Finite Group Databasebeta

← #44561868

Subgroups — #44561868

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×#22548747C211yesmaximal, characteristic, complemented
2C22×#16860797C211yesmaximal, characteristic, complemented
3Q8×C22⋊D251C211yesmaximal, characteristic, complemented
4#30632328C211yesmaximal, characteristic
5#30632080C211yesmaximal, characteristic, complemented
6#30631936C211yesmaximal, characteristic, complemented
7#30631926C211yesmaximal, characteristic, complemented
8#30631911C211yesmaximal, characteristic, complemented
9#30631891C211yesmaximal, characteristic, complemented
10#30631848C211yesmaximal, characteristic, complemented
11#30631840C211yesmaximal, characteristic, complemented
12#30631827C211yesmaximal, characteristic, complemented
13#30631790C211yesmaximal, characteristic, complemented
14#30631755C211yesmaximal, characteristic, complemented
15#451×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic, complemented
18C22×D1004C2211yescharacteristic
19#22548747—24no—
20C2×#16860800C2211yescharacteristic, complemented
21C2×#16860799C2211yescharacteristic
22C2×#16860799C2211yescharacteristic
23C2×#16860797—24no—
24C2×#16860797C2211yescharacteristic
25C2×#16860797C2211yescharacteristic
26C2×#16860797C2211yescharacteristic
27C2×#16860797C2211yescharacteristic
28C2×#16860793C2211yescharacteristic, complemented
29C22×C251⋊C8C2211yescharacteristic
30#22548713—21no—
31#22548713C2211yescharacteristic, complemented
32#22548704C2211yescharacteristic
33#22548700—21no—
34#22548694—21no—
35#22548694C2211yescharacteristic
36#22548691—21no—
37#22548691C2211yescharacteristic, complemented
38#22548690—21no—
39#22548690C2211yescharacteristic
40#22548685—21no—
41Q8×Dic251C2211yescharacteristic, complemented
42#22548677C2211yescharacteristic
43C4×C22⋊D251—21no—
44C4×C22⋊D251C2211yescharacteristic
45C22⋊C4×D251—21no—
46#22548648C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548639C2211yescharacteristic, complemented
49#22548638C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic
51#22548637C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic, complemented
53#22548625C2211yescharacteristic, complemented
54C22×Q8×C251C2211yescharacteristic
55C2×Q16×C251C2211yescharacteristic, complemented
56C2×Q16×C251C2211yescharacteristic, complemented
57C22⋊Q8×C251C2211yescharacteristic
58#103×C251C2211yescharacteristic, complemented
59#103×C251C2211yescharacteristic, complemented
60C22⋊C8×C251C2211yescharacteristic
61C23×D251—21no—
62#16860811—22no—
63#16860811—41no—
64C2×Dic502—21no—
65C2×Dic502C2311yescharacteristic
66Q8×D251—22no—
67Q8×D251—41no—
68C2×D1004—22no—
69C2×D1004—21no—
70C2×D1004—21no—
71C2×D1004—21no—
72C2×D1004—41no—
73C2×D1004C2311yescharacteristic
74C2×D1004C2311yescharacteristic
75C2×C22⋊D251D411yescharacteristic
76C2×C4×D251—21no—
77C2×C4×D251—21no—
78C2×C4×D251—21no—
79C2×C4×D251C2311yescharacteristic
80#16860800—22no—
81#16860799—22no—
82#16860799—41no—
83#16860797—22no—
84#16860797—22no—
85#16860797—22no—
86#16860797D414yescomplemented
87#16860793—22no—
88C2×C251⋊C8—22no—
89C2×C251⋊C8C2311yescharacteristic
90C2×C251⋊C8C2311yescharacteristic
91C4⋊Dic251—21no—
92C4⋊Dic251C2311yescharacteristic
93Dic251⋊C4—21no—
94Dic251⋊C4—21no—
95Dic251⋊C4—21no—
96Dic251⋊C4—21no—
97Dic251⋊C4D411yescharacteristic
98C22⋊Dic251—21no—
99C22⋊Dic251D411yescharacteristic
