Finite Group Databasebeta

← #44551365

Subgroups — #44551365

Order 32128.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#22548691C211yesmaximal, characteristic, complemented
2C2×#22548686C211yesmaximal, characteristic
3C2×#22548639C211yesmaximal, characteristic, complemented
4C2×#22548636C211yesmaximal, characteristic, complemented
5C2×(C2×D251)⋊C8C211yesmaximal, characteristic, complemented
6#30631450C211yesmaximal, characteristic, complemented
7C22×SD16×C251C211yesmaximal, characteristic, complemented
8C22×C22⋊D251—21no—
9C22×C4×D251C2211yescharacteristic
10C22×C251⋊C8C2211yescharacteristic
11#22548691—22no—
12#22548691—22no—
13#22548686—22no—
14#22548686—22no—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×C4⋊Dic251C2211yescharacteristic
17C2×Dic251⋊C4—21no—
18C2×C22⋊Dic251—21no—
19C2×(C2×D251)⋊C4—21no—
20#22548639—22no—
21#22548636—22no—
22(C2×D251)⋊C8—22no—
23C22×Q8×C251C2211yescharacteristic
24C22×D4×C251C2211yescharacteristic
25C2×SD16×C251—22no—
26C2×SD16×C251—22no—
27C2×SD16×C251—22no—
28C22×C8×C251C2211yescharacteristic
29C23×D251D411yescharacteristic
30C2×C22⋊D251—24no—
31C2×C22⋊D251—42no—
32C2×C22⋊D251—42no—
33C22×Dic251—21no—
34C22×Dic251—21no—
35C22×Dic251D411yescharacteristic
36C2×C4×D251—21no—
37C2×C4×D251—21no—
38C2×C4×D251—21no—
39C2×C4×D251—21no—
40C2×C4×D251—21no—
41C2×C4×D251—21no—
42C2×C251⋊C8—21no—
43C2×C251⋊C8—21no—
44C2×C251⋊C8—21no—
45C4⋊Dic251—21no—
46C4⋊Dic251—21no—
47C4⋊Dic251D412yes—
48C4⋊Dic251D412yescomplemented
49Dic251⋊C4—24no—
50C22⋊Dic251—24no—
51(C2×D251)⋊C4—24no—
52C24×C251—21no—
53C2×Q8×C251—21no—
54C2×Q8×C251—21no—
55C2×Q8×C251—21no—
56C2×Q8×C251—21no—
57C2×Q8×C251—21no—
58C2×Q8×C251D412yescomplemented
59C2×D4×C251—21no—
60C2×D4×C251—21no—
61C2×D4×C251—21no—
62C2×D4×C251—21no—
63C2×D4×C251—21no—
64C2×D4×C251D412yes—
65C22×C4×C251—21no—
66C22×C4×C251C2311yescharacteristic
67SD16×C251—28no—
68C2×C8×C251—21no—
69C2×C8×C251—21no—
70C2×C8×C251D412yescomplemented
71C22×D251—21no—
72C22×D251—21no—
73C22×D251—21no—
74C22×D251—21no—
75C22×D251—21no—
76C22×D251—21no—
77C22×D251SD1612yescomplemented
78C22⋊D251—48no—
79C2×Dic251—22no—
80C2×Dic251—21no—
81C2×Dic251—22no—
82C2×Dic251—21no—
83C2×Dic251—21no—
84C2×Dic251—41no—
85C2×Dic251—41no—
86C2×Dic251—41no—
87C2×Dic251—41no—
88C4×D251—41no—
89C4×D251—41no—
90C4×D251—41no—
91C4×D251—41no—
92C251⋊C8—41no—
93C23×C251—22no—
94C23×C251—41no—
95C23×C251—41no—
96C23×C251—41no—
97C23×C251—41no—
98C23×C251—41no—
99C23×C251—41no—
100C23×C251C2×D411yescharacteristic
101Q8×C251—22no—
102Q8×C251—22no—
103Q8×C251—41no—
104Q8×C251—41no—
105D4×C251—22no—
106D4×C251—22no—
107D4×C251—41no—
108D4×C251—41no—
109C2×C4×C251—22no—
110C2×C4×C251—41no—
111C2×C4×C251—41no—
112C2×C4×C251C2×D411yescharacteristic
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251#4111yescharacteristic
118C8×C251—22no—
119C22×C251—42no—
120C22×C251—42no—
121C22×C251—42no—
122C22×C251—81no—
123C22×C251—81no—
124C22×C251—81no—
125C22×C251—81no—
126C22×C251#12011yescharacteristic
127C22×C251#12011yescharacteristic
128C22×C251#12111yescharacteristic
129C22×C251C2×SD1611yescharacteristic
130C22×C251#13511yescharacteristic
131C22×C251AGL(1,ℤ/8)11yescharacteristic
132C22×C251#13711yescharacteristic
133C2×D251—22no—
134C2×D251—22no—
135C2×D251—22no—
136C2×D251—41no—
137C2×D251—41no—
138C2×D251—41no—
139C2×D251—41no—
140C4×C251—42no—
141C4×C251#12011yescharacteristic
142C4×C251#12111yescharacteristic
143C4×C251#12111yescharacteristic
144C4×C251#12711yescharacteristic
145Dic251—41no—
146Dic251—42no—
147Dic251—42no—
148C2×C251—82no—
149C2×C251#39211yescharacteristic
150C2×C251#44811yescharacteristic
151C2×C251#44911yescharacteristic
152C2×C251#45011yescharacteristic
153C2×C251#45211yescharacteristic
154C2×C251#46511yescharacteristic
155C2×C251#46811yescharacteristic
