋊Dic13—22no—
149C4⋊Dic13C2×D412yes—
150C4⋊Dic13C2×D412yes—
151#6101—82no—
152C4×Dic13—82no—
153C2×#1055—21no—
154C2×#1055—21no—
155C2×#1055—22no—
156C2×#1055—22no—
157C2×#1055—22no—
158C2×#1055—22no—
159C2×#1055—22no—
160C2×#1055—44no—
161C2×#481—131no—
162C22×#103—131no—
163#2436D1311yescharacteristic
164#2345—131no—
165#2066—132no—
166#2042—132no—
167#1983—132no—
168#1927—132no—
169#1893—131no—
170#1850—131no—
171#1787—131no—
172C23×C13—22no—
173C23×C13—22no—
174C23×C13—41no—
175C23×C13—42no—
176C23×C13—41no—
177C23×C13—41no—
178C23×C13—42no—
179C23×C13#12112yes—
180C23×C13#13512yes—
181C23×C13#13712yes—
182C23×C13C22×D411yescharacteristic
183Q8×C13—24no—
184Q8×C13—24no—
185Q8×C13—48no—
186Q8×C13—42no—
187Q8×C13—42no—
188D4×C13—41no—
189D4×C13—41no—
190D4×C13—42no—
191D4×C13—84no—
192C2×C4×C13—22no—
193C2×C4×C13—22no—
194C2×C4×C13—22no—
195C2×C4×C13—24no—
196C2×C4×C13—22no—
197C2×C4×C13—22no—
198C2×C4×C13—24no—
199C2×C4×C13—22no—
200C2×C4×C13—22no—
201C2×C4×C13—22no—
202C2×C4×C13—22no—
203C2×C4×C13—48no—
204C2×C4×C13—42no—
205C2×C4×C13—42no—
206C2×C4×C13—41no—
207C2×C4×C13—41no—
208C2×C4×C13—41no—
209C2×C4×C13#12112yes—
210C2×C4×C13C22×D411yescharacteristic
211C2×C4×C13C2×#4112yes—
212C2×C4×C13C2×#4111yescharacteristic
213C2×C4×C13H(5,𝔽2)12yes—
214C2×Dic13—22no—
215C2×Dic13—22no—
216C2×Dic13—42no—
217C2×Dic13—42no—
218C2×Dic13—42no—
219C2×Dic13—42no—
220C2×Dic13—42no—
221C2×Dic13—84no—
222#1055—41no—
223#1055—41no—
224#1055—42no—
225C23×Q8C2×D1311yescharacteristic
226C23×C8—131no—
227C2×#124C2×D1312yes—
228C2×#121C2×D1311yescharacteristic
229C22×C4⋊C4—131no—
230#481—264no—
231C2×#106—131no—
232C2×#106—131no—
233C2×C4⋊C8—132no—
234C2×#103—132no—
235C2×#103—132no—
236C2×#103—132no—
237C2×#103—132no—
238C2×#103—264no—
239C2×#102—131no—
240C2×#102—131no—
241C2×#102—132no—
242C2×C22⋊C8—131no—
243#393—22no—
244#391C2×D1312yes—
245#389—132no—
246#387—262no—
247#386—21no—
248#386—21no—
249#380—131no—
250#380—261no—
251C4×C4⋊C4—132no—
252C4×C22⋊C4—261no—
253C4×C22⋊C4C2×D1311yescharacteristic
254#340—132no—
255#336—132no—
256C22×C13—22no—
257C22×C13—22no—
258C22×C13—24no—
259C22×C13—24no—
260C22×C13—81no—
261C22×C13—81no—
262C22×C13—81no—
263C22×C13—82no—
264C22×C13—82no—
265C22×C13#46414yes—
266C22×C13C2×#12111yescharacteristic
267C22×C13#54612yes—
268C22×C13C2×#13511yescharacteristic
269C22×C13C2×#13711yescharacteristic
270C22×C13#57712yes—
271C4×C13—22no—
272C4×C13—44no—
273C4×C13—42no—
274C4×C13—42no—
275C4×C13—81no—
276C4×C13#52511yescharacteristic
277C4×C13#53511yescharacteristic
278C4×C13#55112yes—
279Dic13—42no—
280Dic13—42no—
281Dic13—82no—
282C22×Q8—21no—
283C22×Q8—24no—
284C22×Q8—22no—
285C22×Q8—22no—
286C22×Q8—42no—
287C22×Q8C22×D1312yes—
288C22×D4—21no—
289C22×D4C22×D1311yescharacteristic
290C23×C4—21no—
291C23×C4—261no—
292C23×C4C22×D1311yescharacteristic
293C22×C8—132no—
294C22×C8—131no—
295C22×C8—131no—
296C22×C8—262no—
297C22×C8—522no—
298#124—24no—
299#124—42no—
300#121—22no—
301#121—22no—
302C2×C4⋊C4—134no—
303C2×C4⋊C4—131no—
304C2×C4⋊C4—131no—
305C2×C4⋊C4—132no—
306C2×C4⋊C4—262no—
307C2×C4⋊C4—261no—
308C2×C4⋊C4—262no—
309C2×C4⋊C4C22×D1311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—261no—
314C2×C22⋊C4C22⋊D1311yescharacteristic
315C2×C22⋊C4C22⋊D1311yescharacteristic
316C2×C42—262no—
317C2×C42C22×D1312yes—
