Finite Group Databasebeta

← #44409102

Subgroups — #44409102

Order 29056.

736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28409867C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D227C212yesmaximal, complemented
3#28409595C211yesmaximal, characteristic, complemented
4#28409206C212yesmaximal, complemented
5#28409203C212yesmaximal, complemented
6#28409202C211yesmaximal, characteristic, complemented
7#28409200C211yesmaximal, characteristic, complemented
8#28409199C211yesmaximal, characteristic, complemented
9#28409047C212yesmaximal, complemented
10#28409013C211yesmaximal, characteristic, complemented
11#453×C227C211yesmaximal, characteristic, complemented
12#20300335—22no—
13#20300335C2212yescomplemented
14#20300333—22no—
15#20300333—22no—
16#20300333C2211yescharacteristic, complemented
17C2×#16813883—22no—
18C2×#16813883—22no—
19C2×#16813883C2211yescharacteristic
20#41×D227—22no—
21#41×D227C2212yes—
22C2×D4×D227C2211yescharacteristic, complemented
23#20300316C2212yescomplemented
24#20300315—22no—
25#20300314C2212yescomplemented
26#20300313C2212yescomplemented
27#20300312—22no—
28SD16×D227—22no—
29D8×D227—22no—
30M16×D227C2212yescomplemented
31#20300284C2211yescharacteristic, complemented
32#20300276C2211yescharacteristic, complemented
33#20300275C2211yescharacteristic, complemented
34#20300260—22no—
35#20300256—22no—
36#20300245C2211yescharacteristic, complemented
37#20300216C2212yescomplemented
38#20300215C2212yescomplemented
39#20300214C2212yescomplemented
40#20300206C2212yescomplemented
41#20300205C2211yescharacteristic, complemented
42H(5,𝔽2)×C227C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C227C2212yescomplemented
44#127×C227C2211yescharacteristic, complemented
45#104×C227C2212yescomplemented
46#100×C227C2211yescharacteristic, complemented
47C23×D227—21no—
48#16813885D412yescomplemented
49#16813884—24no—
50#16813884—22no—
51#16813884—22no—
52#16813884C2312yes—
53#16813883—24no—
54#16813883—24no—
55#16813883—24no—
56#16813883—22no—
57#16813883—22no—
58#16813883—24no—
59#16813883—22no—
60#16813883—22no—
61#16813883—22no—
62#16813883D412yescomplemented
63C2×Dic454—22no—
64C2×Dic454—22no—
65C2×Dic454C2311yescharacteristic
66Q8×D227—22no—
67Q8×D227D412yescomplemented
68C2×D908C2311yescharacteristic
69C2×C22⋊D227—24no—
70C2×C22⋊D227—22no—
71C2×C22⋊D227—21no—
72C2×C22⋊D227—22no—
73C2×C22⋊D227—22no—
74C2×C22⋊D227D412yes—
75D4×D227—24no—
76D4×D227—22no—
77D4×D227—22no—
78D4×D227—22no—
79D4×D227—22no—
80D4×D227D412yescomplemented
81C22×Dic227—22no—
82C22×Dic227D412yes—
83C2×C4×D227—22no—
84C2×C4×D227—22no—
85C2×C4×D227C2311yescharacteristic
86#16813873—22no—
87#16813872—22no—
88#16813871—22no—
89D1816—22no—
90#16813869—22no—
91#16813869—22no—
92#16813868C2312yes—
93#16813867—22no—
94C8×D227—22no—
95Dic227⋊C4—21no—
96C22⋊Dic227—22no—
97(C2×D227)⋊C4—21no—
98(C2×D227)⋊C4—22no—
99C4×Dic227C2311yescharacteristic
100#41×C227—22no—
101#41×C227C2312yes—
102C2×D4×C227—22no—
103C2×D4×C227—22no—
104C2×D4×C227—22no—
105C2×D4×C227C2311yescharacteristic
106C2×D4×C227C2311yescharacteristic
107SD16×C227—22no—
108D8×C227—22no—
109M16×C227C2312yes—
110C42×C227C2311yescharacteristic
111C22×D227—22no—
112C22×D227—22no—
113C22×D227—21no—
114C22×D227—42no—
115C22×D227—82no—
