Finite Group Databasebeta

← #15901338

Subgroups — #15901338

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
1C22×#88050C211yesmaximal, characteristic, complemented
2#13118495C211yesmaximal, characteristic
3#13117713C211yesmaximal, characteristic, complemented
4#13116212C212yesmaximal, complemented
5#13116076C212yesmaximal, complemented
6#13115767C212yesmaximal, complemented
7#13115469C212yesmaximal, complemented
8#13115265C211yesmaximal, characteristic
9#13115013C211yesmaximal, characteristic, complemented
10#13114772C211yesmaximal, characteristic, complemented
11C2×C13×#481C211yesmaximal, characteristic
12C23×Dic26C2211yescharacteristic
13C2×C13×#121C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C2×#88190C2212yes—
16C23×#1055C2211yescharacteristic
17C13×#481—24no—
18C2×C13×#106C2211yescharacteristic
19C2×C13×#106C2211yescharacteristic
20C2×C13×#102C2211yescharacteristic
21C2×C13×#102C2211yescharacteristic
22C2×C13×C22⋊C8C2211yescharacteristic
23#11718881—21no—
24#11718881—21no—
25C2×#88072C2212yes—
26#11718873—21no—
27#11718873C2211yescharacteristic
28C4×C22⋊Dic13—21no—
29C4×C22⋊Dic13C2211yescharacteristic
30C2×#88056C2212yes—
31C2×#88056C2212yes—
32C2×#88054C2212yes—
33#11718646C2212yes—
34#11718642—22no—
35C4⋊C4×Dic13C2212yes—
36#11718635C2212yes—
37C2×#88050—24no—
38C2×#88050C2212yes—
39C2×#88050C2212yes—
40#11718608—22no—
41#11718142C2212yes—
42#11718121C2212yes—
43C22×D4×C13—21no—
44C22×D4×C13C2311yescharacteristic
45C23×C4×C13—21no—
46C23×C4×C13C2311yescharacteristic
47C23×Dic13—21no—
48C22×Dic26—22no—
49C22×Dic26—24no—
50C22×Dic26—22no—
51C22×Dic26—21no—
52C22×Dic26—42no—
53C22×Dic26C2312yes—
54C22×C8×C13C2311yescharacteristic
55C22×C8×C13C2311yescharacteristic
56C13×#121—22no—
57C13×#121—22no—
58C2×C13×C4⋊C4—21no—
59C2×C13×C4⋊C4—22no—
60C2×C13×C4⋊C4D414yes—
61C2×C13×C4⋊C4C2311yescharacteristic
62C2×C13×C4⋊C4C2311yescharacteristic
63C2×C13×C4⋊C4C2311yescharacteristic
64C2×C13×C22⋊C4—21no—
65C2×C22⋊Dic13—21no—
66C2×C22⋊Dic13—22no—
67C2×C22⋊Dic13—22no—
68C2×C22⋊Dic13D411yescharacteristic
69C2×C22⋊Dic13D411yescharacteristic
70#88190—24no—
71#88190—42no—
72C2×C4⋊Dic13—22no—
73C2×C4⋊Dic13C2312yes—
74C2×C4×Dic13—22no—
75C2×C4×Dic13C2312yes—
76C22×#1055—22no—
77C22×#1055—42no—
78C22×#1055C2312yes—
79C13×#106—22no—
80C13×#106—22no—
81C13×#102—22no—
82C13×#102—22no—
83C13×C22⋊C8—22no—
84#88072—42no—
85#88071—21no—
86#88071—22no—
87#88071—22no—
88#88071—22no—
89#88071—22no—
90#88071D412yes—
91#88056—24no—
92#88056—42no—
93#88054—42no—
94#88050—24no—
95#88050—24no—
96#24483—131nomaximal, Sylow
97C24×C13—21no—
98C24×C13C2×D411yescharacteristic
99C2×D4×C13—21no—
100C2×D4×C13—22no—
101C2×D4×C13—21no—
102C2×D4×C13—22no—
103C2×D4×C13—42no—
104C2×D4×C13—42no—
105C2×D4×C13—42no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—21no—
109C22×C4×C13—21no—
110C22×C4×C13—24no—
111C22×C4×C13—21no—
112C22×C4×C13—82no—
113C22×C4×C13C2×D411yescharacteristic
114C22×C4×C13C2×D412yes—
115C22×C4×C13#4112yes—
116C22×C4×C13C2411yescharacteristic
117C22×Dic13—22no—
118C22×Dic13—22no—
119C22×Dic13—44no—
120C22×Dic13—42no—
121C22×Dic13C2×D412yes—
122C22×Dic13C2×D412yes—
123C2×Dic26—28no—
124C2×Dic26—22no—
125C2×Dic26—22no—
126C2×Dic26—22no—
127C2×Dic26—22no—
128C2×Dic26—22no—
129C2×Dic26—48no—
130C2×Dic26—44no—
