Finite Group Databasebeta

← #44550765

Subgroups — #44550765

Order 32128.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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×#22548702C211yesmaximal, characteristic
2#30632164C211yesmaximal, characteristic, complemented
3#30631692C211yesmaximal, characteristic
4C2×#22548633C211yesmaximal, characteristic, complemented
5#30631448C211yesmaximal, characteristic, complemented
6#30631446C211yesmaximal, characteristic, complemented
7#418×C251C211yesmaximal, characteristic
8C22×Dic502C2211yescharacteristic
9C2×#16860794C2211yescharacteristic
10#22548702—22no—
11#22548702C414yescomplemented
12C4×Dic502—22no—
13#22548680—22no—
14#22548674C2211yescharacteristic
15#22548671—22no—
16C2×C22⋊Dic251—22no—
17C2×C4×Dic251C2211yescharacteristic
18#22548636—22no—
19#22548633—22no—
20C22×D4×C251C2211yescharacteristic
21C2×M16×C251C2211yescharacteristic
22#117×C251C2211yescharacteristic
23#102×C251—22no—
24C2×Dic502—21no—
25C2×Dic502—21no—
26C2×Dic502—21no—
27C2×Dic502—21no—
28C2×Dic502C2×C412yes—
29C2×Dic502D412yes—
30C22×Dic251—21no—
31C22×Dic251D412yes—
32#16860794—21no—
33#16860794—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251D412yes—
36Dic251⋊C4—24no—
37C22⋊Dic251—28no—
38C22⋊Dic251—21no—
39C4×Dic251—24no—
40C4×Dic251—21no—
41C4×Dic251—21no—
42C4×Dic251C2×C412yes—
43C24×C251—21no—
44C2×D4×C251—21no—
45C2×D4×C251—21no—
46C2×D4×C251—21no—
47C2×D4×C251—21no—
48C2×D4×C251—21no—
49C2×D4×C251C2×C412yes—
50C22×C4×C251C2311yescharacteristic
51M16×C251—21no—
52M16×C251—21no—
53C2×C8×C251—21no—
54C4⋊C4×C251D412yes—
55C22⋊C4×C251—21no—
56C42×C251—21no—
57Dic502—22no—
58Dic502—22no—
59Dic502—22no—
60Dic502—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—42no—
66C2×Dic251C22⋊C414yes—
67C251⋊C8—41no—
68C23×C251—22no—
69C23×C251—44no—
70C23×C251—41no—
71C23×C251—41no—
72C23×C251C2×D411yescharacteristic
73D4×C251—22no—
74D4×C251—22no—
75D4×C251—41no—
76D4×C251—41no—
77C2×C4×C251—22no—
78C2×C4×C251C22×C411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C2×C4×C251#4111yescharacteristic
83C2×C4×C251#4111yescharacteristic
84C8×C251—41no—
85C22×C251—22no—
86C22×C251—44no—
87C22×C251—42no—
88C22×C251—82no—
89C22×C251—82no—
90C22×C251C2×C22⋊C411yescharacteristic
91C22×C251#12011yescharacteristic
92C22×C251#12311yescharacteristic
93C4×C251—42no—
94C4×C251C4×D411yescharacteristic
95C4×C251C4×D411yescharacteristic
96C4×C251#12111yescharacteristic
97C4×C251#12111yescharacteristic
98Dic251—24no—
99Dic251—42no—
100Dic251—42no—
101C2×C251—21no—
102C2×C251—21no—
103C2×C251—82no—
104C2×C251#38611yescharacteristic
105C2×C251#45412yes—
106C251#188811yescharacteristic, complemented, Sylow
107#1888—2511nomaximal, Sylow
108C2×#124—2511no—
109#519—2511no—
110#418—2511no—
111#418D25111yescharacteristic
112C2×#101—2511no—
113#339—2511no—
114#339—2511no—
115C22×Q8—2511no—
116C22×D4C2×D25111yescharacteristic
117C2×M16—2511no—
118C2×M16C2×D25111yescharacteristic
119#124—5022no—
120#124—2514no—
121C4×Q8—5022no—
122C4×Q8—5022no—
123#117—5022no—
124#117—2511no—
125#117C2×D25111yescharacteristic
126C2×C22⋊C4—5022no—
127C2×C42—2511no—
128#102—22no—
129#102—5022no—
130#101—5022no—
131C24—21no—
132C2×Q8—5021no—
133C2×Q8—5021no—
134C2×Q8—5021no—
135C2×Q8—5021no—
136C2×Q8—2512no—
137C2×Q8—2512no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C4×D25112yes—
144C22×C4—5021no—
145C22×C4—2512no—
146C22×C4C22×D25111yescharacteristic
147M16—21no—
148M16—21no—
149M16—5021no—
150M16—5021no—
151C2×C8—21no—
152C2×C8—5021no—
153C4⋊C4—5024no—
154C4⋊C4—2512no—
155C4⋊C4C22⋊D25112yes—
156C22⋊C4—21no—
157C22⋊C4—5021no—
158C22⋊C4—5028no—
159C42—21no—
160C42—5024no—
161C42—5021no—
162C42—5021no—
163C42—2512no—
164C23—22no—
165C23—44no—
166C23—41no—
167C23—41no—
168C23D4×D25111yescharacteristic
169Q8—5022no—
170Q8—5022no—
171Q8—5022no—
172Q8—5022no—
173D4—22no—
174D4—22no—
175D4—41no—
176D4—41no—
177C2×C4—22no—
178C2×C4—10042no—
179C2×C4—5022no—
180C2×C4—5022no—
181C2×C4—5022no—
182C2×C4—5022no—
183C2×C4—2514no—
184C2×C4C2×C4×D25111yescharacteristic
185C2×C4D4×D25111yescharacteristic
186C2×C4D4×D25111yescharacteristic
187C2×C4C2×C22⋊D25111yescharacteristic
188C2×C4#1686080911yescharacteristic
189C2×C4#1686081011yescharacteristic
190C8—41no—
191C8—10041no—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—82no—
196C22—82no—
197C22C22⋊C4×D25111yescharacteristic
198C22#2254868311yescharacteristic
199C22#2254869711yescharacteristic
200C4—42no—
201C4—10042no—
202C4—10042no—
203C4—5024no—
204C4(C22⋊D251)⋊C411yescharacteristic
205C4C4×C22⋊D25111yescharacteristic
206C4#2254868611yescharacteristic
207C4#2254868811yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#3063156311yescharacteristic
212C2#3063185812yes—
subgroupG/H
centerC22#22548683
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#418×C251C2
socleC22×C251#120
prime radicalC22×D4×C251C22
cosocleC22×C4×C251C23
hypercenter#418D251
hypocenterC251#1888
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #30632164C2semidirect
4. C2×#22548633C2semidirect
5. #30631448C2semidirect
6. #30631446C2semidirect
11. #22548702C4semidirect
106. C251#1888semidirect

Maximal subgroups

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

subgroupindex
C2×#225487022
#306321642
#306316922
C2×#225486332
#306314482
#306314462
#418×C2512
#1888251