Finite Group Databasebeta

← #44549429

Subgroups — #44549429

Order 32128.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632358C211yesmaximal, characteristic, complemented
2#30632063C211yesmaximal, characteristic, complemented
3#30631884C211yesmaximal, characteristic, complemented
4#30631430C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
7#459×C251C211yesmaximal, characteristic, complemented
8C2×#16860810—21no—
9C2×#16860798—21no—
10C2×#16860795—22no—
11#22548712C2211yescharacteristic
12#22548711—21no—
13#22548689—21no—
14#22548688—22no—
15C4×Dic502C2211yescharacteristic
16#22548677—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548636—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×D8×C251—21no—
21#127×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25#16860810—42no—
26#16860810—42no—
27C2×Dic502D411yescharacteristic
28C2×D1004—21no—
29C2×D1004—21no—
30C2×D1004D412yes—
31C2×C22⋊D251—22no—
32C2×C22⋊D251—22no—
33C2×C4×D251—22no—
34#16860798—42no—
35#16860795—44no—
36C2×C251⋊C8—21no—
37C2×C251⋊C8—21no—
38C4⋊Dic251D411yescharacteristic
39Dic251⋊C4—21no—
40(C2×D251)⋊C4—22no—
41C4×Dic251—21no—
42C2×D4×C251—21no—
43C2×D4×C251—21no—
44C2×D4×C251—21no—
45C2×D4×C251D411yescharacteristic
46C22×C4×C251—21no—
47D8×C251—42no—
48C2×C8×C251—21no—
49C4⋊C4×C251D411yescharacteristic
50C22⋊C4×C251—21no—
51C42×C251C2311yescharacteristic
52C22×D251—22no—
53Dic502—21no—
54Dic502D812yescomplemented
55D1004—42no—
56D1004—41no—
57D1004—41no—
58D1004—41no—
59D1004—41no—
60C22⋊D251—44no—
61C22⋊D251—82no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C4×D251—44no—
65C251⋊C8—22no—
66C251⋊C8—41no—
67C23×C251—21no—
68C23×C251—41no—
69D4×C251—21no—
70D4×C251—21no—
71D4×C251—42no—
72D4×C251—41no—
73D4×C251—41no—
74D4×C251—41no—
75D4×C251—41no—
76C2×C4×C251—21no—
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—41no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251—81no—
87C22×C251—81no—
88C22×C251—81no—
89C22×C251#12011yescharacteristic
90C2×D251—42no—
91C2×D251—42no—
92C2×D251—42no—
93C4×C251—21no—
94C4×C251—41no—
95C4×C251C2×D811yescharacteristic
96C4×C251#13511yescharacteristic
97C4×C251AGL(1,ℤ/8)11yescharacteristic
98C4×C251AGL(1,ℤ/8)11yescharacteristic
99Dic251—22no—
100Dic251—41no—
101C2×C251—81no—
102C2×C251—161no—
103C2×C251#44711yescharacteristic
104C2×C251#44911yescharacteristic
105C2×C251#45311yescharacteristic
106D251—82no—
107C251#162111yescharacteristic, complemented, Sylow
108#1621—2511nomaximal, Sylow
109#550—2511no—
110#493—2511no—
111#460—2511no—
112#459D25111yescharacteristic, complemented
113#331—2511no—
114Q8⋊C8—2511no—
115D4⋊C8—2511no—
116C2×#41—5021no—
117C2×SD16—5021no—
118C2×D8—21no—
119C2×D8—5022no—
120#127—5021no—
121#127—2511no—
122#127C2×D25111yescharacteristic
123#121—5022no—
124#121—5021no—
125C4×Q8—2511no—
126C4×D4—5021no—
127C4×D4C2×D25111yescharacteristic
128C4⋊C8—2511no—
129C4⋊C8C2×D25111yescharacteristic
130#102—21no—
131#102—5021no—
132C4×C8—2511no—
133#41—10042no—
134#41—10042no—
135C2×Q8—2511no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—5022no—
140C2×D4—5022no—
141C2×D4—5021no—
142C2×D4—5021no—
143C2×D4—2512no—
144C2×D4C22⋊D25111yescharacteristic
145C22×C4—21no—
146C22×C4—5022no—
147SD16—10042no—
148D8—42no—
149D8—10044no—
150C2×C8—21no—
151C2×C8—5021no—
152C2×C8—5021no—
153C4⋊C4—5021no—
154C4⋊C4—2511no—
155C4⋊C4C22⋊D25111yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—5022no—
158C42—5021no—
159C42C22×D25111yescharacteristic
160C23—21no—
161C23—41no—
162C23—5022no—
163Q8—5021no—
164Q8—2512no—
165D4—21no—
166D4—21no—
167D4—42no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—20082no—
173D4—10042no—
174D4—10044no—
175D4—10041no—
176D4—10041no—
177D4—10041no—
178D4—10041no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—10044no—
183C2×C4—5021no—
184C2×C4—5021no—
185C2×C4D4×D25111yescharacteristic
186C2×C4D4×D25111yescharacteristic
187C2×C4C2×C22⋊D25111yescharacteristic
188C8—41no—
189C8—10041no—
190C8—5022no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—10042no—
198C22—10042no—
199C22—10042no—
200C22#2254868311yescharacteristic
201C4—21no—
202C4—41no—
203C4—10041no—
204C4—5022no—
205C4D8×D25111yescharacteristic
206C4#2254873511yescharacteristic
207C4#2254873911yescharacteristic
208C4#2254874111yescharacteristic
209C2—81no—
210C2—161no—
211C2—20082no—
212C2#3063181911yescharacteristic
213C2#3063183011yescharacteristic
214C2#3063185311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#459×C251C2
socleC22×C251#120
hypercenter#459D251
hypocenterC251#1621
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #30632358C2semidirect
2. #30632063C2semidirect
3. #30631884C2semidirect
4. #30631430C2semidirect
5. #30631417C2semidirect
6. D4⋊(C251⋊C8)C2semidirect
7. #459×C251C2semidirect
54. Dic502D8semidirect
107. C251#1621semidirect
112. #459D251semidirect

Maximal subgroups

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

subgroupindex
#306323582
#306320632
#306318842
#306314302
#306314172
D4⋊(C251⋊C8)2
#459×C2512
#1621251