Finite Group Databasebeta

← #44560531

Subgroups — #44560531

Order 32128.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#30632218C211yesmaximal, characteristic
2C2×Q8×Dic251C211yesmaximal, characteristic, complemented
3#30632023C211yesmaximal, characteristic, complemented
4#30632003C211yesmaximal, characteristic
5#30631996C212yesmaximal, complemented
6#30631907C212yesmaximal, complemented
7#30631843C211yesmaximal, characteristic, complemented
8#30631791C212yesmaximal, complemented
9#30631788C212yesmaximal, complemented
10#30631741C211yesmaximal, characteristic, complemented
11#448×C251C211yesmaximal, characteristic
12C2×#16860800C2212yes—
13#22548708—21no—
14#22548706C2211yescharacteristic
15#22548702C2211yescharacteristic
16#22548696—21no—
17#22548694C2211yescharacteristic
18Q8×Dic251—21no—
19Q8×Dic251—21no—
20Q8×Dic251C2212yes—
21Q8×Dic251C2212yes—
22#22548680C2211yescharacteristic
23D4×Dic251—21no—
24D4×Dic251C2211yescharacteristic
25#22548671—21no—
26C2×C4⋊Dic251—21no—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×C4×Dic251—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548649C2212yes—
31#22548646C2212yes—
32Dic251⋊C8C2212yes—
33#22548640C2212yes—
34#22548639C2212yes—
35#22548636C2212yes—
36C22⋊(C251⋊C8)C2211yescharacteristic
37#22548625C2212yes—
38C22×Q8×C251C2211yescharacteristic
39C2×SD16×C251C2212yes—
40#121×C251C2211yescharacteristic
41#103×C251C2212yes—
42C22⋊C8×C251C2211yescharacteristic
43C2×Dic502C2311yescharacteristic
44C22×Dic251—21no—
45C22×Dic251—21no—
46C22×Dic251D411yescharacteristic
47C22×Dic251D411yescharacteristic
48#16860800—24no—
49C2×C251⋊C8C2312yes—
50C4⋊Dic251—22no—
51C4⋊Dic251—22no—
52C4⋊Dic251—21no—
53C4⋊Dic251—21no—
54C4⋊Dic251C2312yes—
55C4⋊Dic251C2311yescharacteristic
56Dic251⋊C4—21no—
57Dic251⋊C4—21no—
58Dic251⋊C4—21no—
59C22⋊Dic251—21no—
60C22⋊Dic251—21no—
61C22⋊Dic251—21no—
62C22⋊Dic251—21no—
63C4×Dic251—22no—
64C4×Dic251—22no—
65C4×Dic251—21no—
66C4×Dic251D411yescharacteristic
67C4×Dic251D411yescharacteristic
68C4×Dic251C2311yescharacteristic
69C4×Dic251C2311yescharacteristic
70C2×Q8×C251—24no—
71C2×Q8×C251—21no—
72C2×Q8×C251C2312yes—
73C2×D4×C251—21no—
74C2×D4×C251C2311yescharacteristic
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77SD16×C251—24no—
78C2×C8×C251C2312yes—
79C4⋊C4×C251C2311yescharacteristic
80C22⋊C4×C251—21no—
81Dic502—22no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—21no—
85C2×Dic251—21no—
86C2×Dic251—41no—
87C2×Dic251—42no—
88C2×Dic251—42no—
89C2×Dic251C2×D411yescharacteristic
90C2×Dic251C2×D411yescharacteristic
91C2×Dic251C2×D411yescharacteristic
92C2×Dic251C2×D411yescharacteristic
93C251⋊C8—22no—
94C23×C251—21no—
95C23×C251C2×D411yescharacteristic
96Q8×C251—24no—
97Q8×C251—22no—
98Q8×C251#4114yes—
99D4×C251—22no—
100D4×C251—42no—
101C2×C4×C251—24no—
102C2×C4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—21no—
105C2×C4×C251C2×D411yescharacteristic
106C2×C4×C251C2411yescharacteristic
107C8×C251—22no—
108C22×C251—21no—
109C22×C251—21no—
110C22×C251—41no—
111C22×C251—41no—
112C22×C251—41no—
113C22×C251#13712yes—
114C22×C251C22×D411yescharacteristic
115C4×C251—24no—
116C4×C251—21no—
117C4×C251—41no—
118C4×C251C2×#4112yes—
119Dic251—21no—
120Dic251—21no—
121Dic251—21no—
122Dic251—21no—
123Dic251—41no—
