Finite Group Databasebeta

← #28402188

Subgroups — #28402188

Order 14464.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813649C211yesmaximal, characteristic, complemented
2#20299267C211yesmaximal, characteristic, complemented
3D4×C22⋊D113C211yesmaximal, characteristic, complemented
4#20298973C211yesmaximal, characteristic, complemented
5#20298851C211yesmaximal, characteristic, complemented
6#20298848C211yesmaximal, characteristic, complemented
7#20298796C211yesmaximal, characteristic, complemented
8#20298791C211yesmaximal, characteristic, complemented
9#20298743C211yesmaximal, characteristic, complemented
10#20298742C211yesmaximal, characteristic, complemented
11#20298732C211yesmaximal, characteristic, complemented
12#20298730C211yesmaximal, characteristic, complemented
13SD16×Dic113C211yesmaximal, characteristic, complemented
14#20298674C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic, complemented
17C22×C22⋊D113—21no—
18C2×D4×D113C2211yescharacteristic
19#16813649—24no—
20C2×#14268634C2211yescharacteristic, complemented
21C2×D904C2211yescharacteristic, complemented
22C2×#14268632C2211yescharacteristic
23C2×#14268632C2211yescharacteristic, complemented
24C2×#14268632C2211yescharacteristic, complemented
25C2×#14268630C2211yescharacteristic, complemented
26#16813618—21no—
27#16813618C2211yescharacteristic
28#16813618C2211yescharacteristic, complemented
29#16813596—21no—
30#16813596C2211yescharacteristic
31#16813596C2211yescharacteristic, complemented
32#16813594—21no—
33#16813593—21no—
34#16813593C2211yescharacteristic
35#16813593C2211yescharacteristic, complemented
36#16813591—21no—
37#16813590—21no—
38Q8×Dic113C2211yescharacteristic, complemented
39D4×Dic113C2211yescharacteristic
40C4×C22⋊D113—21no—
41C4×C22⋊D113C2211yescharacteristic
42#16813554C2211yescharacteristic, complemented
43#16813551C2211yescharacteristic, complemented
44#16813546C2211yescharacteristic, complemented
45#16813544C2211yescharacteristic, complemented
46#16813544C2211yescharacteristic, complemented
47#16813543C2211yescharacteristic
48#16813543C2211yescharacteristic, complemented
49#16813543C2211yescharacteristic, complemented
50C22⋊#11744751C2211yescharacteristic, complemented
51#16813534C2211yescharacteristic, complemented
52C8×Dic113C2211yescharacteristic, complemented
53C2×#41×C113C2211yescharacteristic, complemented
54C22×D4×C113C2211yescharacteristic
55AGL(1,ℤ/8)×C113—24no—
56C2×SD16×C113C2211yescharacteristic
57C2×SD16×C113C2211yescharacteristic, complemented
58C2×D8×C113C2211yescharacteristic
59C2×D8×C113C2211yescharacteristic, complemented
60C2×M16×C113C2211yescharacteristic, complemented
61C23×D113—21no—
62#14268648—22no—
63#14268648—41no—
64C2×D452—21no—
65C2×D452C2311yescharacteristic
66C2×D452C2311yescharacteristic
67C2×C22⋊D113—21no—
68C2×C22⋊D113—21no—
69C2×C22⋊D113—21no—
70C2×C22⋊D113—21no—
71C2×C22⋊D113—21no—
72C2×C22⋊D113—21no—
73C2×C22⋊D113—21no—
74C2×C22⋊D113—21no—
75C2×C22⋊D113—21no—
76C2×C22⋊D113—44no—
