Finite Group Databasebeta

← #44409103

Subgroups — #44409103

Order 29056.

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

normal subgroups only

This group has 736 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
1#28409868C211yesmaximal, characteristic, complemented
2C2×#20300314C211yesmaximal, characteristic, complemented
3#28409599C211yesmaximal, characteristic, complemented
4#28409203C214yesmaximal, complemented
5#28409198C214yesmaximal, complemented
6#28409049C211yesmaximal, characteristic, complemented
7#28409048C211yesmaximal, characteristic, complemented
8#28409014C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C227C211yesmaximal, characteristic, complemented
10#20300336—21no—
11#20300335—21no—
12#20300335C2212yescomplemented
13#20300334—21no—
14#20300334C2212yescomplemented
15#20300333—22no—
16#20300333—21no—
17C2×#16813884—22no—
18C2×#16813884C2211yescharacteristic
19C2×#16813884C2211yescharacteristic
20#41×D227—22no—
21#41×D227—22no—
22#20300314—22no—
23#20300314C2214yescomplemented
24C2×#16813872—21no—
25C2×#16813869—21no—
26C2×#16813868C2211yescharacteristic, complemented
27#20300284C2212yescomplemented
28#20300275C2212yescomplemented
29#20300259—22no—
30#20300257—22no—
31#20300247C2211yescharacteristic, complemented
32#20300215C2214yescomplemented
33#20300214C2214yescomplemented
34#20300206C2212yescomplemented
35#20300204C2212yescomplemented
36C2×#41×C227C2211yescharacteristic
37C22×D4×C227C2211yescharacteristic, complemented
38AGL(1,ℤ/8)×C227—22no—
39AGL(1,ℤ/8)×C227C2214yescomplemented
40C2×SD16×C227—21no—
41C2×D8×C227—21no—
42C2×M16×C227C2211yescharacteristic, complemented
43#16813885—22no—
44#16813885—22no—
45#16813884—24no—
46#16813884—24no—
47#16813884—22no—
48#16813884—22no—
49#16813884—21no—
50#16813884—21no—
51#16813884—21no—
52#16813884—21no—
53#16813884—21no—
54#16813884—21no—
55#16813884D412yescomplemented
56#16813884C2312yes—
57#16813883—24no—
58#16813883—22no—
59#16813883—24no—
60#16813883—22no—
61C2×Dic454—22no—
62C2×Dic454D411yescharacteristic, complemented
63C2×Dic454C2311yescharacteristic
64Q8×D227—22no—
65Q8×D227—22no—
66C2×D908—22no—
67C2×D908D411yescharacteristic, complemented
68C2×D908C2311yescharacteristic
69C2×C22⋊D227—24no—
70C2×C22⋊D227—24no—
71C2×C22⋊D227—21no—
72C2×C22⋊D227D412yes—
73D4×D227—24no—
74D4×D227—22no—
75D4×D227—24no—
76D4×D227—22no—
77C2×C4×D227—24no—
78C2×C4×D227—21no—
79C2×C4×D227D412yes—
80#16813872—24no—
81#16813869—24no—
82#16813868—21no—
83#16813868C2312yes—
84C2×C227⋊C8—21no—
85C4⋊Dic227—21no—
86C22⋊Dic227—24no—
87C22⋊Dic227—21no—
88C4×Dic227C2312yes—
89C24×C227—21no—
90#41×C227—21no—
91#41×C227—21no—
92#41×C227D412yescomplemented
93#41×C227C2312yes—
94C2×Q8×C227D411yescharacteristic, complemented
95C2×D4×C227—21no—
96C2×D4×C227—21no—
97C2×D4×C227—21no—
98C2×D4×C227—21no—
99C2×D4×C227—21no—
100C2×D4×C227—21no—
101C2×D4×C227D411yescharacteristic, complemented
102C2×D4×C227C2312yes—
103C22×C4×C227—21no—
104C22×C4×C227C2311yescharacteristic
105SD16×C227—24no—
106D8×C227—24no—
107M16×C227—21no—
108M16×C227C2312yes—
109C2×C8×C227—21no—
110C22×D227—24no—
111C22×D227—21no—
112C22×D227C2×D412yes—
113Dic454—24no—
114Dic454—21no—
115Dic454—21no—
116Dic454—21no—
117Dic454C2×D412yes—
118D908—24no—
119D908—21no—
120D908—21no—
121D908—21no—
122D908C2×D412yes—
