Finite Group Databasebeta

← #44549430

Subgroups — #44549430

Order 32128.

278 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 61 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#30632063C211yesmaximal, characteristic, complemented
2#30631887C211yesmaximal, characteristic, complemented
3#30631882C211yesmaximal, characteristic, complemented
4#30631431C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
7#550×C251C211yesmaximal, characteristic, complemented
8C2×#16860797—21no—
9C2×#16860795—22no—
10C2×#16860795—21no—
11#22548712C2211yescharacteristic
12C4⋊(C251⋊C8)C2211yescharacteristic
13C4⋊(C251⋊C8)C2211yescharacteristic
14#22548635—21no—
15#22548635—21no—
16C4×C251⋊C8C2211yescharacteristic
17C2×#41×C251—21no—
18#127×C251—21no—
19#127×C251C2211yescharacteristic
20#121×C251—22no—
21#121×C251—21no—
22C4×Q8×C251C2211yescharacteristic
23C4×D4×C251—21no—
24C4×D4×C251C2211yescharacteristic
25C2×D1004—21no—
26C2×D1004—21no—
27C2×D1004—21no—
28#16860797—42no—
29#16860795—42no—
30#16860795—44no—
31C2×C251⋊C8—21no—
32C2×C251⋊C8—21no—
33C2×C251⋊C8—21no—
34#41×C251—42no—
35#41×C251—42no—
36C2×Q8×C251D411yescharacteristic
37C2×D4×C251—22no—
38C2×D4×C251—22no—
39C2×D4×C251—21no—
40C2×D4×C251—21no—
41C2×D4×C251D411yescharacteristic
42C2×D4×C251D412yes—
43C22×C4×C251—22no—
44C22×C4×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251D411yescharacteristic
47C4⋊C4×C251D411yescharacteristic
48C22⋊C4×C251—22no—
49C22⋊C4×C251—21no—
50C42×C251—21no—
51C42×C251C2311yescharacteristic
52C22×D251—41no—
53D1004—42no—
54D1004—41no—
55D1004—41no—
56D1004—41no—
57D1004—41no—
58C251⋊C8—22no—
59C251⋊C8—41no—
60C251⋊C8—41no—
61C23×C251—21no—
62C23×C251—22no—
63Q8×C251—21no—
64Q8×C251D812yescomplemented
65D4×C251—21no—
66D4×C251—21no—
67D4×C251—42no—
68D4×C251—44no—
69D4×C251—41no—
70D4×C251—41no—
71D4×C251—41no—
72D4×C251—41no—
73D4×C251—82no—
74C2×C4×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—21no—
77C2×C4×C251—44no—
78C2×C4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C22×C251—42no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251—42no—
87C22×C251—42no—
88C22×C251—41no—
89C22×C251#12011yescharacteristic
90C2×D251—81no—
91C2×D251—81no—
92C2×D251—81no—
93C4×C251—22no—
94C4×C251—21no—
95C4×C251—41no—
96C4×C251—41no—
97C4×C251C2×D811yescharacteristic
98C4×C251#13511yescharacteristic
99C4×C251AGL(1,ℤ/8)11yescharacteristic
100C4×C251AGL(1,ℤ/8)11yescharacteristic
101C2×C251—81no—
102C2×C251—82no—
103C2×C251#44711yescharacteristic
104C2×C251#44911yescharacteristic
105C2×C251#45311yescharacteristic
106D251—161no—
107C251#162111yescharacteristic, complemented, Sylow
108#1621—2511nomaximal, Sylow
109#550D25111yescharacteristic, complemented
110#493—2511no—
111#460—2511no—
112#459—2511no—
113#331—2511no—
114Q8⋊C8—2511no—
115D4⋊C8—2511no—
116C2×#41—21no—
117C2×SD16—5021no—
118C2×D8—5021no—
119C2×D8—5022no—
120#127—21no—
121#127—2511no—
122#127C2×D25111yescharacteristic
123#121—22no—
124#121—21no—
125C4×Q8C2×D25111yescharacteristic
126C4×D4—21no—
127C4×D4C2×D25111yescharacteristic
128C4⋊C8—2511no—
129C4⋊C8—2511no—
130#102—5021no—
131#102—5021no—
132C4×C8—2511no—
133#41—42no—
134#41—42no—
135C2×Q8C22⋊D25111yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—5021no—
141C2×D4—5021no—
142C2×D4—5021no—
143C2×D4C22⋊D25111yescharacteristic
144C2×D4C22⋊D25112yes—
145C22×C4—22no—
146C22×C4—21no—
147SD16—10042no—
148D8—10042no—
149D8—10044no—
150C2×C8—5021no—
151C2×C8—5021no—
152C2×C8—5021no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D25111yescharacteristic
155C4⋊C4C22⋊D25111yescharacteristic
156C22⋊C4—22no—
157C22⋊C4—21no—
158C42—21no—
159C42C22×D25111yescharacteristic
160C23—21no—
161C23—22no—
162C23—10041no—
163Q8—21no—
164Q8#1686079512yescomplemented
165D4—21no—
166D4—21no—
167D4—42no—
168D4—44no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—82no—
174D4—10042no—
175D4—10041no—
176D4—10041no—
177D4—10041no—
178D4—10041no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×C22⋊D25111yescharacteristic
186C2×C4C2×C22⋊D25111yescharacteristic
187C2×C4C2×C22⋊D25111yescharacteristic
188C8—10041no—
189C8—10041no—
190C8—5022no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—41no—
197C22—20081no—
198C22—20081no—
199C22—20081no—
200C22#2254868411yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4C2×#1686079511yescharacteristic
206C4#2254873511yescharacteristic
207C4#2254874011yescharacteristic
208C4#2254874111yescharacteristic
209C2—81no—
210C2—82no—
211C2—40161no—
212C2#3063182111yescharacteristic
213C2#3063183211yescharacteristic
214C2#3063185511yescharacteristic
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#550×C251C2
socleC22×C251#120
hypercenter#550D251
hypocenterC251#1621
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #30632063C2semidirect
2. #30631887C2semidirect
3. #30631882C2semidirect
4. #30631431C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
6. D4⋊(C251⋊C8)C2semidirect
7. #550×C251C2semidirect
64. Q8×C251D8semidirect
107. C251#1621semidirect
109. #550D251semidirect
164. Q8#16860795semidirect

Maximal subgroups

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

subgroupindex
#306320632
#306318872
#306318822
#306314312
Q8⋊(C251⋊C8)2
D4⋊(C251⋊C8)2
#550×C2512
#1621251