Finite Group Databasebeta

← #28399812

Subgroups — #28399812

Order 14464.

758 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D113C211yesmaximal, characteristic, complemented
2C2×#16813651C211yesmaximal, characteristic, complemented
3C2×#16813646C211yesmaximal, characteristic, complemented
4#20299108C211yesmaximal, characteristic, complemented
5#20298763C211yesmaximal, characteristic, complemented
6#20298762C211yesmaximal, characteristic, complemented
7#20298761C211yesmaximal, characteristic, complemented
8#20298760C211yesmaximal, characteristic, complemented
9#20298742C211yesmaximal, characteristic, complemented
10#20298741C211yesmaximal, characteristic, complemented
11#20298740C211yesmaximal, characteristic, complemented
12#20298605C211yesmaximal, characteristic, complemented
13#20298601C211yesmaximal, characteristic, complemented
14#20298560C211yesmaximal, characteristic, complemented
15#449×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic
17C2×#14268647—21no—
18C2×#14268647C2211yescharacteristic
19C2×#14268646—21no—
20C2×#14268646C2211yescharacteristic
21C2×#14268646C2211yescharacteristic
22#41×D113—24no—
23#41×D113—24no—
24C2×Q8×D113C2211yescharacteristic
25C2×D4×D113—21no—
26C2×D4×D113C2211yescharacteristic
27C22×C4×D113—21no—
28C22×C4×D113C2211yescharacteristic
29#16813651—24no—
30#16813646—24no—
31C2×Dic452C2211yescharacteristic, complemented
32C2×#14268637C2211yescharacteristic, complemented
33C2×#14268636C2211yescharacteristic, complemented
34C2×#14268635C2211yescharacteristic, complemented
35C2×#14268635C2211yescharacteristic, complemented
36C2×#14268632C2211yescharacteristic, complemented
37C2×#14268630C2211yescharacteristic
38C2×#14268630C2211yescharacteristic
39#16813604—21no—
40#16813602C2211yescharacteristic, complemented
41#16813601C2211yescharacteristic, complemented
42#16813593C2211yescharacteristic, complemented
43#16813590—21no—
44D4×Dic113—21no—
45#16813582—21no—
46#16813580C2211yescharacteristic
47#16813547C2211yescharacteristic, complemented
48#16813546C2211yescharacteristic, complemented
49#16813545C2211yescharacteristic, complemented
50#16813544C2211yescharacteristic, complemented
51#16813543C2211yescharacteristic, complemented
52#16813542C2211yescharacteristic, complemented
53C22⋊#11744751C2211yescharacteristic, complemented
54#16813534C2211yescharacteristic, complemented
55#16813534C2211yescharacteristic, complemented
56C2×#41×C113C2211yescharacteristic
57C2×SD16×C113C2211yescharacteristic, complemented
58C2×D8×C113C2211yescharacteristic, complemented
59#121×C113C2211yescharacteristic, complemented
60#103×C113C2211yescharacteristic, complemented
61#102×C113C2211yescharacteristic, complemented
62C22⋊C8×C113C2211yescharacteristic, complemented
63C23×D113—21no—
64C23×D113D411yescharacteristic
65#14268648—22no—
66#14268648—22no—
67#14268647—28no—
68#14268647—22no—
69#14268647—22no—
70#14268646—28no—
71#14268646—22no—
72#14268646—22no—
73#14268646—22no—
74#14268646—22no—
75C2×Dic226—21no—
76C2×Dic226C2311yescharacteristic
77C2×Dic226C2311yescharacteristic
78Q8×D113—22no—
79Q8×D113D414yescomplemented
80C2×D452—21no—
