Finite Group Databasebeta

← #44551268

Subgroups — #44551268

Order 32128.

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

normal subgroups only

This group has 296 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
1C22×#16860799C211yesmaximal, characteristic
2C2×#22548651C211yesmaximal, characteristic, complemented
3#30631695C211yesmaximal, characteristic, complemented
4#30631691C211yesmaximal, characteristic
5C2×#22548633C211yesmaximal, characteristic, complemented
6C2×#22548631C211yesmaximal, characteristic
7#405×C251C211yesmaximal, characteristic, complemented
8C22×Dic502C2211yescharacteristic
9C22×D1004C2211yescharacteristic
10C2×#16860799—21no—
11C2×#16860799—21no—
12C2×#16860799—22no—
13C2×#16860799C414yescomplemented
14C2×#16860794C2211yescharacteristic
15C2×#16860794C2211yescharacteristic
16#22548651—22no—
17#22548638—22no—
18#22548635—22no—
19#22548633—22no—
20#22548631—22no—
21C2×M16×C251C2211yescharacteristic
22C22×C8×C251C2211yescharacteristic
23#117×C251C2211yescharacteristic
24#97×C251—21no—
25C4×C8×C251—21no—
26C23×D251—21no—
27C2×Dic502—21no—
28C2×Dic502—21no—
29C2×Dic502—21no—
30C2×Dic502—21no—
31C2×Dic502—21no—
32C2×Dic502C2×C412yes—
33C2×D1004—21no—
34C2×D1004—21no—
35C2×D1004—21no—
36C2×D1004—21no—
37C2×D1004—21no—
38C2×D1004C2×C412yes—
39C22×Dic251—21no—
40#16860799—24no—
41#16860799—24no—
42#16860794—21no—
43#16860794—21no—
44#16860794—21no—
45#16860794—21no—
46C2×C251⋊C8—21no—
47C2×C251⋊C8—21no—
48C22×C4×C251C2311yescharacteristic
49M16×C251—21no—
50M16×C251D412yes—
51C2×C8×C251—21no—
52C2×C8×C251C2×C412yes—
53C2×C8×C251D411yescharacteristic
54C2×C8×C251D411yescharacteristic
55C2×C8×C251D412yes—
56C4⋊C4×C251D412yes—
57C22⋊C4×C251—21no—
58C42×C251—21no—
59C22×D251—22no—
60C22×D251—41no—
61C22×D251—41no—
62C22×D251—82no—
63Dic502—22no—
64Dic502—22no—
65Dic502—41no—
66Dic502—41no—
67D1004—22no—
68D1004—22no—
69D1004—41no—
70D1004—41no—
71C2×Dic251—22no—
72C2×Dic251—41no—
73C2×Dic251—41no—
74C251⋊C8—41no—
75C251⋊C8—41no—
76C23×C251#4111yescharacteristic
77C2×C4×C251—22no—
78C2×C4×C251C22×C411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C2×C4×C251#4111yescharacteristic
84C8×C251—22no—
85C8×C251C22⋊C412yes—
86C8×C251C22⋊C412yes—
87C22×C251—22no—
88C22×C251C4×D411yescharacteristic
89C22×C251#12111yescharacteristic
90C22×C251#12411yescharacteristic
91C2×D251—42no—
92C2×D251—82no—
93C2×D251—82no—
94C2×D251—82no—
95C4×C251—42no—
96C4×C251C2×C22⋊C411yescharacteristic
97C4×C251C4×D411yescharacteristic
98C4×C251#12111yescharacteristic
99C4×C251#12711yescharacteristic
100Dic251—42no—
101C2×C251—21no—
102C2×C251—21no—
103C2×C251#39011yescharacteristic
104C2×C251#47112yes—
105D251—82no—
106C251#197711yescharacteristic, complemented, Sylow
107#1977—2511nomaximal, Sylow
108C22×SD16—2511no—
109C2×#108—2511no—
