Finite Group Databasebeta

← #44560654

Subgroups — #44560654

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#30632291C211yesmaximal, characteristic
2#30632163C211yesmaximal, characteristic, complemented
3#30632010C211yesmaximal, characteristic, complemented
4#30632009C211yesmaximal, characteristic, complemented
5#30631995C211yesmaximal, characteristic, complemented
6#30631994C211yesmaximal, characteristic, complemented
7#30631884C211yesmaximal, characteristic, complemented
8#30631824C211yesmaximal, characteristic, complemented
9#30631824C211yesmaximal, characteristic, complemented
10#30631788C211yesmaximal, characteristic, complemented
11#30631784C211yesmaximal, characteristic, complemented
12#30631778C211yesmaximal, characteristic, complemented
13SD16×Dic251C211yesmaximal, characteristic, complemented
14#30631747C211yesmaximal, characteristic, complemented
15#460×C251C211yesmaximal, characteristic, complemented
16C2×Q8×D251C2211yescharacteristic
17C2×#16860798C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19#22548707C2211yescharacteristic
20#22548696—21no—
21#22548696—21no—
22#22548687—21no—
23#22548687—21no—
24#22548686C2211yescharacteristic
25#22548686C2211yescharacteristic
26Q8×Dic251C2211yescharacteristic
27C4×Dic502—21no—
28C4×Dic502C2211yescharacteristic
29#22548680—21no—
30#22548680C2211yescharacteristic
31D4×Dic251C2211yescharacteristic
32D4×Dic251C2211yescharacteristic
33#22548673—21no—
34#22548673C2211yescharacteristic
35#22548672—21no—
36#22548672C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548648C2211yescharacteristic
39#22548647C2211yescharacteristic
40#22548646C2211yescharacteristic
41C4⋊(C251⋊C8)C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48(C2×D251)⋊C8C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50#22548625C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×SD16×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54#127×C251C2211yescharacteristic
55C4×Q8×C251C2211yescharacteristic
56C4⋊C8×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C2×Dic502—21no—
60C2×Dic502C2311yescharacteristic
61Q8×D251—22no—
62Q8×D251—22no—
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C22×Dic251—21no—
66C22×Dic251—21no—
67C2×C4×D251—21no—
68C2×C4×D251C2311yescharacteristic
69#16860798—22no—
70#16860798—22no—
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C4⋊Dic251—21no—
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81Dic251⋊C4D411yescharacteristic
82Dic251⋊C4D411yescharacteristic
83C22⋊Dic251—21no—
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4D411yescharacteristic
89(C2×D251)⋊C4D411yescharacteristic
90C4×Dic251—21no—
91C4×Dic251—21no—
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251C2311yescharacteristic
95C2×Q8×C251C2311yescharacteristic
96C2×D4×C251—21no—
97C2×D4×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C2×D4×C251C2311yescharacteristic
100SD16×C251—22no—
101SD16×C251—22no—
102C2×C8×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C4⋊C4×C251—21no—
105C4⋊C4×C251C2311yescharacteristic
106C42×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251C2×D411yescharacteristic
109Dic502—24no—
110Dic502—21no—
111Dic502#4112yes—
112C22⋊D251—81no—
113C22⋊D251—81no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—81no—
119C2×Dic251—81no—
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C4×D251—24no—
124C4×D251—21no—
125C4×D251—21no—
126C251⋊C8—21no—
127C251⋊C8—21no—
128C23×C251—21no—
129C23×C251—21no—
130Q8×C251—21no—
131Q8×C251#4112yes—
132D4×C251—21no—
133D4×C251—21no—
134D4×C251—21no—
135D4×C251—21no—
136D4×C251—81no—
137D4×C251—81no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—41no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—21no—
153C2×D251—21no—
154C2×D251—21no—
155C4×C251—22no—
156C4×C251—21no—
157C4×C251—21no—
158C4×C251—41no—
