Finite Group Databasebeta

← #44558392

Subgroups — #44558392

Order 32128.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548689C211yesmaximal, characteristic
2#30631622C212yesmaximal, complemented
3#30631617C212yesmaximal, complemented
4#30631604C212yesmaximal, complemented
5#30631602C214yesmaximal, complemented
6#30631598C212yesmaximal, complemented
7#30631550C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C251C211yesmaximal, characteristic, complemented
9C22×D1004C2211yescharacteristic
10C22×C22⋊D251C2212yes—
11#22548689—24no—
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2214yes—
14C2×C22⋊Dic251C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2214yes—
17C2×(C2×D251)⋊C4C2214yes—
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2214yes—
21C22×Q8×C251C2211yescharacteristic
22C23×C4×C251C2211yescharacteristic
23C22⋊Q8×C251—24no—
24C2×C4⋊C4×C251C2212yes—
25C2×C4⋊C4×C251C2211yescharacteristic
26C2×C22⋊C4×C251C2212yes—
27C23×D251C2312yes—
28C2×D1004—22no—
29C2×D1004—42no—
30C2×C22⋊D251—24no—
31C2×C22⋊D251—44no—
32C22×Dic251C2312yes—
33C22×Dic251C2314yes—
34C4⋊Dic251—22no—
35Dic251⋊C4—44no—
36C22⋊Dic251—42no—
37(C2×D251)⋊C4—24no—
38(C2×D251)⋊C4—44no—
39(C2×D251)⋊C4—44no—
40C24×C251C2311yescharacteristic
41C2×Q8×C251—22no—
42C2×Q8×C251—42no—
43C22×C4×C251—42no—
44C22×C4×C251D414yes—
45C22×C4×C251C2312yes—
46C22×C4×C251C2312yes—
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2311yescharacteristic
49C4⋊C4×C251—22no—
50C4⋊C4×C251—24no—
51C22⋊C4×C251—24no—
52C22×D251—22no—
53C22×D251—24no—
54C22×D251—24no—
55C22×D251—24no—
56D1004—44no—
57C22⋊D251—48no—
58C2×Dic251—22no—
59C2×Dic251—24no—
60C2×Dic251—24no—
61C2×Dic251—24no—
62C2×Dic251—24no—
63C23×C251—22no—
64C23×C251—22no—
65C23×C251—22no—
66C23×C251C2×D412yes—
67C23×C251C2411yescharacteristic
68Q8×C251—44no—
69C2×C4×C251—22no—
70C2×C4×C251—22no—
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—24no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—44no—
77C2×C4×C251C2×D412yes—
78C2×C4×C251C2×D412yes—
79C22×C251—22no—
80C22×C251—22no—
81C22×C251—22no—
82C22×C251—44no—
83C22×C251C22×D411yescharacteristic
84C22×C251H(5,𝔽2)12yes—
85C22×C251H(5,𝔽2)12yes—
86C22×C2512−1+412yes—
87C2×D251—42no—
88C2×D251—42no—
89C2×D251—42no—
90C2×D251—48no—
91C4×C251—22no—
92C4×C251—22no—
93C4×C251—42no—
94C4×C251—42no—
95Dic251—42no—
96Dic251—44no—
97C2×C251—42no—
98C2×C251#53411yescharacteristic
99C2×C251#53511yescharacteristic
100C2×C251#53611yescharacteristic
101C2×C251#56214yes—
102D251—82no—
103C251#281611yescharacteristic, complemented, Sylow
104#2816—2511nomaximal, Sylow
105C2×C22⋊Q8D25111yescharacteristic, complemented
106C2×#121—2511no—
107#399—2512no—
108#398—2512no—
109#394—2512no—
110#394—2514no—
111#393—2512no—
112#381—2511no—
113C22×Q8C2×D25111yescharacteristic
114C22×D4—2512no—
115C22×D4—2511no—
116C23×C4C2×D25111yescharacteristic
117C22⋊Q8—24no—
118#121—5024no—
119C2×C4⋊C4—2514no—
120C2×C4⋊C4—2511no—
121C2×C4⋊C4C2×D25112yes—
122C2×C4⋊C4C2×D25111yescharacteristic
123C2×C22⋊C4—2512no—
124C2×C22⋊C4—2514no—
125C2×C22⋊C4—2514no—
126C2×C22⋊C4—2512no—
127C2×C22⋊C4C2×D25112yes—
128#95—2512no—
129#95—2512no—
130#95—2514no—
131C24—2512no—
132C24C22×D25111yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—10042no—
136C2×D4—10044no—
137C2×D4—5024no—
138C2×D4—5022no—
139C22×C4—42no—
140C22×C4—2514no—
141C22×C4—2512no—
142C22×C4C22⋊D25114yes—
143C22×C4C22×D25112yes—
144C22×C4C22×D25112yes—
145C22×C4C22×D25111yescharacteristic
146C22×C4C22×D25111yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—10044no—
150C4⋊C4—5022no—
151C22⋊C4—24no—
152C22⋊C4—10044no—
153C22⋊C4—10044no—
154C22⋊C4—10042no—
155C22⋊C4—5024no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—5022no—
160C23—5024no—
161C23—5024no—
162C23—5024no—
163C23C2×C22⋊D25112yes—
164C23C23×D25111yescharacteristic
165Q8—44no—
166D4—10044no—
167D4—10048no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—5022no—
177C2×C4—5024no—
178C2×C4—5024no—
179C2×C4—5024no—
180C2×C4—5024no—
181C2×C4C2×C22⋊D25112yes—
182C2×C4C2×C22⋊D25112yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—10042no—
188C22—10042no—
189C22—10042no—
190C22—10048no—
191C22C22×C22⋊D25111yescharacteristic
192C22#2254876012yes—
193C22#2254876112yes—
194C22#2254876312yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—10042no—
200C4—10044no—
201C2—42no—
202C2—20082no—
203C2#3063224611yescharacteristic
204C2#3063225111yescharacteristic
205C2#3063226111yescharacteristic
206C2#3063243314yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×C22⋊Q8×C251C2
prime radicalC2×#22548689C2
hypercenterC2×C22⋊Q8D251
hypocenterC251#2816
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #30631622C2semidirect
3. #30631617C2semidirect
4. #30631604C2semidirect
5. #30631602C2semidirect
6. #30631598C2semidirect
7. #30631550C2semidirect
8. C2×C22⋊Q8×C251C2semidirect
103. C251#2816semidirect
105. C2×C22⋊Q8D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486892
#306316222
#306316172
#306316042
#306316022
#306315982
#306315502
C2×C22⋊Q8×C2512
#2816251