Finite Group Databasebeta

← #44560412

Subgroups — #44560412

Order 32128.

454 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 133 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#30632221C211yesmaximal, characteristic, complemented
2#30632022C211yesmaximal, characteristic, complemented
3#30632005C211yesmaximal, characteristic, complemented
4#30631987C212yesmaximal, complemented
5#30631902C212yesmaximal, complemented
6#30631838C211yesmaximal, characteristic, complemented
7#30631821C211yesmaximal, characteristic
8C4×#16860798C212yesmaximal, complemented
9C4×#16860795C212yesmaximal, complemented
10#30631740C211yesmaximal, characteristic, complemented
11C2×C4×D4×C251C211yesmaximal, characteristic, complemented
12C2×#16860798C2212yes—
13C2×#16860795C2212yes—
14#22548709—21no—
15#22548707C2211yescharacteristic, complemented
16#22548703C2211yescharacteristic, complemented
17#22548698—21no—
18#22548694C2211yescharacteristic
19#22548689C2211yescharacteristic
20C4×Dic502C2211yescharacteristic, complemented
21C4×D1004C2211yescharacteristic, complemented
22C4×C22⋊D251—21no—
23#22548672—21no—
24#22548648C2212yescomplemented
25#22548646C2212yescomplemented
26C4⋊(C251⋊C8)C2212yescomplemented
27#22548638C2212yescomplemented
28#22548636C2212yes—
29#22548636C2212yes—
30#22548635C2212yescomplemented
31C22⋊(C251⋊C8)C2211yescharacteristic
32C22⋊(C251⋊C8)C2211yescharacteristic
33C4×C251⋊C8C2212yescomplemented
34C22×D4×C251C2211yescharacteristic
35C23×C4×C251—21no—
36C4×D4×C251—21no—
37C4×D4×C251—21no—
38C4×D4×C251C2212yes—
39C4×D4×C251C2212yes—
40C2×C4⋊C4×C251C2211yescharacteristic
41C2×C22⋊C4×C251—21no—
42C2×C42×C251C2211yescharacteristic
43C2×Dic502C2311yescharacteristic
44C2×D1004C2311yescharacteristic
45C2×C22⋊D251—21no—
46C2×C4×D251—21no—
47#16860798—24no—
48#16860795—24no—
49C2×C251⋊C8C2312yes—
50C2×C251⋊C8C2312yes—
51C4⋊Dic251C2311yescharacteristic
52C4⋊Dic251C2311yescharacteristic
53Dic251⋊C4—21no—
54Dic251⋊C4—21no—
55Dic251⋊C4—21no—
56Dic251⋊C4—21no—
57C22⋊Dic251—21no—
58(C2×D251)⋊C4—21no—
59(C2×D251)⋊C4—21no—
60(C2×D251)⋊C4—21no—
61(C2×D251)⋊C4—21no—
62C4×Dic251—21no—
63C24×C251—21no—
64C2×D4×C251—24no—
65C2×D4×C251—21no—
66C2×D4×C251C2312yes—
67C22×C4×C251—21no—
68C22×C4×C251—22no—
69C22×C4×C251—22no—
70C22×C4×C251—44no—
71C22×C4×C251D411yescharacteristic
72C22×C4×C251D411yescharacteristic
73C22×C4×C251C2311yescharacteristic
74C4⋊C4×C251—21no—
75C4⋊C4×C251C2312yes—
76C22⋊C4×C251—22no—
77C22⋊C4×C251—22no—
78C42×C251D411yescharacteristic
79C42×C251D411yescharacteristic
80C42×C251C2311yescharacteristic
81C42×C251C2311yescharacteristic
82C22×D251—21no—
83Dic502—22no—
84D1004—22no—
85C22⋊D251—42no—
86C2×Dic251—21no—
87C2×Dic251—21no—
88C2×Dic251—21no—
89C4×D251—42no—
90C251⋊C8—22no—
91C251⋊C8—22no—
92C23×C251—24no—
93C23×C251—22no—
94C23×C251—42no—
95C23×C251—42no—
96C23×C251C2×D411yescharacteristic
97D4×C251—24no—
98D4×C251—22no—
99D4×C251#4114yes—
100C2×C4×C251—22no—
101C2×C4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—44no—
105C2×C4×C251—42no—
106C2×C4×C251—44no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2411yescharacteristic
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—24no—
116C22×C251—48no—
117C22×C251—42no—
118C22×C251—42no—
119C22×C251#13512yes—
120C22×C251C22×D411yescharacteristic
121C2×D251—41no—
122C2×D251—41no—
123C2×D251—41no—
124C4×C251—21no—
125C4×C251—21no—
126C4×C251—21no—
127C4×C251—21no—
128C4×C251—42no—
129C4×C251AGL(1,ℤ/8)12yes—
130C4×C251C2×#4112yes—
131Dic251—41no—
132Dic251—41no—
133Dic251—41no—
134C2×C251—22no—
135C2×C251—44no—
136C2×C251#52511yescharacteristic
137C2×C251C2×#13511yescharacteristic
138C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
