Finite Group Databasebeta

← #44561875

Subgroups — #44561875

Order 32128.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548744C211yesmaximal, characteristic, complemented
2Q8×C22⋊D251C211yesmaximal, characteristic, complemented
3#30632331C211yesmaximal, characteristic
4#30632078C211yesmaximal, characteristic, complemented
5#30631930C211yesmaximal, characteristic, complemented
6#30631928C211yesmaximal, characteristic, complemented
7#30631907C211yesmaximal, characteristic, complemented
8#30631890C211yesmaximal, characteristic, complemented
9#30631846C211yesmaximal, characteristic, complemented
10#30631843C211yesmaximal, characteristic, complemented
11#30631838C211yesmaximal, characteristic, complemented
12#30631824C211yesmaximal, characteristic, complemented
13#30631788C211yesmaximal, characteristic, complemented
14#30631758C211yesmaximal, characteristic, complemented
15C22×SD16×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic, complemented
18#22548744—24no—
19C2×Dic1004C2211yescharacteristic, complemented
20C2×#16860800C2211yescharacteristic, complemented
21C2×#16860800C2211yescharacteristic, complemented
22C2×#16860798C2211yescharacteristic
23C2×#16860798C2211yescharacteristic
24C2×#16860793C2211yescharacteristic, complemented
25#22548713—21no—
26#22548713C2211yescharacteristic, complemented
27#22548702C2211yescharacteristic
28#22548696—21no—
29#22548694—21no—
30#22548694C2211yescharacteristic
31#22548694C2211yescharacteristic
32#22548691—21no—
33#22548691C2211yescharacteristic, complemented
34#22548687—21no—
35#22548686C2211yescharacteristic
36#22548684—21no—
37Q8×Dic251C2211yescharacteristic, complemented
38D4×Dic251C2211yescharacteristic
39C4×C22⋊D251—21no—
40C4×C22⋊D251C2211yescharacteristic
41C2×C22⋊Dic251—21no—
42#22548649C2211yescharacteristic, complemented
43Dic251⋊C8C2211yescharacteristic, complemented
44#22548640C2211yescharacteristic, complemented
45#22548640C2211yescharacteristic, complemented
46#22548639C2211yescharacteristic, complemented
47#22548639C2211yescharacteristic, complemented
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic
50C22⋊(C251⋊C8)C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52#22548625C2211yescharacteristic, complemented
53C22×Q8×C251C2211yescharacteristic
54C22×D4×C251C2211yescharacteristic
55C2×SD16×C251—24no—
56C2×SD16×C251C2211yescharacteristic
57C2×SD16×C251C2211yescharacteristic
58C2×SD16×C251C2211yescharacteristic
59C2×SD16×C251C2211yescharacteristic
60C22×C8×C251C2211yescharacteristic
61#16860809—22no—
62#16860809—41no—
63C2×Dic502—21no—
64C2×Dic502C2311yescharacteristic
65C2×Dic502C2311yescharacteristic
66Q8×D251—22no—
67Q8×D251—41no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C22⋊D251D411yescharacteristic
71C22×Dic251—21no—
72C22×Dic251—21no—
73C2×C4×D251—21no—
74C2×C4×D251C2311yescharacteristic
75Dic1004—22no—
76#16860800—22no—
77#16860800—41no—
78#16860798—22no—
79#16860798—41no—
80#16860793—22no—
81C2×C251⋊C8C2311yescharacteristic
82C2×C251⋊C8C2311yescharacteristic
83C4⋊Dic251—21no—
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90Dic251⋊C4—21no—
91Dic251⋊C4D411yescharacteristic
92C22⋊Dic251—21no—
93C22⋊Dic251—21no—
94C22⋊Dic251—21no—
95C22⋊Dic251—21no—
96C22⋊Dic251—21no—
97C22⋊Dic251—21no—
98C22⋊Dic251D411yescharacteristic
99(C2×D251)⋊C4—21no—
100(C2×D251)⋊C4D411yescharacteristic
101C4×Dic251—21no—
102C4×Dic251C2311yescharacteristic
103C24×C251—21no—
104C2×Q8×C251—21no—
105C2×Q8×C251—22no—
106C2×Q8×C251—41no—
107C2×Q8×C251C2311yescharacteristic
