Finite Group Databasebeta

← #44550581

Subgroups — #44550581

Order 32128.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#30632164C211yesmaximal, characteristic
2#30631730C211yesmaximal, characteristic
3#30631729C211yesmaximal, characteristic
4C2×#22548633C211yesmaximal, characteristic
5#30631593C211yesmaximal, characteristic
6#30631447C211yesmaximal, characteristic
7#339×C251C211yesmaximal, characteristic
8C22×Dic502C2211yescharacteristic
9C2×#16860794C2211yescharacteristic
10C4×Dic502—21no—
11C4×Dic502—21no—
12#22548680—21no—
13#22548680—21no—
14#22548674C2211yescharacteristic
15#22548671—22no—
16C2×C4⋊Dic251C2211yescharacteristic
17#22548649—21no—
18#22548648—21no—
19#22548647—21no—
20#22548646—21no—
21#22548633C414yescomplemented
22C251⋊#95—22no—
23C2×M16×C251C2211yescharacteristic
24#117×C251C2211yescharacteristic
25C2×C42×C251C2211yescharacteristic
26C2×Dic502—21no—
27C2×Dic502—21no—
28C2×Dic502—21no—
29C2×Dic502—21no—
30C2×Dic502C2×C412yes—
31C2×Dic502Q812yes—
32C22×Dic251—21no—
33C22×Dic251D412yes—
34#16860794—21no—
35#16860794C2×C412yes—
36C2×C251⋊C8—21no—
37C4⋊Dic251—21no—
38C4⋊Dic251—21no—
39C4⋊Dic251D411yescharacteristic
40C4⋊Dic251D411yescharacteristic
41Dic251⋊C4—22no—
42Dic251⋊C4—22no—
43C22⋊Dic251—21no—
44C4×Dic251—22no—
45C4×Dic251—22no—
46C4×Dic251—21no—
47C22×C4×C251—21no—
48C22×C4×C251C2311yescharacteristic
49M16×C251—21no—
50M16×C251C2×C412yes—
51C2×C8×C251—21no—
52C4⋊C4×C251D411yescharacteristic
53C4⋊C4×C251D411yescharacteristic
54C22⋊C4×C251—21no—
55C42×C251—21no—
56C42×C251—21no—
57C42×C251—21no—
58Dic502—22no—
59Dic502—22no—
60Dic502—22no—
61Dic502—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—21no—
65C2×Dic251—21no—
66C2×Dic251—42no—
67C2×Dic251—41no—
68C2×Dic251C4⋊C414yes—
69C251⋊C8—22no—
70C23×C251C2×D411yescharacteristic
71C2×C4×C251—22no—
72C2×C4×C251—21no—
73C2×C4×C251—21no—
74C2×C4×C251—41no—
75C2×C4×C251—41no—
76C2×C4×C251C22×C411yescharacteristic
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C2×C4×C251C2×Q811yescharacteristic
80C2×C4×C251#4111yescharacteristic
81C2×C4×C251#4111yescharacteristic
82C8×C251—22no—
83C22×C251—21no—
84C22×C251—21no—
85C22×C251C2×C4⋊C411yescharacteristic
86C22×C251#12011yescharacteristic
87C22×C251#12311yescharacteristic
88C4×C251—42no—
89C4×C251—41no—
90C4×C251—41no—
91C4×C251C4×D411yescharacteristic
92C4×C251C4×D411yescharacteristic
93C4×C251C22⋊Q811yescharacteristic
94C4×C251C22⋊Q811yescharacteristic
95Dic251—24no—
96Dic251—41no—
97Dic251—41no—
98Dic251—81no—
99C2×C251—21no—
100C2×C251—21no—
101C2×C251#38511yescharacteristic
102C2×C251#45411yescharacteristic
103C2×C251#45611yescharacteristic
104C251#185711yescharacteristic, complemented, Sylow
105#1857—2511nomaximal, Sylow
106#519—2511no—
107#428—2511no—
108#428—2511no—
109C2×#101—2511no—
110#391—2511no—
111#339—2511no—
112#339D25111yescharacteristic
113C22×Q8—2511no—
114C2×M16—2511no—
115C2×M16C2×D25111yescharacteristic
116C4×Q8—5021no—
117C4×Q8—5021no—
118C4×Q8—5021no—
119C4×Q8—5021no—
120#117—5022no—
121#117—2511no—
122#117C2×D25111yescharacteristic
123C2×C4⋊C4—2511no—
124C2×C42C2×D25111yescharacteristic
125#107—5021no—
126#107—5021no—
127#106—5021no—
128#106—5021no—
129#101—2514no—
130#95—5022no—
131C2×Q8—5021no—
132C2×Q8—5021no—
133C2×Q8—5021no—
134C2×Q8—5021no—
135C2×Q8—2512no—
136C2×Q8—2512no—
137C22×C4—21no—
138C22×C4—5021no—
139C22×C4—2512no—
140C22×C4C22×D25111yescharacteristic
141M16—21no—
142M16—5021no—
143M16—2512no—
144M16C4×D25112yes—
145C2×C8—21no—
146C2×C8—5021no—
147C4⋊C4—5022no—
148C4⋊C4—5022no—
149C4⋊C4—5021no—
150C4⋊C4—5021no—
151C4⋊C4—2511no—
152C4⋊C4—2511no—
153C4⋊C4D100411yescharacteristic
154C4⋊C4D100411yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—5021no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—5022no—
161C42—5022no—
162C42—5021no—
163C23D4×D25111yescharacteristic
164Q8—5022no—
165Q8—5022no—
166Q8—5022no—
167Q8—5022no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—10042no—
174C2×C4—10041no—
175C2×C4—5021no—
176C2×C4—5021no—
177C2×C4—5022no—
178C2×C4—5022no—
179C2×C4—2514no—
180C2×C4C2×C4×D25111yescharacteristic
181C2×C4D4×D25111yescharacteristic
182C2×C4C2×D100411yescharacteristic
183C2×C4Q8×D25111yescharacteristic
184C2×C4#1686080911yescharacteristic
185C2×C4#1686081011yescharacteristic
186C8—22no—
187C8—5022no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D25111yescharacteristic
191C22#2254868511yescharacteristic
192C22#2254869711yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—20081no—
197C4—10041no—
198C4—10041no—
199C4—5024no—
200C4(C22⋊D251)⋊C411yescharacteristic
201C4C4×D100411yescharacteristic
202C4#2254869311yescharacteristic
203C4#2254869511yescharacteristic
204C2—21no—
205C2—21no—
206C2#3063156111yescharacteristic
207C2#3063186111yescharacteristic
208C2#3063187111yescharacteristic
subgroupG/H
centerC22#22548685
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#339×C251C2
socleC22×C251#120
prime radicalC23×C251C2×D4
cosocleC22×C4×C251C23
hypercenter#339D251
hypocenterC251#1857
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
21. #22548633C4semidirect
104. C251#1857semidirect

Maximal subgroups

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

subgroupindex
#306321642
#306317302
#306317292
C2×#225486332
#306315932
#306314472
#339×C2512
#1857251