Finite Group Databasebeta

← #15841703

Subgroups — #15841703

Order 3328.

1138 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125980C211yesmaximal, characteristic, complemented
2#13124844C211yesmaximal, characteristic
3#13124641C211yesmaximal, characteristic, complemented
4#13114099C214yesmaximal, complemented
5#13114001C214yesmaximal, complemented
6#13113508C211yesmaximal, characteristic, complemented
7#13113261C211yesmaximal, characteristic
8#13113231C211yesmaximal, characteristic, complemented
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#11718886—21no—
24#11718879—21no—
25#11718755—22no—
26#11718754C2212yescomplemented
27#11718740C2212yescomplemented
28#11718683—22no—
29#11718682C2212yescomplemented
30#11718681C2212yescomplemented
31#11718680C2212yes—
32C4×#6105—22no—
33#11718669C2212yes—
34#11718632—22no—
35#11718632—21no—
36#11718625—21no—
37#11718586C2211yescharacteristic
38#11718586C2211yescharacteristic
39#11718585C2211yescharacteristic
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—44no—
48C22×Q8×C13—44no—
49C22×Q8×C13D411yescharacteristic
50C22×D4×C13D411yescharacteristic
51C23×C4×C13—22no—
52C23×C4×C13—21no—
53C13×C4⋊Q8—24no—
54C13×C4⋊Q8—24no—
55C13×C4⋊Q8—21no—
56C13×C4⋊Q8—21no—
57C13×C4⋊Q8D412yes—
58C13×C4⋊Q8C2312yes—
59C13×C22⋊Q8—24no—
60C13×C22⋊Q8—28no—
61C13×C22⋊Q8—24no—
62C13×C22⋊Q8—28no—
63C4×Q8×C13—21no—
64C4×Q8×C13—21no—
65C4×Q8×C13D412yes—
66C4×Q8×C13C2312yes—
67C4×D4×C13—24no—
68C4×D4×C13—24no—
69C4×D4×C13—21no—
70C4×D4×C13—21no—
71C4×D4×C13D412yes—
72C4×D4×C13C2312yes—
73C2×C13×C4⋊C4—22no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4D411yescharacteristic
78C2×C13×C4⋊C4D412yes—
79C2×C13×C22⋊C4—22no—
80C2×C13×C22⋊C4—21no—
81C2×C42×C13—21no—
82C2×C42×C13C2311yescharacteristic
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#88075—22no—
95#88072—22no—
96C22⋊#1055—21no—
97C22⋊#1055—21no—
98#88050—22no—
99#88050—24no—
100#88048—22no—
101#88048—22no—
102#88048—24no—
103#88047—22no—
104C4⋊#1055—21no—
105C4⋊#1055—21no—
106C4⋊#1055C2312yes—
107C4⋊#1055C2312yes—
108C4×#1055—21no—
109C4×#1055C2312yes—
110#18777—131nomaximal, Sylow
111C24×C13—21no—
112C2×Q8×C13—28no—
113C2×Q8×C13—21no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13—48no—
118C2×Q8×C13—48no—
119C2×Q8×C13—416no—
120C2×Q8×C13—48no—
121C2×Q8×C13—42no—
122C2×Q8×C13—42no—
123C2×Q8×C13SD1614yescomplemented
124C2×Q8×C13C2×D412yes—
125C2×D4×C13—24no—
126C2×D4×C13—21no—
127C2×D4×C13C2×D412yes—
128C22×C4×C13—21no—
129C22×C4×C13—28no—
130C22×C4×C13—21no—
131C22×C4×C13—22no—
132C22×C4×C13—21no—
133C22×C4×C13—22no—
134C22×C4×C13—22no—
135C22×C4×C13—48no—
136C22×C4×C13—42no—
137C22×C4×C13—42no—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C2×Dic26—21no—
142C2×Dic26—21no—
143C13×C4⋊C4—28no—
144C13×C4⋊C4—22no—
145C13×C4⋊C4—22no—
146C13×C4⋊C4—21no—
147C13×C4⋊C4—21no—
148C13×C4⋊C4—22no—
149C13×C4⋊C4—22no—
150C13×C4⋊C4—22no—
151C13×C4⋊C4C2×D412yes—
152C13×C4⋊C4C2×D412yes—
153C13×C4⋊C4C2×D414yes—
154C13×C22⋊C4—28no—
155C13×C22⋊C4—22no—
156C13×C22⋊C4—22no—
157C42×C13—22no—
158C42×C13—22no—
159C42×C13C2×D411yescharacteristic
160C42×C13C2×D411yescharacteristic
161C42×C13C2×D411yescharacteristic
162C42×C13C2411yescharacteristic
163C22⋊Dic13—41no—
164#6105—44no—
165C4⋊Dic13—21no—
166C4⋊Dic13—21no—
167C4⋊Dic13—21no—
168C4⋊Dic13—21no—
