Finite Group Databasebeta

← #28402072

Subgroups — #28402072

Order 14464.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813647C211yesmaximal, characteristic, complemented
2C2×SD16×D113C211yesmaximal, characteristic, complemented
3#20299255C211yesmaximal, characteristic, complemented
4#20299232C211yesmaximal, characteristic, complemented
5#20298986C211yesmaximal, characteristic, complemented
6#20298842C211yesmaximal, characteristic, complemented
7#20298833C211yesmaximal, characteristic, complemented
8#20298812C211yesmaximal, characteristic, complemented
9#20298796C211yesmaximal, characteristic, complemented
10#20298751C211yesmaximal, characteristic, complemented
11#20298738C211yesmaximal, characteristic, complemented
12#20298730C211yesmaximal, characteristic, complemented
13#20298701C211yesmaximal, characteristic, complemented
14#20298672C211yesmaximal, characteristic, complemented
15#449×C113C211yesmaximal, characteristic, complemented
16C2×Q8×D113C2211yescharacteristic, complemented
17C22×D452C2211yescharacteristic
18C2×D4×D113—21no—
19C2×D4×D113C2211yescharacteristic
20#16813647—24no—
21C2×#14268636C2211yescharacteristic
22C2×#14268636C2211yescharacteristic, complemented
23C2×#14268635C2211yescharacteristic, complemented
24C2×#14268634C2211yescharacteristic
25C2×#14268634C2211yescharacteristic, complemented
26SD16×D113—24no—
27C2×#14268632C2211yescharacteristic
28C2×#14268632C2211yescharacteristic, complemented
29C2×#14268631C2211yescharacteristic, complemented
30C2×C8×D113C2211yescharacteristic, complemented
31#16813618C2211yescharacteristic
32#16813609—21no—
33#16813609C2211yescharacteristic, complemented
34#16813601—21no—
35#16813601C2211yescharacteristic, complemented
36#16813597C2211yescharacteristic
37#16813595—21no—
38#16813593—21no—
39#16813593C2211yescharacteristic, complemented
40#16813592—21no—
41#16813590—21no—
42Q8×Dic113C2211yescharacteristic, complemented
43#16813584C2211yescharacteristic
44C4×C22⋊D113—21no—
45C4×C22⋊D113C2211yescharacteristic
46#16813555C2211yescharacteristic, complemented
47#16813551C2211yescharacteristic, complemented
48#16813545C2211yescharacteristic, complemented
49#16813544C2211yescharacteristic
50#16813544C2211yescharacteristic, complemented
51#16813543C2211yescharacteristic, complemented
52#16813534C2211yescharacteristic, complemented
53#16813532C2211yescharacteristic, complemented
54C2×#41×C113C2211yescharacteristic, complemented
55C2×SD16×C113C2211yescharacteristic, complemented
56C2×D8×C113C2211yescharacteristic, complemented
57#121×C113C2211yescharacteristic
58#103×C113C2211yescharacteristic, complemented
59#102×C113C2211yescharacteristic, complemented
60C22⋊C8×C113C2211yescharacteristic, complemented
61C23×D113—21no—
62C23×D113—21no—
63C2×Dic226—21no—
64C2×Dic226C2311yescharacteristic
65Q8×D113—22no—
66Q8×D113—41no—
67C2×D452—22no—
68C2×D452—21no—
69C2×D452—21no—
70C2×D452—41no—
71C2×D452C2311yescharacteristic
72C2×D452C2311yescharacteristic
