Finite Group Databasebeta

← #451×D251 #44559556

Subgroups — #451×D251 #44559556

Order 32128.

740 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×Q8×D251C211yesmaximal, characteristic
2C2×Q16×D251C212yesmaximal, complemented
3C22⋊Q8×D251C211yesmaximal, characteristic
4#30631845C211yesmaximal, characteristic, complemented
5#30631844C212yesmaximal, complemented
6#30631843C211yesmaximal, characteristic, complemented
7#30631842C211yesmaximal, characteristic, complemented
8#30631841C212yesmaximal, complemented
9#103×D251C212yesmaximal, complemented
10C22⋊C8×D251C211yesmaximal, characteristic
11#451×C251C211yesmaximal, characteristic, complemented
12C22×Dic502—21no—
13C22×Dic502C2211yescharacteristic
14C2×Q8×D251—28no—
15C2×Q8×D251—21no—
16C2×Q8×D251—21no—
17C2×Q8×D251C2211yescharacteristic
18C2×Q8×D251C2212yes—
19C2×Q8×D251C2212yes—
20C22×C4×D251—21no—
21C22×C4×D251C2211yescharacteristic
22C2×Dic1004C2212yescomplemented
23C2×#16860800C2212yescomplemented
24C2×#16860800C2212yescomplemented
25Q16×D251—28no—
26C2×C8×D251C2212yes—
27#22548695C2211yescharacteristic
28#22548694C2211yescharacteristic
29#22548693—21no—
30#22548692—21no—
31#22548691C2211yescharacteristic
32C4⋊C4×D251—21no—
33C4⋊C4×D251C2211yescharacteristic
34C22⋊C4×D251—21no—
35#22548640C2212yescomplemented
36#22548639C2212yescomplemented
37#22548638C2212yescomplemented
38C22⋊(C251⋊C8)C2211yescharacteristic
39(C2×D251)⋊C8C2212yes—
40C22×Q8×C251C2211yescharacteristic
41C2×Q16×C251C2212yescomplemented
42C22⋊Q8×C251C2211yescharacteristic
43#103×C251C2212yescomplemented
44C22⋊C8×C251C2211yescharacteristic
45C23×D251D411yescharacteristic
46C2×Dic502—28no—
47C2×Dic502—24no—
48C2×Dic502—22no—
49C2×Dic502—22no—
50C2×Dic502—21no—
51C2×Dic502—21no—
52C2×Dic502C2311yescharacteristic
53C2×Dic502C2312yes—
54Q8×D251—28no—
55Q8×D251—28no—
56Q8×D251—24no—
57Q8×D251—24no—
58Q8×D251—24no—
59Q8×D251—24no—
60Q8×D251D418yes—
61C22×Dic251—21no—
62C22×Dic251D411yescharacteristic
63C2×C4×D251—28no—
64C2×C4×D251—22no—
65C2×C4×D251—22no—
66C2×C4×D251—24no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251D411yescharacteristic
70C2×C4×D251D411yescharacteristic
71C2×C4×D251C2311yescharacteristic
72C2×C4×D251C2311yescharacteristic
73Dic1004—24no—
74#16860800—24no—
75#16860800—24no—
76C2×C251⋊C8C2312yes—
77C8×D251—24no—
78C4⋊Dic251—21no—
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86C2×Q8×C251—24no—
87C2×Q8×C251—21no—
88C2×Q8×C251C2311yescharacteristic
89C2×Q8×C251C2312yes—
90C22×C4×C251—21no—
91C22×C4×C251C2311yescharacteristic
92Q16×C251—24no—
93C2×C8×C251C2312yes—
94C4⋊C4×C251—21no—
95C4⋊C4×C251C2311yescharacteristic
96C22⋊C4×C251—21no—
97C22×D251—22no—
98C22×D251—22no—
99C22×D251Q1614yescomplemented
100C22×D251C2×D411yescharacteristic
101C22×D251C2×D411yescharacteristic