169#6101—41no—
170#6101—41no—
171C4×Dic13—41no—
172#6099—42no—
173C2×#1055—22no—
174C2×#1055—22no—
175C2×#1055—22no—
176C2×D4×Q8D1311yescharacteristic
177#3229—131no—
178#3101—131no—
179#3078—131no—
180#1651—134no—
181#1637—134no—
182#1546—131no—
183#1486—131no—
184#1479—131no—
185C23×C13—24no—
186C23×C13—22no—
187C23×C13—42no—
188C23×C13—42no—
189C23×C13#12011yescharacteristic
190Q8×C13—22no—
191Q8×C13—24no—
192Q8×C13—416no—
193Q8×C13—42no—
194Q8×C13—42no—
195Q8×C13—44no—
196Q8×C13C2×SD1614yes—
197D4×C13—24no—
198D4×C13—22no—
199D4×C13#13714yes—
200C2×C4×C13—22no—
201C2×C4×C13—24no—
202C2×C4×C13—22no—
203C2×C4×C13—22no—
204C2×C4×C13—24no—
205C2×C4×C13—22no—
206C2×C4×C13—41no—
207C2×C4×C13—416no—
208C2×C4×C13—42no—
209C2×C4×C13—44no—
210C2×C4×C13—42no—
211C2×C4×C13—42no—
212C2×C4×C13—42no—
213C2×C4×C13—44no—
214C2×C4×C13#12011yescharacteristic
215C2×C4×C13#12011yescharacteristic
216C2×C4×C13#12011yescharacteristic
217C2×C4×C13C2×SD1612yes—
218C2×C4×C13AGL(1,ℤ/8)12yes—
219C2×C4×C13C22×D411yescharacteristic
220C2×C4×C13C22×D411yescharacteristic
221C2×C4×C13C22×D411yescharacteristic
222C2×Dic13—41no—
223C2×Dic13—41no—
224Dic26—42no—
225Dic26—42no—
226#1055—42no—
227#1055—42no—
228#1055—42no—
229C23×Q8—21no—
230D4×Q8—22no—
231D4×Q8—22no—
232D4×Q8—22no—
233D4×Q8C2×D1314yes—
234#533—131no—
235C2×C4⋊Q8—21no—
236C2×C4⋊Q8C2×D1311yescharacteristic
237C2×C22⋊Q8—21no—
238C2×C22⋊Q8—22no—
239C2×C4×Q8C2×D1311yescharacteristic
240C2×C4×D4—21no—
241C2×C4×D4C2×D1311yescharacteristic
242#498—132no—
243#487—132no—
244#483—261no—
245#483—262no—
246#479—262no—
247#477—132no—
248#476—132no—
249#475—132no—
250#475—262no—
251#461—132no—
252#450—261no—
253C4×SD16—262no—
254#431—131no—
255#431—131no—
256#428—261no—
257#423—131no—
258#419—261no—
259#332—134no—
260Q8⋊C8—134no—
261D4⋊C8—134no—
262C22×C13—21no—
263C22×C13—21no—
264C22×C13—24no—
265C22×C13—48no—
266C22×C13—42no—
267C22×C13—42no—
268C22×C13#45012yes—
269C22×C13C2×#12011yescharacteristic
270C4×C13—24no—
271C4×C13—21no—
272C4×C13—21no—
273C4×C13—21no—
274C4×C13—44no—
275C4×C13—42no—
276C4×C13—42no—
277C4×C13#45012yes—
278C4×C13C22×SD1611yescharacteristic
279C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
280C4×C13C2×#13712yes—
281Dic13—81no—
282Dic13—81no—
283C22×Q8—22no—
284C22×Q8—24no—
285C22×Q8—24no—
286C22×Q8—21no—
287C22×Q8—44no—
288C22×Q8—44no—
289C22×Q8C22⋊D1311yescharacteristic
290C22×D4C22⋊D1311yescharacteristic
291C23×C4—22no—
292C23×C4—21no—
293C2×SD16—262no—
294C2×M16—261no—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—131no—
300C4⋊Q8—131no—
301C4⋊Q8C22⋊D1312yes—
302C4⋊Q8C22×D1312yes—
303#125—261no—
304C22⋊Q8—24no—
305C22⋊Q8—28no—
306C22⋊Q8—24no—
307C22⋊Q8—28no—
308C22⋊Q8—261no—
309C22⋊Q8—261no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—261no—
313C4×Q8—261no—
314C4×Q8C22⋊D1312yes—
315C4×Q8C22×D1312yes—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4C22⋊D1312yes—
321C4×D4C22×D1312yes—
322#117—261no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D1311yescharacteristic
327C2×C4⋊C4C22⋊D1311yescharacteristic
328C2×C4⋊C4C22⋊D1312yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C42—21no—
332C2×C42C22×D1311yescharacteristic
333#107—262no—
334#106—262no—
335#106—262no—
336#106—264no—
337C4⋊C8—132no—
338C4⋊C8—132no—
339C4⋊C8—261no—
340C4⋊C8—261no—
341#103—262no—
342#103—262no—
