Finite Group Databasebeta

← #44560650

Subgroups — #44560650

Order 32128.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632290C211yesmaximal, characteristic
2#30632165C211yesmaximal, characteristic, complemented
3#30632012C211yesmaximal, characteristic, complemented
4#30632008C211yesmaximal, characteristic, complemented
5#30631996C211yesmaximal, characteristic, complemented
6#30631992C211yesmaximal, characteristic, complemented
7#30631885C211yesmaximal, characteristic, complemented
8#30631885C211yesmaximal, characteristic, complemented
9#30631826C211yesmaximal, characteristic, complemented
10#30631790C211yesmaximal, characteristic, complemented
11#30631788C211yesmaximal, characteristic, complemented
12SD16×Dic251C211yesmaximal, characteristic, complemented
13#30631772C211yesmaximal, characteristic, complemented
14#30631745C211yesmaximal, characteristic, complemented
15#448×C251C211yesmaximal, characteristic, complemented
16C2×#16860797C2211yescharacteristic
17C2×#16860797C2211yescharacteristic
18#22548711C2211yescharacteristic
19#22548706C2211yescharacteristic
20#22548699—21no—
21#22548697—21no—
22#22548689C2211yescharacteristic
23#22548687—21no—
24#22548686—21no—
25Q8×Dic251—21no—
26Q8×Dic251—21no—
27Q8×Dic251C2211yescharacteristic
28Q8×Dic251C2211yescharacteristic
29Q8×Dic251C2211yescharacteristic
30Q8×Dic251C2211yescharacteristic
31D4×Dic251C2211yescharacteristic
32#22548677C2211yescharacteristic
33#22548674—21no—
34#22548674—21no—
35#22548674C2211yescharacteristic
36#22548674C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548648C2211yescharacteristic
39#22548647C2211yescharacteristic
40#22548646C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50#22548625C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C22×Q8×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59C2×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C2×C22⋊D251—21no—
62C2×C22⋊D251—21no—
63C22×Dic251—21no—
64C2×C4×D251—21no—
65#16860797—22no—
66#16860797—22no—
67C2×C251⋊C8C2311yescharacteristic
68C2×C251⋊C8C2311yescharacteristic
69C4⋊Dic251—21no—
70C4⋊Dic251—21no—
71C4⋊Dic251—21no—
72C4⋊Dic251—21no—
73C4⋊Dic251—21no—
74C4⋊Dic251D411yescharacteristic
75C4⋊Dic251D411yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79Dic251⋊C4—21no—
80C22⋊Dic251—21no—
81C22⋊Dic251—21no—
82C22⋊Dic251D411yescharacteristic
83C22⋊Dic251D411yescharacteristic
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86C4×Dic251—21no—
87C4×Dic251—21no—
88C4×Dic251—21no—
89C4×Dic251—21no—
90C4×Dic251—21no—
91C4×Dic251C2311yescharacteristic
92C4×Dic251C2311yescharacteristic
93C2×Q8×C251—22no—
94C2×Q8×C251—22no—
95C2×Q8×C251—21no—
96C2×Q8×C251C2311yescharacteristic
97C2×Q8×C251C2311yescharacteristic
98C2×D4×C251—21no—
99C2×D4×C251C2311yescharacteristic
100C22×C4×C251—21no—
101C22×C4×C251C2311yescharacteristic
102SD16×C251—22no—
103SD16×C251—22no—
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C22×D251—21no—
109D1004—21no—
110D1004—21no—
111C22⋊D251—81no—
112C22⋊D251—81no—
113C22⋊D251—81no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—81no—
118C2×Dic251C2×D411yescharacteristic
119C2×Dic251C2×D411yescharacteristic
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C4×D251—81no—
123C251⋊C8—21no—
124C251⋊C8—21no—
125C23×C251—21no—
126C23×C251C2×D411yescharacteristic
127Q8×C251—24no—
128Q8×C251—21no—
129Q8×C251—21no—
130Q8×C251#4112yes—
131Q8×C251#4112yes—
132D4×C251—21no—
133D4×C251—21no—
134D4×C251—81no—
135C2×C4×C251—24no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—21no—
146C22×C251—21no—
147C22×C251—21no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—41no—
153C2×D251—41no—
154C2×D251—41no—
155C4×C251—22no—
156C4×C251—22no—
157C4×C251—21no—
158C4×C251—41no—
159C4×C251C2×#4111yescharacteristic
160C4×C251C2×#4111yescharacteristic
161Dic251—21no—
162Dic251—21no—
163Dic251—21no—
164Dic251—21no—
165Dic251—41no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—41no—
169C2×C251—81no—