124Dic251—42no—
125Dic251—41no—
126C2×C251—22no—
127C2×C251—81no—
128C2×C251#52511yescharacteristic
129C2×C251C2×#13711yescharacteristic
130C2×C251#57511yescharacteristic
131C251#311711yescharacteristic, complemented, Sylow
132#3117—2511nomaximal, Sylow
133#529—2511no—
134C2×C4×Q8—2511no—
135#484—2511no—
136#481—2511no—
137#479—2512no—
138#464—2512no—
139#451—2511no—
140#448D25111yescharacteristic
141#441—2512no—
142#440—2512no—
143#432—2511no—
144C22×Q8C2×D25111yescharacteristic
145C2×Q16—2512no—
146C2×SD16C2×D25112yes—
147#126—5021no—
148#125—2511no—
149#124—2511no—
150#123—5021no—
151C22⋊Q8—2511no—
152#121C2×D25111yescharacteristic
153C4×Q8—5021no—
154C4×Q8—5021no—
155C4×Q8—2512no—
156C4×Q8—2512no—
157C4×Q8—2511no—
158C4×D4—5021no—
159C4×D4—2511no—
160#117—5021no—
161C2×C4⋊C4—5021no—
162C2×C4⋊C4—2511no—
163C2×C42—5021no—
164C2×C42—2511no—
165#107—2512no—
166#106—2512no—
167C4⋊C8—2512no—
168#103—2512no—
169#103—2512no—
170#103C2×D25112yes—
171#102—2512no—
172C22⋊C8—2511no—
173C22⋊C8C2×D25111yescharacteristic
174#97—2512no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—2511no—
178C2×Q8C22×D25112yes—
179C2×D4—21no—
180C2×D4C22×D25111yescharacteristic
181C22×C4—21no—
182C22×C4—5021no—
183C22×C4—5021no—
184C22×C4—2511no—
185C22×C4—2511no—
186C22×C4C22×D25111yescharacteristic
187Q16—5024no—
188SD16—24no—
189C2×C8—2512no—
190C2×C8C22×D25112yes—
191C4⋊C4—5022no—
192C4⋊C4—5022no—
193C4⋊C4—5021no—
194C4⋊C4—5021no—
195C4⋊C4—5021no—
196C4⋊C4—5021no—
197C4⋊C4—5021no—
198C4⋊C4—2512no—
199C4⋊C4—2511no—
200C4⋊C4C22×D25111yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—5021no—
203C22⋊C4—5021no—
204C22⋊C4—5021no—
205C22⋊C4—5021no—
206C42—5022no—
207C42—5022no—
208C42—5021no—
209C42—2511no—
210C42—2511no—
211C42—2511no—
212C42—2511no—
213C23—21no—
214C23D4×D25111yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—5022no—
218Q8#1686080914yes—
219D4—22no—
220D4—42no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—10042no—
226C2×C4—10042no—
227C2×C4—10041no—
228C2×C4—5022no—
229C2×C4—5021no—
230C2×C4—5021no—
231C2×C4—5022no—
232C2×C4—2511no—
233C2×C4—2511no—
234C2×C4—2511no—
235C2×C4—2511no—
236C2×C4D4×D25111yescharacteristic
237C2×C4C23×D25111yescharacteristic
238C8—22no—
239C8—5022no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#2254874412yes—
246C22C2×D4×D25111yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—10041no—
251C4—10042no—
252C4—10041no—
253C4—5021no—
254C4—5021no—
255C4—5021no—
256C4—5021no—
257C4C2×#1686080912yes—
258C2—22no—
259C2—81no—
260C2#3063219311yescharacteristic
261C2C2×#2254874411yescharacteristic
262C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#448×C251C2
socleC22×C251C22×D4
prime radical#121×C251C22
hypercenter#448D251
hypocenterC251#3117
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. C2×Q8×Dic251C2semidirect
3. #30632023C2semidirect
5. #30631996C2semidirect
6. #30631907C2semidirect
7. #30631843C2semidirect
8. #30631791C2semidirect
9. #30631788C2semidirect
10. #30631741C2semidirect
131. C251#3117semidirect

Maximal subgroups

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

subgroupindex
#306322182
C2×Q8×Dic2512
#306320232
#306320032
#306319962
#306319072
#306318432
#306317912
#306317882
#306317412
#448×C2512
#3117251