343#103—262no—
344#103—264no—
345#102—262no—
346C22⋊C8—261no—
347C22⋊C8—261no—
348C4×C8—132no—
349C4×C8—261no—
350C2×C13—22no—
351C2×C13—44no—
352C2×C13C2×#45011yescharacteristic
353C2×C13#300411yescharacteristic
354C2×C13#302011yescharacteristic
355C24—21no—
356C2×Q8—28no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—416no—
364C2×Q8—48no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—261no—
368C2×Q8—261no—
369C2×Q8#610514yescomplemented
370C2×Q8C2×C22⋊D1312yes—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4C2×C22⋊D1312yes—
374C22×C4—21no—
375C22×C4—28no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C2×C22⋊D1311yescharacteristic
385C22×C4C2×C22⋊D1311yescharacteristic
386C22×C4C2×C22⋊D1311yescharacteristic
387SD16—524no—
388M16—522no—
389C2×C8—262no—
390C2×C8—262no—
391C2×C8—262no—
392C4⋊C4—28no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—261no—
401C4⋊C4—261no—
402C4⋊C4—261no—
403C4⋊C4—261no—
404C4⋊C4—521no—
405C4⋊C4—521no—
406C4⋊C4C2×C22⋊D1312yes—
407C4⋊C4C2×C22⋊D1314yes—
408C4⋊C4C2×C22⋊D1312yes—
409C22⋊C4—28no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—521no—
413C42—22no—
414C42—22no—
415C42—521no—
416C42C2×C22⋊D1311yescharacteristic
417C42C2×C22⋊D1311yescharacteristic
418C42C2×C22⋊D1311yescharacteristic
419C42C23×D1311yescharacteristic
420C13#1877711yescharacteristic, complemented, Sylow
421C23—24no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23#8820711yescharacteristic
426Q8—22no—
427Q8—24no—
428Q8—416no—
429Q8—42no—
430Q8—42no—
431Q8—44no—
432Q8—522no—
433Q8—522no—
434Q8C2×#610514yes—
435D4—24no—
436D4—22no—
437D4#8819614yes—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—416no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—521no—
453C2×C4—521no—
454C2×C4#8818612yes—
455C2×C4C2×#610512yes—
456C2×C4#8820711yescharacteristic
457C2×C4#8820711yescharacteristic
458C2×C4#8820711yescharacteristic
459C2×C4C22×C22⋊D1311yescharacteristic
460C2×C4C22×C22⋊D1311yescharacteristic
461C2×C4C22×C22⋊D1311yescharacteristic
462C8—522no—
463C8—522no—
464C8—522no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22#1171887912yes—
472C22C2×#8820711yescharacteristic
473C4—21no—
474C4—21no—
475C4—24no—
476C4—21no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—1041no—
481C4—1041no—
482C4#1171887912yes—
483C4C2×#8818611yescharacteristic
484C4C22×#610511yescharacteristic
485C4C2×#8819612yes—
486C2—22no—
487C2—44no—
488C2C2×#1171887911yescharacteristic
489C2#1312407011yescharacteristic
490C2#1312420011yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×D4×Q8×C13C2
socleC22×C13C2×#120
prime radicalC22×D4×C13D4
hypercenterC2×D4×Q8D13
hypocenterC13#18777
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. #13125980C2semidirect
3. #13124641C2semidirect
4. #13114099C2semidirect
5. #13114001C2semidirect
6. #13113508C2semidirect
8. #13113231C2semidirect
26. #11718754C22semidirect
27. #11718740C22semidirect
29. #11718682C22semidirect
30. #11718681C22semidirect
40. #11718139C22semidirect
41. Q8⋊#1055C22semidirect
123. C2×Q8×C13SD16semidirect
369. C2×Q8#6105semidirect
420. C13#18777semidirect

Maximal subgroups

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

subgroupindex
#131259802
#131248442
#131246412
#131140992
#131140012
#131135082
#131132612
#131132312
C2×D4×Q8×C132
#1877713