131C2×Dic26—44no—
132C2×Dic26—42no—
133C2×Dic26#4114yes—
134C2×C8×C13—21no—
135C2×C8×C13—21no—
136C2×C8×C13—22no—
137C2×C8×C13—22no—
138C13×C4⋊C4—24no—
139C13×C4⋊C4—21no—
140C13×C4⋊C4—21no—
141C13×C4⋊C4—21no—
142C13×C4⋊C4—21no—
143C13×C4⋊C4C2×D412yes—
144C13×C4⋊C4C2×D412yes—
145C13×C22⋊C4—42no—
146C22⋊Dic13—22no—
147C22⋊Dic13—22no—
148C22⋊Dic13—44no—
149C22⋊Dic13—44no—
150C22⋊Dic13—42no—
151C4⋊Dic13—22no—
152C4⋊Dic13—22no—
153C4⋊Dic13—82no—
154C4×Dic13—22no—
155C4×Dic13—22no—
156C4×Dic13—82no—
157C2×#1055—22no—
158C2×#1055—22no—
159C2×#1055—22no—
160C2×#1055—44no—
161C2×#481D1311yescharacteristic
162C22×#103—131no—
163#2436—131no—
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
183D4×C13—41no—
184D4×C13—41no—
185D4×C13—42no—
186D4×C13—84no—
187C2×C4×C13—22no—
188C2×C4×C13—22no—
189C2×C4×C13—22no—
190C2×C4×C13—24no—
191C2×C4×C13—22no—
192C2×C4×C13—22no—
193C2×C4×C13—42no—
194C2×C4×C13—42no—
195C2×C4×C13—41no—
196C2×C4×C13—41no—
197C2×C4×C13—41no—
198C2×C4×C13—84no—
199C2×C4×C13#12112yes—
200C2×C4×C13C22×D411yescharacteristic
201C2×C4×C13C2×#4112yes—
202C2×C4×C13C2×#4111yescharacteristic
203C2×C4×C13H(5,𝔽2)12yes—
204C2×Dic13—22no—
205C2×Dic13—22no—
206C2×Dic13—24no—
207C2×Dic13—22no—
208C2×Dic13—22no—
209C2×Dic13—22no—
210C2×Dic13—22no—
211C2×Dic13—48no—
212C2×Dic13—42no—
213C2×Dic13—42no—
214C2×Dic13—42no—
215C2×Dic13—42no—
216C2×Dic13—42no—
217Dic26—24no—
218Dic26—24no—
219Dic26—48no—
220Dic26—42no—
221Dic26—42no—
222C8×C13—41no—
223C8×C13—41no—
224#1055—42no—
225C23×Q8—131no—
226C23×C8—131no—
227C2×#124—132no—
228C2×#121C2×D1311yescharacteristic
229C22×C4⋊C4C2×D1311yescharacteristic
230#481—24no—
231C2×#106C2×D1311yescharacteristic
232C2×#106C2×D1311yescharacteristic
233C2×C4⋊C8—132no—
234C2×#103—132no—
235C2×#103—132no—
236C2×#103—132no—
237C2×#103—132no—
238C2×#103—264no—
239C2×#102—132no—
240C2×#102C2×D1311yescharacteristic
241C2×#102C2×D1311yescharacteristic
242C2×C22⋊C8C2×D1311yescharacteristic
243#393—262no—
244#391—132no—
245#389—132no—
246#387—262no—
247#386—261no—
248#386—261no—
249#380—131no—
250#380—261no—
251C4×C4⋊C4—132no—
252C4×C22⋊C4—131no—
253C4×C22⋊C4—261no—
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—42no—
273C4×C13—42no—
274C4×C13—81no—
275C4×C13#52511yescharacteristic
276C4×C13#53511yescharacteristic
277C4×C13#55112yes—
278Dic13—44no—
279Dic13—42no—
280Dic13—42no—
281Dic13—82no—
282C22×Q8—132no—
283C22×Q8—264no—
284C22×Q8—261no—
285C22×Q8—262no—
286C22×Q8—262no—
287C22×Q8—522no—
288C22×D4—21no—
289C22×D4C22×D1311yescharacteristic
290C23×C4—21no—
291C23×C4—261no—
292C23×C4C22×D1311yescharacteristic
293C22×C8—132no—
294C22×C8—262no—
295C22×C8—522no—
296C22×C8C22×D1311yescharacteristic
297C22×C8C22×D1311yescharacteristic
298#124—264no—
299#124—522no—
300#121—22no—
301#121—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—132no—
305C2×C4⋊C4—262no—
306C2×C4⋊C4C22⋊D1314yes—
307C2×C4⋊C4C22×D1311yescharacteristic
308C2×C4⋊C4C22×D1311yescharacteristic
309C2×C4⋊C4C22×D1311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—131no—
312C2×C22⋊C4—131no—
313C2×C22⋊C4—262no—
314C2×C22⋊C4—262no—
315C2×C22⋊C4—261no—
316C2×C42—132no—
317C2×C42—262no—
318#106—22no—
319#106—22no—
320C4⋊C8—522no—
321#103—264no—
322#103—264no—
323#103—264no—
324#103—522no—
325#102—22no—
326#102—22no—
