Finite Group Databasebeta

← #44409089

Subgroups — #44409089

Order 29056.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#20300334C211yesmaximal, characteristic, complemented
2#28409830C211yesmaximal, characteristic, complemented
3#28409596C211yesmaximal, characteristic, complemented
4#28409208C214yesmaximal, complemented
5#28409198C214yesmaximal, complemented
6#28409049C211yesmaximal, characteristic, complemented
7#28409048C211yesmaximal, characteristic, complemented
8C2×#20300204C211yesmaximal, characteristic, complemented
9#575×C227C211yesmaximal, characteristic, complemented
10#20300334—22no—
11#20300334—22no—
12#20300334—22no—
13#20300334C2214yes—
14C2×#16813885—21no—
15C2×#16813884—21no—
16C2×#16813884C2211yescharacteristic
17C22×D908—21no—
18C22×D908C2211yescharacteristic
19C2×D4×D227—22no—
20C2×D4×D227—21no—
21#20300319C2212yescomplemented
22#20300318—21no—
23#20300314C2212yescomplemented
24#20300313—21no—
25#20300308—22no—
26C2×#16813868C2211yescharacteristic, complemented
27#20300284C2212yescomplemented
28#20300277C2212yescomplemented
29#20300262—22no—
30#20300256—22no—
31#20300247C2211yescharacteristic, complemented
32#20300215C2214yescomplemented
33#20300214C2214yescomplemented
34#20300204C2214yescomplemented
35C2×#41×C227C2211yescharacteristic
36C2×#41×C227C2211yescharacteristic, complemented
37#137×C227—21no—
38#137×C227C2212yescomplemented
39AGL(1,ℤ/8)×C227—21no—
40AGL(1,ℤ/8)×C227C2212yescomplemented
41#135×C227—22no—
42C2×M16×C227C2211yescharacteristic, complemented
43C23×D227—22no—
44C23×D227D412yes—
45#16813885—24no—
46#16813885—24no—
47#16813884—24no—
48#16813884—24no—
49#16813884—21no—
50#16813884—21no—
51#16813884D412yescomplemented
52#16813884C2312yes—
53C2×Dic454D411yescharacteristic, complemented
54C2×D908—22no—
55C2×D908—24no—
56C2×D908—24no—
57C2×D908—21no—
58C2×D908—21no—
59C2×D908—21no—
60C2×D908—21no—
61C2×D908D412yes—
62C2×D908D411yescharacteristic, complemented
63C2×D908C2312yes—
64C2×C22⋊D227—22no—
65C2×C22⋊D227—24no—
66C2×C22⋊D227—21no—
67D4×D227—28no—
68D4×D227—24no—
69D4×D227—28no—
70D4×D227—24no—
71C2×C4×D227—22no—
72C2×C4×D227—21no—
73#16813874—22no—
74#16813872—22no—
75#16813871—22no—
76#16813869—22no—
77#16813868—21no—
78#16813868C2312yescomplemented
79C2×C227⋊C8—21no—
80C4⋊Dic227—21no—
81C22⋊Dic227—21no—
82(C2×D227)⋊C4—24no—
83C4×Dic227C2312yescomplemented
84#41×C227—21no—
85#41×C227—21no—
86#41×C227—21no—
87#41×C227—21no—
88#41×C227—21no—
89#41×C227—21no—
90#41×C227D412yescomplemented
91#41×C227C2312yes—
92C2×Q8×C227D411yescharacteristic, complemented
93C2×Q8×C227C2311yescharacteristic, complemented
94C2×D4×C227—21no—
95C2×D4×C227—21no—
96C2×D4×C227D411yescharacteristic, complemented
97C2×D4×C227C2311yescharacteristic, complemented
98C22×C4×C227—21no—
99C22×C4×C227—21no—
100C22×C4×C227C2311yescharacteristic
101Q16×C227—22no—
102SD16×C227—22no—
103SD16×C227—22no—
104D8×C227—22no—
105M16×C227—21no—
106M16×C227C2312yescomplemented
107C2×C8×C227—21no—
108C22×D227—22no—
109C22×D227—21no—
110C22×D227—28no—
111C22×D227—22no—
112C22×D227—82no—
113C22×D227—84no—
114C22×D227C2×D414yes—
115Dic454—21no—
116Dic454C2×D412yes—
117D908—28no—
118D908—22no—
119D908—22no—
120D908—21no—
121D908—22no—
122D908—22no—
123D908C2×D412yes—
124C22⋊D227—28no—
125C22⋊D227—22no—
