Finite Group Databasebeta

← #44409015

Subgroups — #44409015

Order 29056.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300336C211yesmaximal, characteristic, complemented
2C2×#20300315C211yesmaximal, characteristic, complemented
3C2×SD16×D227C211yesmaximal, characteristic, complemented
4#28409598C211yesmaximal, characteristic, complemented
5#28409180C211yesmaximal, characteristic, complemented
6#28409179C211yesmaximal, characteristic, complemented
7#28409176C211yesmaximal, characteristic, complemented
8#28409175C211yesmaximal, characteristic, complemented
9#28409172C211yesmaximal, characteristic, complemented
10#28409171C211yesmaximal, characteristic, complemented
11#28409169C211yesmaximal, characteristic, complemented
12#28409039C211yesmaximal, characteristic, complemented
13#28409027C211yesmaximal, characteristic, complemented
14#28408997C211yesmaximal, characteristic, complemented
15#448×C227C211yesmaximal, characteristic, complemented
16#20300336—24no—
17#20300336—24no—
18C2×#16813885—21no—
19C2×#16813885C2211yescharacteristic
20C2×#16813885C2211yescharacteristic, complemented
21C2×#16813884—21no—
22C2×#16813884—21no—
23C2×#16813884C2211yescharacteristic
24C2×#16813884C2211yescharacteristic, complemented
25C2×Q8×D227—21no—
26C2×Q8×D227C2211yescharacteristic
27C2×Q8×D227C2211yescharacteristic, complemented
28C2×D4×D227C2211yescharacteristic
29#20300315—24no—
30C2×#16813873C2211yescharacteristic
31C2×#16813872C2211yescharacteristic
32C2×#16813871C2211yescharacteristic
33C2×#16813871C2211yescharacteristic
34SD16×D227—24no—
35C2×D1816C2211yescharacteristic
36C2×#16813869C2211yescharacteristic
37C2×#16813867C2211yescharacteristic
38C2×C8×D227C2211yescharacteristic
39#20300284—21no—
40#20300274—21no—
41#20300263C2211yescharacteristic
42#20300262C2211yescharacteristic
43#20300259C2211yescharacteristic
44#20300258—21no—
45#20300256—21no—
46#20300246C2211yescharacteristic
47#20300213C2211yescharacteristic, complemented
48#20300212C2211yescharacteristic, complemented
49#20300212C2211yescharacteristic, complemented
50#20300211C2211yescharacteristic, complemented
51#20300210C2211yescharacteristic, complemented
52#20300208C2211yescharacteristic, complemented
53C22⋊(C227⋊C8)C2211yescharacteristic, complemented
54(C2×D227)⋊C8C2211yescharacteristic, complemented
55(C2×D227)⋊C8C2211yescharacteristic, complemented
56C22×Q8×C227C2211yescharacteristic, complemented
57C2×SD16×C227C2211yescharacteristic
58C2×SD16×C227C2211yescharacteristic
59#121×C227C2211yescharacteristic
60#103×C227C2211yescharacteristic, complemented
61#103×C227C2211yescharacteristic, complemented
62C22⋊C8×C227C2211yescharacteristic, complemented
63C23×D227—21no—
64#16813885—28no—
65#16813885—22no—
66#16813885—22no—
67#16813885—22no—
68#16813885D414yescomplemented
69#16813884—28no—
70#16813884—28no—
71#16813884—22no—
72#16813884—22no—
73#16813884—22no—
74#16813884—22no—
75C2×Dic454—21no—
76C2×Dic454—21no—
77C2×Dic454D411yescharacteristic
78C2×Dic454C2311yescharacteristic
79Q8×D227—28no—
80Q8×D227—22no—
