Finite Group Databasebeta

← #44409104

Subgroups — #44409104

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#28409869C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D227C212yesmaximal, complemented
3#28409600C211yesmaximal, characteristic, complemented
4#28409207C211yesmaximal, characteristic, complemented
5#28409205C211yesmaximal, characteristic, complemented
6#28409204C211yesmaximal, characteristic, complemented
7#28409201C212yesmaximal, complemented
8#28409198C212yesmaximal, complemented
9#28409047C212yesmaximal, complemented
10#28409015C211yesmaximal, characteristic, complemented
11#454×C227C211yesmaximal, characteristic, complemented
12#20300336—22no—
13#20300336C2211yescharacteristic, complemented
14#20300334—22no—
15#20300334—22no—
16#20300334C2212yescomplemented
17C2×#16813885—22no—
18C2×#16813885C2211yescharacteristic
19#41×D227—22no—
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#20300285C2211yescharacteristic, complemented
32#20300284C2211yescharacteristic, complemented
33#20300277C2211yescharacteristic, complemented
34#20300261—22no—
35#20300258—22no—
36#20300245C2211yescharacteristic, complemented
37#20300216C2212yescomplemented
38#20300215C2212yescomplemented
39#20300214C2212yescomplemented
40#20300207C2211yescharacteristic, complemented
41#20300204C2212yescomplemented
422−1+4×C227C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C227C2212yescomplemented
44#124×C227C2211yescharacteristic, complemented
45#104×C227C2212yescomplemented
46#101×C227C2211yescharacteristic, complemented
47C23×D227—21no—
48#16813885—22no—
49#16813885—24no—
50#16813885—22no—
51#16813885—22no—
52#16813885—22no—
53#16813885—22no—
54#16813885D412yescomplemented
55#16813884—24no—
56#16813884—24no—
57#16813884—22no—
58#16813884—22no—
59#16813884—22no—
60#16813884C2312yes—
61#16813883—22no—
62#16813883—22no—
63#16813883D412yescomplemented
64Q8×D227—22no—
65Q8×D227—22no—
66Q8×D227D412yescomplemented
67C2×D908—24no—
68C2×D908—22no—
69C2×D908—22no—
70C2×D908D412yes—
71C2×D908C2311yescharacteristic
72C2×D908C2311yescharacteristic
73C2×C22⋊D227—22no—
74C2×C22⋊D227—21no—
75D4×D227—24no—
76D4×D227—22no—
77D4×D227—24no—
78D4×D227—22no—
79D4×D227—22no—
80D4×D227—22no—
81D4×D227—22no—
82D4×D227D412yescomplemented
83C2×C4×D227—24no—
84C2×C4×D227—22no—
85C2×C4×D227D412yes—
86C2×C4×D227C2311yescharacteristic
87#16813873—22no—
88#16813872—22no—
89#16813871—22no—
90D1816—22no—
91#16813869—22no—
92#16813869—22no—
93#16813868C2312yes—
94#16813867—22no—
95C8×D227—22no—
96Dic227⋊C4—21no—
97(C2×D227)⋊C4—21no—
98(C2×D227)⋊C4—22no—
99C4×Dic227C2311yescharacteristic
100#41×C227—22no—
101#41×C227—22no—
102#41×C227C2312yes—
103C2×Q8×C227—22no—
104C2×Q8×C227C2311yescharacteristic
105C2×D4×C227C2311yescharacteristic
106SD16×C227—22no—
107D8×C227—22no—
108M16×C227C2312yes—
109C22⋊C4×C227—22no—
110C42×C227C2311yescharacteristic
111C22×D227—24no—
112C22×D227—22no—
113C22×D227—21no—
114C22×D227—42no—
115C22×D227—82no—
116C22×D227C2×D412yes—
117C22×D227C2×D411yescharacteristic
118Dic454—22no—
119Dic454—22no—
120Dic454C2×D412yes—
121D908—24no—
122D908—22no—
