Finite Group Databasebeta

← #12078084

Subgroups — #12078084

Order 1280.

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
1C2×C5×D4×Q8C211yesmaximal, characteristic
2#10614954C211yesmaximal, characteristic, complemented
3#10614162C211yesmaximal, characteristic, complemented
4#10614045C211yesmaximal, characteristic
5#10602710C214yesmaximal, complemented
6#10602708C214yesmaximal, complemented
7#10602566C211yesmaximal, characteristic, complemented
8#10602414C211yesmaximal, characteristic, complemented
9#10602391C211yesmaximal, characteristic
10C23×C5×Q8—21no—
11C5×D4×Q8—22no—
12C5×D4×Q8—22no—
13C5×D4×Q8—22no—
14C5×D4×Q8C2214yes—
15C2×C5×C4⋊Q8—21no—
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×C22⋊Q8—21no—
18C2×C5×C22⋊Q8—22no—
19C2×C4×C5×Q8C2211yescharacteristic
20C2×C4×C5×D4—21no—
21C2×C4×C5×D4C2211yescharacteristic
22#65804C2211yescharacteristic
23#65514—21no—
24#65507—21no—
25#65383—22no—
26#65382C2212yescomplemented
27#65368C2212yescomplemented
28#65311—22no—
29#65310C2212yescomplemented
30#65309C2212yescomplemented
31#65308C2212yes—
32C4×#687—22no—
33#65297C2212yes—
34#65260—21no—
35#65260—22no—
36#65253—21no—
37#65214C2211yescharacteristic
38#65214C2211yescharacteristic
39#65213C2211yescharacteristic
40#64767C2214yescomplemented
41Q8⋊#168C2214yescomplemented
42D4⋊#168C2214yes—
43#18777—51nomaximal, Sylow
44C22×C5×Q8—22no—
45C22×C5×Q8—24no—
46C22×C5×Q8—24no—
47C22×C5×Q8—21no—
48C22×C5×Q8—44no—
49C22×C5×Q8—44no—
50C22×C5×Q8D411yescharacteristic
51C22×C5×D4D411yescharacteristic
52C23×C4×C5—22no—
53C23×C4×C5—21no—
54C5×C4⋊Q8—24no—
55C5×C4⋊Q8—24no—
56C5×C4⋊Q8—21no—
57C5×C4⋊Q8—21no—
58C5×C4⋊Q8D412yes—
59C5×C4⋊Q8C2312yes—
60C5×C22⋊Q8—24no—
61C5×C22⋊Q8—28no—
62C5×C22⋊Q8—24no—
63C5×C22⋊Q8—28no—
64C4×C5×Q8—21no—
65C4×C5×Q8—21no—
66C4×C5×Q8D412yes—
67C4×C5×Q8C2312yes—
68C4×C5×D4—24no—
69C4×C5×D4—24no—
70C4×C5×D4—21no—
71C4×C5×D4—21no—
72C4×C5×D4D412yes—
73C4×C5×D4C2312yes—
74C2×C5×C4⋊C4—22no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4D411yescharacteristic
78C2×C5×C4⋊C4D411yescharacteristic
79C2×C5×C4⋊C4D412yes—
80C2×C5×C22⋊C4—22no—
81C2×C5×C22⋊C4—21no—
82C2×C42×C5—21no—
83C2×C42×C5C2311yescharacteristic
84C2×#687—22no—
85#4072—21no—
86#4070—21no—
87#4070—21no—
88C2×#681—21no—
89#4016—21no—
90#4015C2311yescharacteristic
91#4015C2311yescharacteristic
92C4×Dic10—21no—
93C4×Dic10—21no—
94#3967—22no—
95#3967—22no—
96#3964—22no—
97C22⋊#168—21no—
98C22⋊#168—21no—
99#3942—22no—
100#3942—24no—
101#3940—22no—
102#3940—22no—
103#3940—24no—
104#3939—22no—
105C4⋊#168—21no—
106C4⋊#168—21no—
107C4⋊#168C2312yes—
108C4⋊#168C2312yes—
109C4×#168—21no—
110C4×#168C2312yes—
111C2×D4×Q8D511yescharacteristic
112#3229—51no—
113#3101—51no—
114#3078—51no—
115#1651—54no—
116#1637—54no—
117#1546—51no—
118#1486—51no—
119#1479—51no—
120C24×C5—21no—
121C2×C5×Q8—28no—
122C2×C5×Q8—21no—
123C2×C5×Q8—22no—
124C2×C5×Q8—22no—
125C2×C5×Q8—22no—
126C2×C5×Q8—48no—
127C2×C5×Q8—48no—
128C2×C5×Q8—416no—
129C2×C5×Q8—48no—
130C2×C5×Q8—42no—
131C2×C5×Q8—42no—
132C2×C5×Q8SD1614yescomplemented
133C2×C5×Q8C2×D412yes—
134C2×C5×D4—24no—
135C2×C5×D4—21no—
136C2×C5×D4C2×D412yes—
137C22×C4×C5—21no—
138C22×C4×C5—28no—
139C22×C4×C5—21no—
140C22×C4×C5—22no—
141C22×C4×C5—21no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—48no—
145C22×C4×C5—42no—
146C22×C4×C5—42no—
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C2×Dic10—21no—
151C2×Dic10—21no—
152C5×C4⋊C4—28no—
153C5×C4⋊C4—22no—