73C2×C22⋊D113—21no—
74C2×C22⋊D113—21no—
75C2×C22⋊D113—21no—
76C2×C22⋊D113—21no—
77C2×C22⋊D113D411yescharacteristic
78D4×D113—22no—
79D4×D113—41no—
80D4×D113—44no—
81C2×C4×D113—21no—
82C2×C4×D113—21no—
83C2×C4×D113C2311yescharacteristic
84#14268636—22no—
85#14268636—41no—
86#14268635—22no—
87#14268634—22no—
88#14268634D414yescomplemented
89#14268632—22no—
90#14268632—22no—
91#14268631—22no—
92C2×#11744751C2311yescharacteristic
93C2×#11744751C2311yescharacteristic
94C8×D113—22no—
95C4⋊Dic113—21no—
96C4⋊Dic113C2311yescharacteristic
97#14268626—21no—
98#14268626D411yescharacteristic
99C22⋊Dic113—21no—
100C22⋊Dic113—21no—
101C22⋊Dic113—21no—
102C22⋊Dic113—21no—
103C22⋊Dic113D411yescharacteristic
104#14268624—21no—
105#14268624—21no—
106#14268624D411yescharacteristic
107C4×Dic113—21no—
108C4×Dic113C2311yescharacteristic
109#41×C113—22no—
110#41×C113—41no—
111C2×Q8×C113C2311yescharacteristic
112C2×D4×C113—21no—
113C2×D4×C113—21no—
114C2×D4×C113C2311yescharacteristic
115C2×D4×C113C2311yescharacteristic
116C22×C4×C113—21no—
117C22×C4×C113C2311yescharacteristic
118SD16×C113—22no—
119D8×C113—41no—
120C2×C8×C113C2311yescharacteristic
121C2×C8×C113C2311yescharacteristic
122C4⋊C4×C113C2311yescharacteristic
123C22⋊C4×C113—21no—
124C23×C113—21no—
125C23×C113—21no—
126C23×C113C2×D411yescharacteristic
127C22×D113—21no—
128C22×D113—21no—
129C22×D113—42no—
130C22×D113—42no—
131C22×D113—42no—
132C22×D113—42no—
133C22×D113—81no—
134C22×D113—81no—
135C22×D113C2×D411yescharacteristic
136Q8×C113—21no—
137Q8×C113C2×D412yescomplemented
138D4×C113—21no—
139D4×C113—21no—
140D4×C113—21no—
141D4×C113—21no—
142D4×C113—42no—
143D4×C113—42no—
144C2×C4×C113—21no—
145C2×C4×C113—21no—
146C2×C4×C113—21no—
147C2×C4×C113—21no—
148C2×C4×C113—42no—
149C2×C4×C113C2×D411yescharacteristic
150C2×C4×C113C2411yescharacteristic
151C22⋊D113—22no—
152C22⋊D113—42no—
153C22⋊D113—42no—
154C22⋊D113—42no—
155C22⋊D113—81no—
156C2×Dic113—21no—
157C2×Dic113—21no—
158C2×Dic113C2×D411yescharacteristic
159C2×Dic113C2×D411yescharacteristic
160C2×Dic113C2×D411yescharacteristic
161D452—21no—
162D452—21no—
163D452—21no—
164D452—42no—
165D452—42no—
166D452C2×D412yes—
167C4×D113—21no—
168C4×D113—21no—
169C4×D113—42no—
170C4×D113—42no—
171Dic226—21no—
172Dic226—21no—
173Dic226—42no—
174C8×C113—21no—
175C8×C113—21no—
176#11744751—21no—
177#11744751C2×D412yescomplemented
178C22×C113—21no—
179C22×C113—21no—
180C22×C113—21no—
181C22×C113—41no—
182C22×C113—41no—
183C22×C113—41no—
184C22×C113—41no—
185C22×C113—41no—
186C22×C113—41no—
187C22×C113C22×D411yescharacteristic
188C2×D113—21no—
189C2×D113—21no—
190C2×D113—22no—
191C2×D113—41no—
192C2×D113—41no—
193C2×D113—41no—
194C2×D113—41no—
195C2×D113—41no—
196C2×D113—84no—
197C2×D113—84no—
198C2×D113AGL(1,ℤ/8)12yescomplemented
199C4×C113—22no—
200C4×C113—21no—
201C4×C113—41no—
202C4×C113C22×D411yescharacteristic
203C4×C113H(5,𝔽2)11yescharacteristic
204Dic113—21no—
205Dic113—21no—
206Dic113—21no—
207Dic113—41no—
208Dic113—41no—