102Dic502—24no—
103Dic502—28no—
104Dic502—22no—
105Dic502—28no—
106Dic502—22no—
107Dic502—42no—
108Dic502C2×D414yes—
109C2×Dic251—24no—
110C2×Dic251—22no—
111C2×Dic251—22no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251C2×D411yescharacteristic
115C2×Dic251C2×D411yescharacteristic
116C4×D251—28no—
117C4×D251—28no—
118C4×D251—22no—
119C4×D251—22no—
120C4×D251—22no—
121C4×D251—42no—
122C4×D251—42no—
123C4×D251C2×D414yes—
124C251⋊C8—22no—
125C23×C251C2×D411yescharacteristic
126Q8×C251—24no—
127Q8×C251—22no—
128Q8×C251—22no—
129Q8×C251C2×D414yes—
130C2×C4×C251—24no—
131C2×C4×C251—22no—
132C2×C4×C251—22no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251C2×D411yescharacteristic
136C2×C4×C251C2411yescharacteristic
137C8×C251—22no—
138C22×C251—21no—
139C22×C251—21no—
140C22×C251C2×Q1612yescomplemented
141C22×C251C22×D411yescharacteristic
142C2×D251—21no—
143C2×D251—21no—
144C2×D251—28no—
145C2×D251#12012yes—
146C2×D251C2×Q1612yes—
147C2×D251C2×Q1612yescomplemented
148C2×D251#13712yes—
149C4×C251—24no—
150C4×C251—21no—
151C4×C251—41no—
152C4×C251—41no—
153C4×C251C22×D412yes—
154Dic251—24no—
155Dic251—21no—
156Dic251—41no—
157Dic251—41no—
158Dic251#12012yes—
159C2×C251—22no—
160C2×C251C2×#12011yescharacteristic
161C2×C251C22×Q1611yescharacteristic
162C2×C251C2×#13711yescharacteristic
163D251—22no—
164D251#45114yesdirect product factor
165C251C2×#45111yescharacteristic, complemented, Sylow
166C2×#451—2511nomaximal, Sylow
167C23×Q8—2511no—
168C22×Q16—2512no—
169C2×C22⋊Q8—2511no—
170#451—2511no—
171#451—2512no—
172#451—2511no—
173#451—2512no—
174#451—2511no—
175#451D25111yescharacteristic, direct product factor
176C2×#103—2512no—
177C2×C22⋊C8—2511no—
178C22×Q8—5021no—
179C22×Q8—5028no—
180C22×Q8—5021no—
181C22×Q8—5021no—
182C22×Q8—2511no—
183C22×Q8—2512no—
184C22×Q8—2512no—
185C22×Q8—2511no—
186C22×Q8C2×D25111yescharacteristic
187C23×C4—5021no—
188C23×C4—2511no—
189C2×Q16—5028no—
190C2×Q16—2512no—
191C2×Q16—2512no—
192C2×Q16—2512no—
193C2×Q16C2×D25112yescomplemented
194C22×C8—2512no—
195C22⋊Q8—5021no—
196C22⋊Q8—5021no—
197C22⋊Q8—2511no—
198C22⋊Q8—2511no—
199C22⋊Q8—2511no—
200C22⋊Q8C2×D25111yescharacteristic
201C2×C4⋊C4—5021no—
202C2×C4⋊C4—2511no—
203C2×C22⋊C4—5021no—
204#103—2512no—
205#103—2512no—
206#103—2512no—
207#103C2×D25112yescomplemented
208C22⋊C8—2511no—
209C22⋊C8—2512no—
210C22⋊C8C2×D25111yescharacteristic
211C24—2511no—
212C2×Q8—24no—
213C2×Q8—21no—
214C2×Q8—5028no—
215C2×Q8—5024no—
216C2×Q8—5021no—
217C2×Q8—5022no—
218C2×Q8—5022no—
219C2×Q8—5024no—
220C2×Q8—5028no—
221C2×Q8—5028no—
222C2×Q8—5024no—
223C2×Q8—5024no—
224C2×Q8—5024no—
225C2×Q8—5021no—
226C2×Q8—2518no—
227C2×Q8—2511no—
228C2×Q8—2512no—
229C2×Q8C22×D25111yescharacteristic
230C2×Q8C22×D25112yes—
231C22×C4—21no—
232C22×C4—5021no—
