Finite Group Databasebeta

← #44549480

Subgroups — #44549480

Order 32128.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D1004C211yesmaximal, characteristic
2#30632061C211yesmaximal, characteristic, complemented
3#30631974C211yesmaximal, characteristic, complemented
4#30631430C211yesmaximal, characteristic, complemented
5D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#476×C251C211yesmaximal, characteristic, complemented
8C22×D1004—21no—
9C2×D4×D251—21no—
10C2×#16860797—22no—
11#22548712C2211yescharacteristic
12#22548690—21no—
13#22548689—21no—
14#22548685—22no—
15C4×D1004C2211yescharacteristic
16#22548646—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic, complemented
18#22548635—21no—
19C4×C251⋊C8C2211yescharacteristic, complemented
20C4⋊Q8×C251C2211yescharacteristic, complemented
21C4×D4×C251C2211yescharacteristic
22#106×C251—21no—
23C4⋊C8×C251C2211yescharacteristic, complemented
24#102×C251—21no—
25C23×D251—22no—
26C2×D1004—22no—
27C2×D1004—21no—
28C2×D1004—21no—
29C2×D1004—42no—
30C2×D1004—42no—
31C2×D1004D411yescharacteristic
32C2×D1004D412yes—
33C2×C22⋊D251—22no—
34D4×D251—42no—
35D4×D251—42no—
36C2×C4×D251—21no—
37#16860797—44no—
38C2×C251⋊C8—21no—
39C2×C251⋊C8—21no—
40C4⋊Dic251D411yescharacteristic
41(C2×D251)⋊C4—22no—
42(C2×D251)⋊C4—21no—
43C2×Q8×C251—21no—
44C2×D4×C251D411yescharacteristic
45C22×C4×C251—21no—
46C2×C8×C251—21no—
47C4⋊C4×C251—21no—
48C4⋊C4×C251—21no—
49C4⋊C4×C251D411yescharacteristic
50C22⋊C4×C251—21no—
51C42×C251C2311yescharacteristic
52C22×D251—21no—
53C22×D251—22no—
54C22×D251—44no—
55C22×D251—44no—
56C22×D251—82no—
57D1004—21no—
58D1004—42no—
59D1004—44no—
60D1004—41no—
61D1004—41no—
62D1004—41no—
63D1004—41no—
64D1004SD1612yescomplemented
65C22⋊D251—44no—
66C2×Dic251—21no—
67C4×D251—41no—
68C4×D251—41no—
69C251⋊C8—22no—
70C251⋊C8—41no—
71C23×C251—21no—
72Q8×C251—42no—
73D4×C251—21no—
74D4×C251SD1612yescomplemented
75C2×C4×C251—21no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251—41no—
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C8×C251—41no—
83C22×C251—22no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251#12011yescharacteristic
87C2×D251—22no—
88C2×D251—41no—
89C2×D251—41no—
90C2×D251—42no—
91C2×D251—42no—
92C2×D251—42no—
93C2×D251—88no—
94C4×C251—21no—
95C4×C251—41no—
96C4×C251—81no—
97C4×C251C2×SD1611yescharacteristic
98C4×C251C2×SD1611yescharacteristic
99C4×C251AGL(1,ℤ/8)11yescharacteristic
100C4×C251AGL(1,ℤ/8)11yescharacteristic
101Dic251—41no—
102C2×C251—42no—
103C2×C251#45011yescharacteristic
104C2×C251#45011yescharacteristic
105C2×C251#45311yescharacteristic
106D251—42no—
107D251—82no—
108C251#162911yescharacteristic, complemented, Sylow
109#1629—2511nomaximal, Sylow
110D42—2511no—
111#492—2511no—
112#476—2511no—
113#476D25111yescharacteristic, complemented
114#331—2511no—
115D4⋊C8—2511no—
116D4⋊C8—2511no—
117C22×D4—5021no—
118C22×D4—5021no—
119C2×SD16—5022no—
120C4⋊Q8C2×D25111yescharacteristic, complemented
121#127—2511no—
122#121—5021no—
123#121—5021no—
124#120—5022no—
125C4×D4—2511no—
126C4×D4C2×D25111yescharacteristic
127#106—21no—
128#106—5021no—
129C4⋊C8—2511no—
130C4⋊C8C2×D25111yescharacteristic, complemented
131#102—21no—
132#102—5021no—
133C4×C8—2511no—
134C24—5022no—
135C2×Q8—21no—
136C2×D4—10042no—
137C2×D4—10042no—
138C2×D4—10042no—
139C2×D4—10042no—
140C2×D4—5022no—
141C2×D4—5022no—
142C2×D4—5021no—
143C2×D4—5021no—
144C2×D4—2512no—
145C2×D4—2511no—
146C2×D4C22⋊D25111yescharacteristic
147C22×C4—21no—
148C22×C4—5021no—
149SD16—10044no—
150C2×C8—21no—
151C2×C8—5021no—
152C2×C8—5021no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—2511no—
156C4⋊C4C22⋊D25111yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—5022no—
159C22⋊C4—5021no—
160C42C22×D25111yescharacteristic
161C23—21no—
162C23—20082no—
163C23—10044no—
164C23—10044no—
165C23—5022no—
166C23—5021no—
167Q8—42no—
168D4—21no—
169D4—10042no—
170D4—10044no—
171D4—10044no—
172D4—10041no—
173D4—10041no—
174D4—10041no—
175D4—10041no—
176D4—5021no—
177D4—2512no—
178D4#1686079712yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—10041no—
184C2×C4—10041no—
185C2×C4—5021no—
186C2×C4D4×D25111yescharacteristic
187C2×C4D4×D25111yescharacteristic
188C2×C4C2×C22⋊D25111yescharacteristic
189C8—41no—
190C8—10041no—
191C8—5022no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—20088no—
196C22—10042no—
197C22—10042no—
198C22—10042no—
199C22—10041no—
200C22—10041no—
201C22—5022no—
202C22#2254868311yescharacteristic
203C4—21no—
204C4—41no—
205C4—81no—
206C4—10041no—
207C4SD16×D25111yescharacteristic
208C4C2×#1686079711yescharacteristic
209C4#2254874011yescharacteristic
210C4#2254874211yescharacteristic
211C2—42no—
212C2—20082no—
213C2—10042no—
214C2#3063183611yescharacteristic
215C2#3063183711yescharacteristic
216C2#3063185311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#476×C251C2
socleC22×C251#120
prime radicalD4×D1004C2
hypercenter#476D251
hypocenterC251#1629
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #30632061C2semidirect
3. #30631974C2semidirect
4. #30631430C2semidirect
5. D4⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #476×C251C2semidirect
17. C4⋊(C251⋊C8)C22semidirect
19. C4×C251⋊C8C22semidirect
20. C4⋊Q8×C251C22semidirect
23. C4⋊C8×C251C22semidirect
64. D1004SD16semidirect
74. D4×C251SD16semidirect
108. C251#1629semidirect
113. #476D251semidirect
120. C4⋊Q8C2×D251semidirect
130. C4⋊C8C2×D251semidirect
178. D4#16860797semidirect

Maximal subgroups

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

subgroupindex
D4×D10042
#306320612
#306319742
#306314302
D4⋊(C251⋊C8)2
#306314142
#476×C2512
#1629251