209C2×C113—41no—
210C2×C113—81no—
211C2×C113—81no—
212C2×C113D4211yescharacteristic
213C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C113#57711yescharacteristic
215D113—22no—
216D113—42no—
217D113—81no—
218#3265—1131nomaximal, Sylow
219C113#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1131no—
221C22×SD16—1131no—
222#548—1131no—
223D42—1131no—
224#496—1131no—
225#468—1131no—
226#466—1131no—
227#463—1131no—
228#460—1131no—
229#450—1131no—
230#449D11311yescharacteristic, complemented
231#448—1131no—
232#447—1131no—
233#440—1131no—
234#435—1131no—
235C2×#41C2×D11311yescharacteristic, complemented
236C22×Q8—1131no—
237C22×D4—1131no—
238C22×D4—1131no—
239C22×D4—2261no—
240AGL(1,ℤ/8)—2264no—
241C2×SD16—1131no—
242C2×SD16—1131no—
243C2×SD16—1131no—
244C2×SD16—1131no—
245C2×SD16—1131no—
246C2×SD16—2264no—
247C2×SD16C2×D11311yescharacteristic, complemented
248C2×D8—1131no—
249C2×D8—1131no—
250C2×D8C2×D11311yescharacteristic, complemented
251C2×M16—1131no—
252C22×C8—1131no—
253#127—1131no—
254#124—1131no—
255#124—2261no—
256C22⋊Q8—1131no—
257C22⋊Q8—2261no—
258#121—1131no—
259#121—1131no—
260#121—2261no—
261#121—2261no—
262#121C2×D11311yescharacteristic
263#120—2261no—
264#120—2261no—
265C4×Q8—1131no—
266C4×D4—1131no—
267C4×D4—1131no—
268C4×D4—2261no—
269#107—1131no—
270C4⋊C8—1131no—
271#103—1131no—
272#103C2×D11311yescharacteristic, complemented
273#102—1131no—
274#102—1131no—
275#102—1131no—
276#102C2×D11311yescharacteristic, complemented
277C22⋊C8—1131no—
278C22⋊C8C2×D11311yescharacteristic, complemented
279#97—1131no—
280C24—2261no—
281C24—2261no—
282#41—22no—
283#41—41no—
284C2×Q8—1131no—
285C2×Q8—4521no—
286C2×Q8—2261no—
287C2×Q8—2262no—
288C2×Q8C22×D11311yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—1131no—
292C2×D4—1131no—
293C2×D4—1131no—
294C2×D4—4521no—
295C2×D4—4524no—
296C2×D4—4521no—
297C2×D4—2262no—
298C2×D4—2261no—
299C2×D4—2261no—
300C2×D4—2261no—
301C2×D4—2261no—
302C2×D4—2262no—
303C2×D4—2261no—
304C2×D4—2261no—
305C2×D4C22×D11311yescharacteristic
306C2×D4C22×D11311yescharacteristic
307C22×C4—21no—
308C22×C4—1131no—
309C22×C4—2261no—
310C22×C4—2261no—
311C22×C4C22×D11311yescharacteristic
312SD16—22no—
313SD16—1134no—
314SD16—4521no—
315SD16—2262no—
316SD16—2262no—
317SD16—2262no—
318D8—41no—
319D8—2262no—
320D8—2262no—
321M16—2262no—
322C2×C8—1131no—
323C2×C8—1131no—
324C2×C8—2262no—
325C2×C8C22×D11311yescharacteristic
326C2×C8C22×D11311yescharacteristic
327C4⋊C4—1131no—
328C4⋊C4—1131no—
329C4⋊C4—2261no—
330C4⋊C4—2261no—
331C4⋊C4C22×D11311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—1131no—
334C22⋊C4—1131no—
335C22⋊C4—2261no—
336C22⋊C4—2261no—
337C22⋊C4—2261no—
338C22⋊C4—2261no—
339C22⋊C4—2261no—
340C22⋊C4—2261no—
341C42—1131no—
342C42—2261no—
343C23—21no—
344C23—21no—
345C23—1131no—
346C23—9041no—
347C23—9041no—
348C23—4522no—
349C23—4522no—
350C23—4522no—
351C23—4522no—
352C23—2261no—
353C23—2261no—
354C23D4×D11311yescharacteristic
355Q8—21no—
356Q8—4522no—
