Finite Group Databasebeta

← #44548859

Subgroups — #44548859

Order 32128.

284 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 89 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#30632237C211yesmaximal, characteristic
2#30631740C211yesmaximal, characteristic, complemented
3#30631420C214yesmaximal, complemented
4C4×C4⋊C4×C251C211yesmaximal, characteristic, complemented
5C2×#16860810—21no—
6#22548715C411yescharacteristic, complemented
7#22548712C411yescharacteristic, complemented
8#22548703—21no—
9#22548689—21no—
10#22548689C2212yes—
11C4×D1004—21no—
12C4×D1004C412yescomplemented
13C4⋊(C251⋊C8)—21no—
14C22⋊(C251⋊C8)C2212yes—
15C4×C251⋊C8—21no—
16C2×C4⋊C4×C251—21no—
17C2×C42×C251—21no—
18C2×C42×C251C2211yescharacteristic
19#95×C251C2212yes—
20#16860810—42no—
21#16860810—42no—
22C2×Dic502—21no—
23C2×D1004—21no—
24C2×D1004—21no—
25C2×D1004C2×C412yes—
26C2×C22⋊D251—22no—
27C2×C4×D251—22no—
28C2×C251⋊C8—22no—
29C4⋊Dic251—21no—
30C4⋊Dic251C2×C412yes—
31(C2×D251)⋊C4—22no—
32(C2×D251)⋊C4—22no—
33C22×C4×C251—22no—
34C22×C4×C251D411yescharacteristic
35C22×C4×C251D411yescharacteristic
36C22×C4×C251C2311yescharacteristic
37C4⋊C4×C251—24no—
38C42×C251—42no—
39C42×C251C2×C411yescharacteristic
40C42×C251C2×C411yescharacteristic
41C42×C251D411yescharacteristic
42C42×C251D411yescharacteristic
43C22×D251—22no—
44Dic502—42no—
45D1004—42no—
46D1004—42no—
47D1004—42no—
48C22⋊D251—42no—
49C22⋊D251—42no—
50C2×Dic251—22no—
51C4×D251—44no—
52C251⋊C8—42no—
53C23×C251C2×D411yescharacteristic
54C2×C4×C251—22no—
55C2×C4×C251—24no—
56C2×C4×C251—41no—
57C2×C4×C251—42no—
58C2×C4×C251—42no—
59C2×C4×C251C22⋊C411yescharacteristic
60C2×C4×C251C22⋊C411yescharacteristic
61C2×C4×C251C22⋊C411yescharacteristic
62C2×C4×C251C22⋊C411yescharacteristic
63C2×C4×C251D812yes—
64C2×C4×C251SD1612yes—
65C2×C4×C251C22×C411yescharacteristic
66C2×C4×C251C2×D411yescharacteristic
67C22×C251—21no—
68C22×C251—21no—
69C22×C251C2×C22⋊C411yescharacteristic
70C22×C251C2×D811yescharacteristic
71C22×C251C2×SD1611yescharacteristic
72C2×D251—42no—
73C2×D251—42no—
74C2×D251—42no—
75C4×C251—21no—
76C4×C251—21no—
77C4×C251—21no—
78C4×C251—44no—
79C4×C251#10212yes—
80C4×C251#10212yes—
81C4×C251#10412yes—
82Dic251—42no—
83C2×C251—21no—
84C2×C251—21no—
85C2×C251#41011yescharacteristic
86C2×C251C2×#10211yescharacteristic
87C2×C251C2×#10411yescharacteristic
88D251—82no—
89C251#150211yescharacteristic, complemented, Sylow
90#1502—2511nomaximal, Sylow
91#532—2511no—
92#431—2511no—
93C4×C4⋊C4D25111yescharacteristic, complemented
94#327—2514no—
95C2×#41—5021no—
96C4⋊Q8—2511no—
97#127—2511no—
98#124—5021no—
99#121—5021no—
100#121—2512no—
101C4×D4—5021no—
102C4×D4—2512no—
103C2×C4⋊C4—21no—
104C2×C42—21no—
105C2×C42C2×D25111yescharacteristic
106C4⋊C8—5021no—
107C22⋊C8—2512no—
108C4×C8—5021no—
109#95C2×D25112yes—
110#41—10042no—
111#41—10042no—
112C2×Q8—5021no—
113C2×D4—5022no—
114C2×D4—5021no—
115C2×D4—5021no—
116C2×D4—2512no—
117C22×C4—22no—
118C22×C4—5022no—
119C22×C4C22⋊D25111yescharacteristic
120C22×C4C22⋊D25111yescharacteristic
121C22×C4C22×D25111yescharacteristic
122C2×C8—5022no—
123C4⋊C4—24no—
124C4⋊C4—5021no—
125C4⋊C4—2512no—
126C22⋊C4—5022no—
127C22⋊C4—5022no—
128C42—42no—
129C42C4×D25111yescharacteristic
130C42C4×D25111yescharacteristic
131C42D100411yescharacteristic
132C42D100411yescharacteristic
133C23—5022no—
134C23C2×C22⋊D25111yescharacteristic
135Q8—10042no—
136D4—10042no—
137D4—10042no—
138D4—10042no—
139D4—10042no—
140D4—10042no—
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—10044no—
147C2×C4—5022no—
148C2×C4(C2×D251)⋊C411yescharacteristic
149C2×C4(C2×D251)⋊C411yescharacteristic
150C2×C4(C2×D251)⋊C411yescharacteristic
151C2×C4(C2×D251)⋊C411yescharacteristic
152C2×C4#1686079512yes—
153C2×C4#1686079712yes—
154C2×C4C2×C4×D25111yescharacteristic
155C2×C4C2×D100411yescharacteristic
156C8—10042no—
157C22—21no—
158C22—21no—
159C22—10042no—
160C22—10042no—
161C22—10042no—
162C22C2×(C2×D251)⋊C411yescharacteristic
163C22C2×#1686079511yescharacteristic
164C22C2×#1686079711yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—44no—
169C4—10042no—
170C4#2254863512yes—
171C4#2254863512yes—
172C4#2254864312yes—
173C2—21no—
174C2—21no—
175C2—20082no—
176C2#3063166011yescharacteristic
177C2C2×#2254863511yescharacteristic
178C2C2×#2254864311yescharacteristic
subgroupG/H
centerC2×C4(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC4×C4⋊C4×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632237C2
cosocleC22×C4×C251C23
hypercenterC4×C4⋊C4D251
hypocenterC251#1502
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #30631740C2semidirect
3. #30631420C2semidirect
4. C4×C4⋊C4×C251C2semidirect
6. #22548715C4semidirect
7. #22548712C4semidirect
12. C4×D1004C4semidirect
89. C251#1502semidirect
93. C4×C4⋊C4D251semidirect

Maximal subgroups

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

subgroupindex
#306322372
#306317402
#306314202
C4×C4⋊C4×C2512
#1502251