Finite Group Databasebeta

← #28399831

Subgroups — #28399831

Order 14464.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D113C211yesmaximal, characteristic
2C2×#16813654C211yesmaximal, characteristic, complemented
3C2×#16813651C211yesmaximal, characteristic, complemented
4C22⋊Q8×D113C211yesmaximal, characteristic, complemented
5#20298762C211yesmaximal, characteristic, complemented
6#20298760C211yesmaximal, characteristic, complemented
7#20298758C211yesmaximal, characteristic, complemented
8#20298755C211yesmaximal, characteristic, complemented
9#20298752C211yesmaximal, characteristic, complemented
10#20298750C211yesmaximal, characteristic, complemented
11#20298747C211yesmaximal, characteristic, complemented
12#20298598C211yesmaximal, characteristic, complemented
13#20298596C211yesmaximal, characteristic, complemented
14#20298560C211yesmaximal, characteristic, complemented
15#452×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic
17C2×#14268647—21no—
18C2×#14268647C2211yescharacteristic
19C2×#14268646—21no—
20C2×#14268646C2211yescharacteristic
21#41×D113—24no—
22#41×D113—24no—
23C2×Q8×D113C2211yescharacteristic
24C2×Q8×D113C2211yescharacteristic, complemented
25C2×D4×D113—21no—
26C2×D4×D113C2211yescharacteristic
27C22×C4×D113—21no—
28C22×C4×D113C2211yescharacteristic
29#16813654—24no—
30#16813651—24no—
31C2×Dic452C2211yescharacteristic, complemented
32C2×#14268637C2211yescharacteristic, complemented
33C2×#14268637C2211yescharacteristic, complemented
34C2×#14268636C2211yescharacteristic, complemented
35C2×#14268635C2211yescharacteristic, complemented
36C2×#14268634C2211yescharacteristic, complemented
37C2×#14268630C2211yescharacteristic, complemented
38C2×#14268630C2211yescharacteristic, complemented
39#16813602C2211yescharacteristic
40#16813601C2211yescharacteristic, complemented
41#16813600—21no—
42#16813599—21no—
43#16813598C2211yescharacteristic
44C4⋊C4×D113—21no—
45C4⋊C4×D113C2211yescharacteristic, complemented
46C22⋊C4×D113—21no—
47#16813547C2211yescharacteristic, complemented
48#16813546C2211yescharacteristic, complemented
49#16813545C2211yescharacteristic, complemented
50#16813544C2211yescharacteristic, complemented
51#16813543C2211yescharacteristic, complemented
52#16813542C2211yescharacteristic, complemented
53C22⋊#11744751C2211yescharacteristic, complemented
54#16813534C2211yescharacteristic
55#16813534C2211yescharacteristic
56C2×#41×C113C2211yescharacteristic
57C2×Q16×C113C2211yescharacteristic, complemented
58C2×SD16×C113C2211yescharacteristic, complemented
59C22⋊Q8×C113C2211yescharacteristic, complemented
60#103×C113C2211yescharacteristic, complemented
61#102×C113C2211yescharacteristic, complemented
62C22⋊C8×C113C2211yescharacteristic, complemented
63C23×D113—21no—
64C23×D113D411yescharacteristic
65#14268648—22no—
66#14268648D414yes—
67#14268647—28no—
68#14268647—22no—
69#14268647—22no—
70#14268646—28no—
71#14268646—22no—
72#14268646D414yes—
73C2×Dic226—21no—
74C2×Dic226—21no—
75C2×Dic226C2311yescharacteristic
76C2×Dic226C2311yescharacteristic
77Q8×D113—22no—
78Q8×D113—22no—
79Q8×D113—22no—
80Q8×D113—22no—
