Finite Group Databasebeta

← #44561874

Subgroups — #44561874

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
2C2×SD16×D251C211yesmaximal, characteristic, complemented
3#30632352C211yesmaximal, characteristic, complemented
4#30632330C211yesmaximal, characteristic
5#30631930C211yesmaximal, characteristic, complemented
6#30631925C211yesmaximal, characteristic, complemented
7#30631906C211yesmaximal, characteristic, complemented
8#30631889C211yesmaximal, characteristic, complemented
9#30631848C211yesmaximal, characteristic, complemented
10#30631841C211yesmaximal, characteristic, complemented
11#30631836C211yesmaximal, characteristic, complemented
12#30631824C211yesmaximal, characteristic, complemented
13#30631789C211yesmaximal, characteristic, complemented
14#30631760C211yesmaximal, characteristic, complemented
15#497×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×Q8×D251C2211yescharacteristic, complemented
19C2×D4×D251C2211yescharacteristic
20#22548747—24no—
21C2×#16860800C2211yescharacteristic, complemented
22C2×#16860799C2211yescharacteristic
23C2×#16860798C2211yescharacteristic
24C2×#16860797C2211yescharacteristic
25C2×#16860797C2211yescharacteristic
26SD16×D251—24no—
27C2×#16860793C2211yescharacteristic, complemented
28C2×C8×D251C2211yescharacteristic
29#22548714—21no—
30#22548697—21no—
31#22548693—21no—
32#22548693—21no—
33#22548691C2211yescharacteristic
34#22548691C2211yescharacteristic
35#22548691C2211yescharacteristic, complemented
36#22548688—21no—
37#22548686C2211yescharacteristic
38#22548685—21no—
39C4×Dic502C2211yescharacteristic, complemented
40C4×D1004C2211yescharacteristic
41C4×D1004C2211yescharacteristic
42#22548677—21no—
43C22⋊C4×D251—21no—
44#22548649C2211yescharacteristic, complemented
45C4⋊(C251⋊C8)C2211yescharacteristic, complemented
46#22548640C2211yescharacteristic, complemented
47#22548639C2211yescharacteristic, complemented
48#22548639C2211yescharacteristic, complemented
49#22548639C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic
51#22548636C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic
53(C2×D251)⋊C8C2211yescharacteristic, complemented
54C2×Q16×C251C2211yescharacteristic, complemented
55C2×Q16×C251C2211yescharacteristic, complemented
56C2×SD16×C251C2211yescharacteristic
57C2×SD16×C251C2211yescharacteristic
58C4⋊Q8×C251C2211yescharacteristic, complemented
59#124×C251C2211yescharacteristic
60#97×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860811—22no—
63#16860811—41no—
64C2×Dic502—21no—
65C2×Dic502—21no—
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—22no—
69Q8×D251—41no—
70Q8×D251—41no—
71C2×D1004—21no—
72C2×D1004D411yescharacteristic
73C2×D1004C2311yescharacteristic
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76D4×D251—22no—
77D4×D251—41no—
78C2×C4×D251—21no—
79C2×C4×D251—21no—
80C2×C4×D251—21no—
81C2×C4×D251—21no—
82C2×C4×D251C2311yescharacteristic
83C2×C4×D251C2311yescharacteristic
84#16860800—22no—
85#16860799D414yescomplemented
86#16860798—22no—
87#16860797—22no—
88#16860797—22no—
89#16860793—22no—
90C2×C251⋊C8C2311yescharacteristic
91C2×C251⋊C8C2311yescharacteristic
92C8×D251—22no—
93C4⋊Dic251D411yescharacteristic
94C4⋊Dic251C2311yescharacteristic
95C4⋊Dic251C2311yescharacteristic
