Finite Group Databasebeta

← #44560537

Subgroups — #44560537

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#30632223C211yesmaximal, characteristic
2C2×Q8×Dic251C211yesmaximal, characteristic, complemented
3#30632020C211yesmaximal, characteristic, complemented
4#30632006C211yesmaximal, characteristic
5#30631992C212yesmaximal, complemented
6#30631911C212yesmaximal, complemented
7#30631826C211yesmaximal, characteristic
8#30631794C212yesmaximal, complemented
9#30631790C212yesmaximal, complemented
10#30631741C211yesmaximal, characteristic, complemented
11#451×C251C211yesmaximal, characteristic, complemented
12C2×#16860797C2212yes—
13#22548710—21no—
14#22548706C2211yescharacteristic
15#22548704C2211yescharacteristic
16#22548699—21no—
17#22548689C2211yescharacteristic
18Q8×Dic251—21no—
19Q8×Dic251—21no—
20Q8×Dic251C2211yescharacteristic
21Q8×Dic251C2212yes—
22Q8×Dic251C2212yes—
23#22548677C2211yescharacteristic
24(C22⋊D251)⋊C4—21no—
25#22548673—21no—
26C2×C4⋊Dic251—21no—
27C2×C4⋊Dic251C2211yescharacteristic
28C2×C4×Dic251—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548648C2212yes—
31#22548647C2212yes—
32Dic251⋊C8C2212yes—
33#22548639C2212yes—
34#22548639C2212yes—
35#22548637C2212yes—
36C22⋊(C251⋊C8)C2211yescharacteristic
37#22548625C2212yes—
38C22×Q8×C251C2211yescharacteristic
39C2×Q16×C251C2212yes—
40C22⋊Q8×C251C2211yescharacteristic
41#103×C251C2212yes—
42C22⋊C8×C251C2211yescharacteristic
43C2×D1004C2311yescharacteristic
44C2×C22⋊D251—21no—
45C22×Dic251—21no—
46C22×Dic251D411yescharacteristic
47C22×Dic251D411yescharacteristic
48C2×C4×D251—21no—
49#16860797—24no—
50C2×C251⋊C8C2312yes—
51C4⋊Dic251—22no—
52C4⋊Dic251—22no—
53C4⋊Dic251—21no—
54C4⋊Dic251—21no—
55C4⋊Dic251—21no—
56C4⋊Dic251C2312yes—
57C4⋊Dic251C2311yescharacteristic
58Dic251⋊C4—21no—
59(C2×D251)⋊C4—21no—
60(C2×D251)⋊C4—21no—
61(C2×D251)⋊C4—21no—
62(C2×D251)⋊C4—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×C251C2311yescharacteristic
73C2×Q8×C251C2312yes—
74C22×C4×C251—21no—
75C22×C4×C251C2311yescharacteristic
76Q16×C251—24no—
77C2×C8×C251C2312yes—
78C4⋊C4×C251—21no—
79C4⋊C4×C251C2311yescharacteristic
80C22⋊C4×C251—21no—
81C22×D251—21no—
82D1004—22no—
83C22⋊D251—42no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—21no—
87C2×Dic251—41no—
88C2×Dic251—42no—
89C2×Dic251C2×D411yescharacteristic
90C2×Dic251C2×D411yescharacteristic
91C2×Dic251C2×D411yescharacteristic
92C2×Dic251C2×D411yescharacteristic
93C4×D251—42no—
94C251⋊C8—22no—
95C23×C251C2×D411yescharacteristic
96Q8×C251—24no—
97Q8×C251—22no—
98Q8×C251—22no—
99Q8×C251#4114yes—
100C2×C4×C251—24no—
101C2×C4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251—21no—
104C2×C4×C251—21no—
105C2×C4×C251C2×D411yescharacteristic
106C2×C4×C251C2411yescharacteristic
107C8×C251—22no—
108C22×C251—21no—
109C22×C251—21no—
110C22×C251#13712yes—
111C22×C251C22×D411yescharacteristic
112C2×D251—41no—
113C2×D251—41no—
114C2×D251—41no—
115C4×C251—24no—
116C4×C251—21no—
117C4×C251—41no—
118C4×C251—41no—
119C4×C251C2×#4112yes—
120Dic251—21no—
121Dic251—21no—
122Dic251—21no—
123Dic251—21no—
124Dic251—41no—
125Dic251—42no—
126C2×C251—22no—
127C2×C251#52511yescharacteristic
128C2×C251C2×#13711yescharacteristic
129C2×C251#57511yescharacteristic
130D251—81no—
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#451D25111yescharacteristic, complemented
140#448—2511no—
141#441—2512no—
142#440—2512no—
143#432—2511no—
144C22×Q8C2×D25111yescharacteristic
145C2×Q16C2×D25112yes—
146C2×SD16—2512no—
147#126—5021no—
148#125—2511no—
149#124—2511no—
150#123—5021no—
151C22⋊Q8C2×D25111yescharacteristic
152#121—2511no—
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×Q8C22×D25111yescharacteristic
178C2×Q8C22×D25112yes—
179C2×D4—5021no—
180C2×D4—2511no—
181C22×C4—21no—
182C22×C4—5021no—
183C22×C4—5021no—
184C22×C4—2511no—
185C22×C4—2511no—
186C22×C4C22×D25111yescharacteristic
187Q16—24no—
188SD16—5024no—
189C2×C8—2512no—
190C2×C8C22×D25112yes—
191C4⋊C4—21no—
192C4⋊C4—5022no—
193C4⋊C4—5022no—
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—5021no—
214C23D4×D25111yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#1686080914yes—
219D4—10042no—
220D4—5022no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—10042no—
227C2×C4—10042no—
228C2×C4—10041no—
229C2×C4—5022no—
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—10041no—
243C22—10041no—
244C22—10041no—
245C22#2254874712yes—
246C22C2×D4×D25111yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—10041no—
252C4—10042no—
253C4—5021no—
254C4—5021no—
255C4—5021no—
256C4—5021no—
257C4C2×#1686080912yes—
258C2—22no—
259C2—20081no—
260C2#3063219311yescharacteristic
261C2C2×#2254874711yescharacteristic
262C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#451×C251C2
socleC22×C251C22×D4
prime radical#22548689C22
hypercenter#451D251
hypocenterC251#3117
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic251C2semidirect
3. #30632020C2semidirect
5. #30631992C2semidirect
6. #30631911C2semidirect
8. #30631794C2semidirect
9. #30631790C2semidirect
10. #30631741C2semidirect
11. #451×C251C2semidirect
131. C251#3117semidirect
139. #451D251semidirect

Maximal subgroups

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

subgroupindex
#306322232
C2×Q8×Dic2512
#306320202
#306320062
#306319922
#306319112
#306318262
#306317942
#306317902
#306317412
#451×C2512
#3117251