Finite Group Databasebeta

← #44559899

Subgroups — #44559899

Order 32128.

504 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×#22548739C211yesmaximal, characteristic, complemented
2C2×#22548734C211yesmaximal, characteristic, complemented
3#30632250C211yesmaximal, characteristic, complemented
4#30632166C211yesmaximal, characteristic, complemented
5#30631910C211yesmaximal, characteristic, complemented
6#30631908C211yesmaximal, characteristic, complemented
7#30631902C211yesmaximal, characteristic, complemented
8#30631900C211yesmaximal, characteristic, complemented
9#30631886C211yesmaximal, characteristic, complemented
10#30631884C211yesmaximal, characteristic, complemented
11#30631880C211yesmaximal, characteristic, complemented
12#30631712C211yesmaximal, characteristic, complemented
13#30631685C211yesmaximal, characteristic, complemented
14#102×D251C211yesmaximal, characteristic, complemented
15#459×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×#16860809C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548739—24no—
20#22548734—24no—
21C2×Dic1004C2211yescharacteristic, complemented
22C2×#16860799C2211yescharacteristic
23C2×#16860798C2211yescharacteristic
24C2×#16860798C2211yescharacteristic, complemented
25C2×#16860798C2211yescharacteristic, complemented
26C2×#16860795C2211yescharacteristic
27C2×#16860793C2211yescharacteristic
28C2×C8×D251C2211yescharacteristic
29#22548711C2211yescharacteristic
30#22548703C2211yescharacteristic
31#22548702C2211yescharacteristic
32#22548687—21no—
33#22548687—21no—
34#22548683—21no—
35#22548683—21no—
36C4×Dic502C2211yescharacteristic, complemented
37#22548680C2211yescharacteristic, complemented
38D4×Dic251C2211yescharacteristic, complemented
39C4×C22⋊D251—21no—
40(C22⋊D251)⋊C4—21no—
41#22548672C2211yescharacteristic
42#22548671—21no—
43C2×Dic251⋊C4—21no—
44C4⋊C4×D251C2211yescharacteristic
45C4⋊(C251⋊C8)C2211yescharacteristic, complemented
46Dic251⋊C8C2211yescharacteristic, complemented
47Dic251⋊C8C2211yescharacteristic, complemented
48#22548640C2211yescharacteristic, complemented
49#22548638C2211yescharacteristic, complemented
50#22548637C2211yescharacteristic
51#22548636C2211yescharacteristic
52#22548636C2211yescharacteristic, complemented
53#22548635C2211yescharacteristic
54C2×D8×C251C2211yescharacteristic
55C2×D8×C251C2211yescharacteristic, complemented
56#127×C251C2211yescharacteristic
57C4×D4×C251C2211yescharacteristic, complemented
58C4⋊C8×C251C2211yescharacteristic, complemented
59#102×C251C2211yescharacteristic
60#102×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860809—22no—
63#16860809—22no—
64#16860809—22no—
65#16860809D414yescomplemented
66C2×Dic502C2311yescharacteristic
67C2×Dic502C2311yescharacteristic
68C2×D1004C2311yescharacteristic
69C2×C22⋊D251—21no—
70C2×C22⋊D251—21no—
71C2×C22⋊D251—21no—
72C2×C22⋊D251—21no—
73C2×C22⋊D251—21no—
74D4×D251—22no—
75D4×D251—22no—
76C22×Dic251—21no—
77C22×Dic251—21no—
78C22×Dic251—21no—
79C2×C4×D251—21no—
80C2×C4×D251C2311yescharacteristic
81Dic1004—22no—
82#16860799—22no—
83#16860798—22no—
84#16860798—22no—
85#16860798—22no—
86#16860795—22no—
87#16860793—22no—
88C2×C251⋊C8C2311yescharacteristic
89C2×C251⋊C8C2311yescharacteristic
90C8×D251—22no—
91C4⋊Dic251C2311yescharacteristic
92Dic251⋊C4—21no—
93Dic251⋊C4—21no—
94Dic251⋊C4—21no—
95Dic251⋊C4D411yescharacteristic
96Dic251⋊C4D411yescharacteristic
97C22⋊Dic251—21no—
98C22⋊Dic251—21no—
99C22⋊Dic251—21no—
100(C2×D251)⋊C4—21no—
101(C2×D251)⋊C4—21no—
102(C2×D251)⋊C4D411yescharacteristic
103(C2×D251)⋊C4D411yescharacteristic
104C4×Dic251—21no—
105C4×Dic251—21no—