81C2×D452—21no—
82C2×D452C2311yescharacteristic
83C2×C22⋊D113—21no—
84C2×C22⋊D113—21no—
85C2×C22⋊D113—21no—
86D4×D113—28no—
87D4×D113—22no—
88D4×D113—22no—
89C22×Dic113—21no—
90C22×Dic113D411yescharacteristic
91C2×C4×D113—28no—
92C2×C4×D113—21no—
93C2×C4×D113—21no—
94C2×C4×D113—21no—
95C2×C4×D113—22no—
96C2×C4×D113—22no—
97C2×C4×D113—21no—
98C2×C4×D113—21no—
99C2×C4×D113D411yescharacteristic
100C2×C4×D113D411yescharacteristic
101C2×C4×D113C2311yescharacteristic
102C2×C4×D113C2311yescharacteristic
103Dic452—22no—
104#14268637—22no—
105#14268637—22no—
106#14268636—22no—
107#14268635—22no—
108#14268634—22no—
109#14268630—22no—
110#14268630—22no—
111C2×#11744751C2311yescharacteristic
112C2×#11744751C2311yescharacteristic
113C4⋊Dic113—21no—
114C4⋊Dic113C2311yescharacteristic
115#14268626—21no—
116#14268626—21no—
117#14268626—21no—
118C22⋊Dic113—21no—
119#14268624—21no—
120#14268624—21no—
121#41×C113—22no—
122#41×C113—22no—
123C2×Q8×C113C2311yescharacteristic
124C2×Q8×C113C2311yescharacteristic
125C2×D4×C113—21no—
126C2×D4×C113C2311yescharacteristic
127C22×C4×C113—21no—
128C22×C4×C113C2311yescharacteristic
129Q16×C113—22no—
130SD16×C113—22no—
131C2×C8×C113C2311yescharacteristic
132C2×C8×C113C2311yescharacteristic
133C4⋊C4×C113—21no—
134C4⋊C4×C113C2311yescharacteristic
135C22⋊C4×C113—21no—
136C23×C113—21no—
137C23×C113C2×D411yescharacteristic
138C22×D113—24no—
139C22×D113—21no—
140C22×D113—22no—
141C22×D113—22no—
142C22×D113—41no—
143C22×D113—42no—
144C22×D113—42no—
145C22×D113C2×D411yescharacteristic
146C22×D113C2×D411yescharacteristic
147Q8×C113—21no—
148Q8×C113—21no—
149Q8×C113—21no—
150Q8×C113C2×D412yes—
151D4×C113—24no—
152D4×C113—21no—
153D4×C113C2×D412yes—
154C2×C4×C113—24no—
155C2×C4×C113—21no—
156C2×C4×C113—21no—
157C2×C4×C113—21no—
158C2×C4×C113—21no—
159C2×C4×C113—21no—
160C2×C4×C113C2×D411yescharacteristic
161C2×C4×C113C2411yescharacteristic
162C22⋊D113—24no—
163C22⋊D113—24no—
164C22⋊D113—24no—
165C2×Dic113—21no—
166C2×Dic113—24no—
167C2×Dic113—22no—
168C2×Dic113—21no—
169C2×Dic113—21no—
170C2×Dic113C2×D411yescharacteristic
171C2×Dic113C2×D411yescharacteristic
172D452—24no—
173D452—21no—
174D452C2×D412yes—
175C4×D113—24no—
176C4×D113—24no—
177C4×D113—22no—
178C4×D113—21no—
179C4×D113—21no—
180C4×D113—22no—
181C4×D113—24no—
182C4×D113—42no—
183C4×D113—42no—
184C4×D113C2×D412yes—
185C4×D113C2×D412yes—
186Dic226—24no—
187Dic226—21no—
188Dic226—21no—
189Dic226—21no—
190Dic226—42no—
191Dic226C2×D412yes—
192C8×C113—21no—
193C8×C113—21no—
194#11744751—21no—
195#11744751—21no—
196C22×C113—21no—
197C22×C113—22no—
198C22×C113—21no—
199C22×C113—21no—
200C22×C113—41no—
201C22×C113—41no—
202C22×C113C22×D411yescharacteristic
203C2×D113—21no—
204C2×D113—21no—
205C2×D113—21no—
206C2×D113—21no—
207C2×D113—22no—
208C2×D113—44no—
209C2×D113—48no—
210C2×D113—41no—
211C2×D113—41no—
212C2×D113#12012yes—
213C2×D113#13712yescomplemented
214C4×C113—22no—
215C4×C113—21no—
216C4×C113—41no—
217C4×C113—41no—
218C4×C113C22×D411yescharacteristic
219C4×C113C22×D411yescharacteristic
220Dic113—21no—
221Dic113—22no—
222Dic113—41no—
223Dic113—41no—