108C2×Q8×C251C2311yescharacteristic
109C2×D4×C251—22no—
110C2×D4×C251—21no—
111C2×D4×C251—41no—
112C2×D4×C251C2311yescharacteristic
113C2×D4×C251C2311yescharacteristic
114C22×C4×C251—21no—
115C22×C4×C251C2311yescharacteristic
116SD16×C251—22no—
117SD16×C251—22no—
118SD16×C251—22no—
119SD16×C251D414yescomplemented
120C2×C8×C251—22no—
121C2×C8×C251C2311yescharacteristic
122C2×C8×C251C2311yescharacteristic
123C22×D251C2×D411yescharacteristic
124Dic502—21no—
125Dic502—21no—
126Dic502—21no—
127Dic502—21no—
128Dic502—42no—
129C22⋊D251—22no—
130C22⋊D251—42no—
131C22⋊D251—81no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—42no—
137C2×Dic251—81no—
138C2×Dic251C2×D411yescharacteristic
139C2×Dic251C2×D411yescharacteristic
140C2×Dic251C2×D411yescharacteristic
141C4×D251—21no—
142C4×D251—21no—
143C4×D251—42no—
144C251⋊C8—21no—
145C251⋊C8—21no—
146C23×C251—21no—
147C23×C251—21no—
148C23×C251—42no—
149C23×C251—42no—
150C23×C251—81no—
151C23×C251C2×D411yescharacteristic
152Q8×C251—21no—
153Q8×C251—21no—
154Q8×C251—21no—
155Q8×C251—42no—
156Q8×C251C2×D412yes—
157D4×C251—21no—
158D4×C251—21no—
159D4×C251—21no—
160D4×C251—42no—
161D4×C251C2×D412yes—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—42no—
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2411yescharacteristic
169C8×C251—21no—
170C8×C251C2×D412yes—
171C22×C251—21no—
172C22×C251—21no—
173C22×C251—22no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—84no—
180C22×C251#13712yescomplemented
181C22×C251C22×D411yescharacteristic
182C2×D251—21no—
183C2×D251—21no—
184C2×D251—21no—
185C4×C251—21no—
186C4×C251—22no—
187C4×C251—41no—
188C4×C251C22×D411yescharacteristic
189C4×C251H(5,𝔽2)11yescharacteristic
190Dic251—21no—
191Dic251—21no—
192Dic251—21no—
193Dic251—41no—
194Dic251—41no—
195Dic251—41no—
196Dic251—41no—
197C2×C251—22no—
198C2×C251—42no—
199C2×C251—81no—
200C2×C251D4211yescharacteristic
201C2×C251C2×#13711yescharacteristic
202C2×C251#57711yescharacteristic
203D251—41no—
204C251#326611yescharacteristic, complemented, Sylow
205#3266—2511nomaximal, Sylow
206C2×#137—2511no—
207C22×SD16D25111yescharacteristic, complemented
208D4×Q8—2511no—
209#546—2511no—
210#497—2511no—
211#468—2511no—
212#467—2511no—
213#464—2511no—
214#461—2511no—
215#452—2511no—
216#451—2511no—
217#450—2511no—
218#448—2511no—
219#440—2511no—
220#435—2511no—
221C2×#41—2511no—
222C22×Q8—2511no—
223C22×Q8C2×D25111yescharacteristic
224C22×D4C2×D25111yescharacteristic
225#137—5024no—
226C2×Q16—2511no—
227C2×Q16—2511no—
228C2×Q16—2511no—
229C2×SD16—24no—
230C2×SD16—2511no—
231C2×SD16—2511no—
232C2×SD16C2×D25111yescharacteristic
233C2×SD16C2×D25111yescharacteristic
234C2×SD16C2×D25111yescharacteristic
235C2×SD16C2×D25111yescharacteristic
236C2×M16—2511no—
237C22×C8C2×D25111yescharacteristic
238C4⋊Q8—5021no—
239C4⋊Q8—2511no—
240#124—2511no—
241#123—5021no—
242C22⋊Q8—5021no—
243C22⋊Q8—5021no—
244C22⋊Q8—2511no—
245C22⋊Q8—2511no—
246C22⋊Q8—2511no—
247#121—5021no—
248#121—2511no—
249#120—5021no—
250C4×Q8—2511no—
251C4×D4—5021no—
252C4×D4—2511no—
253C4×D4—2511no—
254C2×C22⋊C4—5021no—
255#107—2511no—
256C4⋊C8—2511no—
257#103—2511no—
258#103—2511no—
259#103—2511no—
260#103—2511no—
261#102—2511no—
262#102—2511no—
263C22⋊C8—2511no—
264C22⋊C8—2511no—
265#97—2511no—
266C24—21no—
267#41—10041no—
268#41—5022no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—10041no—
273C2×Q8—5021no—
274C2×Q8—5022no—
275C2×Q8—2511no—
276C2×Q8—2511no—
277C2×Q8C22×D25111yescharacteristic
