Finite Group Databasebeta

← #15839124

Subgroups — #15839124

Order 3328.

1146 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
1#13125378C211yesmaximal, characteristic, complemented
2#13124966C211yesmaximal, characteristic, complemented
3#13124935C211yesmaximal, characteristic
4#13114099C214yesmaximal, complemented
5#13114085C214yesmaximal, complemented
6#13113553C211yesmaximal, characteristic, complemented
7#13113283C211yesmaximal, characteristic, complemented
8#13113279C211yesmaximal, characteristic
9C2×D4×Q8×C13C211yesmaximal, characteristic
10C23×Q8×C13—21no—
11D4×Q8×C13—22no—
12D4×Q8×C13—22no—
13D4×Q8×C13—22no—
14D4×Q8×C13C2214yes—
15C2×C13×C4⋊Q8—21no—
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8—21no—
18C2×C13×C22⋊Q8—22no—
19C2×C4×Q8×C13C2211yescharacteristic
20C2×C4×D4×C13—21no—
21C2×C4×D4×C13C2211yescharacteristic
22#11719175C2211yescharacteristic
23#11718889—21no—
24#11718879—21no—
25#11718741—22no—
26#11718740C2212yescomplemented
27#11718739C2212yescomplemented
28#11718692C2212yescomplemented
29#11718691—22no—
30#11718681C2212yescomplemented
31#11718680C2212yes—
32#11718679—22no—
33#11718668C2212yes—
34#11718632—21no—
35#11718631—21no—
36#11718625—22no—
37#11718587C2211yescharacteristic
38#11718587C2211yescharacteristic
39C4×#6099C2211yescharacteristic
40#11718139C2214yescomplemented
41Q8⋊#1055C2214yescomplemented
42D4⋊#1055C2214yes—
43C22×Q8×C13—22no—
44C22×Q8×C13—24no—
45C22×Q8×C13—24no—
46C22×Q8×C13—21no—
47C22×Q8×C13—48no—
48C22×Q8×C13D411yescharacteristic
49C22×D4×C13D411yescharacteristic
50C23×C4×C13—22no—
51C23×C4×C13—21no—
52C13×C4⋊Q8—24no—
53C13×C4⋊Q8—24no—
54C13×C4⋊Q8—21no—
55C13×C4⋊Q8—21no—
56C13×C4⋊Q8D412yes—
57C13×C4⋊Q8C2312yes—
58C13×C22⋊Q8—24no—
59C13×C22⋊Q8—28no—
60C13×C22⋊Q8—24no—
61C13×C22⋊Q8—28no—
62C4×Q8×C13—21no—
63C4×Q8×C13—21no—
64C4×Q8×C13D412yes—
65C4×Q8×C13C2312yes—
66C4×D4×C13—24no—
67C4×D4×C13—24no—
68C4×D4×C13—21no—
69C4×D4×C13—21no—
70C4×D4×C13D412yes—
71C4×D4×C13C2312yes—
72C2×C13×C4⋊C4—22no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4D411yescharacteristic
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4D412yes—
78C2×C13×C22⋊C4—22no—
79C2×C13×C22⋊C4—21no—
80C2×C42×C13—21no—
81C2×C42×C13C2311yescharacteristic
82C2×#6107—22no—
83C2×#6105—22no—
84#88180—21no—
85#88178—21no—
86#88178—21no—
87C2×#6099—21no—
88#88124—21no—
89#88123C2311yescharacteristic
90#88123C2311yescharacteristic
91C4×Dic26—21no—
92C4×Dic26—21no—
93#88075—22no—
94#88072—22no—
95C22⋊#1055—21no—
96C22⋊#1055—21no—
97#88050—22no—
98#88050—22no—
99#88048—22no—
100#88048—24no—
101#88047—22no—
102#88047—24no—
103C4⋊#1055C2312yes—
104C4⋊#1055C2312yes—
105#88043—21no—
106#88043—21no—
107#88043—21no—
108C4×#1055C2312yes—
109#18540—131nomaximal, Sylow
110C24×C13—21no—
111C2×Q8×C13—24no—
112C2×Q8×C13—28no—
113C2×Q8×C13—21no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13—48no—
117C2×Q8×C13—44no—
118C2×Q8×C13—416no—
119C2×Q8×C13—48no—
120C2×Q8×C13—48no—
121C2×Q8×C13C2×D412yes—
122C2×D4×C13—24no—
123C2×D4×C13—21no—
124C2×D4×C13C2×D412yes—
125C22×C4×C13—21no—
126C22×C4×C13—28no—
127C22×C4×C13—21no—
128C22×C4×C13—22no—
129C22×C4×C13—21no—
130C22×C4×C13—22no—
131C22×C4×C13—22no—
132C22×C4×C13—48no—
133C22×C4×C13—44no—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C2×Dic26—21no—
138C2×Dic26—21no—
139C13×C4⋊C4—28no—
140C13×C4⋊C4—22no—
141C13×C4⋊C4—22no—
142C13×C4⋊C4—21no—
143C13×C4⋊C4—21no—
