Finite Group Databasebeta

← #44552018

Subgroups — #44552018

Order 32128.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#30632258C211yesmaximal, characteristic
2#30631671C211yesmaximal, characteristic
3#30631476C214yesmaximal, complemented
4#410×C251C211yesmaximal, characteristic
5C22×Dic502C411yescharacteristic, complemented
6C2×#16860794—21no—
7#22548713—21no—
8#22548702—21no—
9#22548696C412yescomplemented
10#22548696C2212yes—
11#22548694—21no—
12#22548694—21no—
13#22548674C411yescharacteristic, complemented
14#22548634—21no—
15#22548632C2212yes—
16C2×#41×C251C2211yescharacteristic
17#117×C251—21no—
18#99×C251—21no—
19#99×C251C2212yes—
20C2×Dic502—21no—
21C2×Dic502—21no—
22C2×Dic502—41no—
23C2×Dic502—41no—
24C22×Dic251C2×C412yes—
25#16860794—22no—
26#16860794—41no—
27C2×C251⋊C8—41no—
28C4⋊Dic251—21no—
29Dic251⋊C4—22no—
30Dic251⋊C4—22no—
31C22⋊Dic251—22no—
32C22⋊Dic251—22no—
33C22⋊Dic251C2×C412yes—
34C4×Dic251—21no—
35#41×C251—41no—
36#41×C251—41no—
37C2×Q8×C251D411yescharacteristic
38C2×D4×C251C2×C411yescharacteristic
39C2×D4×C251D411yescharacteristic
40C2×D4×C251C2311yescharacteristic
41C22×C4×C251C2×C411yescharacteristic
42C22×C4×C251D411yescharacteristic
43C22×C4×C251D411yescharacteristic
44C4⋊C4×C251—41no—
45C22⋊C4×C251—22no—
46C42×C251—41no—
47Dic502—42no—
48Dic502—41no—
49Dic502—41no—
50C2×Dic251—22no—
51C2×Dic251—22no—
52C2×Dic251—42no—
53C251⋊C8—42no—
54C23×C251C22⋊C411yescharacteristic
55C23×C251C22×C411yescharacteristic
56C23×C251C2×D411yescharacteristic
57Q8×C251—41no—
58D4×C251—41no—
59D4×C251—41no—
60D4×C251—41no—
61C2×C4×C251—21no—
62C2×C4×C251—21no—
63C2×C4×C251—21no—
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—42no—
68C2×C4×C251C22⋊C411yescharacteristic
69C2×C4×C251C22⋊C411yescharacteristic
70C2×C4×C251C22⋊C411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C22×C251—21no—
73C22×C251—21no—
74C22×C251—21no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251—41no—
78C22×C251C2×C22⋊C411yescharacteristic
79C4×C251—21no—
80C4×C251—21no—
81C4×C251—21no—
82C4×C251—82no—
83C4×C251#9911yescharacteristic
84C4×C251#9911yescharacteristic
85Dic251—42no—
86Dic251—42no—
87C2×C251—21no—
88C2×C251—41no—
89C2×C251—41no—
90C2×C251—41no—
91C2×C251C2×#9911yescharacteristic
92C251#212311yescharacteristic, complemented, Sylow
93#2123—2511nomaximal, Sylow
94#536—2511no—
95#413—2511no—
96#410D25111yescharacteristic
97#352—2514no—
98C2×#41C2×D25111yescharacteristic
99C22×Q8—2511no—
100C2×M16—5021no—
101C4⋊Q8—5021no—
102#124—5021no—
103#123—2512no—
104#123—2512no—
105C22⋊Q8—5021no—
106C22⋊Q8—5021no—
107#117—21no—
108#117—2511no—
109#101—5021no—
110#100—2512no—
111#99—21no—
112#99C2×D25112yes—
113#41—41no—
114#41—41no—
115C2×Q8—10041no—
116C2×Q8—10041no—
117C2×Q8—5021no—
118C2×Q8—5021no—
119C2×Q8D100411yescharacteristic
120C2×D4C4×D25111yescharacteristic
121C2×D4C22⋊D25111yescharacteristic
122C2×D4C22×D25111yescharacteristic
123C22×C4—2512no—
124C22×C4C4×D25111yescharacteristic
125C22×C4C22⋊D25111yescharacteristic
126C22×C4D100411yescharacteristic
127M16—10041no—
128M16—5022no—
129C2×C8—10041no—
130C4⋊C4—41no—
131C4⋊C4—5022no—
132C4⋊C4—5021no—
133C4⋊C4—5022no—
134C22⋊C4—22no—
135C22⋊C4—5022no—
136C22⋊C4—5022no—
137C22⋊C4—2512no—
138C42—41no—
139C42—5021no—
140C23(C2×D251)⋊C411yescharacteristic
141C23C2×C4×D25111yescharacteristic
142C23C2×C22⋊D25111yescharacteristic
143Q8—41no—
144Q8—10042no—
145Q8—10041no—
146Q8—10041no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—10042no—
158C2×C4—5022no—
159C2×C4—5022no—
160C2×C4(C2×D251)⋊C411yescharacteristic
161C2×C4(C2×D251)⋊C411yescharacteristic
162C2×C4(C2×D251)⋊C411yescharacteristic
163C2×C4C2×D100411yescharacteristic
164C8—10042no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×(C2×D251)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—82no—
176C4—10042no—
177C4—10042no—
178C4#2254862911yescharacteristic
179C4#2254862911yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#2254862911yescharacteristic
subgroupG/H
centerC2C2×#22548629
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#410×C251C2
socleC2×C251C2×#99
prime radicalC2×#41×C251C22
cosocleC2×D4×C251C23
hypercenter#410D251
hypocenterC251#2123
LCSG ▷ C23×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #30631476C2semidirect
5. C22×Dic502C4semidirect
9. #22548696C4semidirect
13. #22548674C4semidirect
92. C251#2123semidirect

Maximal subgroups

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

subgroupindex
#306322582
#306316712
#306314762
#410×C2512
#2123251