278C2×Q8C22×D25111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—2511no—
285C2×D4C22×D25111yescharacteristic
286C2×D4C22×D25111yescharacteristic
287C22×C4—21no—
288C22×C4—5021no—
289C22×C4—5021no—
290C22×C4—5021no—
291C22×C4—2511no—
292C22×C4C22×D25111yescharacteristic
293Q16—10041no—
294Q16—5022no—
295Q16—5022no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—10041no—
300SD16—5022no—
301SD16C22⋊D25114yescomplemented
302M16—5022no—
303C2×C8—22no—
304C2×C8—2511no—
305C2×C8—2511no—
306C2×C8C22×D25111yescharacteristic
307C2×C8C22×D25111yescharacteristic
308C4⋊C4—5021no—
309C4⋊C4—5021no—
310C4⋊C4—5021no—
311C4⋊C4—5021no—
312C4⋊C4—5021no—
313C4⋊C4—5021no—
314C4⋊C4—2511no—
315C4⋊C4—2511no—
316C4⋊C4—2511no—
317C22⋊C4—5021no—
318C22⋊C4—5021no—
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—2511no—
325C22⋊C4—2511no—
326C42—5021no—
327C42—2511no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—2511no—
334C23D4×D25111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—10042no—
340Q8—5021no—
341Q8—5021no—
342Q8—5021no—
343Q8—5021no—
344Q8C2×C22⋊D25112yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—20081no—
350D4—10042no—
351D4—5022no—
352D4C2×C22⋊D25112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—20081no—
359C2×C4—10042no—
360C2×C4—10042no—
361C2×C4—5021no—
362C2×C4—5021no—
363C2×C4—5021no—
364C2×C4—5021no—
365C2×C4—5021no—
366C2×C4—5021no—
367C2×C4—2511no—
368C2×C4—2511no—
369C2×C4—2511no—
370C2×C4D4×D25111yescharacteristic
371C2×C4C23×D25111yescharacteristic
372C8—21no—
373C8—5021no—
374C8—5021no—
375C8C2×C22⋊D25112yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—5021no—
386C22—5021no—
387C22—5021no—
388C22#2254874412yescomplemented
389C22C2×D4×D25111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—10041no—
394C4—10041no—
395C4—10041no—
396C4—10041no—
397C4—5021no—
398C4—5021no—
399C4—5021no—
400C4C22×C22⋊D25111yescharacteristic
401C4#2254876011yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—10041no—
406C2D4×C22⋊D25111yescharacteristic
407C2C2×#2254874411yescharacteristic
408C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×SD16×C251C2
socleC22×C251C22×D4
prime radical#30632331C2
hypercenterC22×SD16D251
hypocenterC251#3266
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22548744C2semidirect
2. Q8×C22⋊D251C2semidirect
4. #30632078C2semidirect
5. #30631930C2semidirect
6. #30631928C2semidirect
7. #30631907C2semidirect
8. #30631890C2semidirect
9. #30631846C2semidirect
10. #30631843C2semidirect
11. #30631838C2semidirect
12. #30631824C2semidirect
13. #30631788C2semidirect
14. #30631758C2semidirect
15. C22×SD16×C251C2semidirect
17. C2×Q8×D251C22semidirect
19. C2×Dic1004C22semidirect
20. C2×#16860800C22semidirect
21. C2×#16860800C22semidirect
24. C2×#16860793C22semidirect
26. #22548713C22semidirect
33. #22548691C22semidirect
37. Q8×Dic251C22semidirect
42. #22548649C22semidirect
43. Dic251⋊C8C22semidirect
44. #22548640C22semidirect
45. #22548640C22semidirect
46. #22548639C22semidirect
47. #22548639C22semidirect
51. (C2×D251)⋊C8C22semidirect
52. #22548625C22semidirect
119. SD16×C251D4semidirect
180. C22×C251#137semidirect
204. C251#3266semidirect
207. C22×SD16D251semidirect
301. SD16C22⋊D251semidirect
388. C22#22548744semidirect

Maximal subgroups

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

subgroupindex
C2×#225487442
Q8×C22⋊D2512
#306323312
#306320782
#306319302
#306319282
#306319072
#306318902
#306318462
#306318432
#306318382
#306318242
#306317882
#306317582
C22×SD16×C2512
#3266251