Finite Group Databasebeta

← #44548892

Subgroups — #44548892

Order 32128.

326 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 63 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#121×D251C211yesmaximal, characteristic
2#30631649C211yesmaximal, characteristic, complemented
3#30631549C211yesmaximal, characteristic, complemented
4#30631422C211yesmaximal, characteristic, complemented
5#30631421C211yesmaximal, characteristic, complemented
6#30631420C211yesmaximal, characteristic, complemented
7#327×C251C211yesmaximal, characteristic, complemented
8C2×D4×D251—21no—
9C2×D4×D251C411yescharacteristic, complemented
10C22×C4×D251C2211yescharacteristic
11C2×#16860793—21no—
12#22548690C411yescharacteristic, complemented
13#22548689C2211yescharacteristic
14#22548688—21no—
15#22548687—21no—
16#22548686C411yescharacteristic, complemented
17C4⋊C4×D251C411yescharacteristic, complemented
18C2×(C2×D251)⋊C4—21no—
19C22⋊C4×D251—21no—
20C22⋊(C251⋊C8)C2211yescharacteristic, complemented
21(C2×D251)⋊C8—21no—
22C251⋊#95—21no—
23C251⋊#95C2211yescharacteristic, complemented
24#121×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic, complemented
26#95×C251C2211yescharacteristic, complemented
27C23×D251—21no—
28C23×D251D411yescharacteristic
29C2×D1004—21no—
30C2×D1004C2×C411yescharacteristic
31C2×C22⋊D251—21no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251—21no—
34D4×D251—42no—
35D4×D251—42no—
36D4×D251—42no—
37C22×Dic251—21no—
38C22×Dic251D411yescharacteristic
39C2×C4×D251—21no—
40C2×C4×D251—42no—
41C2×C4×D251C2×C411yescharacteristic
42C2×C4×D251C2×C411yescharacteristic
43C2×C4×D251D411yescharacteristic
44C2×C4×D251D411yescharacteristic
45#16860793—42no—
46C2×C251⋊C8—21no—
47C4⋊Dic251C2×C411yescharacteristic
48Dic251⋊C4—21no—
49C22⋊Dic251—21no—
50(C2×D251)⋊C4—21no—
51(C2×D251)⋊C4—21no—
52(C2×D251)⋊C4—42no—
53C2×D4×C251—21no—
54C2×D4×C251C2×C411yescharacteristic
55C22×C4×C251—21no—
56C22×C4×C251C2311yescharacteristic
57C2×C8×C251—21no—
58C4⋊C4×C251C2×C411yescharacteristic
59C22⋊C4×C251—21no—
60C22×D251—21no—
61C22×D251—21no—
62C22×D251—22no—
63C22×D251—21no—
64C22×D251—41no—
65C22×D251—42no—
66C22×D251—42no—
67C22×D251—42no—
68C22×D251C22⋊C411yescharacteristic
69C22×D251C22⋊C411yescharacteristic
70D1004—41no—
71D1004—42no—
72C22⋊D251—42no—
73C22⋊D251—42no—
74C22⋊D251—41no—
75C22⋊D251—41no—
76C2×Dic251—21no—
77C2×Dic251—41no—
78C2×Dic251—41no—
79C2×Dic251—41no—
80C2×Dic251—41no—
81C2×Dic251C22⋊C411yescharacteristic
82C2×Dic251C22⋊C411yescharacteristic
83C4×D251—22no—
84C4×D251—22no—
85C4×D251—41no—
86C4×D251—41no—
87C4×D251—42no—
88C251⋊C8—41no—
89C23×C251—21no—
90C23×C251C2×D411yescharacteristic
91D4×C251—41no—
92D4×C251—41no—
93D4×C251—41no—
94C2×C4×C251—21no—
95C2×C4×C251—41no—
96C2×C4×C251—41no—
97C2×C4×C251—41no—
98C2×C4×C251—41no—
99C2×C4×C251C22×C411yescharacteristic
100C2×C4×C251C2×D411yescharacteristic
101C8×C251—41no—
102C22×C251—21no—
103C22×C251—21no—
104C22×C251—41no—
105C22×C251—41no—
106C22×C251—41no—
107C22×C251C2×C22⋊C411yescharacteristic
108C22×C251AGL(1,ℤ/8)11yescharacteristic
109C22×C251AGL(1,ℤ/8)11yescharacteristic
110C2×D251—21no—
111C2×D251—21no—
112C2×D251—21no—
113C2×D251—21no—
114C2×D251—21no—
115C2×D251—42no—
116C2×D251—42no—
117C2×D251—41no—
118C2×D251—41no—
119C2×D251—41no—
120C2×D251—84no—
121C2×D251#9912yes—
122C4×C251—21no—
123C4×C251—21no—
124C4×C251—41no—
125C4×C251—81no—
126Dic251—21no—
127Dic251—21no—
128Dic251—41no—
129Dic251—81no—
130C2×C251—21no—
131C2×C251—21no—
132C2×C251—81no—
133C2×C251C2×#9911yescharacteristic
