Finite Group Databasebeta

← #44560002

Subgroups — #44560002

Order 32128.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D251C211yesmaximal, characteristic, complemented
2C2×#22548723C211yesmaximal, characteristic, complemented
3#30632260C211yesmaximal, characteristic, complemented
4#30632253C211yesmaximal, characteristic
5#30631934C211yesmaximal, characteristic, complemented
6#30631933C211yesmaximal, characteristic, complemented
7#30631930C211yesmaximal, characteristic, complemented
8#30631919C211yesmaximal, characteristic, complemented
9#30631918C211yesmaximal, characteristic, complemented
10#30631915C211yesmaximal, characteristic, complemented
11#30631914C211yesmaximal, characteristic, complemented
12#30631724C211yesmaximal, characteristic, complemented
13#30631694C211yesmaximal, characteristic, complemented
14#30631690C211yesmaximal, characteristic, complemented
15#468×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19C2×#16860799C2211yescharacteristic
20C2×#16860798C2211yescharacteristic
21C2×#16860797C2211yescharacteristic
22SD16×D251—24no—
23#22548723—24no—
24C2×#16860793C2211yescharacteristic
25C2×#16860793C2211yescharacteristic, complemented
26C22×C251⋊C8C2211yescharacteristic, complemented
27C2×C8×D251C2211yescharacteristic
28C2×C8×D251C2211yescharacteristic, complemented
29#22548714—21no—
30#22548711—21no—
31#22548701—21no—
32#22548701—21no—
33#22548700—21no—
34#22548697—21no—
35#22548695C2211yescharacteristic, complemented
36#22548694C2211yescharacteristic, complemented
37#22548691C2211yescharacteristic, complemented
38#22548690C2211yescharacteristic
39#22548689C2211yescharacteristic
40#22548686C2211yescharacteristic
41#22548685—21no—
42#22548683—21no—
43#22548673C2211yescharacteristic
44#22548673C2211yescharacteristic
45#22548649C2211yescharacteristic, complemented
46#22548646C2211yescharacteristic, complemented
47#22548646C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548639C2211yescharacteristic, complemented
50#22548638C2211yescharacteristic, complemented
51#22548637C2211yescharacteristic
52#22548636C2211yescharacteristic
53#22548635C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55C2×M16×C251C2211yescharacteristic, complemented
56C22⋊Q8×C251C2211yescharacteristic, complemented
57#121×C251C2211yescharacteristic
58#107×C251C2211yescharacteristic, complemented
59#103×C251C2211yescharacteristic, complemented
60#102×C251C2211yescharacteristic
61C23×D251—21no—
62#16860810—22no—
63#16860810—22no—
64C2×Dic502—21no—
65C2×Dic502D411yescharacteristic
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—22no—
69C2×D1004—21no—
70C2×D1004D411yescharacteristic
71C2×D1004C2311yescharacteristic
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74C2×C22⋊D251—21no—
75C2×C22⋊D251D411yescharacteristic
76C2×C22⋊D251D411yescharacteristic
77D4×D251—22no—
78D4×D251—22no—
79C2×C4×D251—21no—
80C2×C4×D251C2311yescharacteristic
81C2×C4×D251C2311yescharacteristic
82#16860799—22no—
83#16860798—22no—
84#16860797—22no—
85#16860793—22no—
86#16860793—22no—
87C2×C251⋊C8—22no—
88C2×C251⋊C8C2311yescharacteristic
89C2×C251⋊C8C2311yescharacteristic
90C8×D251—22no—
91C8×D251D414yescomplemented
92C4⋊Dic251—21no—
93C4⋊Dic251C2311yescharacteristic
94C4⋊Dic251C2311yescharacteristic
95Dic251⋊C4—21no—
96Dic251⋊C4—21no—
97C22⋊Dic251—21no—
98C22⋊Dic251—21no—
99(C2×D251)⋊C4—21no—
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—
105C4×Dic251—21no—
106C4×Dic251—21no—
107C2×Q8×C251C2311yescharacteristic
