Finite Group Databasebeta

← #44560029

Subgroups — #44560029

Order 32128.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548738C211yesmaximal, characteristic, complemented
2C2×#22548723C211yesmaximal, characteristic, complemented
3#30632267C212yesmaximal, complemented
4#30631942C211yesmaximal, characteristic, complemented
5#30631941C211yesmaximal, characteristic, complemented
6#30631940C211yesmaximal, characteristic, complemented
7#30631919C212yesmaximal, complemented
8#30631918C212yesmaximal, complemented
9#106×D251C211yesmaximal, characteristic, complemented
10#30631687C212yesmaximal, complemented
11#470×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13C2×#16860810C2211yescharacteristic
14#22548738—24no—
15C2×#16860797C2212yes—
16C2×D2008C2211yescharacteristic
17#22548723—24no—
18C2×#16860793C2211yescharacteristic
19C2×#16860793C2211yescharacteristic
20C22×C251⋊C8C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic
23#22548693—22no—
24#22548691C2212yes—
25#22548690—22no—
26#22548690C2212yes—
27#22548689C2212yes—
28#22548688—22no—
29#22548688—22no—
30C4⋊C4×D251C2212yes—
31#22548647C2211yescharacteristic
32#22548646C2212yes—
33#22548639C2212yes—
34#22548637C2212yes—
35#22548635C2212yes—
36C2×Q16×C251C2211yescharacteristic
37C2×M16×C251C2211yescharacteristic
38C22⋊Q8×C251C2212yes—
39#106×C251C2211yescharacteristic
40#103×C251C2212yes—
41#16860811—24no—
42#16860811—24no—
43#16860810—22no—
44#16860810—22no—
45C2×Dic502D411yescharacteristic
46C2×D1004—22no—
47C2×D1004—22no—
48C2×D1004D411yescharacteristic
49C2×D1004C2312yes—
50C2×C22⋊D251—22no—
51C2×C22⋊D251D411yescharacteristic
52C2×C22⋊D251D411yescharacteristic
53C2×C4×D251—22no—
54C2×C4×D251—22no—
55C2×C4×D251C2311yescharacteristic
56C2×C4×D251C2311yescharacteristic
57#16860797—24no—
58D2008—22no—
59#16860793—22no—
60#16860793—22no—
61C2×C251⋊C8—22no—
62C2×C251⋊C8C2311yescharacteristic
63C2×C251⋊C8C2311yescharacteristic
64C8×D251—22no—
65C8×D251D414yescomplemented
66C4⋊Dic251C2312yes—
67Dic251⋊C4—22no—
68Dic251⋊C4—22no—
69(C2×D251)⋊C4—22no—
70(C2×D251)⋊C4—22no—
71(C2×D251)⋊C4—22no—
72C2×Q8×C251C2312yes—
73C22×C4×C251C2311yescharacteristic
74Q16×C251—22no—
75M16×C251—22no—
76C2×C8×C251C2311yescharacteristic
77C2×C8×C251C2311yescharacteristic
78C4⋊C4×C251—22no—
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—22no—
81C22×D251—22no—
82C22×D251C2×D411yescharacteristic
83C22×D251C2×D411yescharacteristic
84Dic502—22no—
85D1004—22no—
86D1004—22no—
87D1004—22no—
88D1004—44no—
89D1004—82no—
90C22⋊D251—22no—
91C22⋊D251—22no—
92C22⋊D251—82no—
93C2×Dic251—22no—
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C4×D251—21no—
97C4×D251—21no—
98C4×D251—44no—
99C4×D251—82no—
100C4×D251C2×D412yes—
101C4×D251#4112yes—
102C251⋊C8—21no—
103C251⋊C8C2×D412yes—
104C23×C251C2×D411yescharacteristic
105Q8×C251—22no—
106Q8×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—21no—
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2411yescharacteristic
113C8×C251—21no—
114C8×C251C2×D412yes—
115C22×C251—21no—
116C22×C251—22no—
117C22×C251C22×D411yescharacteristic
118C2×D251—22no—
119C2×D251—21no—
120C2×D251—22no—
121C2×D251—42no—
122C2×D251—42no—
123C2×D251—42no—
124C2×D251#12112yes—
125C4×C251—21no—
126C4×C251—42no—
127C4×C251—42no—
128C4×C251C22×D411yescharacteristic
129C4×C251C2×#4111yescharacteristic
130Dic251—21no—
131Dic251—42no—
132Dic251#12112yes—
133C2×C251—41no—