123C22⋊D227—24no—
124C22⋊D227—24no—
125C22⋊D227—22no—
126C22⋊D227—22no—
127C22⋊D227—22no—
128C22⋊D227—22no—
129C22⋊D227—84no—
130C2×Dic227—24no—
131C2×Dic227—21no—
132C2×Dic227—22no—
133C2×Dic227C2×D412yes—
134C4×D227—24no—
135C4×D227—24no—
136C4×D227—22no—
137C4×D227—22no—
138C4×D227—22no—
139C4×D227—22no—
140C227⋊C8—22no—
141C23×C227—21no—
142C23×C227—22no—
143C23×C227—41no—
144C23×C227—41no—
145C23×C227—82no—
146C23×C227C2×D411yescharacteristic
147Q8×C227—21no—
148Q8×C227C2×D412yes—
149D4×C227—22no—
150D4×C227—22no—
151D4×C227—21no—
152D4×C227—22no—
153D4×C227—22no—
154D4×C227—41no—
155D4×C227—41no—
156D4×C227C2×D412yes—
157C2×C4×C227—21no—
158C2×C4×C227—22no—
159C2×C4×C227—22no—
160C2×C4×C227C2×D411yescharacteristic
161C2×C4×C227C2×D411yescharacteristic
162C2×C4×C227C2×D411yescharacteristic
163C2×C4×C227C2×D411yescharacteristic
164C2×C4×C227C2×D411yescharacteristic
165C2×C4×C227C2411yescharacteristic
166C8×C227—22no—
167C22×C227—22no—
168C22×C227—41no—
169C22×C227—44no—
170C22×C227—42no—
171C22×C227—81no—
172C22×C227—82no—
173C22×C227—82no—
174C22×C227#12011yescharacteristic
175C22×C227#12011yescharacteristic
176C22×C227C22×D411yescharacteristic
177C2×D227—22no—
178C2×D227—22no—
179C2×D227—42no—
180C2×D227—81no—
181C2×D227—84no—
182C4×C227—22no—
183C4×C227#12011yescharacteristic
184C4×C227#12011yescharacteristic
185C4×C227C22×D411yescharacteristic
186C4×C227C22×D411yescharacteristic
187Dic227—22no—
188Dic227—22no—
189Dic227—42no—
190C2×C227—21no—
191C2×C227—21no—
192C2×C227—21no—
193C2×C227—42no—
194C2×C227—82no—
195C2×C227C2×#12011yescharacteristic
196D227—42no—
197D227—42no—
198C227#302111yescharacteristic, complemented, Sylow
199#3021—2271nomaximal, Sylow
200#585—2271no—
201C2×AGL(1,ℤ/8)—2271no—
202C2×AGL(1,ℤ/8)D22711yescharacteristic, complemented
203#535—2271no—
204#454—2274no—
205#453—2274no—
206#421—2271no—
207#421—2271no—
208#413—2271no—
2092−1+4—4541no—
2102−1+4—4541no—
2112−1+4—2272no—
212H(5,𝔽2)—4542no—
213H(5,𝔽2)—4541no—
214H(5,𝔽2)—4541no—
215H(5,𝔽2)—2272no—
216C2×#41—4542no—
217C2×#41—4542no—
218C2×#41—4542no—
219C2×#41—2271no—
220C2×#41—2271no—
221C2×#41C2×D22711yescharacteristic
222C22×D4C2×D22711yescharacteristic, complemented
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—4542no—
225AGL(1,ℤ/8)—2274no—
226AGL(1,ℤ/8)C2×D22714yescomplemented
227C2×SD16—21no—
228C2×SD16—4541no—
229C2×D8—21no—
230C2×D8—4541no—
231C2×M16—2271no—
232C2×M16C2×D22711yescharacteristic, complemented
233#127—2272no—
234#124—2272no—
235#121—4542no—
236#120—4542no—
237#117—2271no—
238#104—2274no—
239#104—2274no—
240#101—2272no—
241#100—2272no—
242C24—21no—
243#41—21no—
244#41—21no—
245#41—4544no—
246#41—4544no—
247#41—4542no—
248#41—4542no—
249#41—4544no—
250#41—4542no—
251#41—4542no—
252#41—4542no—
253#41—4542no—
254#41—4544no—
255#41—4541no—
256#41—4541no—
257#41—4541no—
258#41—4541no—
259#41—4541no—
260#41—4541no—
261#41—2272no—
262#41—2272no—
263#41C22⋊D22712yescomplemented
264#41C22×D22712yes—
265C2×Q8—4542no—
266C2×Q8—4542no—
267C2×Q8—4542no—
268C2×Q8—2271no—
269C2×Q8—2271no—
270C2×Q8C22⋊D22711yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—4542no—
278C2×D4—4544no—
279C2×D4—4542no—
280C2×D4—4544no—
281C2×D4—4544no—
282C2×D4—4544no—
283C2×D4—4541no—
284C2×D4—4542no—
285C2×D4—2272no—
286C2×D4—2271no—
287C2×D4—2271no—