357Q8—2261no—
358Q8—2261no—
359Q8D4×D11312yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—1132no—
367D4—9041no—
368D4—4522no—
369D4—4522no—
370D4—4522no—
371D4—4522no—
372D4—4522no—
373D4—2262no—
374D4—2261no—
375D4—2261no—
376D4—2261no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—1131no—
383C2×C4—1131no—
384C2×C4—1131no—
385C2×C4—4522no—
386C2×C4—4522no—
387C2×C4—2261no—
388C2×C4—2261no—
389C2×C4—2261no—
390C2×C4—2261no—
391C2×C4D4×D11311yescharacteristic
392C2×C4C23×D11311yescharacteristic
393C8—21no—
394C8—21no—
395C8—1132no—
396C8—2261no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—1132no—
407C22—9044no—
408C22—9044no—
409C22—4521no—
410C22—4521no—
411C22—4521no—
412C22—4521no—
413C22—4521no—
414C22—2262no—
415C22—2261no—
416C22—2261no—
417C22C2×D4×D11311yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—4521no—
422C4—4521no—
423C4—2261no—
424C4—2261no—
425C4—2261no—
426C4C2×D4×D11311yescharacteristic
427C4#1681366711yescharacteristic
428C2—41no—
429C2—81no—
430C2—81no—
431C2—9041no—
432C2—4522no—
433C2—2262no—
434C2#2029923011yescharacteristic
435C2AGL(1,ℤ/8)×D11311yescharacteristic
436C2#2029940511yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#449×C113C2
socleC22×C113C22×D4
hypercenter#449D113
hypocenterC113#3265
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×#16813647C2semidirect
2. C2×SD16×D113C2semidirect
3. #20299255C2semidirect
4. #20299232C2semidirect
5. #20298986C2semidirect
6. #20298842C2semidirect
7. #20298833C2semidirect
8. #20298812C2semidirect
9. #20298796C2semidirect
10. #20298751C2semidirect
11. #20298738C2semidirect
12. #20298730C2semidirect
13. #20298701C2semidirect
14. #20298672C2semidirect
15. #449×C113C2semidirect
16. C2×Q8×D113C22semidirect
22. C2×#14268636C22semidirect
23. C2×#14268635C22semidirect
25. C2×#14268634C22semidirect
28. C2×#14268632C22semidirect
29. C2×#14268631C22semidirect
30. C2×C8×D113C22semidirect
33. #16813609C22semidirect
35. #16813601C22semidirect
39. #16813593C22semidirect
42. Q8×Dic113C22semidirect
46. #16813555C22semidirect
47. #16813551C22semidirect
48. #16813545C22semidirect
50. #16813544C22semidirect
51. #16813543C22semidirect
52. #16813534C22semidirect
53. #16813532C22semidirect
54. C2×#41×C113C22semidirect
55. C2×SD16×C113C22semidirect
56. C2×D8×C113C22semidirect
58. #103×C113C22semidirect
59. #102×C113C22semidirect
60. C22⋊C8×C113C22semidirect
88. #14268634D4semidirect
137. Q8×C113C2×D4semidirect
177. #11744751C2×D4semidirect
198. C2×D113AGL(1,ℤ/8)semidirect
219. C113#3265semidirect
230. #449D113semidirect
235. C2×#41C2×D113semidirect
247. C2×SD16C2×D113semidirect
250. C2×D8C2×D113semidirect
272. #103C2×D113semidirect
276. #102C2×D113semidirect
278. C22⋊C8C2×D113semidirect
359. Q8D4×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136472
C2×SD16×D1132
#202992552
#202992322
#202989862
#202988422
#202988332
#202988122
#202987962
#202987512
#202987382
#202987302
#202987012
#202986722
#449×C1132
#3265113