126C22⋊D227—22no—
127C22⋊D227—84no—
128C2×Dic227—21no—
129C2×Dic227—22no—
130C4×D227—28no—
131C4×D227—22no—
132C4×D227—22no—
133C227⋊C8—22no—
134C23×C227—21no—
135C23×C227—21no—
136C23×C227C2×D411yescharacteristic
137Q8×C227—21no—
138Q8×C227—21no—
139Q8×C227—21no—
140Q8×C227C2×D412yes—
141D4×C227—22no—
142D4×C227—22no—
143D4×C227—21no—
144D4×C227—21no—
145D4×C227—21no—
146D4×C227—41no—
147D4×C227—41no—
148D4×C227C2×D412yes—
149C2×C4×C227—21no—
150C2×C4×C227—22no—
151C2×C4×C227—22no—
152C2×C4×C227—21no—
153C2×C4×C227—41no—
154C2×C4×C227—41no—
155C2×C4×C227C2×D411yescharacteristic
156C2×C4×C227C2×D411yescharacteristic
157C2×C4×C227C2×D411yescharacteristic
158C2×C4×C227C2×D411yescharacteristic
159C2×C4×C227C2×D411yescharacteristic
160C2×C4×C227C2411yescharacteristic
161C8×C227—22no—
162C22×C227—22no—
163C22×C227—41no—
164C22×C227—42no—
165C22×C227—42no—
166C22×C227—41no—
167C22×C227#12011yescharacteristic
168C22×C227#12011yescharacteristic
169C22×C227C22×D411yescharacteristic
170C2×D227—24no—
171C2×D227—22no—
172C2×D227—44no—
173C2×D227—42no—
174C2×D227—44no—
175C2×D227—82no—
176C2×D227—88no—
177C4×C227—22no—
178C4×C227—41no—
179C4×C227#12011yescharacteristic
180C4×C227#12011yescharacteristic
181C4×C227C22×D411yescharacteristic
182C4×C227C22×D411yescharacteristic
183Dic227—22no—
184Dic227—42no—
185C2×C227—21no—
186C2×C227—21no—
187C2×C227—21no—
188C2×C227—42no—
189C2×C227—81no—
190C2×C227C2×#12011yescharacteristic
191D227—44no—
192D227—42no—
193C227#301911yescharacteristic, complemented, Sylow
194#3019—2271nomaximal, Sylow
195C2×H(5,𝔽2)—2271no—
196#575—2271no—
197#575D22711yescharacteristic, complemented
198#535—2271no—
199#455—2274no—
200#453—2274no—
201#421—2271no—
202#421—2271no—
203C2×#100—2271no—
204H(5,𝔽2)—4542no—
205H(5,𝔽2)—4542no—
206H(5,𝔽2)—4542no—
207H(5,𝔽2)—2274no—
208C2×#41—4541no—
209C2×#41—4541no—
210C2×#41—2271no—
211C2×#41C2×D22711yescharacteristic
212C2×#41C2×D22711yescharacteristic, complemented
213C22×D4—4541no—
214C22×D4—4542no—
215C22×D4—4541no—
216C22×D4—2271no—
217#137—21no—
218#137—4541no—
219#137—2272no—
220#137C2×D22712yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—4541no—
223AGL(1,ℤ/8)—2272no—
224AGL(1,ℤ/8)C2×D22712yescomplemented
225#135—22no—
226#135—4542no—
227C2×M16—2271no—
228C2×M16C2×D22711yescharacteristic, complemented
229#127—2272no—
230#124—2272no—
231#121—4542no—
232#120—4542no—
233#117—2271no—
234#104—2274no—
235#104—2274no—
236#100—2274no—
237C24—4542no—
238C24—2272no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—4544no—
246#41—4544no—
247#41—4544no—
248#41—4544no—
249#41—4541no—
250#41—4541no—
251#41—2272no—
252#41—2272no—
253#41C22⋊D22712yescomplemented
254#41C22×D22712yes—
255C2×Q8—2271no—
256C2×Q8C22⋊D22711yescharacteristic, complemented
257C2×Q8C22×D22711yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—4542no—
261C2×D4—4544no—
262C2×D4—4544no—
263C2×D4—4548no—
264C2×D4—4544no—
265C2×D4—4548no—
266C2×D4—4544no—
267C2×D4—4542no—
268C2×D4—4541no—
269C2×D4—4544no—
270C2×D4—4541no—
271C2×D4—4541no—
272C2×D4—4541no—
273C2×D4—4541no—
274C2×D4—2272no—
275C2×D4—2271no—
276C2×D4—2272no—
277C2×D4C22⋊D22711yescharacteristic, complemented
278C2×D4C22×D22711yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—4542no—
282C22×C4—4541no—
283C22×C4C22×D22711yescharacteristic