159C4×C251C2×#4111yescharacteristic
160C4×C251C2×#4111yescharacteristic
161Dic251—22no—
162Dic251—21no—
163Dic251—21no—
164Dic251—21no—
165Dic251—41no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—81no—
169C2×C251—81no—
170C2×C251#52511yescharacteristic
171C2×C251#57711yescharacteristic
172C2×C251#57711yescharacteristic
173D251—41no—
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#460D25111yescharacteristic, complemented
184#448—2511no—
185#448—2511no—
186#440—2511no—
187#440—2511no—
188C4×SD16—2511no—
189C4×SD16—2511no—
190#433—2511no—
191C22×Q8—2511no—
192C2×SD16—2511no—
193C2×SD16—2511no—
194C2×SD16C2×D25111yescharacteristic
195C2×SD16C2×D25111yescharacteristic
196#127C2×D25111yescharacteristic
197#125—2511no—
198#123—5021no—
199#123—5021no—
200#121—5021no—
201#121—5021no—
202#121—2511no—
203#121—2511no—
204C4×Q8—5021no—
205C4×Q8—5021no—
206C4×Q8—2511no—
207C4×Q8—2511no—
208C4×Q8—2511no—
209C4×Q8C2×D25111yescharacteristic
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⋊C8C2×D25111yescharacteristic
222#103—2511no—
223#103—2511no—
224#103—2511no—
225#103—2511no—
226#102—2511no—
227#102—2511no—
228#102C2×D25111yescharacteristic
229#102C2×D25111yescharacteristic
230C22⋊C8—2511no—
231C22⋊C8—2511no—
232#97—2511no—
233C4×C8—2511no—
234C2×Q8—5021no—
235C2×Q8—5022no—
236C2×Q8—5022no—
237C2×Q8—2511no—
238C2×Q8C22×D25111yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—5021no—
242C2×D4—5021no—
243C2×D4C22×D25111yescharacteristic
244C2×D4C22×D25111yescharacteristic
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4—5021no—
248C22×C4—2511no—
249SD16—22no—
250SD16—22no—
251SD16—5022no—
252SD16—5022no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8C22×D25111yescharacteristic
256C2×C8C22×D25111yescharacteristic
257C4⋊C4—21no—
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—5021no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—2511no—
275C22⋊C4—2511no—
276C42—21no—
277C42—5021no—
278C42—5021no—
279C42—5021no—
280C42—5021no—
281C42—2511no—
282C42C22×D25111yescharacteristic
283C23—21no—
284C23—21no—
285C23—2511no—
286Q8—21no—
287Q8—5024no—
288Q8—5021no—
289Q8—2512no—
290Q8#1686080912yes—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4—81no—
297D4—20081no—
298D4—20081no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—20081no—
302C2×C4—20081no—
303C2×C4—5024no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4D4×D25111yescharacteristic
314C2×C4D4×D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5021no—
319C8—5021no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—5021no—
327C22—5021no—
328C22—5021no—
329C22C2×D4×D25111yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—10041no—
335C4—10041no—
336C4—10041no—
337C4—5022no—
338C4—5021no—
339C4—5021no—
340C4—5021no—
341C4#41×D25111yescharacteristic
342C4C2×#1686080911yescharacteristic
343C2—81no—
344C2—81no—
345C2—10041no—
346C2#3063219711yescharacteristic
347C2#3063249311yescharacteristic
348C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#460×C251C2
socleC22×C251C22×D4
prime radical#30632291C2
hypercenter#460D251
hypocenterC251#3126
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #30632163C2semidirect
3. #30632010C2semidirect
4. #30632009C2semidirect
5. #30631995C2semidirect
6. #30631994C2semidirect
7. #30631884C2semidirect
8. #30631824C2semidirect
9. #30631824C2semidirect
10. #30631788C2semidirect
11. #30631784C2semidirect
12. #30631778C2semidirect
13. SD16×Dic251C2semidirect
14. #30631747C2semidirect
15. #460×C251C2semidirect
174. C251#3126semidirect
183. #460D251semidirect

Maximal subgroups

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

subgroupindex
#306322912
#306321632
#306320102
#306320092
#306319952
#306319942
#306318842
#306318242
#306317882
#306317842
#306317782
SD16×Dic2512
#306317472
#460×C2512
#3126251