96Dic251⋊C4—21no—
97Dic251⋊C4—21no—
98Dic251⋊C4—21no—
99Dic251⋊C4—21no—
100C22⋊Dic251—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
106(C2×D251)⋊C4D411yescharacteristic
107C4×Dic251—21no—
108C2×Q8×C251C2311yescharacteristic
109C2×Q8×C251C2311yescharacteristic
110C2×Q8×C251C2311yescharacteristic
111C2×D4×C251C2311yescharacteristic
112Q16×C251—22no—
113Q16×C251—41no—
114SD16×C251—22no—
115SD16×C251—41no—
116C2×C8×C251C2311yescharacteristic
117C2×C8×C251C2311yescharacteristic
118C4⋊C4×C251—21no—
119C4⋊C4×C251—21no—
120C22⋊C4×C251—21no—
121C22⋊C4×C251—21no—
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—42no—
125C22×D251—42no—
126C22×D251—81no—
127C22×D251C2×D411yescharacteristic
128C22×D251C2×D411yescharacteristic
129Dic502—21no—
130Dic502—42no—
131Dic502—42no—
132Dic502C2×D412yes—
133D1004—22no—
134D1004—21no—
135D1004—42no—
136D1004C2×D412yes—
137C22⋊D251—42no—
138C22⋊D251—81no—
139C2×Dic251—21no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251C2×D411yescharacteristic
143C2×Dic251C2×D411yescharacteristic
144C4×D251—21no—
145C4×D251—21no—
146C4×D251—21no—
147C4×D251—21no—
148C4×D251—42no—
149C4×D251—42no—
150C4×D251—42no—
151C4×D251—81no—
152C251⋊C8—21no—
153C251⋊C8—21no—
154C23×C251—21no—
155Q8×C251—21no—
156Q8×C251—21no—
157Q8×C251—21no—
158Q8×C251—21no—
159Q8×C251—21no—
160Q8×C251—21no—
161D4×C251—21no—
162D4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251C2×D411yescharacteristic
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2411yescharacteristic
169C8×C251—21no—
170C8×C251C2×D412yes—
171C22×C251—41no—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251C22×D411yescharacteristic
175C2×D251—21no—
176C2×D251—21no—
177C2×D251—21no—
178C2×D251—22no—
179C2×D251—21no—
180C2×D251—21no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—84no—
184C2×D251#13712yescomplemented
185C4×C251—21no—
186C4×C251—21no—
187C4×C251—41no—
188C4×C251—41no—
189C4×C251—41no—
190C4×C251C22×D411yescharacteristic
191C4×C251H(5,𝔽2)11yescharacteristic
192Dic251—21no—
193Dic251—21no—
194Dic251—22no—
195Dic251—41no—
196Dic251—41no—
197C2×C251—81no—
198C2×C251D4211yescharacteristic
199C2×C251C2×#13711yescharacteristic
200C2×C251#57711yescharacteristic
201D251—22no—
202D251—41no—
203D251—42no—
204C251#326611yescharacteristic, complemented, Sylow
205#3266—2511nomaximal, Sylow
206C2×#137—2511no—
207C22×SD16—2511no—
208D4×Q8—2511no—
209#546—2511no—
210#497D25111yescharacteristic, complemented
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×Q8—2511no—
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×SD16C2×D25111yescharacteristic
235C2×SD16C2×D25111yescharacteristic
236C2×M16—2511no—
237C22×C8—2511no—
238C4⋊Q8—5021no—
239C4⋊Q8C2×D25111yescharacteristic, complemented
240#124C2×D25111yescharacteristic
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#97C2×D25111yescharacteristic, complemented
266C24—5021no—
267#41—10041no—
268#41—5022no—
269C2×Q8—10041no—
270C2×Q8—10041no—
271C2×Q8—5021no—
272C2×Q8—5021no—
273C2×Q8—5022no—
274C2×Q8—5022no—
275C2×Q8—2511no—
276C2×Q8C22×D25111yescharacteristic
277C2×Q8C22×D25111yescharacteristic
278C2×Q8C22×D25111yescharacteristic
279C2×D4—10041no—
280C2×D4—5021no—
281C2×D4—5022no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—2511no—
