Finite Group Databasebeta

← #44550677

Subgroups — #44550677

Order 32128.

360 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#22548694C211yesmaximal, characteristic
2C2×#22548638C212yesmaximal, complemented
3C2×C22⋊(C251⋊C8)C211yesmaximal, characteristic
4#30631449C212yesmaximal, complemented
5C2×#117×C251C211yesmaximal, characteristic
6C22×Dic502C2211yescharacteristic
7C22×C251⋊C8C2212yes—
8#22548694—22no—
9#22548694C414yescomplemented
10C2×C4⋊Dic251C2211yescharacteristic
11C2×Dic251⋊C4—21no—
12C2×C22⋊Dic251—21no—
13#22548638—24no—
14C22⋊(C251⋊C8)—22no—
15C23×C4×C251C2211yescharacteristic
16#117×C251—24no—
17C2×C4⋊C4×C251C2212yes—
18C2×C22⋊C4×C251—21no—
19C2×C42×C251—21no—
20C2×Dic502—21no—
21C2×Dic502—22no—
22C2×Dic502—22no—
23C2×Dic502C2×C412yes—
24C22×Dic251—21no—
25C22×Dic251—21no—
26C2×C251⋊C8—22no—
27C2×C251⋊C8—22no—
28C2×C251⋊C8—22no—
29C4⋊Dic251—21no—
30C4⋊Dic251C2×C412yes—
31Dic251⋊C4—24no—
32C22⋊Dic251—24no—
33C24×C251D411yescharacteristic
34C22×C4×C251—22no—
35C22×C4×C251—22no—
36C22×C4×C251—22no—
37C22×C4×C251C2×C412yes—
38C22×C4×C251D411yescharacteristic
39C22×C4×C251D412yes—
40C22×C4×C251C2311yescharacteristic
41C4⋊C4×C251—22no—
42C4⋊C4×C251D414yes—
43C22⋊C4×C251—24no—
44C42×C251—24no—
45Dic502—24no—
46Dic502—42no—
47C2×Dic251—22no—
48C2×Dic251—22no—
49C2×Dic251—41no—
50C2×Dic251—41no—
51C2×Dic251—42no—
52C251⋊C8—42no—
53C23×C251—21no—
54C23×C251—21no—
55C23×C251—22no—
56C23×C251—22no—
57C23×C251C22⋊C412yes—
58C23×C251C2×D411yescharacteristic
59C2×C4×C251—21no—
60C2×C4×C251—21no—
61C2×C4×C251—24no—
62C2×C4×C251—24no—
63C2×C4×C251—24no—
64C2×C4×C251—42no—
65C2×C4×C251—42no—
66C2×C4×C251C22⋊C412yes—
67C2×C4×C251C22×C411yescharacteristic
68C2×C4×C251C2×D412yes—
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251#4112yes—
71C22×C251—22no—
72C22×C251—22no—
73C22×C251—22no—
74C22×C251—42no—
75C22×C251—42no—
76C22×C251C2×C22⋊C411yescharacteristic
77C22×C251#12011yescharacteristic
78C22×C251#12311yescharacteristic
79C22×C251#13512yes—
80C22×C251#13712yes—
81C4×C251—22no—
82C4×C251—44no—
83C4×C251C4×D412yes—
84C4×C251#12112yes—
85Dic251—42no—
86Dic251—42no—
87C2×C251—42no—
88C2×C251#38611yescharacteristic
89C2×C251#41611yescharacteristic
90C2×C251#41911yescharacteristic
91C2×C251#45212yes—
92C2×C251#48512yes—
93C251#187311yescharacteristic, complemented, Sylow
94#1873—2511nomaximal, Sylow
95C2×C22⋊Q8—2511no—
96C2×#117D25111yescharacteristic
97C2×#103—2512no—
98C2×C22⋊C8—2511no—
99#340—2512no—
100C22×Q8—2511no—
101C23×C4C2×D25111yescharacteristic
102C22×C8—2512no—
103C22⋊Q8—5022no—
104C22⋊Q8—2514no—
105#117—24no—
106C2×C4⋊C4—5021no—
107C2×C4⋊C4—2511no—
108C2×C4⋊C4C2×D25112yes—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—5021no—
111C2×C42—21no—
112#103—5024no—
113C22⋊C8—5022no—
114C24C22⋊D25111yescharacteristic
115C2×Q8—5022no—
116C2×Q8—5022no—
117C2×Q8—5021no—
118C2×Q8—2512no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—5021no—
123C22×C4—5021no—
124C22×C4C4×D25112yes—
125C22×C4C22⋊D25111yescharacteristic
126C22×C4D100412yes—
127C22×C4C22×D25111yescharacteristic
128C2×C8—5022no—
129C2×C8—5022no—
130C2×C8—5022no—
131C4⋊C4—22no—
132C4⋊C4—5024no—
133C4⋊C4—5021no—
134C4⋊C4—2512no—
135C4⋊C4C22⋊D25114yes—
136C22⋊C4—24no—
137C22⋊C4—5024no—
138C42—24no—
139C23—22no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23(C2×D251)⋊C412yes—
144C23C2×C22⋊D25111yescharacteristic
145Q8—10042no—
146Q8—5024no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—10041no—
155C2×C4—10041no—
156C2×C4—10042no—
157C2×C4—5022no—
158C2×C4—5022no—
159C2×C4(C2×D251)⋊C412yes—
160C2×C4C2×C4×D25111yescharacteristic
161C2×C4C2×C22⋊D25112yes—
162C2×C4C2×D100411yescharacteristic
163C2×C4#1686081012yes—
164C8—10042no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22C2×(C2×D251)⋊C411yescharacteristic
171C22#2254868411yescharacteristic
172C22#2254869811yescharacteristic
173C22#2254873512yes—
174C22#2254874512yes—
175C4—22no—
176C4—44no—
177C4—10042no—
178C4—10042no—
179C4C4×C22⋊D25112yes—
180C4#2254868912yes—
181C2—42no—
182C2#3063156611yescharacteristic
183C2#3063168111yescharacteristic
184C2#3063169511yescharacteristic
185C2#3063184912yes—
186C2#3063202812yes—
subgroupG/H
centerC23C2×C22⋊D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC2×#117×C251C2
socleC23×C251C2×D4
prime radicalC24×C251D4
cosocleC22×C4×C251C23
hypercenterC2×#117D251
hypocenterC251#1873
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#22548638C2semidirect
4. #30631449C2semidirect
9. #22548694C4semidirect
93. C251#1873semidirect

Maximal subgroups

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

subgroupindex
C2×#225486942
C2×#225486382
C2×C22⋊(C251⋊C8)2
#306314492
C2×#117×C2512
#1873251