100(C2×D251)⋊C4—21no—
101(C2×D251)⋊C4—21no—
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4D411yescharacteristic
106C4×Dic251—21no—
107C4×Dic251C2311yescharacteristic
108C2×Q8×C251—21no—
109C2×Q8×C251—22no—
110C2×Q8×C251—41no—
111C2×Q8×C251C2311yescharacteristic
112C2×Q8×C251C2311yescharacteristic
113C2×Q8×C251C2311yescharacteristic
114C22×C4×C251—21no—
115C22×C4×C251C2311yescharacteristic
116Q16×C251—22no—
117Q16×C251—41no—
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251—21no—
121C4⋊C4×C251C2311yescharacteristic
122C22⋊C4×C251—21no—
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—42no—
126C22×D251—42no—
127C22×D251—81no—
128C22×D251C2×D411yescharacteristic
129Dic502—21no—
130Dic502—21no—
131Dic502—42no—
132D1004—21no—
133D1004—21no—
134D1004—21no—
135D1004—42no—
136D1004—42no—
137D1004—81no—
138D1004C2×D412yes—
139C22⋊D251—22no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251C2×D411yescharacteristic
143C2×Dic251C2×D411yescharacteristic
144C2×Dic251C2×D411yescharacteristic
145C4×D251—21no—
146C4×D251—21no—
147C4×D251—42no—
148C4×D251—42no—
149C4×D251—81no—
150C251⋊C8—21no—
151C251⋊C8C2×D412yes—
152C23×C251C2×D411yescharacteristic
153Q8×C251—21no—
154Q8×C251—21no—
155Q8×C251—21no—
156Q8×C251—21no—
157Q8×C251—21no—
158Q8×C251—42no—
159Q8×C251C2×D412yes—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
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×C251—21no—
171C22×C251—21no—
172C22×C251—21no—
173C22×C251#13712yescomplemented
174C22×C251C22×D411yescharacteristic
175C2×D251—21no—
176C2×D251—22no—
177C2×D251—21no—
178C2×D251—21no—
179C2×D251—41no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—41no—
184C2×D251—84no—
185C4×C251—21no—
186C4×C251—22no—
187C4×C251—41no—
188C4×C251—41no—
189C4×C251—41no—
190C4×C251C22×D411yescharacteristic
191C4×C251H(5,𝔽2)11yescharacteristic
192Dic251—21no—
193Dic251—21no—
194Dic251—21no—
195Dic251—41no—
196Dic251—41no—
197C2×C251—22no—
198C2×C251D4211yescharacteristic
199C2×C251C2×#13711yescharacteristic
200C2×C251#57711yescharacteristic
201D251—41no—
202D251—42no—
203D251—81no—
204C251#326611yescharacteristic, complemented, Sylow
205#3266—2511nomaximal, Sylow
206C2×#137—2511no—
207C22×SD16—2511no—
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#451D25111yescharacteristic, complemented
217#450—2511no—
218#448—2511no—
219#440—2511no—
220#435—2511no—
221C2×#41—2511no—
222C22×Q8—2511no—
223C22×Q8C2×D25111yescharacteristic
224C22×D4—2511no—
225#137—5024no—
226C2×Q16—2511no—
227C2×Q16C2×D25111yescharacteristic, complemented
228C2×Q16C2×D25111yescharacteristic, complemented
229C2×SD16—5024no—
230C2×SD16—2511no—
231C2×SD16—2511no—
232C2×SD16—2511no—
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×M16—2511no—
237C22×C8—2511no—
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⋊Q8C2×D25111yescharacteristic
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#103C2×D25111yescharacteristic, complemented
260#103C2×D25111yescharacteristic, complemented
261#102—2511no—
262#102—2511no—
263C22⋊C8—2511no—
264C22⋊C8C2×D25111yescharacteristic
265#97—2511no—
266C24—5021no—
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×Q8C22×D25111yescharacteristic
277C2×Q8C22×D25111yescharacteristic
278C2×Q8C22×D25111yescharacteristic
279C2×D4—10041no—
280C2×D4—5021no—
281C2×D4—5021no—