134C2×C251#41811yescharacteristic
135C2×C251#42111yescharacteristic
136D251—22no—
137D251—41no—
138D251—81no—
139C251#151011yescharacteristic, complemented, Sylow
140#1510—2511nomaximal, Sylow
141C2×#121—2511no—
142#407—2511no—
143#381—2511no—
144#327—2511no—
145#327—2511no—
146#327—2511no—
147#327D25111yescharacteristic, complemented
148C22×D4—5021no—
149C22×D4—2511no—
150C23×C4—2511no—
151C2×M16—5021no—
152#121—5021no—
153#121—5021no—
154#121—2511no—
155#121—2511no—
156#121—2511no—
157#121C2×D25111yescharacteristic
158C2×C4⋊C4—2511no—
159C2×C22⋊C4—5021no—
160C2×C22⋊C4—5021no—
161C22⋊C8—5021no—
162C22⋊C8—2511no—
163C22⋊C8C2×D25111yescharacteristic, complemented
164#95—5021no—
165#95—2511no—
166#95C2×D25111yescharacteristic, complemented
167C24—5021no—
168C24—2511no—
169C2×D4—21no—
170C2×D4—10042no—
171C2×D4—10042no—
172C2×D4—10042no—
173C2×D4—5021no—
174C2×D4—5021no—
175C2×D4—5021no—
176C2×D4—5021no—
177C2×D4—2511no—
178C2×D4C4×D25111yescharacteristic
179C22×C4—21no—
180C22×C4—10042no—
181C22×C4—5021no—
182C22×C4—5021no—
183C22×C4—2511no—
184C22×C4—2511no—
185C22×C4—2511no—
186C22×C4—2511no—
187C22×C4—2511no—
188C22×C4C22×D25111yescharacteristic
189M16—10042no—
190C2×C8—21no—
191C2×C8—5021no—
192C4⋊C4—5021no—
193C4⋊C4—2511no—
194C4⋊C4C4×D25111yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—10042no—
197C22⋊C4—5021no—
198C22⋊C4—5021no—
199C22⋊C4—5021no—
200C23—21no—
201C23—10041no—
202C23—10042no—
203C23—10042no—
204C23—10042no—
205C23—5022no—
206C23—5021no—
207C23—5021no—
208C23—5021no—
209C23—2511no—
210C23—2511no—
211C23D4×D25111yescharacteristic
212D4—41no—
213D4—41no—
214D4—41no—
215D4—10041no—
216D4—10042no—
217D4—10042no—
218D4—10041no—
219D4—10041no—
220D4—10042no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—10041no—
227C2×C4—10041no—
228C2×C4—10042no—
229C2×C4—10041no—
230C2×C4—10041no—
231C2×C4—10041no—
232C2×C4—10041no—
233C2×C4—5022no—
234C2×C4—5022no—
235C2×C4—5021no—
236C2×C4—2511no—
237C2×C4—2511no—
238C2×C4C2×C4×D25111yescharacteristic
239C2×C4D4×D25111yescharacteristic
240C8—41no—
241C8—10041no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—20084no—
248C22—10042no—
249C22—10042no—
250C22—10041no—
251C22—10041no—
252C22—10041no—
253C22—5021no—
254C22—5021no—
255C22—5021no—
256C22—5021no—
257C22—5021no—
258C22—2512no—
259C22C22⋊C4×D25111yescharacteristic
260C22#2254873911yescharacteristic
261C22#2254874211yescharacteristic
262C4—21no—
263C4—21no—
264C4—41no—
265C4—81no—
266C4—20081no—
267C4—10041no—
268C4—5021no—
269C4—5021no—
270C2—21no—
271C2—21no—
272C2—81no—
273C2—20081no—
274C2—10041no—
275C2—5022no—
276C2#99×D25111yescharacteristic
277C2#3063169011yescharacteristic
278C2#3063170211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#327×C251C2
socleC22×C251C2×C22⋊C4
prime radical#121×D251C2
cosocleC22×C4×C251C23
hypercenter#327D251
hypocenterC251#1510
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #30631649C2semidirect
3. #30631549C2semidirect
4. #30631422C2semidirect
5. #30631421C2semidirect
6. #30631420C2semidirect
7. #327×C251C2semidirect
9. C2×D4×D251C4semidirect
12. #22548690C4semidirect
16. #22548686C4semidirect
17. C4⋊C4×D251C4semidirect
20. C22⋊(C251⋊C8)C22semidirect
23. C251⋊#95C22semidirect
25. C22⋊C8×C251C22semidirect
26. #95×C251C22semidirect
139. C251#1510semidirect
147. #327D251semidirect
163. C22⋊C8C2×D251semidirect
166. #95C2×D251semidirect

Maximal subgroups

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

subgroupindex
#121×D2512
#306316492
#306315492
#306314222
#306314212
#306314202
#327×C2512
#1510251