Finite Group Databasebeta

← #44549325

Subgroups — #44549325

Order 32128.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#30632282C211yesmaximal, characteristic
2#30631843C211yesmaximal, characteristic, complemented
3#30631606C211yesmaximal, characteristic, complemented
4#30631425C211yesmaximal, characteristic, complemented
5#30631421C211yesmaximal, characteristic
6#30631412C211yesmaximal, characteristic, complemented
7#448×C251C211yesmaximal, characteristic
8C2×#16860800—21no—
9#22548708—21no—
10#22548705—21no—
11#22548696—21no—
12#22548696—21no—
13#22548694C2211yescharacteristic
14#22548692—21no—
15D4×Dic251—21no—
16C2×Dic251⋊C4—21no—
17C2×Dic251⋊C4C2211yescharacteristic
18#22548639—21no—
19C22⋊(C251⋊C8)C2211yescharacteristic
20C251⋊#95—21no—
21C251⋊#95C2211yescharacteristic
22C22×Q8×C251C2211yescharacteristic
23C2×SD16×C251—21no—
24#121×C251C2211yescharacteristic
25#103×C251—21no—
26C22⋊C8×C251C2211yescharacteristic
27C2×Dic502D411yescharacteristic
28C22×Dic251—21no—
29C22×Dic251—21no—
30C22×Dic251D411yescharacteristic
31C22×Dic251D411yescharacteristic
32#16860800—42no—
33C2×C251⋊C8—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251D411yescharacteristic
36Dic251⋊C4—21no—
37Dic251⋊C4—21no—
38Dic251⋊C4—21no—
39Dic251⋊C4—21no—
40Dic251⋊C4—21no—
41Dic251⋊C4—42no—
42C22⋊Dic251—21no—
43C22⋊Dic251—21no—
44C22⋊Dic251—21no—
45C22⋊Dic251—21no—
46C4×Dic251—21no—
47C2×Q8×C251—21no—
48C2×Q8×C251—21no—
49C2×Q8×C251—42no—
50C2×D4×C251—21no—
51C2×D4×C251D411yescharacteristic
52C22×C4×C251—21no—
53C22×C4×C251C2311yescharacteristic
54SD16×C251—42no—
55C2×C8×C251—21no—
56C4⋊C4×C251D411yescharacteristic
57C22⋊C4×C251—21no—
58Dic502—41no—
59C2×Dic251—22no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—21no—
66C2×Dic251—41no—
67C2×Dic251—41no—
68C2×Dic251—41no—
69C2×Dic251—81no—
70C251⋊C8—41no—
71C23×C251—21no—
72C23×C251C2×D411yescharacteristic
73Q8×C251—22no—
74Q8×C251—22no—
75Q8×C251—41no—
76Q8×C251—41no—
77D4×C251—41no—
78D4×C251—41no—
79D4×C251—41no—
80C2×C4×C251—22no—
81C2×C4×C251—21no—
82C2×C4×C251—41no—
83C2×C4×C251—41no—
84C2×C4×C251—41no—
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C8×C251—41no—
88C22×C251—21no—
89C22×C251—21no—
90C22×C251—41no—
91C22×C251—41no—
92C22×C251—41no—
93C22×C251#12011yescharacteristic
94C22×C251#13511yescharacteristic
95C22×C251#13711yescharacteristic
96C4×C251—21no—
97C4×C251—21no—
98C4×C251—41no—
99C4×C251—42no—
100Dic251—42no—
101Dic251—41no—
102Dic251—41no—
103Dic251—41no—
104Dic251—41no—
105C2×C251—21no—
106C2×C251—21no—
107C2×C251—81no—
108C2×C251#45211yescharacteristic
109C2×C251#45611yescharacteristic
110C2×C251B(4,𝔽2)11yescharacteristic
111C251#160311yescharacteristic, complemented, Sylow
112#1603—2511nomaximal, Sylow
113#539—2511no—
114#451—2511no—
115#448D25111yescharacteristic
116#395—2511no—
117#328—2511no—
118#327—2511no—
119#324—2511no—
120C22×Q8C2×D25111yescharacteristic
121C2×Q16—5021no—
122C2×SD16—21no—
123#126—5021no—
124#125—5021no—
125#123—5021no—
126#123—5021no—
127C22⋊Q8—5021no—
128C22⋊Q8—2511no—
129#121C2×D25111yescharacteristic
130C4×D4—5021no—
131C2×C4⋊C4—5021no—
132C2×C4⋊C4—2511no—
133#103—21no—
134#103—5021no—
135C22⋊C8—2511no—
136C22⋊C8C2×D25111yescharacteristic
137#95—5021no—
138#95—2511no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8—2511no—
143C2×D4—21no—
144C2×D4C22⋊D25111yescharacteristic
145C22×C4—21no—
146C22×C4—5021no—
147C22×C4—5021no—
148C22×C4—2511no—
149C22×C4—2511no—
150C22×C4C22×D25111yescharacteristic
151Q16—10042no—
152SD16—42no—
153C2×C8—21no—
154C2×C8—5021no—
155C4⋊C4—10042no—
156C4⋊C4—5021no—
157C4⋊C4—5021no—
158C4⋊C4—5021no—
159C4⋊C4—5021no—
160C4⋊C4—5021no—
161C4⋊C4—5021no—
162C4⋊C4—2511no—
163C4⋊C4C22⋊D25111yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—5021no—
166C22⋊C4—5021no—
167C22⋊C4—5021no—
168C22⋊C4—5021no—
169C42—5021no—
170C23—21no—
171C23D4×D25111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—10041no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—20081no—
186C2×C4—10041no—
187C2×C4—10041no—
188C2×C4—10041no—
189C2×C4—5022no—
190C2×C4—5021no—
191C2×C4—5021no—
192C2×C4—5021no—
193C2×C4—5021no—
194C2×C4—5021no—
195C2×C4—5021no—
196C2×C4D4×D25111yescharacteristic
197C2×C4C2×C22⋊D25111yescharacteristic
198C8—41no—
199C8—10041no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#2254868311yescharacteristic
206C22#2254873611yescharacteristic
207C22#2254874411yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—10042no—
213C4—10041no—
214C4—10041no—
215C4—10041no—
216C4—10041no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#3063184611yescharacteristic
221C2#3063186811yescharacteristic
222C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#448×C251C2
socleC22×C251#120
prime radical#121×C251C22
hypercenter#448D251
hypocenterC251#1603
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #30631843C2semidirect
3. #30631606C2semidirect
4. #30631425C2semidirect
6. #30631412C2semidirect
111. C251#1603semidirect

Maximal subgroups

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

subgroupindex
#306322822
#306318432
#306316062
#306314252
#306314212
#306314122
#448×C2512
#1603251