81Q8×D227—22no—
82Q8×D227—22no—
83Q8×D227D414yescomplemented
84C2×D908—21no—
85C2×D908—21no—
86C2×D908—21no—
87C2×D908D411yescharacteristic
88C2×D908C2311yescharacteristic
89C2×D908C2311yescharacteristic
90C2×C22⋊D227—21no—
91C2×C22⋊D227—21no—
92C2×C22⋊D227—21no—
93C2×C22⋊D227—21no—
94C2×C22⋊D227D411yescharacteristic
95C2×C22⋊D227D411yescharacteristic
96D4×D227—22no—
97D4×D227—22no—
98C2×C4×D227—21no—
99C2×C4×D227—21no—
100C2×C4×D227—21no—
101C2×C4×D227—21no—
102C2×C4×D227—21no—
103C2×C4×D227C2311yescharacteristic
104C2×C4×D227C2311yescharacteristic
105#16813873—22no—
106#16813872—22no—
107#16813871—22no—
108#16813871—22no—
109D1816—22no—
110#16813869—22no—
111#16813867—22no—
112C2×C227⋊C8C2311yescharacteristic
113C2×C227⋊C8C2311yescharacteristic
114C8×D227—22no—
115C4⋊Dic227C2311yescharacteristic
116C22⋊Dic227—21no—
117(C2×D227)⋊C4—21no—
118(C2×D227)⋊C4—21no—
119(C2×D227)⋊C4—21no—
120C4×Dic227—21no—
121C2×Q8×C227—22no—
122C2×Q8×C227—22no—
123C2×Q8×C227—21no—
124C2×Q8×C227C2311yescharacteristic
125C2×Q8×C227C2311yescharacteristic
126C2×D4×C227—21no—
127C2×D4×C227C2311yescharacteristic
128C22×C4×C227—21no—
129C22×C4×C227C2311yescharacteristic
130SD16×C227—22no—
131SD16×C227—22no—
132C2×C8×C227C2311yescharacteristic
133C2×C8×C227C2311yescharacteristic
134C4⋊C4×C227C2311yescharacteristic
135C22⋊C4×C227—21no—
136C22×D227—21no—
137C22×D227—21no—
138C22×D227—42no—
139C22×D227—81no—
140C22×D227—81no—
141C22×D227C2×D411yescharacteristic
142C22×D227C2×D411yescharacteristic
143Dic454—24no—
144Dic454—21no—
145Dic454—22no—
146Dic454—24no—
147Dic454C2×D412yes—
148D908—24no—
149D908—24no—
150D908—22no—
151D908—21no—
152D908—21no—
153D908—21no—
154D908—81no—
155D908C2×D412yes—
156C22⋊D227—24no—
157C22⋊D227—22no—
158C22⋊D227—22no—
159C22⋊D227—42no—
160C22⋊D227—81no—
161C22⋊D227—81no—
162C2×Dic227—21no—
163C2×Dic227—21no—
164C2×Dic227C2×D411yescharacteristic
165C2×Dic227C2×D411yescharacteristic
166C4×D227—24no—
167C4×D227—24no—
168C4×D227—24no—
169C4×D227—24no—
170C4×D227—21no—
171C4×D227—21no—
172C4×D227—24no—
173C4×D227C2×D412yes—
174C4×D227C2×D412yes—
175C227⋊C8—21no—
176C227⋊C8—21no—
177C23×C227—21no—
178C23×C227C2×D411yescharacteristic
179Q8×C227—24no—
180Q8×C227—21no—
181Q8×C227—21no—
182Q8×C227C2×D412yes—
183Q8×C227C2×D412yes—
184D4×C227—21no—
185D4×C227—21no—
186D4×C227—81no—
187C2×C4×C227—24no—
188C2×C4×C227—21no—
189C2×C4×C227—21no—
190C2×C4×C227—21no—
191C2×C4×C227—21no—
192C2×C4×C227—21no—
193C2×C4×C227C2×D411yescharacteristic
194C2×C4×C227C2411yescharacteristic
195C8×C227—21no—
196C8×C227—21no—
197C22×C227—21no—
198C22×C227—21no—
199C22×C227—21no—
200C22×C227—41no—
201C22×C227—41no—
202C22×C227—41no—
203C22×C227C22×D411yescharacteristic
204C2×D227—21no—
205C2×D227—21no—
206C2×D227—21no—
207C2×D227—22no—
208C2×D227—21no—
209C2×D227—21no—
210C2×D227—41no—
211C2×D227—41no—
212C2×D227—41no—
213C2×D227—41no—
214C2×D227—41no—
215C2×D227—84no—
216C2×D227#12012yes—
217C4×C227—22no—
218C4×C227—22no—
219C4×C227—21no—
220C4×C227—41no—
221C4×C227C22×D411yescharacteristic