224Dic113#12012yes—
225C2×C113—41no—
226C2×C113—42no—
227C2×C113C2×#12011yescharacteristic
228C2×C113C2×#13711yescharacteristic
229C2×C113#57511yescharacteristic
230D113—22no—
231D113—22no—
232D113—42no—
233#3007—1131nomaximal, Sylow
234C113#300711yescharacteristic, complemented, Sylow
235C22×#41—1131no—
236C2×#137—1131no—
237C2×#137—1131no—
238C2×C22⋊Q8—1131no—
239#452—1131no—
240#452—1131no—
241#452—1131no—
242#452D11311yescharacteristic, complemented
243#451—1131no—
244#451—1131no—
245#450—1131no—
246#450—1131no—
247#417—1131no—
248#417—1131no—
249#407—1131no—
250C2×#41—1131no—
251C2×#41—1131no—
252C2×#41—1131no—
253C2×#41—2261no—
254C2×#41—2261no—
255C2×#41—2264no—
256C2×#41—2264no—
257C2×#41C2×D11311yescharacteristic
258C22×Q8—1131no—
259C22×Q8—1131no—
260C22×D4—1131no—
261C22×D4—2261no—
262C23×C4—1131no—
263C23×C4—2261no—
264#137—2264no—
265#137—2264no—
266C2×Q16—1131no—
267C2×Q16—1131no—
268C2×Q16—1131no—
269C2×Q16C2×D11311yescharacteristic, complemented
270C2×SD16—1131no—
271C2×SD16—1131no—
272C2×SD16—1131no—
273C2×SD16C2×D11311yescharacteristic, complemented
274C2×M16—1131no—
275C2×M16—1131no—
276C22⋊Q8—1131no—
277C22⋊Q8—1131no—
278C22⋊Q8—1131no—
279C22⋊Q8—2261no—
280C22⋊Q8—2261no—
281C22⋊Q8C2×D11311yescharacteristic, complemented
282C2×C4⋊C4—1131no—
283C2×C4⋊C4—2261no—
284C2×C22⋊C4—2261no—
285#103—1131no—
286#103—1131no—
287#103—1131no—
288#103C2×D11311yescharacteristic, complemented
289#102—1131no—
290#102—1131no—
291#102—1131no—
292#102C2×D11311yescharacteristic, complemented
293C22⋊C8—1131no—
294C22⋊C8—1131no—
295C22⋊C8—1131no—
296C22⋊C8C2×D11311yescharacteristic, complemented
297C24—1131no—
298C24—2261no—
299#41—22no—
300#41—22no—
301#41—1134no—
302#41—1134no—
303#41—2262no—
304#41—2262no—
305#41—2268no—
306#41—2262no—
307#41—2262no—
308#41—2268no—
309C2×Q8—1131no—
310C2×Q8—1131no—
311C2×Q8—2261no—
312C2×Q8—2261no—
313C2×Q8—2262no—
314C2×Q8—2262no—
315C2×Q8—2262no—
316C2×Q8—2262no—
317C2×Q8C22×D11311yescharacteristic
318C2×Q8C22×D11311yescharacteristic
319C2×D4—21no—
320C2×D4—1131no—
321C2×D4—2261no—
322C2×D4—2261no—
323C2×D4—2262no—
324C2×D4—2268no—
325C2×D4—2261no—
326C2×D4—2261no—
327C2×D4—2262no—
328C2×D4C22×D11311yescharacteristic
329C22×C4—21no—
330C22×C4—1131no—
331C22×C4—1131no—
332C22×C4—1131no—
333C22×C4—1131no—
334C22×C4—1131no—
335C22×C4—2262no—
336C22×C4—2262no—
337C22×C4—2261no—
338C22×C4—2261no—
339C22×C4—2261no—
340C22×C4—2261no—
341C22×C4—2268no—
342C22×C4—2261no—
343C22×C4—2261no—
344C22×C4C22×D11311yescharacteristic
345Q16—22no—
346Q16—2262no—
347Q16—2262no—
348Q16—2262no—
349SD16—22no—
350SD16—2262no—
351SD16—2262no—
352SD16—2262no—
353M16—2262no—
354M16—2262no—
355C2×C8—1131no—
356C2×C8—1131no—
357C2×C8C22×D11311yescharacteristic
358C2×C8C22×D11311yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—1131no—
361C4⋊C4—2261no—
362C4⋊C4—2261no—
363C4⋊C4—2261no—
364C4⋊C4—2261no—
365C4⋊C4C22×D11311yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—2261no—
368C22⋊C4—2261no—
369C22⋊C4—2261no—
370C23—21no—
371C23—1131no—
372C23—1131no—
373C23—4521no—
374C23—4522no—
375C23—4522no—
