Finite Group Databasebeta

← #28402205

Subgroups — #28402205

Order 14464.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813651C211yesmaximal, characteristic, complemented
2#20299255C211yesmaximal, characteristic, complemented
3#20299242C211yesmaximal, characteristic, complemented
4#20298981C211yesmaximal, characteristic, complemented
5#20298856C211yesmaximal, characteristic, complemented
6#20298848C211yesmaximal, characteristic, complemented
7#20298818C211yesmaximal, characteristic, complemented
8#20298811C211yesmaximal, characteristic, complemented
9#20298760C211yesmaximal, characteristic, complemented
10#20298749C211yesmaximal, characteristic, complemented
11#20298740C211yesmaximal, characteristic, complemented
12#20298735C211yesmaximal, characteristic, complemented
13SD16×Dic113C211yesmaximal, characteristic, complemented
14#20298674C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
16C2×#14268646C2211yescharacteristic
17C2×Q8×D113C2211yescharacteristic, complemented
18#16813651—24no—
19C2×Dic452C2211yescharacteristic, complemented
20C2×#14268637C2211yescharacteristic, complemented
21C2×#14268635C2211yescharacteristic
22C2×#14268635C2211yescharacteristic
23C2×#14268635C2211yescharacteristic, complemented
24C2×#14268630C2211yescharacteristic, complemented
25#16813609—21no—
26#16813609C2211yescharacteristic
27#16813609C2211yescharacteristic, complemented
28#16813603—21no—
29#16813601—21no—
30#16813601C2211yescharacteristic
31#16813601C2211yescharacteristic
32#16813594—21no—
33#16813593—21no—
34#16813593C2211yescharacteristic
35#16813593C2211yescharacteristic, complemented
36#16813591—21no—
37Q8×Dic113C2211yescharacteristic, complemented
38D4×Dic113C2211yescharacteristic
39C4×C22⋊D113—21no—
40C4×C22⋊D113C2211yescharacteristic
41C2×C22⋊Dic113—21no—
42#16813554C2211yescharacteristic, complemented
43#16813551C2211yescharacteristic, complemented
44#16813547C2211yescharacteristic, complemented
45#16813547C2211yescharacteristic, complemented
46#16813546C2211yescharacteristic, complemented
47#16813543C2211yescharacteristic
48#16813543C2211yescharacteristic
49#16813543C2211yescharacteristic, complemented
50C22⋊#11744751C2211yescharacteristic
51#16813534C2211yescharacteristic, complemented
52C8×Dic113C2211yescharacteristic, complemented
53C2×#41×C113C2211yescharacteristic
54C22×D4×C113C2211yescharacteristic
55AGL(1,ℤ/8)×C113—24no—
56C2×SD16×C113C2211yescharacteristic
57C2×SD16×C113C2211yescharacteristic
58C2×D8×C113C2211yescharacteristic
59C2×D8×C113C2211yescharacteristic
60C2×M16×C113C2211yescharacteristic
61#14268646—22no—
62#14268646—41no—
63C2×Dic226—21no—
64C2×Dic226C2311yescharacteristic
65C2×Dic226C2311yescharacteristic
66Q8×D113—22no—
67Q8×D113—41no—
68C2×C22⋊D113—21no—
69C2×C22⋊D113—21no—
70C2×C22⋊D113—21no—
71C2×C22⋊D113D411yescharacteristic
72C22×Dic113—21no—
73C22×Dic113—21no—
74C2×C4×D113—21no—
75C2×C4×D113C2311yescharacteristic
76Dic452—22no—
77#14268637—22no—
78#14268635—22no—
79#14268635—41no—
80#14268635—41no—
81#14268630—22no—
82C2×#11744751C2311yescharacteristic
83C2×#11744751C2311yescharacteristic
84C4⋊Dic113—21no—
85C4⋊Dic113C2311yescharacteristic
86C4⋊Dic113C2311yescharacteristic
87#14268626—21no—
88#14268626—21no—
89#14268626D411yescharacteristic
90C22⋊Dic113—21no—
91C22⋊Dic113—21no—
92C22⋊Dic113—21no—
93C22⋊Dic113—21no—
94C22⋊Dic113—21no—
95C22⋊Dic113—21no—
96C22⋊Dic113—21no—
97C22⋊Dic113—21no—
98C22⋊Dic113—21no—
99C22⋊Dic113D411yescharacteristic
100#14268624D411yescharacteristic
101C4×Dic113—21no—
102C4×Dic113C2311yescharacteristic
103C24×C113—21no—