327#102—522no—
328C22⋊C8—22no—
329#95—132no—
330#95—262no—
331#95—262no—
332#95—262no—
333#95—262no—
334#95—261no—
335C2×C13—24no—
336C2×C13—161no—
337C2×C13#241111yescharacteristic
338C2×C13C2×#46411yescharacteristic
339C2×C13#304211yescharacteristic
340C2×C13#330112yes—
341C2×C13#331612yes—
342C24—21no—
343C24D4×D1311yescharacteristic
344C2×Q8—134no—
345C2×Q8—268no—
346C2×Q8—262no—
347C2×Q8—262no—
348C2×Q8—262no—
349C2×Q8—262no—
350C2×Q8—262no—
351C2×Q8—524no—
352C2×Q8—528no—
353C2×Q8—524no—
354C2×Q8—522no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—82no—
369C22×C4—132no—
370C22×C4—132no—
371C22×C4—262no—
372C22×C4—262no—
373C22×C4—524no—
374C22×C4—522no—
375C22×C4D4×D1311yescharacteristic
376C22×C4#612912yes—
377C22×C4C2×C22⋊D1312yes—
378C22×C4C23×D1311yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—262no—
384C2×C8—262no—
385C2×C8—262no—
386C2×C8—524no—
387C4⋊C4—24no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—262no—
393C4⋊C4—262no—
394C4⋊C4—1042no—
395C4⋊C4C2×C22⋊D1312yes—
396C4⋊C4C2×C22⋊D1312yes—
397C22⋊C4—42no—
398C22⋊C4—262no—
399C22⋊C4—262no—
400C22⋊C4—524no—
401C22⋊C4—524no—
402C22⋊C4—522no—
403C42—262no—
404C42—262no—
405C42—1042no—
406C13#2448311yescharacteristic, complemented, Sylow
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—41no—
413C23—42no—
414C23#8816912yes—
415C23#8817012yes—
416C23#8819312yes—
417C23C2×D4×D1311yescharacteristic
418Q8—264no—
419Q8—264no—
420Q8—528no—
421Q8—522no—
422Q8—522no—
423D4—41no—
424D4—41no—
425D4—42no—
426D4—84no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—84no—
439C2×C4—262no—
440C2×C4—262no—
441C2×C4—262no—
442C2×C4—262no—
443C2×C4—262no—
444C2×C4—262no—
445C2×C4—264no—
446C2×C4—528no—
447C2×C4—522no—
448C2×C4—522no—
449C2×C4—522no—
450C2×C4—522no—
451C2×C4—522no—
452C2×C4#8819312yes—
453C2×C4C2×#612911yescharacteristic
454C2×C4#41×D1312yes—
455C2×C4#8825612yes—
456C2×C4C22×C22⋊D1311yescharacteristic
457C8—41no—
458C8—41no—
459C8—522no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—81no—
465C22—82no—
466C22—82no—
467C22—81no—
468C22—81no—
469C22#1171882714yes—
470C22#1171935912yes—
471C22C2×#8816911yescharacteristic
472C22C2×#8817011yescharacteristic
473C22#1171948012yes—
474C22C2×#8819311yescharacteristic
475C4—22no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—524no—
480C4—522no—
481C4—522no—
482C4—1042no—
483C4#1171936312yes—
484C4#1171953311yescharacteristic
485C4#1171953511yescharacteristic
486C2—24no—
487C2—161no—
488C2#1311822211yescharacteristic
489C2C2×#1171882711yescharacteristic
490C2#1312435011yescharacteristic
491C2#1312673612yes—
492C2#1312694512yes—
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#481C2
socleC23×C13C22×D4
prime radicalC2×C13×#121C22
hypercenterC2×#481D13
hypocenterC13#24483
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C22×#88050C2semidirect
3. #13117713C2semidirect
4. #13116212C2semidirect
5. #13116076C2semidirect
6. #13115767C2semidirect
7. #13115469C2semidirect
9. #13115013C2semidirect
10. #13114772C2semidirect
406. C13#24483semidirect

Maximal subgroups

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

subgroupindex
C22×#880502
#131184952
#131177132
#131162122
#131160762
#131157672
#131154692
#131152652
#131150132
#131147722
C2×C13×#4812
#2448313