108C2×D4×C251—21no—
109C2×D4×C251C2311yescharacteristic
110C22×C4×C251C2311yescharacteristic
111SD16×C251—22no—
112M16×C251—22no—
113C2×C8×C251C2311yescharacteristic
114C2×C8×C251C2311yescharacteristic
115C4⋊C4×C251—21no—
116C4⋊C4×C251C2311yescharacteristic
117C4⋊C4×C251C2311yescharacteristic
118C22⋊C4×C251—21no—
119C22⋊C4×C251—21no—
120C22×D251—21no—
121C22×D251—42no—
122C22×D251—81no—
123C22×D251—81no—
124C22×D251C2×D411yescharacteristic
125C22×D251C2×D411yescharacteristic
126Dic502—22no—
127Dic502—21no—
128Dic502—21no—
129Dic502—42no—
130D1004—22no—
131D1004—21no—
132D1004—21no—
133D1004—81no—
134C22⋊D251—22no—
135C22⋊D251—22no—
136C22⋊D251—42no—
137C22⋊D251—81no—
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—42no—
147C4×D251C2×D412yes—
148C4×D251#4112yes—
149C251⋊C8—21no—
150C251⋊C8C2×D412yes—
151C23×C251—21no—
152C23×C251C2×D411yescharacteristic
153Q8×C251—21no—
154Q8×C251—21no—
155D4×C251—21no—
156D4×C251—21no—
157D4×C251—81no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251C2×D411yescharacteristic
164C2×C4×C251C2411yescharacteristic
165C8×C251—21no—
166C8×C251C2×D412yes—
167C22×C251—21no—
168C22×C251—21no—
169C22×C251—21no—
170C22×C251—41no—
171C22×C251—41no—
172C22×C251—41no—
173C22×C251C22×D411yescharacteristic
174C2×D251—21no—
175C2×D251—21no—
176C2×D251—21no—
177C2×D251—21no—
178C2×D251—21no—
179C2×D251—41no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—84no—
183C2×D251#12112yes—
184C4×C251—21no—
185C4×C251—41no—
186C4×C251—41no—
187C4×C251—41no—
188C4×C251C22×D411yescharacteristic
189C4×C251C2×#4111yescharacteristic
190Dic251—21no—
191Dic251—41no—
192Dic251—41no—
193Dic251—41no—
194Dic251#12112yes—
195C2×C251—41no—
196C2×C251—81no—
197C2×C251C2×#12111yescharacteristic
198C2×C251#57711yescharacteristic
199C2×C251#57711yescharacteristic
200D251—22no—
201D251—41no—
202D251—81no—
203C251#305711yescharacteristic, complemented, Sylow
204#3057—2511nomaximal, Sylow
205C22×SD16—2511no—
206C2×#131—2511no—
207#536—2511no—
208#535—2511no—
209#468—2511no—
210#468—2511no—
211#468—2511no—
212#468D25111yescharacteristic, complemented
213#465—2511no—
214#465—2511no—
215#465—2511no—
216#465—2511no—
217#427—2511no—
218#419—2511no—
219#418—2511no—
220C2×#41—2511no—
221C22×Q8—2511no—
222C22×D4—2511no—
223C2×SD16—5024no—
224C2×SD16—2511no—
225C2×SD16—2511no—
226C2×SD16—2511no—
227C2×SD16C2×D25111yescharacteristic
228#131—5024no—
229C2×M16—2511no—
230C2×M16—2511no—
231C2×M16C2×D25111yescharacteristic, complemented
232C22×C8—2511no—
233C22×C8—2511no—
234C22×C8—2511no—
235C4⋊Q8—5021no—
236#127—5021no—
237#124—5021no—
238#124—5021no—
239#123—5021no—
240#123—5021no—
241C22⋊Q8—2511no—
242C22⋊Q8—2511no—
243C22⋊Q8—2511no—
244C22⋊Q8C2×D25111yescharacteristic, complemented
245#121—2511no—
246#121—2511no—
247#121—2511no—
248#121C2×D25111yescharacteristic
249#120—5021no—
250#120—5021no—
251#117—2511no—
252#117—2511no—
253#107—2511no—
254#107C2×D25111yescharacteristic, complemented
255#106—2511no—
256#106—2511no—
257#103—2511no—
258#103—2511no—
259#103—2511no—
260#103C2×D25111yescharacteristic, complemented
261#102—2511no—
262#102—2511no—
263#102—2511no—
264#102C2×D25111yescharacteristic
265C24—5021no—
266#41—5022no—
267#41—5022no—
268C2×Q8—5021no—
269C2×Q8—5022no—
270C2×Q8—5022no—
271C2×Q8—2511no—
272C2×Q8—2511no—
273C2×Q8C22×D25111yescharacteristic
274C2×D4—21no—
275C2×D4—5021no—
276C2×D4—5021no—
277C2×D4—5021no—
278C2×D4—5022no—
279C2×D4—5022no—
280C2×D4—5021no—
281C2×D4—2511no—
282C2×D4—2511no—
283C2×D4—2511no—