144C13×C4⋊C4—22no—
145C13×C4⋊C4—22no—
146C13×C4⋊C4—22no—
147C13×C4⋊C4C2×D412yes—
148C13×C4⋊C4C2×D412yes—
149C13×C4⋊C4C2×D414yes—
150C13×C22⋊C4—28no—
151C13×C22⋊C4—22no—
152C13×C22⋊C4—22no—
153C42×C13—22no—
154C42×C13—22no—
155C42×C13C2×D411yescharacteristic
156C42×C13C2×D411yescharacteristic
157C42×C13C2×D411yescharacteristic
158C42×C13C2411yescharacteristic
159C22⋊Dic13—41no—
160#6107—44no—
161#6105—44no—
162C4⋊Dic13—21no—
163C4⋊Dic13—21no—
164C4⋊Dic13—21no—
165C4⋊Dic13—21no—
166#6101—41no—
167#6101—41no—
168C4×Dic13—41no—
169#6099—24no—
170C2×#1055—22no—
171C2×#1055—22no—
172C2×#1055—22no—
173C2×D4×Q8D1311yescharacteristic
174#3165—131no—
175#3113—131no—
176#3110—131no—
177#1651—134no—
178#1649—134no—
179#1556—131no—
180#1491—131no—
181#1490—131no—
182C23×C13—24no—
183C23×C13—22no—
184C23×C13—42no—
185C23×C13—42no—
186C23×C13#12011yescharacteristic
187Q8×C13—24no—
188Q8×C13—22no—
189Q8×C13—416no—
190Q8×C13—44no—
191Q8×C13—44no—
192Q8×C13AGL(1,ℤ/8)14yes—
193D4×C13—24no—
194D4×C13—22no—
195D4×C13#13714yes—
196C2×C4×C13—24no—
197C2×C4×C13—22no—
198C2×C4×C13—22no—
199C2×C4×C13—22no—
200C2×C4×C13—22no—
201C2×C4×C13—22no—
202C2×C4×C13—24no—
203C2×C4×C13—22no—
204C2×C4×C13—416no—
205C2×C4×C13—44no—
206C2×C4×C13—44no—
207C2×C4×C13—42no—
208C2×C4×C13—42no—
209C2×C4×C13—44no—
210C2×C4×C13#12011yescharacteristic
211C2×C4×C13#12011yescharacteristic
212C2×C4×C13#12011yescharacteristic
213C2×C4×C13C22×D411yescharacteristic
214C2×C4×C13C22×D411yescharacteristic
215C2×C4×C13C22×D411yescharacteristic
216C2×Dic13—41no—
217C2×Dic13—41no—
218Dic26—42no—
219Dic26—42no—
220#1055—24no—
221#1055—42no—
222#1055—42no—
223C23×Q8—21no—
224D4×Q8—22no—
225D4×Q8—22no—
226D4×Q8—22no—
227D4×Q8C2×D1314yes—
228#533—131no—
229C2×C4⋊Q8—21no—
230C2×C4⋊Q8C2×D1311yescharacteristic
231C2×C22⋊Q8—21no—
232C2×C22⋊Q8—22no—
233C2×C4×Q8C2×D1311yescharacteristic
234C2×C4×D4—21no—
235C2×C4×D4C2×D1311yescharacteristic
236#501—262no—
237#500—132no—
238#498—132no—
239#484—261no—
240#483—261no—
241#475—132no—
242#474—132no—
243#462—132no—
244#461—132no—
245#451—261no—
246#450—261no—
247#441—262no—
248#440—262no—
249#432—131no—
250#432—131no—
251#428—262no—
252C4×M16—131no—
253#332—134no—
254Q8⋊C8—134no—
255D4⋊C8—134no—
256C22×C13—21no—
257C22×C13—21no—
258C22×C13—24no—
259C22×C13—48no—
260C22×C13—42no—
261C22×C13—42no—
262C22×C13#45612yes—
263C22×C13C2×#12011yescharacteristic
264C4×C13—24no—
265C4×C13—21no—
266C4×C13—21no—
267C4×C13—21no—
268C4×C13—21no—
269C4×C13—44no—
270C4×C13—42no—
271C4×C13—42no—
272C4×C13C2×AGL(1,ℤ/8)12yes—
273C4×C13C2×#13712yes—
274Dic13—81no—
275Dic13—81no—
276C22×Q8—22no—
277C22×Q8—24no—
278C22×Q8—24no—
279C22×Q8—21no—
280C22×Q8—48no—
281C22×Q8C22⋊D1311yescharacteristic
282C22×D4C22⋊D1311yescharacteristic
283C23×C4—22no—
284C23×C4—21no—
285C2×Q16—262no—
286C2×SD16—262no—
287C2×M16—261no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—131no—
293C4⋊Q8—131no—
294C4⋊Q8C22⋊D1312yes—
295C4⋊Q8C22×D1312yes—
296#125—261no—
297C22⋊Q8—24no—
298C22⋊Q8—28no—
299C22⋊Q8—24no—
300C22⋊Q8—28no—
301C22⋊Q8—261no—
302C22⋊Q8—261no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—261no—
306C4×Q8—261no—
307C4×Q8C22⋊D1312yes—
308C4×Q8C22×D1312yes—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4C22⋊D1312yes—
314C4×D4C22×D1312yes—
315#117—261no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C22⋊D1311yescharacteristic
320C2×C4⋊C4C22⋊D1311yescharacteristic