139D251—81no—
140C251#310511yescharacteristic, complemented, Sylow
141#3105—2511nomaximal, Sylow
142#529—2511no—
143C2×C4×D4D25111yescharacteristic, complemented
144#484—2511no—
145#481—2511no—
146#478—2512no—
147#463—2512no—
148#450—2511no—
149#447—2511no—
150C4×SD16—2512no—
151C4×D8—2512no—
152#431—2511no—
153C22×D4C2×D25111yescharacteristic
154C23×C4—21no—
155C2×SD16—2512no—
156C2×D8—2512no—
157#126—5021no—
158#125—2511no—
159#124—2511no—
160#123—5021no—
161C22⋊Q8—2511no—
162#121—2511no—
163C4×Q8—2511no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—5021no—
167C4×D4—2511no—
168C4×D4C2×D25112yes—
169C4×D4C2×D25112yes—
170#117—5021no—
171C2×C4⋊C4C2×D25111yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D25111yescharacteristic
174#107—2512no—
175#106—2512no—
176C4⋊C8—2512no—
177#103—2512no—
178#102—2512no—
179#102—2512no—
180#102—2512no—
181C22⋊C8—2511no—
182C22⋊C8—2511no—
183C4×C8—2512no—
184C24—21no—
185C2×Q8—2511no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—5021no—
189C2×D4—2511no—
190C2×D4C22×D25112yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—5021no—
196C22×C4C22⋊D25111yescharacteristic
197C22×C4C22⋊D25111yescharacteristic
198C22×C4C22×D25111yescharacteristic
199SD16—5024no—
200D8—5024no—
201C2×C8—2512no—
202C2×C8—2512no—
203C4⋊C4—21no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—5021no—
207C4⋊C4—5021no—
208C4⋊C4—2511no—
209C4⋊C4—2511no—
210C4⋊C4C22×D25112yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—5021no—
214C22⋊C4—5021no—
215C22⋊C4—5021no—
216C22⋊C4—5021no—
217C22⋊C4—5021no—
218C42—5021no—
219C42C22⋊D25111yescharacteristic
220C42C22⋊D25111yescharacteristic
221C42C22×D25111yescharacteristic
222C42C22×D25111yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—5021no—
228C23C2×C22⋊D25111yescharacteristic
229Q8—5022no—
230D4—24no—
231D4—22no—
232D4—10042no—
233D4—5022no—
234D4#1686081014yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—10042no—
243C2×C4—5021no—
244C2×C4—5021no—
245C2×C4—5021no—
246C2×C4C2×C22⋊D25111yescharacteristic
247C2×C4C2×C22⋊D25111yescharacteristic
248C2×C4C2×C22⋊D25111yescharacteristic
249C2×C4C2×C22⋊D25111yescharacteristic
250C2×C4C2×C22⋊D25111yescharacteristic
251C2×C4C23×D25111yescharacteristic
252C8—5022no—
253C8—5022no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—10041no—
261C22—10041no—
262C22—10041no—
263C22#2254873512yes—
264C22C22×C22⋊D25111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—10041no—
271C4—10041no—
272C4—10041no—
273C4#2254874112yes—
274C4C2×#1686081012yes—
275C2—22no—
276C2—44no—
277C2—20081no—
278C2#3063219811yescharacteristic
279C2C2×#2254873511yescharacteristic
280C2C2×#2254874111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C4×D4×C251C2
socleC22×C251C22×D4
prime radical#30631821C2
hypercenterC2×C4×D4D251
hypocenterC251#3105
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #30632221C2semidirect
2. #30632022C2semidirect
3. #30632005C2semidirect
4. #30631987C2semidirect
5. #30631902C2semidirect
6. #30631838C2semidirect
8. C4×#16860798C2semidirect
9. C4×#16860795C2semidirect
10. #30631740C2semidirect
11. C2×C4×D4×C251C2semidirect
15. #22548707C22semidirect
16. #22548703C22semidirect
20. C4×Dic502C22semidirect
21. C4×D1004C22semidirect
24. #22548648C22semidirect
25. #22548646C22semidirect
26. C4⋊(C251⋊C8)C22semidirect
27. #22548638C22semidirect
30. #22548635C22semidirect
33. C4×C251⋊C8C22semidirect
140. C251#3105semidirect
143. C2×C4×D4D251semidirect

Maximal subgroups

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

subgroupindex
#306322212
#306320222
#306320052
#306319872
#306319022
#306318382
#306318212
C4×#168607982
C4×#168607952
#306317402
C2×C4×D4×C2512
#3105251