170C2×C251#52511yescharacteristic
171C2×C251#57711yescharacteristic
172C2×C251#57711yescharacteristic
173D251—81no—
174C251#312611yescharacteristic, complemented, Sylow
175#3126—2511nomaximal, Sylow
176#540—2511no—
177#519—2511no—
178#482—2511no—
179#482—2511no—
180#479—2511no—
181#479—2511no—
182#460—2511no—
183#460—2511no—
184#448—2511no—
185#448D25111yescharacteristic, complemented
186#440—2511no—
187#440—2511no—
188C4×SD16—2511no—
189C4×SD16—2511no—
190#433—2511no—
191C22×Q8C2×D25111yescharacteristic
192C2×SD16—2511no—
193C2×SD16—2511no—
194C2×SD16C2×D25111yescharacteristic
195C2×SD16C2×D25111yescharacteristic
196#127—2511no—
197#125—2511no—
198#123—5021no—
199#123—5021no—
200#121—5021no—
201#121—5021no—
202#121—2511no—
203#121C2×D25111yescharacteristic
204C4×Q8—5021no—
205C4×Q8—5021no—
206C4×Q8—2511no—
207C4×Q8—2511no—
208C4×Q8—2511no—
209C4×Q8—2511no—
210C4×D4—2511no—
211C4×D4—2511no—
212#117—5021no—
213#117—5021no—
214#117—2511no—
215#117—2511no—
216#107—2511no—
217#107—2511no—
218#106—2511no—
219#106—2511no—
220C4⋊C8—2511no—
221C4⋊C8—2511no—
222#103—2511no—
223#103—2511no—
224#103C2×D25111yescharacteristic
225#103C2×D25111yescharacteristic
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102—2511no—
230C22⋊C8—2511no—
231C22⋊C8C2×D25111yescharacteristic
232#97—2511no—
233C4×C8—2511no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D25111yescharacteristic
238C2×Q8C22×D25111yescharacteristic
239C2×D4—21no—
240C2×D4—5021no—
241C2×D4—5021no—
242C2×D4—5021no—
243C2×D4—2511no—
244C2×D4C22×D25111yescharacteristic
245C22×C4—21no—
246C22×C4—5021no—
247C22×C4—5021no—
248C22×C4C22×D25111yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—5022no—
252SD16—5022no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8C22×D25111yescharacteristic
256C2×C8C22×D25111yescharacteristic
257C4⋊C4—5021no—
258C4⋊C4—5021no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4—2511no—
265C4⋊C4—2511no—
266C4⋊C4—2511no—
267C4⋊C4—2511no—
268C4⋊C4C22×D25111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—2511no—
275C22⋊C4—2511no—
276C42—5021no—
277C42—5021no—
278C42—5021no—
279C42—5021no—
280C42—5021no—
281C42—2511no—
282C42—2511no—
283C23—21no—
284C23—5021no—
285C23D4×D25111yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1686080912yes—
290Q8#1686080912yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—20081no—
295D4—20081no—
296D4—20081no—
297D4—5021no—
298D4—5021no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—20081no—
306C2×C4—20081no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4—2511no—
314C2×C4D4×D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5021no—
319C8—5021no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—10041no—
327C22—10041no—
328C22—10041no—
329C22C2×D4×D25111yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—10041no—
335C4—10041no—
336C4—10041no—
337C4—5021no—
338C4—5021no—
339C4—5021no—
340C4—5021no—
341C4C2×#1686080911yescharacteristic
342C4C2×#1686080911yescharacteristic
343C2—41no—
344C2—81no—
345C2—20081no—
346C2#3063219311yescharacteristic
347C2#3063249311yescharacteristic
348C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#448×C251C2
socleC22×C251C22×D4
prime radical#30632290C2
hypercenter#448D251
hypocenterC251#3126
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #30632165C2semidirect
3. #30632012C2semidirect
4. #30632008C2semidirect
5. #30631996C2semidirect
6. #30631992C2semidirect
7. #30631885C2semidirect
8. #30631885C2semidirect
9. #30631826C2semidirect
10. #30631790C2semidirect
11. #30631788C2semidirect
12. SD16×Dic251C2semidirect
13. #30631772C2semidirect
14. #30631745C2semidirect
15. #448×C251C2semidirect
174. C251#3126semidirect
185. #448D251semidirect

Maximal subgroups

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

subgroupindex
#306322902
#306321652
#306320122
#306320082
#306319962
#306319922
#306318852
#306318262
#306317902
#306317882
SD16×Dic2512
#306317722
#306317452
#448×C2512
#3126251