110#419—2511no—
111#418—2511no—
112C2×#101—2511no—
113C2×#100—2511no—
114#405D25111yescharacteristic, complemented
115C22×Q8—2511no—
116C22×D4—2511no—
117C2×SD16—5021no—
118C2×SD16—5021no—
119C2×SD16—5022no—
120C2×SD16—2514no—
121C2×M16—2511no—
122C2×M16—2511no—
123C2×M16C2×D25111yescharacteristic
124C22×C8C2×D25111yescharacteristic
125#117C2×D25111yescharacteristic
126#108—5022no—
127#103—5022no—
128#102—5022no—
129#101—5022no—
130#100—5022no—
131#97—21no—
132C4×C8—21no—
133C24—5021no—
134C2×Q8—5021no—
135C2×Q8—5021no—
136C2×Q8—5021no—
137C2×Q8—5021no—
138C2×Q8—5021no—
139C2×Q8—2512no—
140C2×D4—5021no—
141C2×D4—5021no—
142C2×D4—5021no—
143C2×D4—5021no—
144C2×D4—5021no—
145C2×D4—2512no—
146C22×C4—5021no—
147C22×C4C22×D25111yescharacteristic
148SD16—5024no—
149SD16—5024no—
150M16—21no—
151M16—5021no—
152M16—5021no—
153M16—5021no—
154M16—5021no—
155M16D100412yes—
156C2×C8—21no—
157C2×C8—5021no—
158C2×C8—5021no—
159C2×C8C4×D25112yes—
160C2×C8C22⋊D25111yescharacteristic
161C2×C8C22⋊D25111yescharacteristic
162C2×C8D100412yes—
163C4⋊C4D100412yes—
164C22⋊C4—21no—
165C42—21no—
166C23—20082no—
167C23—10041no—
168C23—10041no—
169C23—5022no—
170C23#1686081011yescharacteristic
171Q8—10041no—
172Q8—10041no—
173Q8—5022no—
174Q8—5022no—
175D4—10041no—
176D4—10041no—
177D4—5022no—
178D4—5022no—
179C2×C4—22no—
180C2×C4—10041no—
181C2×C4—10041no—
182C2×C4—5022no—
183C2×C4C2×C4×D25111yescharacteristic
184C2×C4C2×C22⋊D25111yescharacteristic
185C2×C4C2×D100411yescharacteristic
186C2×C4C2×D100411yescharacteristic
187C2×C4C2×D100411yescharacteristic
188C2×C4#1686081011yescharacteristic
189C8—22no—
190C8—10041no—
191C8—10041no—
192C8(C2×D251)⋊C412yes—
193C8(C2×D251)⋊C412yes—
194C22—22no—
195C22—20082no—
196C22—20082no—
197C22—20082no—
198C22—10042no—
199C22C4×D100411yescharacteristic
200C22#2254868911yescharacteristic
201C22#2254870311yescharacteristic
202C4—42no—
203C4—10042no—
204C4C2×(C2×D251)⋊C411yescharacteristic
205C4C4×D100411yescharacteristic
206C4#2254868911yescharacteristic
207C4#2254871211yescharacteristic
208C2—21no—
209C2—21no—
210C2—20082no—
211C2#3063158911yescharacteristic
212C2#3063195012yes—
subgroupG/H
centerC22#22548689
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#405×C251C2
socleC22×C251#121
prime radicalC22×D1004C22
cosocleC22×C4×C251C23
hypercenter#405D251
hypocenterC251#1977
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405

Complements

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

NH ≅ G/Ntype
2. C2×#22548651C2semidirect
3. #30631695C2semidirect
5. C2×#22548633C2semidirect
7. #405×C251C2semidirect
13. C2×#16860799C4semidirect
106. C251#1977semidirect
114. #405D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607992
C2×#225486512
#306316952
#306316912
C2×#225486332
C2×#225486312
#405×C2512
#1977251