282C2×D4—5022no—
283C2×D4—5021no—
284C2×D4—2511no—
285C2×D4—2511no—
286C2×D4—2511no—
287C22×C4—21no—
288C22×C4—5021no—
289C22×C4—5021no—
290C22×C4—5021no—
291C22×C4—2511no—
292C22×C4C22×D25111yescharacteristic
293Q16—22no—
294Q16—41no—
295Q16—5022no—
296SD16—10041no—
297SD16—5022no—
298SD16—5022no—
299SD16—5022no—
300SD16—5022no—
301SD16—2514no—
302M16—5022no—
303C2×C8—5022no—
304C2×C8—2511no—
305C2×C8—2511no—
306C2×C8C22×D25111yescharacteristic
307C2×C8C22×D25111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—5021no—
310C4⋊C4—5021no—
311C4⋊C4—5021no—
312C4⋊C4—5021no—
313C4⋊C4—5021no—
314C4⋊C4—2511no—
315C4⋊C4—2511no—
316C4⋊C4C22×D25111yescharacteristic
317C22⋊C4—21no—
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—20081no—
329C23—10042no—
330C23—10042no—
331C23—5021no—
332C23—5021no—
333C23—2511no—
334C23D4×D25111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—10042no—
342Q8—5021no—
343Q8—5021no—
344Q8D4×D25112yes—
345D4—20081no—
346D4—10042no—
347D4—10042no—
348D4—5022no—
349D4—5021no—
350D4—5021no—
351D4—5021no—
352D4—2512no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—20081no—
361C2×C4—10042no—
362C2×C4—10042no—
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—21no—
374C8—5021no—
375C8—2512no—
376C22—21no—
377C22—21no—
378C22—20084no—
379C22—10041no—
380C22—10041no—
381C22—10041no—
382C22—10041no—
383C22—10041no—
384C22—5021no—
385C22—5021no—
386C22—5021no—
387C22—5022no—
388C22#2254874712yescomplemented
389C22C2×D4×D25111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—10041no—
396C4—10041no—
397C4—5021no—
398C4—5021no—
399C4—5021no—
400C4C2×D4×D25111yescharacteristic
401C4#2254876011yescharacteristic
402C2—22no—
403C2—20081no—
404C2—10042no—
405C2—10041no—
406C2#3063231911yescharacteristic
407C2C2×#2254874711yescharacteristic
408C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#451×C251C2
socleC22×C251C22×D4
prime radical#30632328C2
hypercenter#451D251
hypocenterC251#3266
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#22548747C2semidirect
2. C22×#16860797C2semidirect
3. Q8×C22⋊D251C2semidirect
5. #30632080C2semidirect
6. #30631936C2semidirect
7. #30631926C2semidirect
8. #30631911C2semidirect
9. #30631891C2semidirect
10. #30631848C2semidirect
11. #30631840C2semidirect
12. #30631827C2semidirect
13. #30631790C2semidirect
14. #30631755C2semidirect
15. #451×C251C2semidirect
17. C2×Q8×D251C22semidirect
20. C2×#16860800C22semidirect
28. C2×#16860793C22semidirect
31. #22548713C22semidirect
37. #22548691C22semidirect
41. Q8×Dic251C22semidirect
46. #22548648C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548639C22semidirect
49. #22548638C22semidirect
52. (C2×D251)⋊C8C22semidirect
53. #22548625C22semidirect
55. C2×Q16×C251C22semidirect
56. C2×Q16×C251C22semidirect
58. #103×C251C22semidirect
59. #103×C251C22semidirect
86. #16860797D4semidirect
173. C22×C251#137semidirect
204. C251#3266semidirect
216. #451D251semidirect
227. C2×Q16C2×D251semidirect
228. C2×Q16C2×D251semidirect
259. #103C2×D251semidirect
260. #103C2×D251semidirect
388. C22#22548747semidirect

Maximal subgroups

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

subgroupindex
C2×#225487472
C22×#168607972
Q8×C22⋊D2512
#306323282
#306320802
#306319362
#306319262
#306319112
#306318912
#306318482
#306318402
#306318272
#306317902
#306317552
#451×C2512
#3266251