285C2×D4—2511no—
286C2×D4C22×D25111yescharacteristic
287C22×C4—5021no—
288C22×C4—5021no—
289C22×C4—5021no—
290C22×C4—5021no—
291C22×C4—2511no—
292C22×C4—2511no—
293Q16—22no—
294Q16—41no—
295Q16—5022no—
296SD16—22no—
297SD16—41no—
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—21no—
310C4⋊C4—5021no—
311C4⋊C4—5021no—
312C4⋊C4—5021no—
313C4⋊C4—5021no—
314C4⋊C4—2511no—
315C4⋊C4—2511no—
316C4⋊C4—2511no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C22⋊C4—5021no—
323C22⋊C4—5021no—
324C22⋊C4—2511no—
325C22⋊C4—2511no—
326C42—5021no—
327C42C22×D25111yescharacteristic
328C23—21no—
329C23—20081no—
330C23—10042no—
331C23—10042no—
332C23—5021no—
333C23—2511no—
334C23—2511no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—10042no—
342Q8—10042no—
343Q8—5021no—
344Q8—2512no—
345D4—21no—
346D4—21no—
347D4—20081no—
348D4—10042no—
349D4—10042no—
350D4—5022no—
351D4—5021no—
352D4—2512no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—20081no—
357C2×C4—10042no—
358C2×C4—10042no—
359C2×C4—10042no—
360C2×C4—5021no—
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×C4D4×D25111yescharacteristic
370C2×C4D4×D25111yescharacteristic
371C2×C4C23×D25111yescharacteristic
372C8—21no—
373C8—5021no—
374C8—5021no—
375C8D4×D25112yes—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—20084no—
380C22—10041no—
381C22—10041no—
382C22—5021no—
383C22—5021no—
384C22—5021no—
385C22—5022no—
386C22—5021no—
387C22—5021no—
388C22—2512no—
389C22C2×D4×D25111yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—10041no—
396C4—10041no—
397C4—5021no—
398C4—5022no—
399C4—5021no—
400C4C2×D4×D25111yescharacteristic
401C4#2254876011yescharacteristic
402C2—81no—
403C2—10042no—
404C2—10041no—
405C2—5022no—
406C2#3063232211yescharacteristic
407C2#137×D25111yescharacteristic
408C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#497×C251C2
socleC22×C251C22×D4
prime radical#30632330C2
hypercenter#497D251
hypocenterC251#3266
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22548747C2semidirect
2. C2×SD16×D251C2semidirect
3. #30632352C2semidirect
5. #30631930C2semidirect
6. #30631925C2semidirect
7. #30631906C2semidirect
8. #30631889C2semidirect
9. #30631848C2semidirect
10. #30631841C2semidirect
11. #30631836C2semidirect
12. #30631824C2semidirect
13. #30631789C2semidirect
14. #30631760C2semidirect
15. #497×C251C2semidirect
18. C2×Q8×D251C22semidirect
21. C2×#16860800C22semidirect
27. C2×#16860793C22semidirect
35. #22548691C22semidirect
39. C4×Dic502C22semidirect
44. #22548649C22semidirect
45. C4⋊(C251⋊C8)C22semidirect
46. #22548640C22semidirect
47. #22548639C22semidirect
48. #22548639C22semidirect
49. #22548639C22semidirect
53. (C2×D251)⋊C8C22semidirect
54. C2×Q16×C251C22semidirect
55. C2×Q16×C251C22semidirect
58. C4⋊Q8×C251C22semidirect
60. #97×C251C22semidirect
85. #16860799D4semidirect
184. C2×D251#137semidirect
204. C251#3266semidirect
210. #497D251semidirect
227. C2×Q16C2×D251semidirect
228. C2×Q16C2×D251semidirect
239. C4⋊Q8C2×D251semidirect
265. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487472
C2×SD16×D2512
#306323522
#306323302
#306319302
#306319252
#306319062
#306318892
#306318482
#306318412
#306318362
#306318242
#306317892
#306317602
#497×C2512
#3266251