156D251—42no—
157C251#200011yescharacteristic, complemented, Sylow
158#2000—2511nomaximal, Sylow
159C22×SD16D25111yescharacteristic, complemented
160C2×C22⋊Q8—2511no—
161C2×#121—2511no—
162C2×#103—2511no—
163C2×#102—2511no—
164C2×C22⋊C8—2511no—
165#340—2511no—
166C22×Q8C2×D25111yescharacteristic
167C22×D4—5021no—
168C22×D4C2×D25111yescharacteristic
169C23×C4—2511no—
170C2×SD16—22no—
171C2×SD16—22no—
172C2×SD16—22no—
173C22×C8—2511no—
174C22×C8C2×D25111yescharacteristic
175C22⋊Q8—5022no—
176C22⋊Q8—5022no—
177#121—5022no—
178#121—5022no—
179C2×C4⋊C4—5021no—
180C2×C4⋊C4—2511no—
181C2×C4⋊C4—2511no—
182C2×C22⋊C4—5021no—
183C2×C22⋊C4—5021no—
184#103—5022no—
185#102—5022no—
186C22⋊C8—5022no—
187C24—21no—
188C24—2511no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8C22⋊D25112yescomplemented
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—10042no—
201C2×D4—10042no—
202C2×D4—5024no—
203C2×D4C22⋊D25112yes—
204C22×C4—21no—
205C22×C4—5021no—
206C22×C4—5021no—
207C22×C4—5021no—
208C22×C4—5021no—
209C22×C4—5021no—
210C22×C4—5021no—
211C22×C4—5021no—
212C22×C4—5021no—
213C22×C4—2511no—
214C22×C4C22×D25111yescharacteristic
215SD16—28no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—5021no—
219C2×C8—5021no—
220C2×C8—5021no—
221C2×C8C22⋊D25112yescomplemented
222C4⋊C4—5024no—
223C4⋊C4—5021no—
224C4⋊C4—5021no—
225C4⋊C4—2512no—
226C4⋊C4—2512no—
227C22⋊C4—5024no—
228C22⋊C4—5024no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—5021no—
237C23—5021no—
238C23—5021no—
239C23—5021no—
240C23—5021no—
241C23—5021no—
242C23—2512no—
243C23D4×D25111yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—10048no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—10041no—
257C2×C4—10041no—
258C2×C4—10041no—
259C2×C4—10041no—
260C2×C4—10041no—
261C2×C4—10041no—
262C2×C4—10041no—
263C2×C4—10041no—
264C2×C4—5021no—
265C2×C4—5022no—
266C2×C4—5022no—
267C2×C4—5021no—
268C2×C4—5021no—
269C2×C4D4×D25111yescharacteristic
270C2×C4D4×D25111yescharacteristic
271C2×C4C2×C22⋊D25111yescharacteristic
272C2×C4C2×C22⋊D25111yescharacteristic
273C2×C4C2×C22⋊D25111yescharacteristic
274C2×C4#1686080911yescharacteristic
275C8—22no—
276C8—10041no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—10041no—
285C22—10041no—
286C22—10041no—
287C22—10041no—
288C22—5022no—
289C22—5022no—
290C22—5022no—
291C22#2254868311yescharacteristic
292C22#2254868311yescharacteristic
293C22#2254868611yescharacteristic
294C22SD16×D25111yescharacteristic
295C22#2254873611yescharacteristic
296C22#2254874211yescharacteristic
297C22#2254874411yescharacteristic
298C4—42no—
299C4—10042no—
300C4—10042no—
301C4—10041no—
302C4#2254868411yescharacteristic
303C4#2254868711yescharacteristic
304C4#2254868711yescharacteristic
305C4#2254871111yescharacteristic
306C2—82no—
307C2—10042no—
308C2#3063159511yescharacteristic
309C2#3063182411yescharacteristic
310C2#3063183111yescharacteristic
311C2#3063183611yescharacteristic
312C2#3063184611yescharacteristic
313C2#3063191311yescharacteristic
314C2#3063193011yescharacteristic
subgroupG/H
centerC23D4×D251
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
FittingC22×SD16×C251C2
socleC23×C251C2×D4
prime radicalC2×#22548686C2
hypercenterC22×SD16D251
hypocenterC251#2000
LCSG ▷ C22×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22548691C2semidirect
3. C2×#22548639C2semidirect
4. C2×#22548636C2semidirect
5. C2×(C2×D251)⋊C8C2semidirect
6. #30631450C2semidirect
7. C22×SD16×C251C2semidirect
48. C4⋊Dic251D4semidirect
58. C2×Q8×C251D4semidirect
70. C2×C8×C251D4semidirect
77. C22×D251SD16semidirect
157. C251#2000semidirect
159. C22×SD16D251semidirect
194. C2×Q8C22⋊D251semidirect
221. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486912
C2×#225486862
C2×#225486392
C2×#225486362
C2×(C2×D251)⋊C82
#306314502
C22×SD16×C2512
#2000251