Finite Group Databasebeta

← #15901964

Subgroups — #15901964

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718889C211yesmaximal, characteristic
2#13119310C211yesmaximal, characteristic
3Q8×C22⋊Dic13C211yesmaximal, characteristic
4#13116247C212yesmaximal, complemented
5#13116157C212yesmaximal, complemented
6#13115259C211yesmaximal, characteristic
7#13115175C212yesmaximal, complemented
8#13115014C211yesmaximal, characteristic
9#13114926C211yesmaximal, characteristic
10#13114839C212yesmaximal, complemented
11C22×C13×#103C211yesmaximal, characteristic
12C23×Q8×C13C2211yescharacteristic
13C23×C8×C13C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C2×Q8×Dic13C2212yes—
16C22×#6107C2211yescharacteristic
17C22×#6107C2211yescharacteristic
18C2×#88178C2211yescharacteristic
19C2×C13×#103—24no—
20C2×C13×#103C2212yes—
21C2×C13×#103C2212yes—
22#11718891—22no—
23#11718891C2212yes—
24#11718889—24no—
25C2×#88075C2211yescharacteristic
26C2×#88075C2211yescharacteristic
27#11718873—21no—
28#11718873C2211yescharacteristic
29#11718871—21no—
30#11718871—21no—
31C4×C22⋊Dic13—21no—
32C4×C22⋊Dic13C2211yescharacteristic
33C2×C22⋊#1055C2211yescharacteristic
34C2×#88056C2212yes—
35C2×#88054C2212yes—
36#11718641—22no—
37#11718637C2212yes—
38#11718635C2212yes—
39#11718143C2212yes—
40#11718142C2212yes—
41#11718121C2212yes—
42#11718121C2212yes—
43C22×Q8×C13—21no—
44C22×Q8×C13—22no—
45C22×Q8×C13—24no—
46C22×Q8×C13—42no—
47C22×Q8×C13D414yes—
48C22×Q8×C13C2311yescharacteristic
49C22×Q8×C13C2311yescharacteristic
50C23×C4×C13—21no—
51C23×C4×C13—21no—
52C23×C4×C13C2311yescharacteristic
53C22×Dic26C2311yescharacteristic
54C22×C8×C13—22no—
55C22×C8×C13—42no—
56C22×C8×C13C2312yes—
57C2×C13×C4⋊C4—22no—
58C2×C13×C4⋊C4—22no—
59C2×C13×C4⋊C4—21no—
60C2×C13×C4⋊C4C2312yes—
61C2×C22⋊Dic13—21no—
62C2×C22⋊Dic13—21no—
63C2×C22⋊Dic13D411yescharacteristic
64C2×C22⋊Dic13D411yescharacteristic
65Q8×Dic13—24no—
66Q8×Dic13—42no—
67C2×#6107—22no—
68C2×#6107—22no—
69C2×#6107—24no—
70C2×#6107—42no—
71C2×C4⋊Dic13—22no—
72C2×C4⋊Dic13C2311yescharacteristic
73C2×C4⋊Dic13C2312yes—
74#88178—22no—
75#88178—22no—
76C2×#6101—21no—
77C2×#6101—22no—
78C2×#6101—22no—
79C2×C4×Dic13—22no—
80C2×C4×Dic13C2312yes—
81C22×#1055C2311yescharacteristic
82C22×#1055C2311yescharacteristic
83C13×#103—24no—
84C13×#103—24no—
85#88075—22no—
86#88075—22no—
87#88071—21no—
88#88071—22no—
89#88071—22no—
90#88071—22no—
91#88071—22no—
92#88071D412yes—
93C22⋊#1055—22no—
94#88056—42no—
95#88054—42no—
96#24532—131nomaximal, Sylow
97C24×C13C2×D411yescharacteristic
98C2×Q8×C13—24no—
99C2×Q8×C13—21no—
100C2×Q8×C13—21no—
101C2×Q8×C13—22no—
102C2×Q8×C13—28no—
103C2×Q8×C13—22no—
104C2×Q8×C13—22no—
105C2×Q8×C13—44no—
106C2×Q8×C13—44no—
107C2×Q8×C13—48no—
108C2×Q8×C13C2×D412yes—
109C2×Q8×C13C2×D412yes—
110C2×Q8×C13#4114yes—
111C22×C4×C13—21no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—24no—
115C22×C4×C13—21no—
116C22×C4×C13—22no—
117C22×C4×C13—44no—
118C22×C4×C13—42no—
119C22×C4×C13—82no—
120C22×C4×C13C2×D411yescharacteristic
121C22×C4×C13C2×D412yes—
122C22×C4×C13#4112yes—
123C22×C4×C13C2411yescharacteristic
124C22×Dic13—21no—
125C22×Dic13—22no—
126C22×Dic13—21no—
127C22×Dic13C2×D412yes—
128C22×Dic13C2×D412yes—
129C2×Dic26—21no—
130C2×Dic26—21no—
131C2×Dic26—22no—
132C2×Dic26—22no—
133C2×C8×C13—22no—
134C2×C8×C13—22no—
135C2×C8×C13—22no—
136C2×C8×C13—44no—
137C13×C4⋊C4—22no—