116C22×D227C2×D411yescharacteristic
117Dic454—24no—
118Dic454—22no—
119Dic454—22no—
120Dic454C2×D412yes—
121D908—22no—
122D908—22no—
123D908C2×D412yes—
124C22⋊D227—24no—
125C22⋊D227—24no—
126C22⋊D227—22no—
127C22⋊D227—22no—
128C22⋊D227—24no—
129C22⋊D227—22no—
130C22⋊D227—41no—
131C22⋊D227—41no—
132C22⋊D227—82no—
133C2×Dic227—22no—
134C2×Dic227—24no—
135C2×Dic227—22no—
136C2×Dic227—22no—
137C2×Dic227—22no—
138C2×Dic227—21no—
139C2×Dic227C2×D412yes—
140C2×Dic227C2×D411yescharacteristic
141C4×D227—24no—
142C4×D227—22no—
143C4×D227—22no—
144C4×D227—22no—
145C4×D227C2×D412yes—
146C4×D227C2×D412yes—
147C227⋊C8—22no—
148C23×C227—22no—
149C23×C227—21no—
150C23×C227C2×D412yes—
151Q8×C227C2×D412yes—
152D4×C227—24no—
153D4×C227—22no—
154D4×C227—22no—
155D4×C227—22no—
156D4×C227—82no—
157D4×C227C2×D412yes—
158C2×C4×C227—22no—
159C2×C4×C227—22no—
160C2×C4×C227—21no—
161C2×C4×C227C2411yescharacteristic
162C8×C227—22no—
163C22×C227—22no—
164C22×C227—22no—
165C22×C227—42no—
166C22×C227—42no—
167C22×C227—41no—
168C22×C227—82no—
169C22×C227C22×D411yescharacteristic
170C2×D227—22no—
171C2×D227—41no—
172C2×D227—42no—
173C2×D227—41no—
174C2×D227—82no—
175C2×D227—84no—
176C2×D227—82no—
177C2×D227#12012yes—
178C4×C227—22no—
179C4×C227—41no—
180C4×C227C22×D412yes—
181Dic227—22no—
182Dic227—22no—
183Dic227—41no—
184Dic227#12011yescharacteristic
185Dic227#12011yescharacteristic
186C2×C227—21no—
187C2×C227—42no—
188C2×C227—42no—
189C2×C227—81no—
190C2×C227C2×#12011yescharacteristic
191D227—22no—
192D227—42no—
193D227—81no—
194C227#302111yescharacteristic, complemented, Sylow
195#3021—2271nomaximal, Sylow
196#585—2271no—
197C2×AGL(1,ℤ/8)—2272no—
198#535—2271no—
199#454—2272no—
200#454—2272no—
201#453—2271no—
202#453—2271no—
203#453—2271no—
204#453D22711yescharacteristic, complemented
205#421—2272no—
206#413—2271no—
2072−1+4—4542no—
2082−1+4—2272no—
209H(5,𝔽2)—4542no—
210H(5,𝔽2)—4542no—
211H(5,𝔽2)—2271no—
212H(5,𝔽2)C2×D22711yescharacteristic, complemented
213C2×#41—4542no—
214C2×#41—4542no—
215C2×#41—4542no—
216C2×#41—2272no—
217C2×#41—2271no—
218C22×D4—2271no—
219AGL(1,ℤ/8)—4542no—
220AGL(1,ℤ/8)—4542no—
221AGL(1,ℤ/8)—2272no—
222AGL(1,ℤ/8)—2272no—
223AGL(1,ℤ/8)—2272no—
224AGL(1,ℤ/8)C2×D22712yescomplemented
225C2×SD16—4542no—
226C2×D8—4542no—
227C2×M16—2272no—
228#127—2271no—
229#127C2×D22711yescharacteristic, complemented
230#124—2271no—
231#124—2271no—
232#121—4542no—
233#120—4542no—
234#117—2271no—
235#104—2272no—
236#104—2272no—
237#104—2272no—
238#104C2×D22712yescomplemented
239#101—2272no—
240#100—2271no—
241#100C2×D22711yescharacteristic, complemented
242C24—4541no—
243#41—22no—
244#41—4544no—
245#41—4542no—
246#41—4542no—
247#41—4544no—
248#41—4544no—
249#41—4544no—
250#41—4542no—
251#41—4542no—
252#41—4542no—
253#41—4542no—
254#41—4544no—
255#41—4542no—
256#41—2272no—
257#41—2272no—
258#41—2272no—
259#41C22×D22712yes—
260C2×Q8—4542no—
261C2×Q8—4542no—
262C2×Q8—4542no—
263C2×Q8—2272no—
264C2×Q8—2271no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—4544no—
269C2×D4—4542no—
270C2×D4—4542no—
271C2×D4—4542no—
272C2×D4—4544no—
273C2×D4—4542no—
274C2×D4—4542no—
275C2×D4—4542no—
276C2×D4—4542no—
277C2×D4—4541no—
278C2×D4—2272no—
279C2×D4—2272no—
280C2×D4—2271no—
281C2×D4C22×D22711yescharacteristic