123D908—22no—
124D908—24no—
125D908—22no—
126D908—22no—
127D908—82no—
128D908C2×D412yes—
129C22⋊D227—24no—
130C22⋊D227—22no—
131C22⋊D227—22no—
132C22⋊D227—22no—
133C22⋊D227—41no—
134C22⋊D227—41no—
135C22⋊D227—82no—
136C2×Dic227—22no—
137C2×Dic227—21no—
138C2×Dic227C2×D411yescharacteristic
139C4×D227—24no—
140C4×D227—24no—
141C4×D227—22no—
142C4×D227—22no—
143C4×D227—22no—
144C4×D227—22no—
145C4×D227—22no—
146C4×D227C2×D412yes—
147C4×D227C2×D412yes—
148C227⋊C8—22no—
149C23×C227—21no—
150Q8×C227—22no—
151Q8×C227—22no—
152Q8×C227C2×D412yes—
153D4×C227—22no—
154D4×C227—22no—
155D4×C227—22no—
156D4×C227C2×D412yes—
157C2×C4×C227—24no—
158C2×C4×C227—22no—
159C2×C4×C227—21no—
160C2×C4×C227C2×D412yes—
161C2×C4×C227C2411yescharacteristic
162C8×C227—22no—
163C22×C227—22no—
164C22×C227—42no—
165C22×C227—41no—
166C22×C227C22×D411yescharacteristic
167C2×D227—22no—
168C2×D227—22no—
169C2×D227—41no—
170C2×D227—42no—
171C2×D227—42no—
172C2×D227—41no—
173C2×D227—82no—
174C2×D227—84no—
175C2×D227—84no—
176C2×D227#12012yes—
177C4×C227—22no—
178C4×C227—22no—
179C4×C227—41no—
180C4×C227C22×D412yes—
181Dic227—22no—
182Dic227—41no—
183Dic227#12011yescharacteristic
184Dic227#12011yescharacteristic
185C2×C227—21no—
186C2×C227—42no—
187C2×C227—81no—
188C2×C227C2×#12011yescharacteristic
189D227—22no—
190D227—42no—
191D227—42no—
192D227—81no—
193C227#302111yescharacteristic, complemented, Sylow
194#3021—2271nomaximal, Sylow
195#585—2271no—
196C2×AGL(1,ℤ/8)—2272no—
197#535—2271no—
198#454—2271no—
199#454—2271no—
200#454—2271no—
201#454D22711yescharacteristic, complemented
202#453—2272no—
203#453—2272no—
204#421—2272no—
205#413—2271no—
2062−1+4—4542no—
2072−1+4—2271no—
2082−1+4C2×D22711yescharacteristic, complemented
209H(5,𝔽2)—4542no—
210H(5,𝔽2)—4542no—
211H(5,𝔽2)—2272no—
212C2×#41—4542no—
213C2×#41—4542no—
214C2×#41—4542no—
215C2×#41—2272no—
216C2×#41—2271no—
217C22×D4—2271no—
218AGL(1,ℤ/8)—4542no—
219AGL(1,ℤ/8)—4542no—
220AGL(1,ℤ/8)—2272no—
221AGL(1,ℤ/8)—2272no—
222AGL(1,ℤ/8)—2272no—
223AGL(1,ℤ/8)C2×D22712yescomplemented
224C2×SD16—4542no—
225C2×D8—4542no—
226C2×M16—2272no—
227#127—2271no—
228#127—2271no—
229#124—2271no—
230#124C2×D22711yescharacteristic, complemented
231#121—4542no—
232#120—4542no—
233#117—2271no—
234#104—2272no—
235#104—2272no—
236#104—2272no—
237#104C2×D22712yescomplemented
238#101—2271no—
239#101C2×D22711yescharacteristic, complemented
240#100—2272no—
241C24—4541no—
242#41—22no—
243#41—22no—
244#41—4544no—
245#41—4544no—
246#41—4542no—
247#41—4542no—
248#41—4542no—
249#41—4542no—
250#41—4544no—
251#41—4542no—
252#41—4542no—
253#41—4542no—
254#41—4542no—
255#41—4542no—
256#41—4542no—
257#41—2272no—
258#41—2272no—
259#41—2272no—
260#41C22×D22712yes—
261C2×Q8—22no—
262C2×Q8—4542no—
263C2×Q8—4542no—
264C2×Q8—2272no—
265C2×Q8C22×D22711yescharacteristic
266C2×D4—4542no—
267C2×D4—4544no—
268C2×D4—4542no—
269C2×D4—4544no—
270C2×D4—4542no—
271C2×D4—4544no—
272C2×D4—4542no—
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×D4—2271no—
282C2×D4C22×D22711yescharacteristic
283C22×C4—4544no—
284C22×C4—4542no—
285C22×C4—2272no—
286C22×C4—2271no—
287SD16—22no—