104#41×C113—22no—
105#41×C113—41no—
106C2×Q8×C113C2311yescharacteristic
107C2×D4×C113—22no—
108C2×D4×C113—21no—
109C2×D4×C113—21no—
110C2×D4×C113—41no—
111C2×D4×C113C2311yescharacteristic
112C2×D4×C113C2311yescharacteristic
113C2×D4×C113C2311yescharacteristic
114C22×C4×C113—21no—
115C22×C4×C113C2311yescharacteristic
116SD16×C113—22no—
117SD16×C113D414yescomplemented
118D8×C113—22no—
119D8×C113—22no—
120M16×C113—22no—
121C2×C8×C113C2311yescharacteristic
122C2×C8×C113C2311yescharacteristic
123C23×C113—21no—
124C23×C113—21no—
125C23×C113—21no—
126C23×C113—42no—
127C23×C113—42no—
128C23×C113—81no—
129C23×C113C2×D411yescharacteristic
130C22×D113C2×D411yescharacteristic
131Q8×C113—21no—
132Q8×C113C2×D412yes—
133D4×C113—21no—
134D4×C113—21no—
135D4×C113—21no—
136D4×C113—21no—
137D4×C113—21no—
138D4×C113—42no—
139D4×C113—42no—
140D4×C113C2×D412yes—
141C2×C4×C113—21no—
142C2×C4×C113—21no—
143C2×C4×C113—21no—
144C2×C4×C113—42no—
145C2×C4×C113C2×D411yescharacteristic
146C2×C4×C113C2411yescharacteristic
147C22⋊D113—41no—
148C22⋊D113—42no—
149C22⋊D113—42no—
150C22⋊D113—81no—
151C2×Dic113—21no—
152C2×Dic113—21no—
153C2×Dic113—21no—
154C2×Dic113—21no—
155C2×Dic113—42no—
156C2×Dic113—81no—
157C2×Dic113C2×D411yescharacteristic
158C2×Dic113C2×D411yescharacteristic
159C2×Dic113C2×D411yescharacteristic
160C4×D113—21no—
161C4×D113—21no—
162C4×D113—42no—
163Dic226—21no—
164Dic226—21no—
165Dic226—21no—
166Dic226—21no—
167Dic226—42no—
168C8×C113—21no—
169C8×C113C2×D412yes—
170#11744751—21no—
171#11744751—21no—
172C22×C113—22no—
173C22×C113—21no—
174C22×C113—21no—
175C22×C113—21no—
176C22×C113—41no—
177C22×C113—41no—
178C22×C113—41no—
179C22×C113—41no—
180C22×C113—41no—
181C22×C113—41no—
182C22×C113—41no—
183C22×C113—41no—
184C22×C113—84no—
185C22×C113C22×D411yescharacteristic
186C2×D113—21no—
187C2×D113—21no—
188C2×D113—21no—
189C4×C113—22no—
190C4×C113—21no—
191C4×C113C22×D411yescharacteristic
192C4×C113H(5,𝔽2)11yescharacteristic
193Dic113—21no—
194Dic113—21no—
195Dic113—21no—
196Dic113—41no—
197Dic113—41no—
198Dic113—41no—
199Dic113—41no—
200C2×C113—41no—
201C2×C113—42no—
202C2×C113—81no—
203C2×C113—81no—
204C2×C113D4211yescharacteristic
205C2×C113#57511yescharacteristic
206C2×C113#57711yescharacteristic
207D113—41no—
208#3275—1131nomaximal, Sylow
209C113#327511yescharacteristic, complemented, Sylow
210C2×#137—1131no—
211C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
212#548—1131no—
213#546—1131no—
214#495—1131no—
215#470—1131no—
216#469—1131no—
217#464—1131no—
218#463—1131no—
219#452—1131no—
220#450—1131no—
221#449—1131no—
222#448—1131no—
223C4×SD16—1131no—
224#436—1131no—
225C2×#41—1131no—
226C2×#41C2×D11311yescharacteristic
227C22×Q8—1131no—
228C22×D4C2×D11311yescharacteristic
229#137—2264no—
230AGL(1,ℤ/8)—24no—
231C2×Q16—1131no—
232C2×Q16—1131no—
233C2×SD16—1131no—
234C2×SD16—1131no—
235C2×SD16—1131no—
236C2×SD16C2×D11311yescharacteristic
237C2×SD16C2×D11311yescharacteristic
238C2×D8C2×D11311yescharacteristic
239C2×D8C2×D11311yescharacteristic
240C2×M16—1131no—
241C2×M16C2×D11311yescharacteristic
242#124—1131no—
243#124—1131no—
244#124—2261no—
245#123—2261no—
246C22⋊Q8—1131no—
247C22⋊Q8—1131no—
248C22⋊Q8—2261no—
249#121—1131no—
250#121—1131no—
251#121—2261no—
252#121—2261no—
253#120—2261no—
254C4×Q8—1131no—
255C4×D4—1131no—
256C4×D4—1131no—
257C4×D4—2261no—
258C2×C22⋊C4—2261no—
259#106—1131no—
260C4⋊C8—1131no—
261#103—1131no—
262#103—1131no—
263#103—1131no—
264#102—1131no—
265#102—1131no—
266#102—1131no—
267C22⋊C8—1131no—
268C22⋊C8—1131no—
269C4×C8—1131no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—4521no—
