Finite Group Databasebeta

← #44562096

Subgroups — #44562096

Order 32128.

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

normal subgroups only

This group has 494 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
1C2×#22548736C212yesmaximal, complemented
2#30632359C211yesmaximal, characteristic, complemented
3#30632341C211yesmaximal, characteristic, complemented
4#30632060C211yesmaximal, characteristic, complemented
5#30631917C212yesmaximal, complemented
6#30631897C211yesmaximal, characteristic, complemented
7#30631882C211yesmaximal, characteristic, complemented
8#30631852C212yesmaximal, complemented
9#30631833C212yesmaximal, complemented
10C8×D1004C211yesmaximal, characteristic, complemented
11C4×SD16×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13C2×#16860809C2212yes—
14#22548736—28no—
15C2×#16860800C2212yes—
16C2×#16860799C2212yes—
17C2×#16860799C2212yes—
18C2×#16860795C2212yes—
19C2×C8×D251C2212yes—
20#22548715C2211yescharacteristic
21#22548712—21no—
22#22548712C2211yescharacteristic
23#22548699—22no—
24#22548695C2212yes—
25#22548690—22no—
26#22548690C2212yes—
27#22548689—21no—
28C4×D1004—21no—
29C4×D1004C2211yescharacteristic
30C2×C4⋊Dic251—21no—
31C4⋊(C251⋊C8)C2211yescharacteristic
32#22548638C2212yes—
33#22548635C2212yes—
34(C2×D251)⋊C8C2212yes—
35C2×SD16×C251C2211yescharacteristic
36C4×Q8×C251C2211yescharacteristic
37C4×D4×C251C2211yescharacteristic
38#106×C251C2211yescharacteristic
39#103×C251C2211yescharacteristic
40#102×C251C2211yescharacteristic
41C4×C8×C251C2211yescharacteristic
42#16860811—24no—
43#16860811—42no—
44#16860809—24no—
45#16860809—42no—
46C2×Dic502C2312yes—
47C2×D1004—22no—
48C2×D1004—22no—
49C2×D1004—21no—
50C2×D1004—21no—
51C2×D1004D411yescharacteristic
52C2×D1004C2312yes—
53C2×C22⋊D251—22no—
54C22×Dic251—22no—
55C2×C4×D251—22no—
56C2×C4×D251C2312yes—
57#16860800—24no—
58#16860799—24no—
59#16860799—42no—
60#16860795—24no—
61C2×C251⋊C8C2312yes—
62C8×D251—24no—
63C4⋊Dic251—22no—
64C4⋊Dic251—21no—
65C4⋊Dic251—21no—
66C4⋊Dic251D411yescharacteristic
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—22no—
69(C2×D251)⋊C4D412yes—
70C2×Q8×C251C2311yescharacteristic
71C2×D4×C251C2311yescharacteristic
72C22×C4×C251—21no—
73SD16×C251D414yescomplemented
74C2×C8×C251C2311yescharacteristic
75C2×C8×C251C2311yescharacteristic
76C4⋊C4×C251—21no—
77C4⋊C4×C251C2311yescharacteristic
78C4⋊C4×C251C2311yescharacteristic
79C22⋊C4×C251—21no—
80C42×C251—21no—
81C42×C251C2311yescharacteristic
82C22×D251—22no—
83C22×D251C2×D412yes—
84Dic502—22no—
85Dic502—22no—
86D1004—22no—
87D1004—22no—
88D1004—22no—
89D1004—44no—
90D1004—42no—
91D1004—81no—
92D1004—82no—
93C22⋊D251—44no—
94C2×Dic251—22no—
95C2×Dic251—44no—
96C2×Dic251C2×D412yes—
97C4×D251—22no—
98C4×D251—22no—
99C4×D251—44no—
100C251⋊C8—22no—
101C23×C251—21no—
102Q8×C251—21no—
103Q8×C251C2×D412yes—
104D4×C251—21no—
105D4×C251C2×D412yes—
106C2×C4×C251—21no—
107C2×C4×C251—21no—
108C2×C4×C251—21no—
109C2×C4×C251—42no—
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2411yescharacteristic
113C8×C251—21no—
114C8×C251C2×D412yes—
115C22×C251—22no—
116C22×C251—41no—
117C22×C251—41no—
118C22×C251C22×D411yescharacteristic
119C2×D251—22no—
120C2×D251—22no—
121C2×D251—22no—
122C2×D251—42no—
123C2×D251—42no—
124C2×D251—42no—
125C4×C251—22no—
126C4×C251—21no—
127C4×C251—41no—
128C4×C251—41no—
129C4×C251#13512yescomplemented
130C4×C251C22×D411yescharacteristic
131C4×C251H(5,𝔽2)11yescharacteristic
132Dic251—22no—
133Dic251—42no—
134C2×C251—42no—
135C2×C251D4211yescharacteristic
136C2×C251C2×#13511yescharacteristic
137C2×C251#57711yescharacteristic
138D251—42no—
139D251—82no—