106C4×Dic251C2311yescharacteristic
107C2×D4×C251—21no—
108C2×D4×C251—21no—
109C2×D4×C251C2311yescharacteristic
110C2×D4×C251C2311yescharacteristic
111C2×D4×C251C2311yescharacteristic
112C22×C4×C251—21no—
113D8×C251—22no—
114D8×C251—22no—
115C2×C8×C251C2311yescharacteristic
116C2×C8×C251C2311yescharacteristic
117C4⋊C4×C251C2311yescharacteristic
118C22⋊C4×C251—21no—
119C42×C251C2311yescharacteristic
120C22×D251—21no—
121C22×D251—42no—
122C22×D251—81no—
123C22×D251—81no—
124C22×D251C2×D411yescharacteristic
125Dic502—21no—
126Dic502—21no—
127Dic502—21no—
128Dic502C2×D412yes—
129D1004—21no—
130D1004—21no—
131C22⋊D251—24no—
132C22⋊D251—42no—
133C22⋊D251—42no—
134C22⋊D251—81no—
135C22⋊D251—81no—
136C2×Dic251—21no—
137C2×Dic251—24no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—42no—
141C2×Dic251—81no—
142C2×Dic251C2×D411yescharacteristic
143C2×Dic251C2×D411yescharacteristic
144C2×Dic251C2×D411yescharacteristic
145C4×D251—81no—
146C4×D251C2×D412yes—
147C4×D251#4112yes—
148C251⋊C8—21no—
149C251⋊C8—21no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153D4×C251—21no—
154D4×C251—21no—
155D4×C251—21no—
156D4×C251—21no—
157D4×C251—21no—
158D4×C251—81no—
159D4×C251—81no—
160D4×C251C2×D412yes—
161C2×C4×C251—21no—
162C2×C4×C251—81no—
163C2×C4×C251C2×D411yescharacteristic
164C2×C4×C251C2×D411yescharacteristic
165C2×C4×C251C2411yescharacteristic
166C8×C251—21no—
167C8×C251—21no—
168C22×C251—22no—
169C22×C251—41no—
170C22×C251—41no—
171C22×C251—41no—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251C22×D411yescharacteristic
178C2×D251—21no—
179C2×D251—21no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—84no—
184C2×D251#12112yes—
185C4×C251—21no—
186C4×C251—21no—
187C4×C251—41no—
188C4×C251C22×D411yescharacteristic
189C4×C251C2×#4111yescharacteristic
190Dic251—22no—
191Dic251—21no—
192Dic251—21no—
193Dic251—41no—
194Dic251—41no—
195Dic251#12112yes—
196C2×C251—42no—
197C2×C251—81no—
198C2×C251—81no—
199C2×C251C2×#12111yescharacteristic
200C2×C251#57611yescharacteristic
201C2×C251#57711yescharacteristic
202D251—22no—
203D251—81no—
204C251#304511yescharacteristic, complemented, Sylow
205#3045—2511nomaximal, Sylow
206C2×AGL(1,ℤ/8)—2511no—
207C2×#135—2511no—
208#535—2511no—
209#520—2511no—
210#464—2511no—
211#464—2511no—
212#463—2511no—
213#463—2511no—
214#460—2511no—
215#460—2511no—
216#459—2511no—
217#459D25111yescharacteristic, complemented
218#424—2511no—
219#417—2511no—
220C2×#102—2511no—
221C2×#41—2511no—
222C2×#41—2511no—
223C22×D4—2511no—
224AGL(1,ℤ/8)—5024no—
225#135—5024no—
226C2×Q16—2511no—
227C2×SD16—2511no—
228C2×SD16—2511no—
229C2×SD16—2511no—
230C2×SD16—2511no—
231C2×D8—2511no—
232C2×D8C2×D25111yescharacteristic
233C2×D8C2×D25111yescharacteristic, complemented
234C2×M16—2511no—
235C22×C8—2511no—
236#127—2511no—
237#127C2×D25111yescharacteristic
238#124—2511no—
239#124—2511no—
240#121—5021no—
241#121—5021no—
242#120—5021no—
243#120—5021no—
244C4×Q8—2511no—
245C4×Q8—2511no—
246C4×D4—5021no—
247C4×D4—5021no—
248C4×D4—2511no—
249C4×D4C2×D25111yescharacteristic, complemented
250#117—5021no—
251#117—2511no—
252C2×C4⋊C4—5021no—
253C2×C4⋊C4—2511no—
254C4⋊C8—2511no—
255C4⋊C8—2511no—
256C4⋊C8—2511no—
257C4⋊C8C2×D25111yescharacteristic, complemented
258#103—2511no—
259#103—2511no—
260#102—2511no—
261#102—2511no—
262#102—2511no—
263#102—2511no—
264#102C2×D25111yescharacteristic
265#102C2×D25111yescharacteristic, complemented
266C24—5021no—
267#41—5022no—
268#41—5022no—
269#41—5022no—
270#41—2514no—
271C2×Q8—2511no—
272C2×Q8—2511no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—5021no—
276C2×D4—5021no—
277C2×D4—5022no—