288C2×D4C22⋊D22711yescharacteristic, complemented
289C2×D4C22×D22712yes—
290C22×C4—21no—
291C22×C4—4544no—
292C22×C4—4541no—
293C22×C4—2272no—
294C22×C4C22×D22711yescharacteristic
295SD16—24no—
296SD16—4544no—
297D8—24no—
298D8—4544no—
299M16—21no—
300M16—4541no—
301M16—2272no—
302M16C22×D22712yes—
303C2×C8—21no—
304C2×C8—4541no—
305C4⋊C4—4541no—
306C22⋊C4—4541no—
307C22⋊C4—4544no—
308C42—2272no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—4544no—
315C23—4541no—
316C23—2272no—
317C23D4×D22711yescharacteristic
318Q8—21no—
319Q8—4544no—
320Q8—4541no—
321Q8—4541no—
322Q8—4541no—
323Q8—2272no—
324Q8C2×C22⋊D22712yes—
325D4—22no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4—21no—
330D4—41no—
331D4—41no—
332D4—18164no—
333D4—4544no—
334D4—4544no—
335D4—4544no—
336D4—4542no—
337D4—4542no—
338D4—4542no—
339D4—4542no—
340D4—4541no—
341D4—4541no—
342D4—4541no—
343D4—2272no—
344D4C2×C22⋊D22712yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—4544no—
349C2×C4—4544no—
350C2×C4—4542no—
351C2×C4—4544no—
352C2×C4—4541no—
353C2×C4—4542no—
354C2×C4—4542no—
355C2×C4—4542no—
356C2×C4—4542no—
357C2×C4—2272no—
358C2×C4D4×D22711yescharacteristic
359C2×C4D4×D22711yescharacteristic
360C2×C4D4×D22711yescharacteristic
361C2×C4C2×C22⋊D22711yescharacteristic
362C2×C4C2×C22⋊D22711yescharacteristic
363C2×C4C23×D22711yescharacteristic
364C8—22no—
365C8—4542no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—18161no—
374C22—18164no—
375C22—9082no—
376C22—4542no—
377C22—4542no—
378C22#2030025611yescharacteristic
379C22#2030025611yescharacteristic
380C22C2×D4×D22711yescharacteristic
381C4—22no—
382C4—9082no—
383C4—4542no—
384C4—4542no—
385C4#2030025611yescharacteristic
386C4#2030025611yescharacteristic
387C4C2×D4×D22711yescharacteristic
388C4C22×C22⋊D22711yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—9082no—
395C2—9082no—
396C2C2×#2030025611yescharacteristic
subgroupG/H
centerC2C2×#20300256
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
FittingC2×AGL(1,ℤ/8)×C227C2
socleC2×C227C2×#120
hypercenterC2×AGL(1,ℤ/8)D227
hypocenterC227#3021
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #28409868C2semidirect
2. C2×#20300314C2semidirect
3. #28409599C2semidirect
4. #28409203C2semidirect
5. #28409198C2semidirect
6. #28409049C2semidirect
7. #28409048C2semidirect
8. #28409014C2semidirect
9. C2×AGL(1,ℤ/8)×C227C2semidirect
12. #20300335C22semidirect
14. #20300334C22semidirect
23. #20300314C22semidirect
26. C2×#16813868C22semidirect
27. #20300284C22semidirect
28. #20300275C22semidirect
31. #20300247C22semidirect
32. #20300215C22semidirect
33. #20300214C22semidirect
34. #20300206C22semidirect
35. #20300204C22semidirect
37. C22×D4×C227C22semidirect
39. AGL(1,ℤ/8)×C227C22semidirect
42. C2×M16×C227C22semidirect
55. #16813884D4semidirect
62. C2×Dic454D4semidirect
67. C2×D908D4semidirect
92. #41×C227D4semidirect
94. C2×Q8×C227D4semidirect
101. C2×D4×C227D4semidirect
198. C227#3021semidirect
202. C2×AGL(1,ℤ/8)D227semidirect
222. C22×D4C2×D227semidirect
226. AGL(1,ℤ/8)C2×D227semidirect
232. C2×M16C2×D227semidirect
263. #41C22⋊D227semidirect
270. C2×Q8C22⋊D227semidirect
288. C2×D4C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
#284098682
C2×#203003142
#284095992
#284092032
#284091982
#284090492
#284090482
#284090142
C2×AGL(1,ℤ/8)×C2272
#3021227