134C2×C251C2×#12111yescharacteristic
135C2×C251#57712yes—
136D251—22no—
137D251—41no—
138D251—82no—
139C251#306011yescharacteristic, complemented, Sylow
140#3060—2511nomaximal, Sylow
141C2×#135—2511no—
142C2×#131—2511no—
143#537—2512no—
144#470—2511no—
145#470D25111yescharacteristic, complemented
146#469—2511no—
147#469—2511no—
148#465—2512no—
149#465—2512no—
150C2×#106—2511no—
151#417—2512no—
152C2×#41—2512no—
153C2×#41—2511no—
154#135—5024no—
155C2×Q16C2×D25111yescharacteristic
156C2×SD16—2512no—
157C2×D8—2511no—
158#131—5024no—
159C2×M16—2511no—
160C2×M16—2511no—
161C2×M16C2×D25111yescharacteristic
162C22×C8—2511no—
163C22×C8—2511no—
164C22×C8—2511no—
165C22⋊Q8—5022no—
166C22⋊Q8—2512no—
167C22⋊Q8C2×D25112yes—
168#121—5022no—
169#121—5022no—
170#121—5022no—
171#121—2512no—
172#121—2512no—
173C2×C4⋊C4—2512no—
174#106—2511no—
175#106—2512no—
176#106C2×D25111yescharacteristic
177#103—2512no—
178#103C2×D25112yes—
179#102—2512no—
180#102—2512no—
181#41—5022no—
182#41—5022no—
183#41—5024no—
184#41—5024no—
185C2×Q8—2511no—
186C2×Q8C22×D25112yes—
187C2×D4—5022no—
188C2×D4—5022no—
189C2×D4—5022no—
190C2×D4—2511no—
191C2×D4—2511no—
192C2×D4—2511no—
193C2×D4—2512no—
194C22×C4—5022no—
195C22×C4—5022no—
196C22×C4—2511no—
197C22×C4—2511no—
198C22×C4C22×D25111yescharacteristic
199Q16—22no—
200SD16—5024no—
201D8—5022no—
202M16—22no—
203M16—5022no—
204M16—5022no—
205C2×C8—5022no—
206C2×C8—5022no—
207C2×C8—2511no—
208C2×C8—2511no—
209C2×C8—2514no—
210C2×C8C22×D25111yescharacteristic
211C2×C8C22×D25111yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—5022no—
214C4⋊C4—5022no—
215C4⋊C4—2512no—
216C4⋊C4C22×D25112yes—
217C22⋊C4—22no—
218C22⋊C4—5022no—
219C22⋊C4—5022no—
220C22⋊C4—5022no—
221C23—5022no—
222C23—2511no—
223C23—2511no—
224C23D4×D25111yescharacteristic
225Q8—22no—
226Q8—22no—
227Q8—5022no—
228D4—20082no—
229D4—20082no—
230D4—10044no—
231D4—5022no—
232D4—5022no—
233D4—5022no—
234D4—5022no—
235D4—5022no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—20082no—
241C2×C4—10044no—
242C2×C4—5022no—
243C2×C4—5021no—
244C2×C4—5021no—
245C2×C4—2511no—
246C2×C4—2511no—
247C2×C4—2512no—
248C2×C4—2512no—
249C2×C4D4×D25111yescharacteristic
250C2×C4C23×D25111yescharacteristic
251C8—21no—
252C8—5021no—
253C8—2512no—
254C8D4×D25112yes—
255C22—22no—
256C22—21no—
257C22—10042no—
258C22—10042no—
259C22—10042no—
260C22—5022no—
261C22—5021no—
262C22—5022no—
263C22—2512no—
264C22C2×D4×D25111yescharacteristic
265C4—21no—
266C4—42no—
267C4—42no—
268C4—10042no—
269C4—5021no—
270C4—2512no—
271C4C2×D4×D25111yescharacteristic
272C4#41×D25111yescharacteristic
273C2—41no—
274C2—20082no—
275C2—10041no—
276C2—5022no—
277C2#121×D25111yescharacteristic
278C2#3063249612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#470×C251C2
socleC22×C251C22×D4
hypercenter#470D251
hypocenterC251#3060
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#22548738C2semidirect
2. C2×#22548723C2semidirect
3. #30632267C2semidirect
4. #30631942C2semidirect
5. #30631941C2semidirect
6. #30631940C2semidirect
7. #30631919C2semidirect
8. #30631918C2semidirect
9. #106×D251C2semidirect
10. #30631687C2semidirect
11. #470×C251C2semidirect
65. C8×D251D4semidirect
139. C251#3060semidirect
145. #470D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487382
C2×#225487232
#306322672
#306319422
#306319412
#306319402
#306319192
#306319182
#106×D2512
#306316872
#470×C2512
#3060251