154C5×C4⋊C4—22no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4—21no—
157C5×C4⋊C4—22no—
158C5×C4⋊C4—22no—
159C5×C4⋊C4—22no—
160C5×C4⋊C4C2×D412yes—
161C5×C4⋊C4C2×D412yes—
162C5×C4⋊C4C2×D414yes—
163C5×C22⋊C4—28no—
164C5×C22⋊C4—22no—
165C5×C22⋊C4—22no—
166C42×C5—22no—
167C42×C5—22no—
168C42×C5C2×D411yescharacteristic
169C42×C5C2×D411yescharacteristic
170C42×C5C2×D411yescharacteristic
171C42×C5C2411yescharacteristic
172C22⋊Dic5—41no—
173#687—44no—
174C4⋊Dic5—21no—
175C4⋊Dic5—21no—
176C4⋊Dic5—21no—
177C4⋊Dic5—21no—
178#683—41no—
179#683—41no—
180C4×Dic5—41no—
181#681—42no—
182C2×#168—22no—
183C2×#168—22no—
184C2×#168—22no—
185C23×Q8—21no—
186D4×Q8—22no—
187D4×Q8—22no—
188D4×Q8—22no—
189D4×Q8C2×D514yes—
190#533—51no—
191C2×C4⋊Q8—21no—
192C2×C4⋊Q8C2×D511yescharacteristic
193C2×C22⋊Q8—21no—
194C2×C22⋊Q8—22no—
195C2×C4×Q8C2×D511yescharacteristic
196C2×C4×D4—21no—
197C2×C4×D4C2×D511yescharacteristic
198#498—52no—
199#487—52no—
200#483—101no—
201#483—102no—
202#479—102no—
203#477—52no—
204#476—52no—
205#475—52no—
206#475—102no—
207#461—52no—
208#450—101no—
209C4×SD16—102no—
210#431—51no—
211#431—51no—
212#428—101no—
213#423—51no—
214#419—101no—
215#332—54no—
216Q8⋊C8—54no—
217D4⋊C8—54no—
218C23×C5—24no—
219C23×C5—22no—
220C23×C5—42no—
221C23×C5—42no—
222C23×C5#12011yescharacteristic
223C5×Q8—22no—
224C5×Q8—24no—
225C5×Q8—416no—
226C5×Q8—42no—
227C5×Q8—42no—
228C5×Q8—44no—
229C5×Q8C2×SD1614yes—
230C5×D4—24no—
231C5×D4—22no—
232C5×D4#13714yes—
233C2×C4×C5—22no—
234C2×C4×C5—24no—
235C2×C4×C5—22no—
236C2×C4×C5—22no—
237C2×C4×C5—24no—
238C2×C4×C5—22no—
239C2×C4×C5—41no—
240C2×C4×C5—416no—
241C2×C4×C5—42no—
242C2×C4×C5—44no—
243C2×C4×C5—42no—
244C2×C4×C5—42no—
245C2×C4×C5—42no—
246C2×C4×C5—44no—
247C2×C4×C5#12011yescharacteristic
248C2×C4×C5#12011yescharacteristic
249C2×C4×C5#12011yescharacteristic
250C2×C4×C5C2×SD1612yes—
251C2×C4×C5AGL(1,ℤ/8)12yes—
252C2×C4×C5C22×D411yescharacteristic
253C2×C4×C5C22×D411yescharacteristic
254C2×C4×C5C22×D411yescharacteristic
255C2×Dic5—41no—
256C2×Dic5—41no—
257Dic10—42no—
258Dic10—42no—
259#168—42no—
260#168—42no—
261#168—42no—
262C22×Q8—22no—
263C22×Q8—24no—
264C22×Q8—24no—
265C22×Q8—21no—
266C22×Q8—44no—
267C22×Q8—44no—
268C22×Q8C22⋊D511yescharacteristic
269C22×D4C22⋊D511yescharacteristic
270C23×C4—22no—
271C23×C4—21no—
272C2×SD16—102no—
273C2×M16—101no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—51no—
279C4⋊Q8—51no—
280C4⋊Q8C22⋊D512yes—
281C4⋊Q8C22×D512yes—
282#125—101no—
283C22⋊Q8—24no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—28no—
287C22⋊Q8—101no—
288C22⋊Q8—101no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—101no—
292C4×Q8—101no—
293C4×Q8C22⋊D512yes—
294C4×Q8C22×D512yes—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4C22⋊D512yes—
300C4×D4C22×D512yes—
301#117—101no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C22⋊D511yescharacteristic
306C2×C4⋊C4C22⋊D511yescharacteristic
307C2×C4⋊C4C22⋊D512yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C42—21no—
311C2×C42C22×D511yescharacteristic
312#107—102no—
313#106—102no—
314#106—102no—
315#106—104no—
316C4⋊C8—52no—
317C4⋊C8—52no—
318C4⋊C8—101no—
319C4⋊C8—101no—
320#103—102no—
321#103—102no—
322#103—102no—
323#103—104no—
324#102—102no—
325C22⋊C8—101no—
326C22⋊C8—101no—
327C4×C8—52no—
328C4×C8—101no—
329C22×C5—21no—
330C22×C5—21no—
331C22×C5—24no—
332C22×C5—48no—
333C22×C5—42no—
334C22×C5—42no—
335C22×C5#45012yes—