278C2×D4—5022no—
279C2×D4—5021no—
280C2×D4—5021no—
281C2×D4—5021no—
282C2×D4—2511no—
283C2×D4C22×D25111yescharacteristic
284C2×D4C22×D25111yescharacteristic
285C2×D4C22×D25111yescharacteristic
286C22×C4—21no—
287C22×C4—5021no—
288C22×C4—5021no—
289C22×C4—5021no—
290C22×C4—5021no—
291C22×C4—2511no—
292Q16—5022no—
293SD16—5022no—
294SD16—5022no—
295SD16—5022no—
296SD16—5022no—
297D8—22no—
298D8—22no—
299D8—5022no—
300M16—5022no—
301C2×C8—5022no—
302C2×C8—2511no—
303C2×C8—2511no—
304C2×C8C22×D25111yescharacteristic
305C2×C8C22×D25111yescharacteristic
306C4⋊C4—5021no—
307C4⋊C4—5021no—
308C4⋊C4—5021no—
309C4⋊C4—2511no—
310C4⋊C4—2511no—
311C4⋊C4—2511no—
312C4⋊C4C22×D25111yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—5021no—
315C22⋊C4—5021no—
316C22⋊C4—5021no—
317C22⋊C4—5021no—
318C22⋊C4—5021no—
319C22⋊C4—2511no—
320C22⋊C4—2511no—
321C42—5021no—
322C42—5021no—
323C42—2511no—
324C42C22×D25111yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—20081no—
329C23—20081no—
330C23—10042no—
331C23—5021no—
332C23—2511no—
333Q8—5021no—
334Q8—5021no—
335Q8—5021no—
336Q8—2512no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—81no—
343D4—81no—
344D4—20081no—
345D4—20081no—
346D4—10042no—
347D4—10042no—
348D4—5024no—
349D4—5021no—
350D4—5021no—
351D4D4×D25112yes—
352C2×C4—21no—
353C2×C4—81no—
354C2×C4—20081no—
355C2×C4—20081no—
356C2×C4—10042no—
357C2×C4—5021no—
358C2×C4—5021no—
359C2×C4—5021no—
360C2×C4—5024no—
361C2×C4—2511no—
362C2×C4—2511no—
363C2×C4—2511no—
364C2×C4—2512no—
365C2×C4—2512no—
366C2×C4D4×D25111yescharacteristic
367C2×C4D4×D25111yescharacteristic
368C2×C4C23×D25111yescharacteristic
369C8—21no—
370C8—21no—
371C8—5021no—
372C8—5021no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—20084no—
383C22—10041no—
384C22—10041no—
385C22—10041no—
386C22—5021no—
387C22—5021no—
388C22—2512no—
389C22C2×D4×D25111yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—10041no—
394C4—10041no—
395C4—5021no—
396C4—5021no—
397C4—5022no—
398C4—2512no—
399C4C2×D4×D25111yescharacteristic
400C4#41×D25111yescharacteristic
401C2—42no—
402C2—81no—
403C2—81no—
404C2—20081no—
405C2—5022no—
406C2#121×D25111yescharacteristic
407C2#3063248811yescharacteristic
408C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#459×C251C2
socleC22×C251C22×D4
hypercenter#459D251
hypocenterC251#3045
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22548739C2semidirect
2. C2×#22548734C2semidirect
3. #30632250C2semidirect
4. #30632166C2semidirect
5. #30631910C2semidirect
6. #30631908C2semidirect
7. #30631902C2semidirect
8. #30631900C2semidirect
9. #30631886C2semidirect
10. #30631884C2semidirect
11. #30631880C2semidirect
12. #30631712C2semidirect
13. #30631685C2semidirect
14. #102×D251C2semidirect
15. #459×C251C2semidirect
21. C2×Dic1004C22semidirect
24. C2×#16860798C22semidirect
25. C2×#16860798C22semidirect
36. C4×Dic502C22semidirect
37. #22548680C22semidirect
38. D4×Dic251C22semidirect
45. C4⋊(C251⋊C8)C22semidirect
46. Dic251⋊C8C22semidirect
47. Dic251⋊C8C22semidirect
48. #22548640C22semidirect
49. #22548638C22semidirect
52. #22548636C22semidirect
55. C2×D8×C251C22semidirect
57. C4×D4×C251C22semidirect
58. C4⋊C8×C251C22semidirect
60. #102×C251C22semidirect
65. #16860809D4semidirect
204. C251#3045semidirect
217. #459D251semidirect
233. C2×D8C2×D251semidirect
249. C4×D4C2×D251semidirect
257. C4⋊C8C2×D251semidirect
265. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487392
C2×#225487342
#306322502
#306321662
#306319102
#306319082
#306319022
#306319002
#306318862
#306318842
#306318802
#306317122
#306316852
#102×D2512
#459×C2512
#3045251