222C4×C227C22×D411yescharacteristic
223Dic227—22no—
224Dic227—21no—
225Dic227—41no—
226Dic227#12012yes—
227C2×C227—41no—
228C2×C227—81no—
229C2×C227C2×#12011yescharacteristic
230C2×C227#57711yescharacteristic
231C2×C227#57711yescharacteristic
232D227—22no—
233D227—41no—
234D227—42no—
235D227—81no—
236C227#301011yescharacteristic, complemented, Sylow
237#3010—2271nomaximal, Sylow
238C2×2−1+4—2271no—
239C2×AGL(1,ℤ/8)—2271no—
240C22×SD16—2271no—
241#535—2271no—
242#449—2271no—
243#449—2271no—
244#449—2271no—
245#449—2271no—
246#448—2271no—
247#448—2271no—
248#448—2271no—
249#448D22711yescharacteristic, complemented
250#419—2271no—
251#416—2271no—
252#408—2271no—
2532−1+4—4544no—
2542−1+4—4544no—
255C2×#41—4541no—
256C2×#41—4541no—
257C2×#41—4541no—
258C2×#41—2271no—
259C2×#41—2271no—
260C2×#41—2271no—
261C2×#41—2271no—
262C22×Q8—4541no—
263C22×Q8—2271no—
264C22×Q8—2271no—
265C22×Q8C2×D22711yescharacteristic, complemented
266C22×D4—2271no—
267AGL(1,ℤ/8)—4544no—
268C2×SD16—4544no—
269C2×SD16—2271no—
270C2×SD16—2271no—
271C2×SD16—2271no—
272C2×SD16—2271no—
273C2×SD16C2×D22711yescharacteristic
274C2×SD16C2×D22711yescharacteristic
275C2×D8—2271no—
276C2×D8—2271no—
277C2×M16—2271no—
278C22×C8—2271no—
279#127—4541no—
280#124—4541no—
281#121—2271no—
282#121—2271no—
283#121—2271no—
284#121C2×D22711yescharacteristic
285#120—4541no—
286#120—4541no—
287#117—2271no—
288#103—2271no—
289#103—2271no—
290#103—2271no—
291#103—2271no—
292#103C2×D22711yescharacteristic, complemented
293#103C2×D22711yescharacteristic, complemented
294#102—2271no—
295#102—2271no—
296C22⋊C8—2271no—
297C22⋊C8—2271no—
298C22⋊C8—2271no—
299C22⋊C8C2×D22711yescharacteristic, complemented
300C24—4541no—
301#41—4548no—
302#41—4548no—
303#41—4542no—
304#41—4542no—
305#41—4542no—
306#41—4542no—
307#41—4548no—
308#41—4542no—
309#41—4542no—
310#41—4542no—
311#41—2274no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—21no—
315C2×Q8—4541no—
316C2×Q8—4541no—
317C2×Q8—4548no—
318C2×Q8—4542no—
319C2×Q8—4542no—
320C2×Q8—4542no—
321C2×Q8—2274no—
322C2×Q8—2271no—
323C2×Q8—2271no—
324C2×Q8C22×D22711yescharacteristic
325C2×Q8C22×D22711yescharacteristic
326C2×D4—21no—
327C2×D4—4541no—
328C2×D4—4541no—
329C2×D4—4541no—
330C2×D4—4541no—
331C2×D4—4541no—
332C2×D4—4541no—
333C2×D4—4542no—
334C2×D4—4542no—
335C2×D4—4541no—
336C2×D4—2271no—
337C2×D4—2271no—
338C2×D4—2271no—
339C2×D4—2271no—
340C2×D4—2271no—
341C2×D4C22×D22711yescharacteristic
342C22×C4—21no—
343C22×C4—4541no—
344C22×C4—4541no—
345C22×C4—4541no—
346C22×C4—4541no—
347C22×C4—4541no—
348C22×C4—2271no—
349C22×C4—2271no—
350C22×C4C22×D22711yescharacteristic
351SD16—22no—
352SD16—22no—
353SD16—4542no—
354SD16—4542no—
355SD16—4542no—
356SD16—4542no—
357D8—4542no—
358D8—4542no—
359M16—4542no—
360C2×C8—4542no—
361C2×C8—2271no—
362C2×C8—2271no—
363C2×C8C22×D22711yescharacteristic
364C2×C8C22×D22711yescharacteristic
365C4⋊C4—2271no—
366C4⋊C4C22×D22711yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—4541no—
369C22⋊C4—4541no—
370C22⋊C4—4541no—