77C2×C22⋊D113D411yescharacteristic
78D4×D113—22no—
79D4×D113—41no—
80C22×Dic113—21no—
81C2×C4×D113—21no—
82C2×C4×D113C2311yescharacteristic
83#14268634—22no—
84D904—22no—
85#14268632—22no—
86#14268632—41no—
87#14268632—41no—
88#14268630—22no—
89C2×#11744751C2311yescharacteristic
90C2×#11744751C2311yescharacteristic
91C4⋊Dic113—21no—
92C4⋊Dic113C2311yescharacteristic
93C4⋊Dic113C2311yescharacteristic
94#14268626D411yescharacteristic
95C22⋊Dic113—21no—
96C22⋊Dic113—21no—
97C22⋊Dic113—21no—
98C22⋊Dic113D411yescharacteristic
99#14268624—21no—
100#14268624—21no—
101#14268624D411yescharacteristic
102C4×Dic113—21no—
103C4×Dic113C2311yescharacteristic
104C24×C113—21no—
105#41×C113—22no—
106#41×C113—41no—
107C2×Q8×C113C2311yescharacteristic
108C2×D4×C113—22no—
109C2×D4×C113—21no—
110C2×D4×C113—21no—
111C2×D4×C113—41no—
112C2×D4×C113C2311yescharacteristic
113C2×D4×C113C2311yescharacteristic
114C2×D4×C113C2311yescharacteristic
115C22×C4×C113—21no—
116C22×C4×C113C2311yescharacteristic
117SD16×C113—22no—
118SD16×C113D414yescomplemented
119D8×C113—22no—
120D8×C113—22no—
121M16×C113—22no—
122C2×C8×C113C2311yescharacteristic
123C2×C8×C113C2311yescharacteristic
124C23×C113—21no—
125C23×C113—21no—
126C23×C113—21no—
127C23×C113—42no—
128C23×C113—42no—
129C23×C113—81no—
130C23×C113C2×D411yescharacteristic
131C22×D113—21no—
132C22×D113—21no—
133C22×D113—42no—
134C22×D113—42no—
135C22×D113—81no—
136C22×D113C2×D411yescharacteristic
137Q8×C113—21no—
138Q8×C113C2×D412yescomplemented
139D4×C113—21no—
140D4×C113—21no—
141D4×C113—21no—
142D4×C113—21no—
143D4×C113—21no—
144D4×C113—42no—
145D4×C113—42no—
146D4×C113C2×D412yes—
147C2×C4×C113—21no—
148C2×C4×C113—21no—
149C2×C4×C113—21no—
150C2×C4×C113—42no—
151C2×C4×C113C2×D411yescharacteristic
152C2×C4×C113C2411yescharacteristic
153C22⋊D113—22no—
154C22⋊D113—42no—
155C22⋊D113—42no—
156C22⋊D113—42no—
157C22⋊D113—42no—
158C22⋊D113—42no—
159C22⋊D113—81no—
160C22⋊D113—81no—
161C22⋊D113—81no—
162C22⋊D113—81no—
163C2×Dic113—21no—
164C2×Dic113—21no—
165C2×Dic113—42no—
166C2×Dic113C2×D411yescharacteristic
167C2×Dic113C2×D411yescharacteristic
168C2×Dic113C2×D411yescharacteristic
169D452—21no—
170D452—21no—
171D452—21no—
172D452—21no—
173D452—42no—
174C4×D113—21no—
175C4×D113—21no—
176C4×D113—42no—
177C8×C113—21no—
178C8×C113C2×D412yescomplemented
179#11744751—21no—
180#11744751—21no—
181C22×C113—22no—
182C22×C113—21no—
183C22×C113—21no—
184C22×C113—21no—
185C22×C113—41no—
186C22×C113—41no—
187C22×C113—41no—
188C22×C113—41no—
189C22×C113—41no—
190C22×C113—41no—
191C22×C113—41no—
192C22×C113—41no—
193C22×C113—84no—
194C22×C113C22×D411yescharacteristic
195C2×D113—21no—
196C2×D113—21no—
197C2×D113—21no—
198C2×D113—41no—
199C2×D113—41no—
200C2×D113—41no—
201C2×D113—41no—
202C2×D113—41no—
203C2×D113—41no—
204C2×D113—84no—
205C4×C113—22no—
206C4×C113—21no—
207C4×C113C22×D411yescharacteristic
208C4×C113H(5,𝔽2)11yescharacteristic
209Dic113—21no—
210Dic113—21no—
211Dic113—41no—
212Dic113—41no—
213Dic113AGL(1,ℤ/8)12yescomplemented