138C13×C4⋊C4—22no—
139C13×C4⋊C4—42no—
140C22⋊Dic13—22no—
141C22⋊Dic13—42no—
142C22⋊Dic13—42no—
143C22⋊Dic13Q1614yescomplemented
144#6107—44no—
145#6107—44no—
146C4⋊Dic13—21no—
147C4⋊Dic13—21no—
148C4⋊Dic13—22no—
149C4⋊Dic13—22no—
150C4⋊Dic13—44no—
151#6101—42no—
152#6101—82no—
153#6101—82no—
154C4×Dic13—22no—
155C4×Dic13—22no—
156C4×Dic13—44no—
157C2×#1055—21no—
158C2×#1055—21no—
159C2×#1055—22no—
160C2×#1055—22no—
161C2×#451—131no—
162C22×#103D1311yescharacteristic
163#2502—131no—
164Q8×C22⋊C4—131no—
165#2072—132no—
166#2056—132no—
167#1892—131no—
168#1878—132no—
169#1850—131no—
170#1824—131no—
171#1803—132no—
172C23×C13—22no—
173C23×C13—22no—
174C23×C13#12112yes—
175C23×C13C2×Q1612yes—
176C23×C13#13712yes—
177C23×C13C22×D411yescharacteristic
178Q8×C13—22no—
179Q8×C13—24no—
180Q8×C13—22no—
181Q8×C13—22no—
182Q8×C13—48no—
183Q8×C13—42no—
184Q8×C13#12314yes—
185C2×C4×C13—22no—
186C2×C4×C13—24no—
187C2×C4×C13—22no—
188C2×C4×C13—22no—
189C2×C4×C13—22no—
190C2×C4×C13—24no—
191C2×C4×C13—22no—
192C2×C4×C13—48no—
193C2×C4×C13—42no—
194C2×C4×C13—42no—
195C2×C4×C13—42no—
196C2×C4×C13—42no—
197C2×C4×C13—84no—
198C2×C4×C13#12112yes—
199C2×C4×C13C22×D411yescharacteristic
200C2×C4×C13C2×#4112yes—
201C2×C4×C13C2×#4111yescharacteristic
202C2×C4×C13H(5,𝔽2)12yes—
203C2×Dic13—22no—
204C2×Dic13—22no—
205C2×Dic13—22no—
206C2×Dic13—22no—
207C2×Dic13—22no—
208C2×Dic13—41no—
209C2×Dic13—41no—
210C2×Dic13—42no—
211C2×Dic13—42no—
212C2×Dic13—42no—
213C2×Dic13—42no—
214C2×Dic13—42no—
215C2×Dic13C2×Q1614yes—
216Dic26—41no—
217Dic26—41no—
218Dic26—42no—
219C8×C13—42no—
220#1055—41no—
221#1055—41no—
222C23×Q8C2×D1311yescharacteristic
223C22×Q16—131no—
224C22×Q16—131no—
225C23×C8C2×D1311yescharacteristic
226C2×C22⋊Q8—131no—
227C2×C4×Q8—132no—
228C22×C4⋊C4C2×D1311yescharacteristic
229#451—264no—
230C2×C4⋊C8—132no—
231C2×#103—24no—
232C2×#103—131no—
233C2×#103—131no—
234C2×#103—132no—
235C2×#103C2×D1312yes—
236C2×#103C2×D1312yes—
237C2×C22⋊C8—131no—
238#398—262no—
239#391—132no—
240#391—132no—
241#391—262no—
242#385—261no—
243#385—261no—
244#384—132no—
245#380—131no—
246#380—261no—
247C4×C22⋊C4—131no—
248C4×C22⋊C4—261no—
249#340—132no—
250#340—132no—
251#340—132no—
252#336—132no—
253C22×C13—22no—
254C22×C13—22no—
255C22×C13—24no—
256C22×C13—24no—
257C22×C13#46214yes—
258C22×C13C2×#12111yescharacteristic
259C22×C13#54612yes—
260C22×C13C22×Q1611yescharacteristic
261C22×C13C2×#13711yescharacteristic
262C22×C13#57712yes—
263C4×C13—22no—
264C4×C13—24no—
265C4×C13—42no—
266C4×C13—82no—
267C4×C13C2×#12311yescharacteristic
268C4×C13#53411yescharacteristic
269C4×C13#55112yes—
270Dic13—24no—
271Dic13—42no—
272Dic13—81no—
273Dic13—81no—
274Dic13—82no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8—24no—
278C22×Q8—42no—
279C22×Q8—131no—
280C22×Q8C22⋊D1314yes—
281C22×Q8C22×D1311yescharacteristic
282C22×Q8C22×D1311yescharacteristic
283C23×C4—21no—
284C23×C4—21no—
285C23×C4C22×D1311yescharacteristic
286C2×Q16—264no—
287C2×Q16—262no—
288C2×Q16—262no—
289C2×Q16—522no—
290C22×C8—22no—
291C22×C8—42no—
292C22×C8—131no—
293C22×C8—131no—
294C22×C8C22×D1312yes—
295C22⋊Q8—262no—
296C22⋊Q8—262no—
297C4×Q8—264no—
298C4×Q8—522no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—132no—
303C2×C4⋊C4—131no—
304C2×C4⋊C4—262no—
305C2×C4⋊C4—261no—
306C2×C4⋊C4—262no—
307C2×C4⋊C4—262no—
308C2×C4⋊C4C22×D1312yes—
309C2×C22⋊C4—131no—