318#106—262no—
319#106—262no—
320C4⋊C8—522no—
321#103—264no—
322#103—264no—
323#103—264no—
324#103—522no—
325#102—262no—
326#102—262no—
327#102—522no—
328C22⋊C8—262no—
329#95—22no—
330#95—22no—
331#95—262no—
332#95—262no—
333#95—261no—
334#95C22⋊D1312yes—
335C2×C13—24no—
336C2×C13—161no—
337C2×C13#241111yescharacteristic
338C2×C13C2×#46411yescharacteristic
339C2×C13#304211yescharacteristic
340C2×C13#330112yes—
341C2×C13#331612yes—
342C24—21no—
343C24C2×C22⋊D1311yescharacteristic
344C2×Q8—22no—
345C2×Q8—28no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—42no—
354C2×Q8#612914yes—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—261no—
369C22×C4—261no—
370C22×C4—264no—
371C22×C4—262no—
372C22×C4—1042no—
373C22×C4D4×D1312yes—
374C22×C4#612912yes—
375C22×C4C2×C22⋊D1312yes—
376C22×C4C2×C22⋊D1312yes—
377C22×C4C2×C22⋊D1311yescharacteristic
378C22×C4C23×D1311yescharacteristic
379C2×C8—261no—
380C2×C8—261no—
381C2×C8—262no—
382C2×C8—262no—
383C2×C8—262no—
384C2×C8—262no—
385C2×C8—262no—
386C2×C8—524no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—132no—
390C4⋊C4—132no—
391C4⋊C4—264no—
392C4⋊C4—261no—
393C4⋊C4—261no—
394C4⋊C4—262no—
395C4⋊C4—262no—
396C4⋊C4—1042no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—42no—
402C22⋊C4—522no—
403C42—22no—
404C42—22no—
405C42—1042no—
406C13#2448311yescharacteristic, complemented, Sylow
407C23—22no—
408C23—22no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23#8819212yes—
415C23#8820412yes—
416C23#8820512yes—
417C23C22×C22⋊D1311yescharacteristic
418Q8—24no—
419Q8—24no—
420Q8—48no—
421Q8—42no—
422Q8—42no—
423D4—41no—
424D4—42no—
425D4—41no—
426D4—84no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—262no—
445C2×C4—262no—
446C2×C4—522no—
447C2×C4—522no—
448C2×C4—522no—
449C2×C4—522no—
450C2×C4—522no—
451C2×C4—1044no—
452C2×C4#8819212yes—
453C2×C4C2×D4×D1311yescharacteristic
454C2×C4C2×#612912yes—
455C2×C4C2×#612911yescharacteristic
456C2×C4#8825112yes—
457C8—521no—
458C8—521no—
459C8—522no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—81no—
465C22—82no—
466C22—82no—
467C22—81no—
468C22—81no—
469C22#1171871514yes—
470C22C2×#8819211yescharacteristic
471C22#1171951112yes—
472C22C2×#8820411yescharacteristic
473C22C2×#8820511yescharacteristic
474C22#1171954212yes—
475C4—22no—
476C4—44no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—522no—
481C4—522no—
482C4—1042no—
483C4#1171929111yescharacteristic
484C4#1171929212yes—
485C4#1171929711yescharacteristic
486C2—24no—
487C2—161no—
488C2#1311821711yescharacteristic
489C2C2×#1171871511yescharacteristic
490C2#1312434411yescharacteristic
491C2#1312673812yes—
492C2#1312694712yes—
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2436C2
socleC23×C13C22×D4
prime radicalC2×C13×#121C22
hypercenter#2436D13
hypocenterC13#24483
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
2. C22×#88075C2semidirect
3. #13117711C2semidirect
4. #13116217C2semidirect
5. #13116072C2semidirect
6. #13115769C2semidirect
7. #13115467C2semidirect
9. #13115016C2semidirect
10. #13114771C2semidirect
406. C13#24483semidirect

Maximal subgroups

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

subgroupindex
C2×#117186952
C22×#880752
#131177112
#131162172
#131160722
#131157692
#131154672
#131152652
#131150162
#131147712
C13×#24362
#2448313