Finite Group Databasebeta

← #44549541

Subgroups — #44549541

Order 32128.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#30632322C211yesmaximal, characteristic
2#30631964C211yesmaximal, characteristic, complemented
3#30631889C211yesmaximal, characteristic, complemented
4#30631431C211yesmaximal, characteristic, complemented
5#30631416C211yesmaximal, characteristic, complemented
6#30631416C211yesmaximal, characteristic, complemented
7#486×C251C211yesmaximal, characteristic, complemented
8C2×D4×D251—21no—
9C2×D4×D251—21no—
10C2×#16860797—21no—
11#22548686—21no—
12#22548686—21no—
13#22548683—22no—
14C4×D1004C2211yescharacteristic
15C4×D1004C2211yescharacteristic
16#22548648—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic, complemented
18C4⋊(C251⋊C8)C2211yescharacteristic, complemented
19#22548639—21no—
20#22548635—21no—
21C4⋊Q8×C251C2211yescharacteristic, complemented
22#127×C251C2211yescharacteristic
23#102×C251—22no—
24C4×C8×C251C2211yescharacteristic, complemented
25C23×D251—22no—
26C2×D1004D411yescharacteristic
27C2×D1004D411yescharacteristic
28C2×C22⋊D251—22no—
29C2×C22⋊D251—22no—
30D4×D251—42no—
31D4×D251—42no—
32D4×D251—42no—
33D4×D251—42no—
34C2×C4×D251—21no—
35C2×C4×D251—21no—
36#16860797—42no—
37C2×C251⋊C8—21no—
38C2×C251⋊C8—21no—
39C4⋊Dic251D411yescharacteristic
40C4⋊Dic251D411yescharacteristic
41C22⋊Dic251—22no—
42(C2×D251)⋊C4—21no—
43(C2×D251)⋊C4—21no—
44C2×Q8×C251—21no—
45C2×D4×C251—21no—
46C2×D4×C251—21no—
47C2×D4×C251D412yes—
48C2×C8×C251—21no—
49C4⋊C4×C251—21no—
50C4⋊C4×C251—21no—
51C42×C251C2311yescharacteristic
52C22×D251—21no—
53C22×D251—21no—
54C22×D251—44no—
55C22×D251—44no—
56C22×D251—82no—
57D1004—21no—
58D1004—21no—
59D1004D812yes—
60D1004SD1612yescomplemented
61C22⋊D251—44no—
62C22⋊D251—44no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C4×D251—41no—
66C4×D251—41no—
67C4×D251—41no—
68C4×D251—41no—
69C251⋊C8—41no—
70C251⋊C8—41no—
71C23×C251—22no—
72Q8×C251—41no—
73Q8×C251—41no—
74D4×C251—42no—
75D4×C251—41no—
76D4×C251—41no—
77D4×C251—41no—
78D4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C8×C251—41no—
84C22×C251—42no—
85C22×C251—42no—
86C22×C251—42no—
87C22×C251#12011yescharacteristic
88C2×D251—22no—
89C2×D251—22no—
90C2×D251—41no—
91C2×D251—41no—
92C2×D251—41no—
93C2×D251—41no—
94C2×D251—88no—
95C4×C251—21no—
96C4×C251—81no—
97C4×C251C2×D811yescharacteristic
98C4×C251C2×SD1611yescharacteristic
99C4×C251AGL(1,ℤ/8)11yescharacteristic
100C4×C251AGL(1,ℤ/8)11yescharacteristic
101Dic251—41no—
102Dic251—41no—
103C2×C251—82no—
104C2×C251#44711yescharacteristic
105C2×C251#45011yescharacteristic
106C2×C251#45511yescharacteristic
107D251—42no—
108D251—42no—
109C251#163911yescharacteristic, complemented, Sylow
110#1639—2511nomaximal, Sylow
111D42—2511no—
112#486D25111yescharacteristic, complemented
113#474—2511no—
114#461—2511no—
115#331—2511no—
116D4⋊C8—2511no—
117D4⋊C8—2511no—
118C22×D4—5021no—
119C22×D4—5021no—
120C2×SD16—5021no—
121C4⋊Q8C2×D25111yescharacteristic, complemented
122#127C2×D25111yescharacteristic
123#121—5021no—
124#121—5021no—
125#120—5022no—
126C4×D4—2511no—
127C4×D4—2511no—
128#107—5021no—
129C4⋊C8—2511no—
130C4⋊C8—2511no—
131#103—5021no—
132#102—22no—
133#102—5021no—
134C4×C8C2×D25111yescharacteristic, complemented
135C24—5022no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—10042no—
140C2×D4—10042no—
141C2×D4—10042no—
142C2×D4—10042no—
143C2×D4—5022no—
144C2×D4—5022no—
145C2×D4—2511no—
146C2×D4—2511no—
147C2×D4C22⋊D25112yes—
148C22×C4—5021no—
149C22×C4—5021no—
150SD16—10042no—
151C2×C8—21no—
152C2×C8—5021no—
153C2×C8—5021no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2511no—
157C4⋊C4—2511no—
158C22⋊C4—5022no—
159C22⋊C4—5021no—
160C22⋊C4—5021no—
161C42C22×D25111yescharacteristic
162C23—22no—
163C23—20082no—
164C23—10044no—
165C23—10044no—
166C23—5021no—
167C23—5021no—
168Q8—41no—
169Q8—41no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—10044no—
176D4—10044no—
177D4—5021no—
178D4—5021no—
179D4—2512no—
180D4—2512no—
181C2×C4—41no—
182C2×C4—10041no—
183C2×C4—10041no—
184C2×C4—10041no—
185C2×C4—10041no—
186C2×C4—5021no—
187C2×C4—5021no—
188C2×C4D4×D25111yescharacteristic
189C2×C4D4×D25111yescharacteristic
190C2×C4C2×C22⋊D25111yescharacteristic
191C8—41no—
192C8—10041no—
193C8—10041no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—20088no—
198C22—10041no—
199C22—10041no—
200C22—10041no—
201C22—10041no—
202C22—5022no—
203C22—5022no—
204C22#2254868311yescharacteristic
205C4—21no—
206C4—81no—
207C4—10041no—
208C4—10041no—
209C4D8×D25111yescharacteristic
210C4SD16×D25111yescharacteristic
211C4#2254873911yescharacteristic
212C4#2254874211yescharacteristic
213C2—82no—
214C2—10042no—
215C2—10042no—
216C2#3063181911yescharacteristic
217C2#3063183611yescharacteristic
218C2#3063186411yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#486×C251C2
socleC22×C251#120
prime radical#30632322C2
hypercenter#486D251
hypocenterC251#1639
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #30631964C2semidirect
3. #30631889C2semidirect
4. #30631431C2semidirect
5. #30631416C2semidirect
6. #30631416C2semidirect
7. #486×C251C2semidirect
17. C4⋊(C251⋊C8)C22semidirect
18. C4⋊(C251⋊C8)C22semidirect
21. C4⋊Q8×C251C22semidirect
24. C4×C8×C251C22semidirect
60. D1004SD16semidirect
109. C251#1639semidirect
112. #486D251semidirect
121. C4⋊Q8C2×D251semidirect
134. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306323222
#306319642
#306318892
#306314312
#306314162
#486×C2512
#1639251