214C2×C113—42no—
215C2×C113—41no—
216C2×C113—81no—
217C2×C113—81no—
218C2×C113D4211yescharacteristic
219C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C113#57611yescharacteristic
221D113—41no—
222D113—81no—
223D113—81no—
224#3274—1131nomaximal, Sylow
225C113#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—1131no—
227C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
228#550—1131no—
229D42—1131no—
230#493—1131no—
231#469—1131no—
232#469—1131no—
233#460—1131no—
234#459—1131no—
235#449—1131no—
236#449—1131no—
237#447—1131no—
238#447—1131no—
239C4×SD16—1131no—
240#436—1131no—
241C2×#41—1131no—
242C2×#41C2×D11311yescharacteristic, complemented
243C22×D4—1131no—
244C22×D4—2261no—
245C22×D4C2×D11311yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—2264no—
248C2×SD16—1131no—
249C2×SD16C2×D11311yescharacteristic
250C2×SD16C2×D11311yescharacteristic, complemented
251C2×D8—1131no—
252C2×D8—1131no—
253C2×D8—1131no—
254C2×D8—1131no—
255C2×D8C2×D11311yescharacteristic
256C2×D8C2×D11311yescharacteristic, complemented
257C2×M16—1131no—
258C2×M16C2×D11311yescharacteristic, complemented
259#127—1131no—
260#127—1131no—
261#127—2261no—
262#121—1131no—
263#121—1131no—
264#121—1131no—
265#121—1131no—
266#121—2261no—
267#121—2261no—
268#121—2261no—
269#120—2261no—
270#120—2261no—
271C4×Q8—1131no—
272C4×D4—1131no—
273C4×D4—1131no—
274C4×D4—2261no—
275#106—1131no—
276C4⋊C8—1131no—
277#103—1131no—
278#102—1131no—
279#102—1131no—
280#102—1131no—
281#102—1131no—
282#102—1131no—
283C22⋊C8—1131no—
284C22⋊C8—1131no—
285C4×C8—1131no—
286C24—21no—
287C24—2261no—
288#41—22no—
289#41—41no—
290#41—4521no—
291#41—2262no—
292C2×Q8C22×D11311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1131no—
298C2×D4—1131no—
299C2×D4—1131no—
300C2×D4—4524no—
301C2×D4—4521no—
302C2×D4—2261no—
303C2×D4—2261no—
304C2×D4—2261no—
305C2×D4—2261no—
306C2×D4—2261no—
307C2×D4—2261no—
308C2×D4—2261no—
309C2×D4—2261no—
310C2×D4—2261no—
311C2×D4—2261no—
312C2×D4—2262no—
313C2×D4C22×D11311yescharacteristic
314C2×D4C22×D11311yescharacteristic
315C2×D4C22×D11311yescharacteristic
316C22×C4—21no—
317C22×C4—1131no—
318C22×C4—2261no—
319C22×C4—2261no—
320C22×C4C22×D11311yescharacteristic
321SD16—22no—
322SD16—2262no—
323SD16C22⋊D11314yescomplemented
324D8—22no—
325D8—22no—
326D8—4521no—
327D8—4521no—
328D8—2262no—
329D8—2262no—
330M16—22no—
331M16—2262no—
332C2×C8—1131no—
333C2×C8—1131no—
334C2×C8C22×D11311yescharacteristic
335C2×C8C22×D11311yescharacteristic
336C4⋊C4—1131no—
337C4⋊C4—1131no—
338C4⋊C4—1131no—
339C4⋊C4—2261no—
340C22⋊C4—1131no—
341C22⋊C4—1131no—
342C22⋊C4—2261no—
343C22⋊C4—2261no—
344C22⋊C4—2261no—
345C22⋊C4—2261no—
346C22⋊C4—2261no—
347C42—1131no—
348C42—2261no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—1131no—
356C23—9041no—
357C23—4522no—
358C23—4522no—
359C23—2261no—
360C23—2261no—
361C23D4×D11311yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D11312yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—9041no—