376C23—2262no—
377C23—2262no—
378C23—2264no—
379C23—2261no—
380C23D4×D11311yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—1132no—
385Q8—4522no—
386Q8—2264no—
387Q8—2261no—
388Q8—2261no—
389Q8—2261no—
390Q8D4×D11312yes—
391D4—24no—
392D4—21no—
393D4—1132no—
394D4—2264no—
395D4—2264no—
396D4—2264no—
397D4—2264no—
398D4—2261no—
399D4D4×D11312yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—1131no—
407C2×C4—1132no—
408C2×C4—1132no—
409C2×C4—1131no—
410C2×C4—4522no—
411C2×C4—4522no—
412C2×C4—2262no—
413C2×C4—2264no—
414C2×C4—2262no—
415C2×C4—2264no—
416C2×C4—2264no—
417C2×C4—2262no—
418C2×C4—2264no—
419C2×C4—2261no—
420C2×C4—2261no—
421C2×C4—2261no—
422C2×C4—2261no—
423C2×C4—2261no—
424C2×C4D4×D11311yescharacteristic
425C2×C4C23×D11311yescharacteristic
426C8—21no—
427C8—21no—
428C8—2261no—
429C8—2261no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—1132no—
437C22—1132no—
438C22—4528no—
439C22—4524no—
440C22—4521no—
441C22—4521no—
442C22—2262no—
443C22—2261no—
444C22—2261no—
445C22—2261no—
446C22—2261no—
447C22C2×D4×D11311yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—1132no—
453C4—4521no—
454C4—4521no—
455C4—2261no—
456C4—2262no—
457C4C2×D4×D11311yescharacteristic
458C4C2×D4×D11311yescharacteristic
459C2—41no—
460C2—42no—
461C2—4522no—
462C2—2262no—
463C2—2262no—
464C2#120×D11311yescharacteristic
465C2#137×D11311yescharacteristic
466C2#2029939511yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#452×C113C2
socleC22×C113C22×D4
prime radicalC2×#41×D113C2
hypercenter#452D113
hypocenterC113#3007
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#16813654C2semidirect
3. C2×#16813651C2semidirect
4. C22⋊Q8×D113C2semidirect
5. #20298762C2semidirect
6. #20298760C2semidirect
7. #20298758C2semidirect
8. #20298755C2semidirect
9. #20298752C2semidirect
10. #20298750C2semidirect
11. #20298747C2semidirect
12. #20298598C2semidirect
13. #20298596C2semidirect
14. #20298560C2semidirect
15. #452×C113C2semidirect
24. C2×Q8×D113C22semidirect
31. C2×Dic452C22semidirect
32. C2×#14268637C22semidirect
33. C2×#14268637C22semidirect
34. C2×#14268636C22semidirect
35. C2×#14268635C22semidirect
36. C2×#14268634C22semidirect
37. C2×#14268630C22semidirect
38. C2×#14268630C22semidirect
40. #16813601C22semidirect
45. C4⋊C4×D113C22semidirect
47. #16813547C22semidirect
48. #16813546C22semidirect
49. #16813545C22semidirect
50. #16813544C22semidirect
51. #16813543C22semidirect
52. #16813542C22semidirect
53. C22⋊#11744751C22semidirect
57. C2×Q16×C113C22semidirect
58. C2×SD16×C113C22semidirect
59. C22⋊Q8×C113C22semidirect
60. #103×C113C22semidirect
61. #102×C113C22semidirect
62. C22⋊C8×C113C22semidirect
213. C2×D113#137semidirect
234. C113#3007semidirect
242. #452D113semidirect
269. C2×Q16C2×D113semidirect
273. C2×SD16C2×D113semidirect
281. C22⋊Q8C2×D113semidirect
288. #103C2×D113semidirect
292. #102C2×D113semidirect
296. 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×#168136542
C2×#168136512
C22⋊Q8×D1132
#202987622
#202987602
#202987582
#202987552
#202987522
#202987502
#202987472
#202985982
#202985962
#202985602
#452×C1132
#3007113