284C2×D4—2511no—
285C2×D4C22×D25111yescharacteristic
286C22×C4—5021no—
287C22×C4—2511no—
288C22×C4—2511no—
289C22×C4C22×D25111yescharacteristic
290SD16—22no—
291SD16—5022no—
292SD16—5022no—
293SD16—5022no—
294M16—22no—
295M16—5022no—
296M16—5022no—
297C2×C8—5022no—
298C2×C8—5022no—
299C2×C8—2511no—
300C2×C8—2511no—
301C2×C8—2514no—
302C2×C8C22×D25111yescharacteristic
303C2×C8C22×D25111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—5021no—
306C4⋊C4—5021no—
307C4⋊C4—5021no—
308C4⋊C4—2511no—
309C4⋊C4—2511no—
310C4⋊C4C22×D25111yescharacteristic
311C4⋊C4C22×D25111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—5021no—
315C22⋊C4—5021no—
316C22⋊C4—5021no—
317C22⋊C4—5021no—
318C22⋊C4—5021no—
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4—5021no—
322C42—5021no—
323C42—5021no—
324C23—21no—
325C23—20081no—
326C23—20081no—
327C23—10042no—
328C23—5021no—
329C23—2511no—
330C23—2511no—
331C23D4×D25111yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—10042no—
335Q8—5022no—
336Q8—5021no—
337Q8—5021no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—20081no—
342D4—20081no—
343D4—20081no—
344D4—10042no—
345D4—5022no—
346D4—5022no—
347D4—5022no—
348D4—5021no—
349D4—5021no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—10042no—
356C2×C4—5021no—
357C2×C4—5021no—
358C2×C4—5021no—
359C2×C4—5021no—
360C2×C4—5021no—
361C2×C4—2511no—
362C2×C4—2511no—
363C2×C4—2512no—
364C2×C4—2512no—
365C2×C4D4×D25111yescharacteristic
366C2×C4C23×D25111yescharacteristic
367C8—21no—
368C8—5021no—
369C8—2512no—
370C8D4×D25112yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—20084no—
378C22—10041no—
379C22—10041no—
380C22—10041no—
381C22—5021no—
382C22—5021no—
383C22—5021no—
384C22—5021no—
385C22—5021no—
386C22—2512no—
387C22C2×D4×D25111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—10041no—
393C4—10041no—
394C4—10041no—
395C4—5021no—
396C4—2512no—
397C4C2×D4×D25111yescharacteristic
398C4#41×D25111yescharacteristic
399C2—41no—
400C2—81no—
401C2—20081no—
402C2—10041no—
403C2—5022no—
404C2#121×D25111yescharacteristic
405C2#3063249611yescharacteristic
406C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#468×C251C2
socleC22×C251C22×D4
prime radical#30632253C2
hypercenter#468D251
hypocenterC251#3057
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D251C2semidirect
2. C2×#22548723C2semidirect
3. #30632260C2semidirect
5. #30631934C2semidirect
6. #30631933C2semidirect
7. #30631930C2semidirect
8. #30631919C2semidirect
9. #30631918C2semidirect
10. #30631915C2semidirect
11. #30631914C2semidirect
12. #30631724C2semidirect
13. #30631694C2semidirect
14. #30631690C2semidirect
15. #468×C251C2semidirect
25. C2×#16860793C22semidirect
26. C22×C251⋊C8C22semidirect
28. C2×C8×D251C22semidirect
35. #22548695C22semidirect
36. #22548694C22semidirect
37. #22548691C22semidirect
45. #22548649C22semidirect
46. #22548646C22semidirect
47. #22548646C22semidirect
48. #22548640C22semidirect
49. #22548639C22semidirect
50. #22548638C22semidirect
55. C2×M16×C251C22semidirect
56. C22⋊Q8×C251C22semidirect
58. #107×C251C22semidirect
59. #103×C251C22semidirect
91. C8×D251D4semidirect
203. C251#3057semidirect
212. #468D251semidirect
231. C2×M16C2×D251semidirect
244. C22⋊Q8C2×D251semidirect
254. #107C2×D251semidirect
260. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2512
C2×#225487232
#306322602
#306322532
#306319342
#306319332
#306319302
#306319192
#306319182
#306319152
#306319142
#306317242
#306316942
#306316902
#468×C2512
#3057251