284Q16—22no—
285Q16—4542no—
286SD16—22no—
287SD16—22no—
288SD16—4542no—
289SD16—4542no—
290D8—22no—
291D8—4542no—
292M16—21no—
293M16—4541no—
294M16—2272no—
295M16C22×D22712yescomplemented
296C2×C8—21no—
297C2×C8—4541no—
298C4⋊C4—4541no—
299C22⋊C4—4544no—
300C22⋊C4—4541no—
301C42—2272no—
302C23—21no—
303C23—21no—
304C23—18164no—
305C23—18162no—
306C23—4548no—
307C23—4542no—
308C23—4542no—
309C23—4541no—
310C23—2274no—
311C23D4×D22711yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—4541no—
316Q8—2272no—
317Q8C2×C22⋊D22712yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—18164no—
326D4—4548no—
327D4—4548no—
328D4—4542no—
329D4—4542no—
330D4—4542no—
331D4—4542no—
332D4—4542no—
333D4—4541no—
334D4—4542no—
335D4—2272no—
336D4C2×C22⋊D22712yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—4548no—
344C2×C4—4542no—
345C2×C4—4541no—
346C2×C4—4542no—
347C2×C4—4542no—
348C2×C4D4×D22711yescharacteristic
349C2×C4D4×D22711yescharacteristic
350C2×C4D4×D22711yescharacteristic
351C2×C4C2×C22⋊D22711yescharacteristic
352C2×C4C2×C22⋊D22711yescharacteristic
353C2×C4C23×D22711yescharacteristic
354C8—22no—
355C8—4542no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—18162no—
362C22—18168no—
363C22—9084no—
364C22—9082no—
365C22—9084no—
366C22—4544no—
367C22—4542no—
368C22#2030025611yescharacteristic
369C22#2030025611yescharacteristic
370C22C2×D4×D22711yescharacteristic
371C4—22no—
372C4—41no—
373C4—9082no—
374C4—4542no—
375C4#2030025611yescharacteristic
376C4#2030025611yescharacteristic
377C4C2×D4×D22711yescharacteristic
378C4C22×C22⋊D22711yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—9082no—
385C2—9084no—
386C2C2×#2030025611yescharacteristic
subgroupG/H
centerC2C2×#20300256
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#575×C227C2
socleC2×C227C2×#120
hypercenter#575D227
hypocenterC227#3019
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#20300334C2semidirect
2. #28409830C2semidirect
3. #28409596C2semidirect
4. #28409208C2semidirect
5. #28409198C2semidirect
6. #28409049C2semidirect
7. #28409048C2semidirect
8. C2×#20300204C2semidirect
9. #575×C227C2semidirect
21. #20300319C22semidirect
23. #20300314C22semidirect
26. C2×#16813868C22semidirect
27. #20300284C22semidirect
28. #20300277C22semidirect
31. #20300247C22semidirect
32. #20300215C22semidirect
33. #20300214C22semidirect
34. #20300204C22semidirect
36. C2×#41×C227C22semidirect
38. #137×C227C22semidirect
40. AGL(1,ℤ/8)×C227C22semidirect
42. C2×M16×C227C22semidirect
51. #16813884D4semidirect
53. C2×Dic454D4semidirect
62. C2×D908D4semidirect
78. #16813868C23semidirect
83. C4×Dic227C23semidirect
90. #41×C227D4semidirect
92. C2×Q8×C227D4semidirect
93. C2×Q8×C227C23semidirect
96. C2×D4×C227D4semidirect
97. C2×D4×C227C23semidirect
106. M16×C227C23semidirect
193. C227#3019semidirect
197. #575D227semidirect
212. C2×#41C2×D227semidirect
220. #137C2×D227semidirect
224. AGL(1,ℤ/8)C2×D227semidirect
228. C2×M16C2×D227semidirect
253. #41C22⋊D227semidirect
256. C2×Q8C22⋊D227semidirect
257. C2×Q8C22×D227semidirect
277. C2×D4C22⋊D227semidirect
278. C2×D4C22×D227semidirect
295. M16C22×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003342
#284098302
#284095962
#284092082
#284091982
#284090492
#284090482
C2×#203002042
#575×C2272
#3019227