Finite Group Databasebeta

← #44549319

Subgroups — #44549319

Order 32128.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632249C211yesmaximal, characteristic
2#30631828C212yesmaximal, complemented
3#30631599C211yesmaximal, characteristic, complemented
4#30631422C212yesmaximal, complemented
5#323×C251C211yesmaximal, characteristic, complemented
6C22×Dic502C2211yescharacteristic, complemented
7C22×C22⋊D251—21no—
8C2×#16860799—22no—
9#22548699—22no—
10#22548689—21no—
11#22548689C2212yes—
12#22548685—22no—
13C2×C22⋊Dic251—21no—
14#22548640—22no—
15C251⋊#95—21no—
16C251⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C251C2211yescharacteristic
18C22⋊C8×C251C2212yescomplemented
19C23×D251—21no—
20C2×Dic502—22no—
21C2×Dic502—42no—
22C2×D1004—21no—
23C2×D1004D412yes—
24C2×C22⋊D251—22no—
25C2×C22⋊D251—22no—
26C2×C22⋊D251—42no—
27C2×C22⋊D251—42no—
28C22×Dic251—21no—
29C22×Dic251—21no—
30C22×Dic251—21no—
31#16860799—44no—
32C4⋊Dic251—21no—
33C4⋊Dic251D412yes—
34C22⋊Dic251—42no—
35(C2×D251)⋊C4—22no—
36(C2×D251)⋊C4—22no—
37C24×C251D411yescharacteristic
38C22×C4×C251D411yescharacteristic
39C22×C4×C251C2311yescharacteristic
40C2×C8×C251—22no—
41C22⋊C4×C251—22no—
42C22×D251—22no—
43C22×D251—42no—
44C22×D251—81no—
45C22×D251—81no—
46Dic502—24no—
47Dic502—42no—
48D1004—42no—
49D1004—81no—
50C22⋊D251—44no—
51C22⋊D251—42no—
52C22⋊D251—42no—
53C2×Dic251—22no—
54C2×Dic251—22no—
55C2×Dic251—42no—
56C2×Dic251—41no—
57C2×Dic251—41no—
58C2×Dic251—41no—
59C2×Dic251—41no—
60C2×Dic251—41no—
61C23×C251—21no—
62C23×C251—21no—
63C23×C251—41no—
64C23×C251—41no—
65C23×C251SD1612yescomplemented
66C23×C251C2×D411yescharacteristic
67C2×C4×C251—21no—
68C2×C4×C251—41no—
69C2×C4×C251—41no—
70C2×C4×C251C2×D412yes—
71C8×C251—42no—
72C22×C251—21no—
73C22×C251—21no—
74C22×C251—22no—
75C22×C251—42no—
76C22×C251—41no—
77C22×C251—41no—
78C22×C251—81no—
79C22×C251#12011yescharacteristic
80C22×C251C2×SD1611yescharacteristic
81C22×C251AGL(1,ℤ/8)11yescharacteristic
82C2×D251—42no—
83C2×D251—42no—
84C2×D251—42no—
85C2×D251—84no—
86C4×C251—22no—
87C4×C251—41no—
88Dic251—42no—
89Dic251—42no—
90Dic251—81no—
91C2×C251—21no—
92C2×C251—21no—
93C2×C251—42no—
94C2×C251#45011yescharacteristic
95C2×C251#45511yescharacteristic
96C2×C251B(4,𝔽2)11yescharacteristic
97D251—82no—
98C251#160111yescharacteristic, complemented, Sylow
99#1601—2511nomaximal, Sylow
100#534—2511no—
101#448—2512no—
102#393—2511no—
103#327—2512no—
104#323D25111yescharacteristic, complemented
105C22×Q8—2511no—
106C22×D4—5021no—
107C2×SD16—5022no—
108#123—5022no—
109#121—5021no—
110#121—2512no—
111#120—5022no—
112C2×C22⋊C4—5021no—
113C2×C22⋊C4C2×D25111yescharacteristic
114#103—5022no—
115C22⋊C8C2×D25112yescomplemented
116#95—5021no—
117#95—2511no—
118C24—5021no—
119C24D100411yescharacteristic
120C2×Q8—10042no—
121C2×Q8—5022no—
122C2×D4—10042no—
123C2×D4—10042no—
124C2×D4—5021no—
125C2×D4—5022no—
126C2×D4—5022no—
127C2×D4—2512no—
128C22×C4—5021no—
129C22×C4—5021no—
130C22×C4—5021no—
131C22×C4D100411yescharacteristic
132C22×C4C22×D25111yescharacteristic
133SD16—10044no—
134C2×C8—22no—
135C4⋊C4—5021no—
136C4⋊C4—2512no—
137C22⋊C4—22no—
138C22⋊C4—10042no—
139C22⋊C4—5022no—
140C22⋊C4—5022no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—20081no—
146C23—20081no—
147C23—10042no—
148C23—5022no—
149C23#1686079912yescomplemented
150C23C2×D100411yescharacteristic
151Q8—10042no—
152Q8—5024no—
153D4—20081no—
154D4—10042no—
155D4—10044no—
156D4—10042no—
157D4—10042no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—10041no—
162C2×C4—10041no—
163C2×C4—10041no—
164C2×C4—10042no—
165C2×C4—10041no—
166C2×C4—10041no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4D4×D25112yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—20084no—
179C22—10042no—
180C22—10042no—
181C22—10042no—
182C22#2254868511yescharacteristic
183C22C2×#1686079911yescharacteristic
184C22#2254874311yescharacteristic
185C4—22no—
186C4—41no—
187C4—20081no—
188C4—10042no—
189C4—10042no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—20082no—
194C2#3063184011yescharacteristic
195C2#3063186711yescharacteristic
196C2#3063187511yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#323×C251C2
socleC22×C251#120
prime radical#30632249C2
hypercenter#323D251
hypocenterC251#1601
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #30631828C2semidirect
3. #30631599C2semidirect
4. #30631422C2semidirect
5. #323×C251C2semidirect
6. C22×Dic502C22semidirect
16. C251⋊#95C22semidirect
18. C22⋊C8×C251C22semidirect
65. C23×C251SD16semidirect
98. C251#1601semidirect
104. #323D251semidirect
115. C22⋊C8C2×D251semidirect
149. C23#16860799semidirect

Maximal subgroups

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

subgroupindex
#306322492
#306318282
#306315992
#306314222
#323×C2512
#1601251