81C2×D452C2311yescharacteristic
82C2×C22⋊D113—21no—
83C2×C22⋊D113—21no—
84C2×C22⋊D113—21no—
85C2×C22⋊D113—21no—
86C2×C22⋊D113—21no—
87D4×D113—28no—
88D4×D113—22no—
89D4×D113D414yescomplemented
90C22×Dic113—21no—
91C22×Dic113—21no—
92C22×Dic113D411yescharacteristic
93C2×C4×D113—28no—
94C2×C4×D113—21no—
95C2×C4×D113—21no—
96C2×C4×D113—21no—
97C2×C4×D113—22no—
98C2×C4×D113—22no—
99C2×C4×D113D411yescharacteristic
100C2×C4×D113D411yescharacteristic
101C2×C4×D113C2311yescharacteristic
102C2×C4×D113C2311yescharacteristic
103Dic452—22no—
104#14268637—22no—
105#14268636—22no—
106#14268635—22no—
107#14268635—22no—
108#14268632—22no—
109#14268630—22no—
110#14268630—22no—
111C2×#11744751C2311yescharacteristic
112C2×#11744751C2311yescharacteristic
113C4⋊Dic113—21no—
114C4⋊Dic113C2311yescharacteristic
115#14268626—21no—
116C22⋊Dic113—21no—
117C22⋊Dic113—21no—
118#14268624—21no—
119#14268624—21no—
120C4×Dic113—21no—
121#41×C113—22no—
122#41×C113—22no—
123C2×Q8×C113C2311yescharacteristic
124C2×D4×C113—21no—
125C2×D4×C113—21no—
126C2×D4×C113C2311yescharacteristic
127C2×D4×C113C2311yescharacteristic
128C22×C4×C113—21no—
129C22×C4×C113C2311yescharacteristic
130SD16×C113—22no—
131D8×C113—22no—
132C2×C8×C113C2311yescharacteristic
133C2×C8×C113C2311yescharacteristic
134C4⋊C4×C113C2311yescharacteristic
135C22⋊C4×C113—21no—
136C23×C113—21no—
137C23×C113—21no—
138C23×C113C2×D411yescharacteristic
139C22×D113—21no—
140C22×D113—22no—
141C22×D113—24no—
142C22×D113—41no—
143C22×D113—41no—
144C22×D113—42no—
145C22×D113—42no—
146C22×D113C2×D411yescharacteristic
147C22×D113C2×D411yescharacteristic
148Q8×C113—21no—
149Q8×C113C2×D412yes—
150D4×C113—24no—
151D4×C113—21no—
152D4×C113—21no—
153D4×C113—21no—
154D4×C113—42no—
155D4×C113C2×D412yes—
156C2×C4×C113—24no—
157C2×C4×C113—21no—
158C2×C4×C113—21no—
159C2×C4×C113—21no—
160C2×C4×C113—21no—
161C2×C4×C113C2×D411yescharacteristic
162C2×C4×C113C2411yescharacteristic
163C22⋊D113—24no—
164C22⋊D113—24no—
165C22⋊D113—24no—
166C22⋊D113—42no—
167C22⋊D113—42no—
168C2×Dic113—24no—
169C2×Dic113—21no—
170C2×Dic113—22no—
171C2×Dic113—21no—
172C2×Dic113—21no—
173C2×Dic113—42no—
174C2×Dic113C2×D411yescharacteristic
175C2×Dic113C2×D411yescharacteristic
176D452—24no—
177D452—21no—
178D452C2×D412yes—
179C4×D113—24no—
180C4×D113—24no—
181C4×D113—22no—
182C4×D113—24no—
183C4×D113—21no—
184C4×D113—21no—
185C4×D113—22no—
186C4×D113C2×D412yes—
187C4×D113C2×D412yes—
188Dic226—21no—
189Dic226—24no—
190Dic226—21no—
191Dic226—21no—
192Dic226C2×D412yes—
193C8×C113—21no—
194C8×C113—21no—
195#11744751—21no—
196#11744751—21no—
197C22×C113—21no—
198C22×C113—22no—
199C22×C113—21no—
200C22×C113—21no—
201C22×C113—41no—
202C22×C113—41no—
203C22×C113—41no—
204C22×C113—41no—
205C22×C113—41no—
206C22×C113C22×D411yescharacteristic
207C2×D113—21no—
208C2×D113—21no—
209C2×D113—21no—
210C2×D113—22no—
211C2×D113—21no—
212C2×D113—21no—
213C2×D113—44no—
214C2×D113—48no—
215C2×D113—41no—
216C2×D113—41no—
217C2×D113#12012yes—
218C4×C113—22no—
219C4×C113—21no—
220C4×C113—41no—
221C4×C113C22×D411yescharacteristic
222C4×C113C22×D411yescharacteristic
223Dic113—22no—
224Dic113—21no—