233C22×C4—5021no—
234C22×C4—5028no—
235C22×C4—5022no—
236C22×C4—5022no—
237C22×C4—5024no—
238C22×C4—5021no—
239C22×C4—2511no—
240C22×C4—2511no—
241C22×C4—2511no—
242C22×C4—2511no—
243C22×C4—2511no—
244C22×C4C22×D25111yescharacteristic
245Q16—24no—
246Q16—5024no—
247Q16—5024no—
248Q16—5024no—
249C2×C8—5024no—
250C2×C8—2512no—
251C2×C8C22×D25112yes—
252C4⋊C4—21no—
253C4⋊C4—5021no—
254C4⋊C4—5021no—
255C4⋊C4—5021no—
256C4⋊C4—5021no—
257C4⋊C4—2511no—
258C4⋊C4C22×D25111yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—5021no—
261C22⋊C4—5021no—
262C22⋊C4—5021no—
263C23—5022no—
264C23—5022no—
265C23—2511no—
266C23—2514no—
267C23—2511no—
268C23D4×D25111yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—22no—
272Q8—10042no—
273Q8—5028no—
274Q8—5024no—
275Q8—5028no—
276Q8—5022no—
277Q8—5022no—
278Q8—2514no—
279Q8D4×D25114yes—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—10042no—
286C2×C4—10042no—
287C2×C4—5028no—
288C2×C4—5028no—
289C2×C4—5022no—
290C2×C4—5022no—
291C2×C4—5021no—
292C2×C4—5021no—
293C2×C4—5024no—
294C2×C4—5022no—
295C2×C4—5022no—
296C2×C4—5022no—
297C2×C4—2511no—
298C2×C4—2511no—
299C2×C4—2514no—
300C2×C4D4×D25111yescharacteristic
301C2×C4C23×D25111yescharacteristic
302C8—22no—
303C8—5022no—
304C22—21no—
305C22—21no—
306C22—5028no—
307C22—5021no—
308C22—5021no—
309C22—2512no—
310C22—2512no—
311C22—2512no—
312C22—2512no—
313C22Q16×D25112yescomplemented
314C22C2×D4×D25111yescharacteristic
315C4—24no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—10041no—
320C4—10041no—
321C4—5024no—
322C4—5021no—
323C4—2512no—
324C4C2×D4×D25112yes—
325C2—22no—
326C2—5022no—
327C2—2514no—
328C2#120×D25111yescharacteristic
329C2C2×Q16×D25111yescharacteristic
330C2#137×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#451×C251C2
socleC22×C251C22×D4
prime radicalC23×D251D4
hypercenter#451D251
hypocenterC251C2×#451
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×Q16×D251C2semidirect
4. #30631845C2semidirect
5. #30631844C2semidirect
6. #30631843C2semidirect
7. #30631842C2semidirect
8. #30631841C2semidirect
9. #103×D251C2semidirect
11. #451×C251C2semidirect
22. C2×Dic1004C22semidirect
23. C2×#16860800C22semidirect
24. C2×#16860800C22semidirect
35. #22548640C22semidirect
36. #22548639C22semidirect
37. #22548638C22semidirect
41. C2×Q16×C251C22semidirect
43. #103×C251C22semidirect
99. C22×D251Q16semidirect
140. C22×C251C2×Q16semidirect
147. C2×D251C2×Q16semidirect
164. D251#451direct (N × H)
165. C251C2×#451semidirect
175. #451D251direct (N × H)
193. C2×Q16C2×D251semidirect
207. #103C2×D251semidirect
313. C22Q16×D251semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D2512
C2×Q16×D2512
C22⋊Q8×D2512
#306318452
#306318442
#306318432
#306318422
#306318412
#103×D2512
C22⋊C8×D2512
#451×C2512
C2×#451251