371C22⋊C4—4541no—
372C42—4541no—
373C23—21no—
374C23—18161no—
375C23—18161no—
376C23—9082no—
377C23—4541no—
378C23—4541no—
379C23—2271no—
380C23—2271no—
381C23D4×D22711yescharacteristic
382Q8—24no—
383Q8—21no—
384Q8—21no—
385Q8—4544no—
386Q8—4544no—
387Q8—4542no—
388Q8—4541no—
389Q8—2272no—
390Q8D4×D22712yes—
391Q8D4×D22712yes—
392D4—21no—
393D4—21no—
394D4—81no—
395D4—18161no—
396D4—18161no—
397D4—18161no—
398D4—9082no—
399D4—4544no—
400D4—4544no—
401D4—4542no—
402D4—4544no—
403D4—4542no—
404D4—4542no—
405D4—4541no—
406D4—4541no—
407D4—4541no—
408D4—2272no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—4544no—
416C2×C4—4544no—
417C2×C4—4544no—
418C2×C4—4544no—
419C2×C4—4544no—
420C2×C4—4541no—
421C2×C4—4541no—
422C2×C4—4541no—
423C2×C4—4541no—
424C2×C4—2271no—
425C2×C4—2271no—
426C2×C4—2272no—
427C2×C4—2272no—
428C2×C4D4×D22711yescharacteristic
429C2×C4C23×D22711yescharacteristic
430C8—21no—
431C8—21no—
432C8—4541no—
433C8—4541no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—18164no—
441C22—9081no—
442C22—9081no—
443C22—9081no—
444C22—9081no—
445C22—9081no—
446C22—4541no—
447C22—4542no—
448C22—4541no—
449C22—4541no—
450C22—4541no—
451C22—4541no—
452C22—2272no—
453C22C2×D4×D22711yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—41no—
458C4—9081no—
459C4—4542no—
460C4—4541no—
461C4—2272no—
462C4C2×D4×D22711yescharacteristic
463C4C2×D4×D22711yescharacteristic
464C2—41no—
465C2—81no—
466C2—18161no—
467C2—9082no—
468C2—9081no—
469C2—4542no—
470C2#120×D22711yescharacteristic
471C2#2840983811yescharacteristic
472C2#2840984111yescharacteristic
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#448×C227C2
socleC22×C227C22×D4
hypercenter#448D227
hypocenterC227#3010
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#20300336C2semidirect
2. C2×#20300315C2semidirect
3. C2×SD16×D227C2semidirect
4. #28409598C2semidirect
5. #28409180C2semidirect
6. #28409179C2semidirect
7. #28409176C2semidirect
8. #28409175C2semidirect
9. #28409172C2semidirect
10. #28409171C2semidirect
11. #28409169C2semidirect
12. #28409039C2semidirect
13. #28409027C2semidirect
14. #28408997C2semidirect
15. #448×C227C2semidirect
20. C2×#16813885C22semidirect
24. C2×#16813884C22semidirect
27. C2×Q8×D227C22semidirect
47. #20300213C22semidirect
48. #20300212C22semidirect
49. #20300212C22semidirect
50. #20300211C22semidirect
51. #20300210C22semidirect
52. #20300208C22semidirect
53. C22⋊(C227⋊C8)C22semidirect
54. (C2×D227)⋊C8C22semidirect
55. (C2×D227)⋊C8C22semidirect
56. C22×Q8×C227C22semidirect
60. #103×C227C22semidirect
61. #103×C227C22semidirect
62. C22⋊C8×C227C22semidirect
68. #16813885D4semidirect
83. Q8×D227D4semidirect
236. C227#3010semidirect
249. #448D227semidirect
265. C22×Q8C2×D227semidirect
292. #103C2×D227semidirect
293. #103C2×D227semidirect
299. C22⋊C8C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003362
C2×#203003152
C2×SD16×D2272
#284095982
#284091802
#284091792
#284091762
#284091752
#284091722
#284091712
#284091692
#284090392
#284090272
#284089972
#448×C2272
#3010227