336C22×C5C2×#12011yescharacteristic
337C4×C5—24no—
338C4×C5—21no—
339C4×C5—21no—
340C4×C5—21no—
341C4×C5—44no—
342C4×C5—42no—
343C4×C5—42no—
344C4×C5#45012yes—
345C4×C5C22×SD1611yescharacteristic
346C4×C5C2×AGL(1,ℤ/8)11yescharacteristic
347C4×C5C2×#13712yes—
348Dic5—81no—
349Dic5—81no—
350C24—21no—
351C2×Q8—28no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—416no—
359C2×Q8—48no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—101no—
363C2×Q8—101no—
364C2×Q8#68714yescomplemented
365C2×Q8C2×C22⋊D512yes—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4C2×C22⋊D512yes—
369C22×C4—21no—
370C22×C4—28no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C2×C22⋊D511yescharacteristic
380C22×C4C2×C22⋊D511yescharacteristic
381C22×C4C2×C22⋊D511yescharacteristic
382SD16—204no—
383M16—202no—
384C2×C8—102no—
385C2×C8—102no—
386C2×C8—102no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—101no—
396C4⋊C4—101no—
397C4⋊C4—101no—
398C4⋊C4—101no—
399C4⋊C4—201no—
400C4⋊C4—201no—
401C4⋊C4C2×C22⋊D512yes—
402C4⋊C4C2×C22⋊D514yes—
403C4⋊C4C2×C22⋊D512yes—
404C22⋊C4—28no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—201no—
408C42—22no—
409C42—22no—
410C42—201no—
411C42C2×C22⋊D511yescharacteristic
412C42C2×C22⋊D511yescharacteristic
413C42C2×C22⋊D511yescharacteristic
414C42C23×D511yescharacteristic
415C2×C5—22no—
416C2×C5—44no—
417C2×C5C2×#45011yescharacteristic
418C2×C5#300411yescharacteristic
419C2×C5#302011yescharacteristic
420C23—24no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23#409911yescharacteristic
425Q8—22no—
426Q8—24no—
427Q8—416no—
428Q8—42no—
429Q8—42no—
430Q8—44no—
431Q8—202no—
432Q8—202no—
433Q8C2×#68714yes—
434D4—24no—
435D4—22no—
436D4#408814yes—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—416no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—201no—
452C2×C4—201no—
453C2×C4#407812yes—
454C2×C4C2×#68712yes—
455C2×C4#409911yescharacteristic
456C2×C4#409911yescharacteristic
457C2×C4#409911yescharacteristic
458C2×C4C22×C22⋊D511yescharacteristic
459C2×C4C22×C22⋊D511yescharacteristic
460C2×C4C22×C22⋊D511yescharacteristic
461C8—202no—
462C8—202no—
463C8—202no—
464C5#1877711yescharacteristic, complemented, Sylow
465C22—21no—
466C22—21no—
467C22—24no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22#6550712yes—
472C22C2×#409911yescharacteristic
473C4—21no—
474C4—21no—
475C4—24no—
476C4—21no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—401no—
481C4—401no—
482C4#6550712yes—
483C4C2×#407811yescharacteristic
484C4C22×#68711yescharacteristic
485C4C2×#408812yes—
486C2—22no—
487C2—44no—
488C2C2×#6550711yescharacteristic
489C2#1061579011yescharacteristic
490C2#1061584711yescharacteristic
subgroupG/H
centerC22C2×#4099
commutator, cosocleC42×C5C24
FrattiniC42C23×D5
FittingC2×C5×D4×Q8C2
socleC22×C5C2×#120
prime radicalC22×C5×D4D4
hypercenterC2×D4×Q8D5
hypocenterC5#18777
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. #10614954C2semidirect
3. #10614162C2semidirect
5. #10602710C2semidirect
6. #10602708C2semidirect
7. #10602566C2semidirect
8. #10602414C2semidirect
26. #65382C22semidirect
27. #65368C22semidirect
29. #65310C22semidirect
30. #65309C22semidirect
40. #64767C22semidirect
41. Q8⋊#168C22semidirect
132. C2×C5×Q8SD16semidirect
364. C2×Q8#687semidirect
464. C5#18777semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×Q82
#106149542
#106141622
#106140452
#106027102
#106027082
#106025662
#106024142
#106023912
#187775