288SD16—4542no—
289SD16—4542no—
290SD16—4542no—
291D8—22no—
292D8—4542no—
293D8—4542no—
294D8—4542no—
295M16—4542no—
296M16—2272no—
297M16C22×D22712yes—
298C2×C8—4542no—
299C4⋊C4—4541no—
300C22⋊C4—22no—
301C22⋊C4—4541no—
302C22⋊C4—4542no—
303C42—2271no—
304C42C22×D22711yescharacteristic
305C23—21no—
306C23—18162no—
307C23—9082no—
308C23—4544no—
309C23—4542no—
310C23—4541no—
311C23—2272no—
312C23—2271no—
313Q8—22no—
314Q8—22no—
315Q8—4542no—
316Q8—4542no—
317Q8—2272no—
318Q8D4×D22712yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—18162no—
323D4—18162no—
324D4—9081no—
325D4—9081no—
326D4—4544no—
327D4—4542no—
328D4—4542no—
329D4—4544no—
330D4—4544no—
331D4—4542no—
332D4—4542no—
333D4—4542no—
334D4—4542no—
335D4—4542no—
336D4—2272no—
337D4D4×D22712yes—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—4544no—
342C2×C4—4544no—
343C2×C4—4542no—
344C2×C4—4542no—
345C2×C4—4542no—
346C2×C4—4542no—
347C2×C4—4541no—
348C2×C4—4542no—
349C2×C4—4542no—
350C2×C4—2271no—
351C2×C4—2272no—
352C2×C4—2272no—
353C2×C4D4×D22712yes—
354C2×C4C23×D22711yescharacteristic
355C8—22no—
356C8—4542no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—18164no—
361C22—18162no—
362C22—18164no—
363C22—9082no—
364C22—9082no—
365C22—9081no—
366C22—9081no—
367C22—4542no—
368C22—4542no—
369C22—2272no—
370C22C2×D4×D22711yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—9081no—
375C4—4542no—
376C4—2271no—
377C4—2271no—
378C4C2×D4×D22712yes—
379C2—21no—
380C2—42no—
381C2—81no—
382C2—18161no—
383C2—9082no—
384C2—9082no—
385C2—4542no—
386C2#120×D22711yescharacteristic
subgroupG/H
centerC2#120×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#454×C227C2
socleC2×C227C2×#120
hypercenter#454D227
hypocenterC227#3021
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #28409869C2semidirect
2. AGL(1,ℤ/8)×D227C2semidirect
3. #28409600C2semidirect
4. #28409207C2semidirect
5. #28409205C2semidirect
6. #28409204C2semidirect
7. #28409201C2semidirect
8. #28409198C2semidirect
9. #28409047C2semidirect
10. #28409015C2semidirect
11. #454×C227C2semidirect
13. #20300336C22semidirect
16. #20300334C22semidirect
22. C2×D4×D227C22semidirect
23. #20300316C22semidirect
25. #20300314C22semidirect
26. #20300313C22semidirect
30. M16×D227C22semidirect
31. #20300285C22semidirect
32. #20300284C22semidirect
33. #20300277C22semidirect
36. #20300245C22semidirect
37. #20300216C22semidirect
38. #20300215C22semidirect
39. #20300214C22semidirect
40. #20300207C22semidirect
41. #20300204C22semidirect
42. 2−1+4×C227C22semidirect
43. AGL(1,ℤ/8)×C227C22semidirect
44. #124×C227C22semidirect
45. #104×C227C22semidirect
46. #101×C227C22semidirect
54. #16813885D4semidirect
63. #16813883D4semidirect
66. Q8×D227D4semidirect
82. D4×D227D4semidirect
193. C227#3021semidirect
201. #454D227semidirect
208. 2−1+4C2×D227semidirect
223. AGL(1,ℤ/8)C2×D227semidirect
230. #124C2×D227semidirect
237. #104C2×D227semidirect
239. #101C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284098692
AGL(1,ℤ/8)×D2272
#284096002
#284092072
#284092052
#284092042
#284092012
#284091982
#284090472
#284090152
#454×C2272
#3021227