Finite Group Databasebeta

← #44550847

Subgroups — #44550847

Order 32128.

324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 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#30632454C211yesmaximal, characteristic, complemented
2#30631704C211yesmaximal, characteristic, complemented
3#30631692C211yesmaximal, characteristic, complemented
4C2×#22548632C211yesmaximal, characteristic, complemented
5#30631458C211yesmaximal, characteristic, complemented
6C251⋊#338C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C251C211yesmaximal, characteristic
8#22548722—22no—
9#22548722—22no—
10C2×#16860794—22no—
11C2×#16860794C2211yescharacteristic
12C2×#16860794C2211yescharacteristic
13C2×#16860794C2211yescharacteristic
14#22548674C2211yescharacteristic
15#22548642—22no—
16#22548636—22no—
17#22548632—22no—
18C2×#41×C251C2211yescharacteristic
19C22×D4×C251C2211yescharacteristic
20AGL(1,ℤ/8)×C251—22no—
21AGL(1,ℤ/8)×C251—22no—
22C2×SD16×C251C412yescomplemented
23C2×D8×C251C412yescomplemented
24C2×M16×C251C2211yescharacteristic
25#16860794—24no—
26#16860794—21no—
27#16860794—21no—
28#16860794—21no—
29#16860794—21no—
30#16860794—21no—
31#16860794D412yes—
32C2×C251⋊C8—24no—
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C2×C251⋊C8D412yes—
36C4⋊Dic251—21no—
37C22⋊Dic251—21no—
38C4×Dic251—21no—
39C24×C251—21no—
40#41×C251—21no—
41#41×C251—21no—
42#41×C251D412yes—
43#41×C251D412yes—
44C2×Q8×C251C2×C411yescharacteristic
45C2×D4×C251—21no—
46C2×D4×C251—21no—
47C2×D4×C251—21no—
48C2×D4×C251—21no—
49C2×D4×C251—21no—
50C2×D4×C251—21no—
51C2×D4×C251C2×C412yes—
52C2×D4×C251C2×C411yescharacteristic
53C22×C4×C251—21no—
54C22×C4×C251C2311yescharacteristic
55SD16×C251—24no—
56D8×C251—24no—
57M16×C251—21no—
58M16×C251—21no—
59C2×C8×C251C2×C412yes—
60C2×Dic251—41no—
61C251⋊C8—22no—
62C251⋊C8—22no—
63C251⋊C8—41no—
64C23×C251—21no—
65C23×C251—22no—
66C23×C251—41no—
67C23×C251—41no—
68C23×C251—82no—
69C23×C251#4111yescharacteristic
70Q8×C251—21no—
71Q8×C251C22⋊C412yes—
72D4×C251—22no—
73D4×C251—22no—
74D4×C251—22no—
75D4×C251—21no—
76D4×C251—22no—
77D4×C251—41no—
78D4×C251—41no—
79D4×C251C22⋊C412yes—
80C2×C4×C251—21no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251C22×C411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251#4111yescharacteristic
89C8×C251—22no—
90C22×C251—22no—
91C22×C251—41no—
92C22×C251—42no—
93C22×C251—81no—
94C22×C251—82no—
95C22×C251—82no—
96C22×C251—82no—
97C22×C251C4×D411yescharacteristic
98C22×C251#12111yescharacteristic
99C22×C251#12311yescharacteristic
100C4×C251—22no—
101C4×C251C2×C22⋊C411yescharacteristic
102C4×C251C4×D411yescharacteristic
103C4×C251#12011yescharacteristic
104C4×C251#12111yescharacteristic
105Dic251—81no—
106C2×C251—21no—
107C2×C251—21no—
108C2×C251—21no—
109C2×C251—42no—
110C2×C251—82no—
111C2×C251#38611yescharacteristic
112C251#190311yescharacteristic, complemented, Sylow
113#1903—2511nomaximal, Sylow
114C2×AGL(1,ℤ/8)D25111yescharacteristic
115#568—2511no—
116#421—2511no—
117#418—2511no—
118C2×#100—2511no—
119#344—2511no—
120#338—2511no—
121C2×#41C2×D25111yescharacteristic
122C22×D4C2×D25111yescharacteristic
123AGL(1,ℤ/8)—22no—
124AGL(1,ℤ/8)—22no—
125C2×SD16Dic25112yescomplemented
126C2×D8Dic25112yescomplemented
127#131—5022no—
128#131—5022no—
129C2×M16—5022no—
130C2×M16—2511no—
131C2×M16—2511no—
132C2×M16—2511no—
133C2×M16C2×D25111yescharacteristic
134#117—2511no—
135#104—5022no—
136#102—5022no—
137#100—5022no—
138C24—21no—
139#41—21no—
140#41—21no—
141#41C22⋊D25112yes—
142#41C22⋊D25112yes—
143C2×Q8C2×Dic25111yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×Dic25112yes—
151C2×D4C2×Dic25111yescharacteristic
152C22×C4—21no—
153C22×C4C22×D25111yescharacteristic
154SD16—24no—
155D8—24no—
156M16—21no—
157M16—21no—
158M16—5021no—
159M16—5021no—
160M16—5024no—
161M16—5021no—
162M16—5021no—
163M16—5021no—
164M16—2512no—
165C2×C8—5021no—
166C2×C8—5024no—
167C2×C8—5021no—
168C2×C8—2512no—
169C2×C8C2×Dic25112yes—
170C4⋊C4—5021no—
171C22⋊C4—5021no—
172C42—5021no—
173C23—21no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—82no—
178C23#1686080911yescharacteristic
179Q8—21no—
180Q8C22⋊Dic25112yes—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—21no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4C22⋊Dic25112yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—10041no—
193C2×C4C22×Dic25111yescharacteristic
194C2×C4D4×D25111yescharacteristic
195C2×C4D4×D25111yescharacteristic
196C2×C4C2×C22⋊D25111yescharacteristic
197C2×C4C2×C22⋊D25111yescharacteristic
198C2×C4#1686080911yescharacteristic
199C8—22no—
200C8—10041no—
201C8—5022no—
202C8—5022no—
203C22—22no—
204C22—41no—
205C22—42no—
206C22—81no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22D4×Dic25111yescharacteristic
211C22#2254868711yescharacteristic
212C22#2254869611yescharacteristic
213C4—22no—
214C4—20081no—
215C4C2×C22⋊Dic25111yescharacteristic
216C4D4×Dic25111yescharacteristic
217C4#2254868311yescharacteristic
218C4#2254868711yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—82no—
224C2#3063156711yescharacteristic
subgroupG/H
centerC2#30631567
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radicalC2×AGL(1,ℤ/8)×C251C2
socleC2×C251#386
cosocleC22×C4×C251C23
hypercenterC2×AGL(1,ℤ/8)D251
hypocenterC251#1903
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #30632454C2semidirect
2. #30631704C2semidirect
3. #30631692C2semidirect
4. C2×#22548632C2semidirect
5. #30631458C2semidirect
6. C251⋊#338C2semidirect
22. C2×SD16×C251C4semidirect
23. C2×D8×C251C4semidirect
112. C251#1903semidirect
125. C2×SD16Dic251semidirect
126. C2×D8Dic251semidirect

Maximal subgroups

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

subgroupindex
#306324542
#306317042
#306316922
C2×#225486322
#306314582
C251⋊#3382
C2×AGL(1,ℤ/8)×C2512
#1903251