140C251#328411yescharacteristic, complemented, Sylow
141#3284—2511nomaximal, Sylow
142C2×#135—2512no—
143#550—2511no—
144#547—2511no—
145#492—2511no—
146#465—2512no—
147#462—2511no—
148#459—2511no—
149#452—2512no—
150#449—2512no—
151C4×SD16D25111yescharacteristic, complemented
152C8×D4—2511no—
153C2×#41—2512no—
154C2×#41—2512no—
155#135—5028no—
156C2×Q16—2512no—
157C2×SD16—2512no—
158C2×SD16—2512no—
159C2×SD16C2×D25111yescharacteristic
160C2×D8—2512no—
161C22×C8—2512no—
162C4⋊Q8—2511no—
163#127—5021no—
164#127—2511no—
165#123—5022no—
166C22⋊Q8—2512no—
167#121—5021no—
168#121—5022no—
169#121—2512no—
170C4×Q8C2×D25111yescharacteristic
171C4×D4—5021no—
172C4×D4—2511no—
173C4×D4C2×D25111yescharacteristic
174C2×C4⋊C4—5021no—
175#106C2×D25111yescharacteristic
176C4⋊C8—2511no—
177#103—2512no—
178#103C2×D25111yescharacteristic
179#102—2512no—
180#102C2×D25111yescharacteristic
181C22⋊C8—2512no—
182C4×C8C2×D25111yescharacteristic
183#41—10042no—
184#41—10042no—
185#41—5024no—
186#41—5024no—
187C2×Q8—2512no—
188C2×Q8C22×D25111yescharacteristic
189C2×D4—5022no—
190C2×D4—5022no—
191C2×D4—5022no—
192C2×D4—5021no—
193C2×D4—5021no—
194C2×D4—2511no—
195C2×D4—2512no—
196C2×D4C22×D25111yescharacteristic
197C22×C4—21no—
198C22×C4—5022no—
199C22×C4—5022no—
200C22×C4—2512no—
201Q16—5024no—
202SD16—10042no—
203SD16—5024no—
204SD16D100414yescomplemented
205D8—5024no—
206C2×C8—5024no—
207C2×C8—2512no—
208C2×C8C22×D25111yescharacteristic
209C2×C8C22×D25111yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—5022no—
212C4⋊C4—5021no—
213C4⋊C4—5021no—
214C4⋊C4—2511no—
215C4⋊C4C22×D25111yescharacteristic
216C4⋊C4C22×D25111yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—5022no—
219C22⋊C4—5022no—
220C22⋊C4—2512no—
221C42—21no—
222C42C22×D25111yescharacteristic
223C23—21no—
224C23—5022no—
225C23—2512no—
226Q8—21no—
227Q8—5022no—
228Q8—5022no—
229Q8C2×D100412yes—
230D4—21no—
231D4—20081no—
232D4—20082no—
233D4—10044no—
234D4—10044no—
235D4—10042no—
236D4—5022no—
237D4—5022no—
238D4—5022no—
239D4C2×D100412yes—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—10044no—
245C2×C4—10044no—
246C2×C4—5022no—
247C2×C4—5022no—
248C2×C4—5022no—
249C2×C4—2512no—
250C2×C4D4×D25111yescharacteristic
251C2×C4D4×D25111yescharacteristic
252C2×C4C23×D25111yescharacteristic
253C8—21no—
254C8—5022no—
255C8C2×D100412yes—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—10042no—
260C22—10042no—
261C22—10042no—
262C22—5022no—
263C22—5022no—
264C22—5022no—
265C22C2×D4×D25111yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—41no—
270C4—10042no—
271C4—5022no—
272C4#2254873612yescomplemented
273C4C22×D100411yescharacteristic
274C4#2254876111yescharacteristic
275C2—42no—
276C2—20082no—
277C2—10042no—
278C2D4×D100411yescharacteristic
279C2C2×#2254873611yescharacteristic
280C2#3063249411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×SD16×C251C2
socleC22×C251C22×D4
hypercenterC4×SD16D251
hypocenterC251#3284
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22548736C2semidirect
2. #30632359C2semidirect
3. #30632341C2semidirect
4. #30632060C2semidirect
5. #30631917C2semidirect
6. #30631897C2semidirect
7. #30631882C2semidirect
8. #30631852C2semidirect
9. #30631833C2semidirect
10. C8×D1004C2semidirect
11. C4×SD16×C251C2semidirect
73. SD16×C251D4semidirect
129. C4×C251#135semidirect
140. C251#3284semidirect
151. C4×SD16D251semidirect
204. SD16D1004semidirect
272. C4#22548736semidirect

Maximal subgroups

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

subgroupindex
C2×#225487362
#306323592
#306323412
#306320602
#306319172
#306318972
#306318822
#306318522
#306318332
C8×D10042
C4×SD16×C2512
#3284251