Finite Group Databasebeta

← #44550444

Subgroups — #44550444

Order 32128.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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×C4⋊C4×D251C211yesmaximal, characteristic
2#30631650C212yesmaximal
3#30631456C211yesmaximal, characteristic, complemented
4#30631455C212yesmaximal, complemented
5#343×C251C211yesmaximal, characteristic, complemented
6C22×C4×D251C412yes—
7C22×C4×D251C2211yescharacteristic
8C2×#16860794C2212yes—
9C2×C4⋊Dic251C2211yescharacteristic
10C2×Dic251⋊C4C412yes—
11C4⋊C4×D251—22no—
12C4⋊C4×D251—22no—
13(C2×D251)⋊C8—22no—
14(C2×D251)⋊C8—22no—
15C2×M16×C251C2212yes—
16C2×C4⋊C4×C251C2211yescharacteristic
17C23×D251C2×C411yescharacteristic
18C22×Dic251C2×C411yescharacteristic
19C22×Dic251C2×C412yes—
20C2×C4×D251—22no—
21C2×C4×D251—22no—
22C2×C4×D251—21no—
23C2×C4×D251—22no—
24C2×C4×D251—21no—
25C2×C4×D251—44no—
26C2×C4×D251D412yes—
27C2×C4×D251D412yes—
28#16860794—22no—
29#16860794—22no—
30C2×C251⋊C8—22no—
31C4⋊Dic251—22no—
32Dic251⋊C4—24no—
33C22×C4×C251C2×C412yes—
34C22×C4×C251C2311yescharacteristic
35M16×C251—22no—
36M16×C251—22no—
37C2×C8×C251—22no—
38C4⋊C4×C251—22no—
39C22×D251—21no—
40C22×D251—21no—
41C22×D251—22no—
42C22×D251—22no—
43C22×D251C22⋊C411yescharacteristic
44C22×D251C22⋊C411yescharacteristic
45C2×Dic251—21no—
46C2×Dic251—21no—
47C2×Dic251—22no—
48C2×Dic251—42no—
49C2×Dic251C22⋊C412yes—
50C4×D251—22no—
51C4×D251—22no—
52C4×D251—22no—
53C4×D251—22no—
54C4×D251—44no—
55C4×D251—44no—
56C251⋊C8—42no—
57C23×C251C22×C411yescharacteristic
58C2×C4×C251—22no—
59C2×C4×C251—42no—
60C2×C4×C251C2×D412yes—
61C2×C4×C251#4111yescharacteristic
62C2×C4×C251#4112yes—
63C2×C4×C251#4111yescharacteristic
64C8×C251—42no—
65C22×C251—21no—
66C22×C251—21no—
67C22×C251C2×C22⋊C411yescharacteristic
68C22×C251#11711yescharacteristic
69C22×C251#11711yescharacteristic
70C2×D251—24no—
71C2×D251—24no—
72C2×D251—21no—
73C2×D251—21no—
74C2×D251—21no—
75C2×D251—21no—
76C2×D251#10012yes—
77C2×D251#10112yes—
78C4×C251—42no—
79C4×C251#12312yes—
80C4×C251#12312yes—
81Dic251—22no—
82Dic251—42no—
83C2×C251—21no—
84C2×C251—21no—
85C2×C251#38111yescharacteristic
86C2×C251C2×#10011yescharacteristic
87C2×C251C2×#10111yescharacteristic
88D251—22no—
89D251—22no—
90C251#182111yescharacteristic, complemented, Sylow
91#1821—2511nomaximal, Sylow
92C22×C4⋊C4—2511no—
93#407—2512no—
94#343—2512no—
95#343—2511no—
96#343D25111yescharacteristic, complemented
97C23×C4—2512no—
98C23×C4—2511no—
99C2×M16—2512no—
100C2×M16C2×D25112yes—
101C2×C4⋊C4—5022no—
102C2×C4⋊C4—5022no—
103C2×C4⋊C4—2512no—
104C2×C4⋊C4—2511no—
105C2×C4⋊C4C2×D25111yescharacteristic
106C22⋊C8—5022no—
107C22⋊C8—5022no—
108C24—2511no—
109C22×C4—10044no—
110C22×C4—5022no—
111C22×C4—5022no—
112C22×C4—5021no—
113C22×C4—5022no—
114C22×C4—5021no—
115C22×C4—2512no—
116C22×C4—2511no—
117C22×C4—2512no—
118C22×C4—2512no—
119C22×C4C4×D25112yes—
120C22×C4C22×D25111yescharacteristic
121M16—22no—
122M16—22no—
123M16—5022no—
124M16—5022no—
125C2×C8—22no—
126C2×C8—5022no—
127C4⋊C4—22no—
128C4⋊C4—5024no—
129C4⋊C4—5022no—
130C23—5022no—
131C23—5022no—
132C23—5021no—
133C23—5021no—
134C23—2511no—
135C23—2511no—
136C23C2×C4×D25111yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—10044no—
140C2×C4—10044no—
141C2×C4—10042no—
142C2×C4—5022no—
143C2×C4—5021no—
144C2×C4—5021no—
145C2×C4—5022no—
146C2×C4—5022no—
147C2×C4—5022no—
148C2×C4—5022no—
149C2×C4—2512no—
150C2×C4D4×D25112yes—
151C2×C4#1686080911yescharacteristic
152C2×C4#1686081012yes—
153C2×C4#1686081111yescharacteristic
154C8—42no—
155C8—10042no—
156C22—21no—
157C22—21no—
158C22—5024no—
159C22—5024no—
160C22—5021no—
161C22—5021no—
162C22—5021no—
163C22—5021no—
164C22—2512no—
165C22—2512no—
166C22C22⋊C4×D25111yescharacteristic
167C22#2254867211yescharacteristic
168C22#2254867311yescharacteristic
169C4—42no—
170C4—10042no—
171C4—5022no—
172C4#2254869712yes—
173C4#2254870012yes—
174C2—21no—
175C2—21no—
176C2—5022no—
177C2—5022no—
178C2#3063154911yescharacteristic
179C2#100×D25111yescharacteristic
180C2#101×D25111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#343×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC23×D251C2×C4
cosocleC22×C4×C251C23
hypercenter#343D251
hypocenterC251#1821
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #30631456C2semidirect
4. #30631455C2semidirect
5. #343×C251C2semidirect
90. C251#1821semidirect
96. #343D251semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2512
#306316502
#306314562
#306314552
#343×C2512
#1821251