321C2×C4⋊C4C22⋊D1312yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C42—21no—
325C2×C42C22×D1311yescharacteristic
326#107—264no—
327#107—262no—
328#106—264no—
329#106—262no—
330C4⋊C8—132no—
331C4⋊C8—132no—
332#103—262no—
333#103—262no—
334#103—262no—
335#102—262no—
336C22⋊C8—261no—
337C22⋊C8—261no—
338#97—261no—
339#97—261no—
340#97—261no—
341C4×C8—132no—
342C2×C13—22no—
343C2×C13—44no—
344C2×C13#300311yescharacteristic
345C2×C13#300411yescharacteristic
346C2×C13C2×#45611yescharacteristic
347C24—21no—
348C2×Q8—24no—
349C2×Q8—28no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—48no—
354C2×Q8—44no—
355C2×Q8—416no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—261no—
359C2×Q8—261no—
360C2×Q8C2×C22⋊D1312yes—
361C2×D4—24no—
362C2×D4—21no—
363C2×D4C2×C22⋊D1312yes—
364C22×C4—21no—
365C22×C4—28no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4C2×C22⋊D1311yescharacteristic
374C22×C4C2×C22⋊D1311yescharacteristic
375C22×C4C2×C22⋊D1311yescharacteristic
376Q16—524no—
377SD16—524no—
378M16—264no—
379C2×C8—262no—
380C2×C8—262no—
381C2×C8—262no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—261no—
391C4⋊C4—261no—
392C4⋊C4—261no—
393C4⋊C4—261no—
394C4⋊C4—521no—
395C4⋊C4—521no—
396C4⋊C4C2×C22⋊D1312yes—
397C4⋊C4C2×C22⋊D1312yes—
398C4⋊C4C2×C22⋊D1314yes—
399C22⋊C4—28no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—521no—
403C42—22no—
404C42—22no—
405C42—521no—
406C42C2×C22⋊D1311yescharacteristic
407C42C2×C22⋊D1311yescharacteristic
408C42C2×C22⋊D1311yescharacteristic
409C42C23×D1311yescharacteristic
410C13#1854011yescharacteristic, complemented, Sylow
411C23—24no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23#8820711yescharacteristic
416Q8—24no—
417Q8—22no—
418Q8—416no—
419Q8—44no—
420Q8—44no—
421Q8—522no—
422Q8—522no—
423Q8#8818614yes—
424D4—24no—
425D4—22no—
426D4#8819614yes—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—416no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—521no—
442C2×C4—521no—
443C2×C4#8820711yescharacteristic
444C2×C4#8820711yescharacteristic
445C2×C4#8820711yescharacteristic
446C2×C4C22×C22⋊D1311yescharacteristic
447C2×C4C22×C22⋊D1311yescharacteristic
448C2×C4C22×C22⋊D1311yescharacteristic
449C8—264no—
450C8—522no—
451C8—522no—
452C22—21no—
453C22—21no—
454C22—24no—
455C22—48no—
456C22—42no—
457C22—42no—
458C22#1171890712yes—
459C22C2×#8820711yescharacteristic
460C4—21no—
461C4—21no—
462C4—24no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—1041no—
469C4—1041no—
470C4C2×#8818612yes—
471C4C2×#8819612yes—
472C2—22no—
473C2—44no—
474C2#1312406311yescharacteristic
475C2#1312407011yescharacteristic
476C2C2×#1171890711yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×D4×Q8×C13C2
socleC22×C13C2×#120
prime radicalC22×D4×C13D4
hypercenterC2×D4×Q8D13
hypocenterC13#18540
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #13125378C2semidirect
2. #13124966C2semidirect
4. #13114099C2semidirect
5. #13114085C2semidirect
6. #13113553C2semidirect
7. #13113283C2semidirect
26. #11718740C22semidirect
27. #11718739C22semidirect
28. #11718692C22semidirect
30. #11718681C22semidirect
40. #11718139C22semidirect
41. Q8⋊#1055C22semidirect
410. C13#18540semidirect

Maximal subgroups

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

subgroupindex
#131253782
#131249662
#131249352
#131140992
#131140852
#131135532
#131132832
#131132792
C2×D4×Q8×C132
#1854013