310C2×C22⋊C4—131no—
311C2×C22⋊C4—261no—
312C2×C22⋊C4—261no—
313C2×C42—132no—
314C2×C42—262no—
315C4⋊C8—522no—
316#103—24no—
317#103—24no—
318#103—262no—
319#103—262no—
320#103—522no—
321C22⋊C8—262no—
322#95—132no—
323#95—262no—
324#95—262no—
325#95—262no—
326#95—262no—
327#95—261no—
328C2×C13—24no—
329C2×C13#241211yescharacteristic
330C2×C13C2×#46211yescharacteristic
331C2×C13#304311yescharacteristic
332C2×C13#329912yes—
333C2×C13#331612yes—
334C24D4×D1311yescharacteristic
335C2×Q8—24no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—28no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—262no—
346C2×Q8—261no—
347C2×Q8—262no—
348C2×Q8—261no—
349C2×Q8#612714yes—
350C2×Q8C2×C22⋊D1312yes—
351C2×Q8C2×C22⋊D1312yes—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—82no—
361C22×C4—132no—
362C22×C4—132no—
363C22×C4—261no—
364C22×C4—262no—
365C22×C4—261no—
366C22×C4D4×D1311yescharacteristic
367C22×C4#612912yes—
368C22×C4C2×C22⋊D1312yes—
369C22×C4C23×D1311yescharacteristic
370Q16—524no—
371Q16—524no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—44no—
376C2×C8—262no—
377C2×C8—262no—
378C2×C8—261no—
379C2×C8—261no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C4⋊C4—262no—
384C4⋊C4—261no—
385C4⋊C4—262no—
386C4⋊C4—261no—
387C4⋊C4—524no—
388C4⋊C4—522no—
389C4⋊C4—1042no—
390C4⋊C4—1042no—
391C22⋊C4—134no—
392C22⋊C4—262no—
393C22⋊C4—522no—
394C22⋊C4—522no—
395C42—262no—
396C42—262no—
397C42—524no—
398C13#2453211yescharacteristic, complemented, Sylow
399C23—22no—
400C23—22no—
401C23Q16×D1312yes—
402C23#8817212yes—
403C23#8819312yes—
404C23C2×D4×D1311yescharacteristic
405Q8—24no—
406Q8—22no—
407Q8—22no—
408Q8—22no—
409Q8—48no—
410Q8—42no—
411Q8—522no—
412Q8—521no—
413Q8—521no—
414Q8#8818314yes—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—84no—
428C2×C4—134no—
429C2×C4—262no—
430C2×C4—262no—
431C2×C4—262no—
432C2×C4—262no—
433C2×C4—262no—
434C2×C4—522no—
435C2×C4—522no—
436C2×C4—522no—
437C2×C4—522no—
438C2×C4—522no—
439C2×C4—521no—
440C2×C4—521no—
441C2×C4#8819312yes—
442C2×C4C2×#612712yes—
443C2×C4C2×#612911yescharacteristic
444C2×C4#8825112yes—
445C2×C4C22×C22⋊D1311yescharacteristic
446C8—42no—
447C8—521no—
448C8—521no—
449C22—22no—
450C22—22no—
451C22—24no—
452C22—24no—
453C22#1171884314yes—
454C22#1171919512yes—
455C22#1171946712yes—
456C22C2×Q16×D1311yescharacteristic
457C22C2×#8817211yescharacteristic
458C22C2×#8819311yescharacteristic
459C4—22no—
460C4—24no—
461C4—42no—
462C4—82no—
463C4—264no—
464C4—522no—
465C4—1041no—
466C4—1041no—
467C4—1042no—
468C4#1171920012yes—
469C4C2×#8818311yescharacteristic
470C4#1171950911yescharacteristic
471C2—24no—
472C2#1311823211yescharacteristic
473C2C2×#1171884311yescharacteristic
474C2#1312435411yescharacteristic
475C2#1312670812yes—
476C2#1312694312yes—
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC22×C13×#103C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenterC22×#103D13
hypocenterC13#24532
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
4. #13116247C2semidirect
5. #13116157C2semidirect
7. #13115175C2semidirect
10. #13114839C2semidirect
143. C22⋊Dic13Q16semidirect
398. C13#24532semidirect

Maximal subgroups

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

subgroupindex
C2×#117188892
#131193102
Q8×C22⋊Dic132
#131162472
#131161572
#131152592
#131151752
#131150142
#131149262
#131148392
C22×C13×#1032
#2453213