225Dic113—41no—
226Dic113—41no—
227Dic113#12012yes—
228C2×C113—41no—
229C2×C113—42no—
230C2×C113—81no—
231C2×C113C2×#12011yescharacteristic
232C2×C113#57511yescharacteristic
233C2×C113#57511yescharacteristic
234D113—22no—
235D113—41no—
236D113—42no—
237#3005—1131nomaximal, Sylow
238C113#300511yescharacteristic, complemented, Sylow
239C22×#41—1131no—
240C2×#137—1131no—
241C2×AGL(1,ℤ/8)—1131no—
242#525—1131no—
243#452—1131no—
244#452—1131no—
245#452—1131no—
246#452—1131no—
247#449—1131no—
248#449—1131no—
249#449—1131no—
250#449D11311yescharacteristic, complemented
251#419—1131no—
252#418—1131no—
253#407—1131no—
254C2×#41—1131no—
255C2×#41—1131no—
256C2×#41—1131no—
257C2×#41—1131no—
258C2×#41—2261no—
259C2×#41—2261no—
260C2×#41—2264no—
261C2×#41—2264no—
262C2×#41C2×D11311yescharacteristic
263C22×Q8—1131no—
264C22×D4—1131no—
265C22×D4—2261no—
266C23×C4—1131no—
267C23×C4—2261no—
268#137—2264no—
269AGL(1,ℤ/8)—2264no—
270C2×Q16—1131no—
271C2×Q16—1131no—
272C2×SD16—1131no—
273C2×SD16—1131no—
274C2×SD16—1131no—
275C2×SD16C2×D11311yescharacteristic, complemented
276C2×D8—1131no—
277C2×D8C2×D11311yescharacteristic, complemented
278C2×M16—1131no—
279C2×M16—1131no—
280#123—2261no—
281C22⋊Q8—1131no—
282C22⋊Q8—1131no—
283#121—1131no—
284#121C2×D11311yescharacteristic, complemented
285#120—2261no—
286C4×D4—2261no—
287C4×D4—2261no—
288#117—1131no—
289#103—1131no—
290#103—1131no—
291#103—1131no—
292#103C2×D11311yescharacteristic, complemented
293#102—1131no—
294#102—1131no—
295#102—1131no—
296#102C2×D11311yescharacteristic, complemented
297C22⋊C8—1131no—
298C22⋊C8—1131no—
299C22⋊C8—1131no—
300C22⋊C8C2×D11311yescharacteristic, complemented
301C24—1131no—
302C24—2261no—
303#41—22no—
304#41—22no—
305#41—2262no—
306#41—2262no—
307#41—2268no—
308#41—2262no—
309#41—2262no—
310#41—2262no—
311#41—2268no—
312#41—2262no—
313#41—2262no—
314#41—2262no—
315C2×Q8—1134no—
316C2×Q8—1131no—
317C2×Q8—1131no—
318C2×Q8—2261no—
319C2×Q8—2262no—
320C2×Q8C22×D11311yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—1134no—
324C2×D4—1131no—
325C2×D4—2261no—
326C2×D4—2262no—
327C2×D4—2261no—
328C2×D4—2268no—
329C2×D4—2261no—
330C2×D4—2261no—
331C2×D4—2261no—
332C2×D4—2261no—
333C2×D4C22×D11311yescharacteristic
334C2×D4C22×D11311yescharacteristic
335C22×C4—21no—
336C22×C4—1131no—
337C22×C4—1131no—
338C22×C4—1131no—
339C22×C4—1131no—
340C22×C4—1131no—
341C22×C4—2262no—
342C22×C4—2262no—
343C22×C4—2261no—
344C22×C4—2261no—
345C22×C4—2268no—
346C22×C4—2261no—
347C22×C4—2261no—
348C22×C4—2261no—
349C22×C4C22×D11311yescharacteristic
350Q16—2262no—
351Q16—2262no—
352SD16—22no—
353SD16—2262no—
354SD16—2262no—
355SD16—2262no—
356D8—22no—
357D8—2262no—
358M16—2262no—
359M16—2262no—
360C2×C8—1131no—
361C2×C8—1131no—
362C2×C8C22×D11311yescharacteristic
363C2×C8C22×D11311yescharacteristic
364C4⋊C4—1131no—
365C4⋊C4—2261no—
366C4⋊C4—2261no—
367C4⋊C4C22×D11311yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—2261no—
370C22⋊C4—2261no—
371C22⋊C4—2261no—
372C22⋊C4—2261no—
373C42—2261no—
374C23—21no—
375C23—21no—
376C23—1131no—
377C23—1131no—
378C23—4521no—
379C23—4521no—
380C23—4522no—
381C23—4522no—