274#41—2262no—
275C2×Q8—1131no—
276C2×Q8—1131no—
277C2×Q8—4521no—
278C2×Q8—2261no—
279C2×Q8—2262no—
280C2×Q8C22×D11311yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—1131no—
286C2×D4—2261no—
287C2×D4—2261no—
288C2×D4—2261no—
289C2×D4C22×D11311yescharacteristic
290C2×D4C22×D11311yescharacteristic
291C2×D4C22×D11311yescharacteristic
292C22×C4—21no—
293C22×C4—1131no—
294C22×C4—2261no—
295C22×C4—2261no—
296C22×C4—2261no—
297C22×C4C22×D11311yescharacteristic
298Q16—2262no—
299Q16—2262no—
300SD16—22no—
301SD16—4521no—
302SD16—4521no—
303SD16—2262no—
304SD16C22⋊D11314yescomplemented
305D8—22no—
306D8—22no—
307M16—22no—
308M16—2262no—
309C2×C8—1131no—
310C2×C8—1131no—
311C2×C8C22×D11311yescharacteristic
312C2×C8C22×D11311yescharacteristic
313C4⋊C4—1131no—
314C4⋊C4—1131no—
315C4⋊C4—1131no—
316C4⋊C4—2261no—
317C4⋊C4—2261no—
318C4⋊C4—2261no—
319C22⋊C4—1131no—
320C22⋊C4—1131no—
321C22⋊C4—2261no—
322C22⋊C4—2261no—
323C22⋊C4—2261no—
324C22⋊C4—2261no—
325C22⋊C4—2261no—
326C22⋊C4—2261no—
327C22⋊C4—2261no—
328C22⋊C4—2261no—
329C22⋊C4—2261no—
330C42—1131no—
331C42—2261no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1131no—
339C23D4×D11311yescharacteristic
340Q8—21no—
341Q8—4522no—
342Q8—2261no—
343Q8—2261no—
344Q8—2261no—
345Q8—2261no—
346Q8C2×C22⋊D11312yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—9041no—
355D4—4522no—
356D4—4522no—
357D4—4521no—
358D4C2×C22⋊D11312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—1131no—
364C2×C4—1131no—
365C2×C4—1131no—
366C2×C4—9041no—
367C2×C4—4522no—
368C2×C4—4522no—
369C2×C4—2261no—
370C2×C4—2261no—
371C2×C4—2261no—
372C2×C4—2261no—
373C2×C4—2261no—
374C2×C4—2261no—
375C2×C4D4×D11311yescharacteristic
376C2×C4C23×D11311yescharacteristic
377C8—21no—
378C8—2261no—
379C8—2261no—
380C8C2×C22⋊D11312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—2261no—
395C22—2261no—
396C22—2261no—
397C22C2×D4×D11311yescharacteristic
398C4—21no—
399C4—22no—
400C4—4521no—
401C4—4521no—
402C4—4521no—
403C4—4521no—
404C4—2261no—
405C4—2261no—
406C4—2261no—
407C4C22×C22⋊D11311yescharacteristic
408C4#1681366711yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—4521no—
414C2D4×C22⋊D11311yescharacteristic
415C2#2029939411yescharacteristic
416C2#2029940811yescharacteristic
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#3275
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×#16813651C2semidirect
2. #20299255C2semidirect
3. #20299242C2semidirect
4. #20298981C2semidirect
5. #20298856C2semidirect
6. #20298848C2semidirect
7. #20298818C2semidirect
8. #20298811C2semidirect
9. #20298760C2semidirect
10. #20298749C2semidirect
11. #20298740C2semidirect
12. #20298735C2semidirect
13. SD16×Dic113C2semidirect
14. #20298674C2semidirect
15. C2×AGL(1,ℤ/8)×C113C2semidirect
17. C2×Q8×D113C22semidirect
19. C2×Dic452C22semidirect
20. C2×#14268637C22semidirect
23. C2×#14268635C22semidirect
24. C2×#14268630C22semidirect
27. #16813609C22semidirect
35. #16813593C22semidirect
37. Q8×Dic113C22semidirect
42. #16813554C22semidirect
43. #16813551C22semidirect
44. #16813547C22semidirect
45. #16813547C22semidirect
46. #16813546C22semidirect
49. #16813543C22semidirect
51. #16813534C22semidirect
52. C8×Dic113C22semidirect
117. SD16×C113D4semidirect
209. C113#3275semidirect
211. C2×AGL(1,ℤ/8)D113semidirect
304. SD16C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136512
#202992552
#202992422
#202989812
#202988562
#202988482
#202988182
#202988112
#202987602
#202987492
#202987402
#202987352
SD16×Dic1132
#202986742
C2×AGL(1,ℤ/8)×C1132
#3275113