282C2×D4C22×D22711yescharacteristic
283C22×C4—4542no—
284C22×C4—4542no—
285C22×C4—4542no—
286C22×C4—2272no—
287C22×C4—2271no—
288SD16—22no—
289SD16—4542no—
290SD16—4542no—
291SD16—4542no—
292D8—22no—
293D8—4542no—
294D8—4542no—
295D8—4542no—
296M16—4542no—
297M16—2272no—
298M16C22×D22712yes—
299C2×C8—4542no—
300C4⋊C4—4541no—
301C22⋊C4—4541no—
302C22⋊C4—4542no—
303C22⋊C4—4542no—
304C42—2271no—
305C42C22×D22711yescharacteristic
306C23—22no—
307C23—21no—
308C23—18162no—
309C23—9082no—
310C23—4542no—
311C23—4542no—
312C23—4541no—
313C23—2271no—
314C23D4×D22712yes—
315Q8—4544no—
316Q8—4542no—
317Q8—4542no—
318Q8—2272no—
319Q8D4×D22712yes—
320D4—24no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—82no—
325D4—18162no—
326D4—9081no—
327D4—9081no—
328D4—4542no—
329D4—4544no—
330D4—4544no—
331D4—4542no—
332D4—4544no—
333D4—4542no—
334D4—4542no—
335D4—4542no—
336D4—2272no—
337D4D4×D22712yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—4544no—
342C2×C4—4542no—
343C2×C4—4542no—
344C2×C4—4541no—
345C2×C4—4542no—
346C2×C4—4544no—
347C2×C4—4542no—
348C2×C4—4542no—
349C2×C4—4542no—
350C2×C4—4542no—
351C2×C4—2272no—
352C2×C4—2271no—
353C2×C4—2272no—
354C2×C4—2272no—
355C2×C4C23×D22711yescharacteristic
356C8—22no—
357C8—4542no—
358C22—22no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—82no—
364C22—18162no—
365C22—18162no—
366C22—18164no—
367C22—9082no—
368C22—9081no—
369C22—9081no—
370C22—4542no—
371C22—2272no—
372C22C2×D4×D22711yescharacteristic
373C4—22no—
374C4—41no—
375C4—9081no—
376C4—4542no—
377C4—4542no—
378C4—2271no—
379C4—2271no—
380C4C2×D4×D22712yes—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—81no—
385C2—18161no—
386C2—9082no—
387C2—4542no—
388C2#120×D22711yescharacteristic
subgroupG/H
centerC2#120×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#453×C227C2
socleC2×C227C2×#120
hypercenter#453D227
hypocenterC227#3021
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #28409867C2semidirect
2. AGL(1,ℤ/8)×D227C2semidirect
3. #28409595C2semidirect
4. #28409206C2semidirect
5. #28409203C2semidirect
6. #28409202C2semidirect
7. #28409200C2semidirect
8. #28409199C2semidirect
9. #28409047C2semidirect
10. #28409013C2semidirect
11. #453×C227C2semidirect
13. #20300335C22semidirect
16. #20300333C22semidirect
22. C2×D4×D227C22semidirect
23. #20300316C22semidirect
25. #20300314C22semidirect
26. #20300313C22semidirect
30. M16×D227C22semidirect
31. #20300284C22semidirect
32. #20300276C22semidirect
33. #20300275C22semidirect
36. #20300245C22semidirect
37. #20300216C22semidirect
38. #20300215C22semidirect
39. #20300214C22semidirect
40. #20300206C22semidirect
41. #20300205C22semidirect
42. H(5,𝔽2)×C227C22semidirect
43. AGL(1,ℤ/8)×C227C22semidirect
44. #127×C227C22semidirect
45. #104×C227C22semidirect
46. #100×C227C22semidirect
48. #16813885D4semidirect
62. #16813883D4semidirect
67. Q8×D227D4semidirect
80. D4×D227D4semidirect
194. C227#3021semidirect
204. #453D227semidirect
212. H(5,𝔽2)C2×D227semidirect
224. AGL(1,ℤ/8)C2×D227semidirect
229. #127C2×D227semidirect
238. #104C2×D227semidirect
241. #100C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284098672
AGL(1,ℤ/8)×D2272
#284095952
#284092062
#284092032
#284092022
#284092002
#284091992
#284090472
#284090132
#453×C2272
#3021227