382C23—2264no—
383C23—2262no—
384C23—2261no—
385C23D4×D11311yescharacteristic
386Q8—21no—
387Q8—1132no—
388Q8—2264no—
389Q8—2261no—
390Q8—2261no—
391Q8—2261no—
392Q8D4×D11312yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—1132no—
399D4—4522no—
400D4—4522no—
401D4—2264no—
402D4—2264no—
403D4—2264no—
404D4—2264no—
405D4—2261no—
406D4D4×D11312yes—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—1131no—
413C2×C4—1132no—
414C2×C4—1132no—
415C2×C4—1131no—
416C2×C4—4522no—
417C2×C4—2262no—
418C2×C4—2262no—
419C2×C4—2264no—
420C2×C4—2264no—
421C2×C4—2264no—
422C2×C4—2262no—
423C2×C4—2261no—
424C2×C4—2261no—
425C2×C4—2264no—
426C2×C4—2261no—
427C2×C4—2261no—
428C2×C4—2261no—
429C2×C4D4×D11311yescharacteristic
430C2×C4C23×D11311yescharacteristic
431C8—21no—
432C8—21no—
433C8—2261no—
434C8—2261no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—1132no—
445C22—4524no—
446C22—4528no—
447C22—4521no—
448C22—4521no—
449C22—2262no—
450C22—2261no—
451C22—2261no—
452C22—2261no—
453C22—2261no—
454C22—2261no—
455C22C2×D4×D11311yescharacteristic
456C4—21no—
457C4—22no—
458C4—41no—
459C4—1132no—
460C4—4521no—
461C4—4521no—
462C4—2261no—
463C4—2262no—
464C4C2×D4×D11311yescharacteristic
465C4C2×D4×D11311yescharacteristic
466C2—41no—
467C2—42no—
468C2—81no—
469C2—4522no—
470C2—4521no—
471C2—2262no—
472C2#120×D11311yescharacteristic
473C2#2029939411yescharacteristic
474C2#2029939511yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#449×C113C2
socleC22×C113C22×D4
hypercenter#449D113
hypocenterC113#3005
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D113C2semidirect
2. C2×#16813651C2semidirect
3. C2×#16813646C2semidirect
4. #20299108C2semidirect
5. #20298763C2semidirect
6. #20298762C2semidirect
7. #20298761C2semidirect
8. #20298760C2semidirect
9. #20298742C2semidirect
10. #20298741C2semidirect
11. #20298740C2semidirect
12. #20298605C2semidirect
13. #20298601C2semidirect
14. #20298560C2semidirect
15. #449×C113C2semidirect
31. C2×Dic452C22semidirect
32. C2×#14268637C22semidirect
33. C2×#14268636C22semidirect
34. C2×#14268635C22semidirect
35. C2×#14268635C22semidirect
36. C2×#14268632C22semidirect
40. #16813602C22semidirect
41. #16813601C22semidirect
42. #16813593C22semidirect
47. #16813547C22semidirect
48. #16813546C22semidirect
49. #16813545C22semidirect
50. #16813544C22semidirect
51. #16813543C22semidirect
52. #16813542C22semidirect
53. C22⋊#11744751C22semidirect
54. #16813534C22semidirect
55. #16813534C22semidirect
57. C2×SD16×C113C22semidirect
58. C2×D8×C113C22semidirect
59. #121×C113C22semidirect
60. #103×C113C22semidirect
61. #102×C113C22semidirect
62. C22⋊C8×C113C22semidirect
79. Q8×D113D4semidirect
89. D4×D113D4semidirect
238. C113#3005semidirect
250. #449D113semidirect
275. C2×SD16C2×D113semidirect
277. C2×D8C2×D113semidirect
284. #121C2×D113semidirect
292. #103C2×D113semidirect
296. #102C2×D113semidirect
300. C22⋊C8C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1132
C2×#168136512
C2×#168136462
#202991082
#202987632
#202987622
#202987612
#202987602
#202987422
#202987412
#202987402
#202986052
#202986012
#202985602
#449×C1132
#3005113