372D4—9041no—
373D4—9041no—
374D4—9041no—
375D4—4522no—
376D4—4522no—
377D4—4522no—
378D4—4522no—
379D4—4522no—
380D4—4522no—
381D4—2262no—
382D4—2261no—
383D4—2261no—
384D4—2261no—
385D4—2261no—
386D4C2×C22⋊D11312yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—1131no—
392C2×C4—1131no—
393C2×C4—1131no—
394C2×C4—4522no—
395C2×C4—4522no—
396C2×C4—2261no—
397C2×C4—2261no—
398C2×C4—2261no—
399C2×C4—2261no—
400C2×C4D4×D11311yescharacteristic
401C2×C4C23×D11311yescharacteristic
402C8—21no—
403C8—2261no—
404C8—2261no—
405C8C2×C22⋊D11312yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—9044no—
420C22—4521no—
421C22—4521no—
422C22—4521no—
423C22—4521no—
424C22—4521no—
425C22—4521no—
426C22—2261no—
427C22—2261no—
428C22—2261no—
429C22C2×D4×D11311yescharacteristic
430C4—21no—
431C4—22no—
432C4—1132no—
433C4—4521no—
434C4—4521no—
435C4—2261no—
436C4—2261no—
437C4C22×C22⋊D11311yescharacteristic
438C4#1681366711yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—9041no—
444C2—9041no—
445C2—4521no—
446C2D4×C22⋊D11311yescharacteristic
447C2AGL(1,ℤ/8)×D11311yescharacteristic
448C2#2029940211yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×AGL(1,ℤ/8)×C113C2
socleC22×C113C22×D4
hypercenterC2×AGL(1,ℤ/8)D113
hypocenterC113#3274
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16813649C2semidirect
2. #20299267C2semidirect
3. D4×C22⋊D113C2semidirect
4. #20298973C2semidirect
5. #20298851C2semidirect
6. #20298848C2semidirect
7. #20298796C2semidirect
8. #20298791C2semidirect
9. #20298743C2semidirect
10. #20298742C2semidirect
11. #20298732C2semidirect
12. #20298730C2semidirect
13. SD16×Dic113C2semidirect
14. #20298674C2semidirect
15. C2×AGL(1,ℤ/8)×C113C2semidirect
16. C2×#14268648C22semidirect
20. C2×#14268634C22semidirect
21. C2×D904C22semidirect
23. C2×#14268632C22semidirect
24. C2×#14268632C22semidirect
25. C2×#14268630C22semidirect
28. #16813618C22semidirect
31. #16813596C22semidirect
35. #16813593C22semidirect
38. Q8×Dic113C22semidirect
42. #16813554C22semidirect
43. #16813551C22semidirect
44. #16813546C22semidirect
45. #16813544C22semidirect
46. #16813544C22semidirect
48. #16813543C22semidirect
49. #16813543C22semidirect
50. C22⋊#11744751C22semidirect
51. #16813534C22semidirect
52. C8×Dic113C22semidirect
53. C2×#41×C113C22semidirect
57. C2×SD16×C113C22semidirect
59. C2×D8×C113C22semidirect
60. C2×M16×C113C22semidirect
118. SD16×C113D4semidirect
138. Q8×C113C2×D4semidirect
178. C8×C113C2×D4semidirect
213. Dic113AGL(1,ℤ/8)semidirect
225. C113#3274semidirect
227. C2×AGL(1,ℤ/8)D113semidirect
242. C2×#41C2×D113semidirect
250. C2×SD16C2×D113semidirect
256. C2×D8C2×D113semidirect
258. C2×M16C2×D113semidirect
323. SD16C22⋊D113semidirect
363. Q8C2×C22⋊D113semidirect
405. C8C2×C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136492
#202992672
D4×C22⋊D1132
#202989732
#202988512
#202988482
#202987962
#202987912
#202987432
#202987422
#202987322
#202987302
SD16×Dic1132
#202986742
C2×AGL(1,ℤ/8)×C1132
#3274113