Finite Group Databasebeta

← #44558424

Subgroups — #44558424

Order 32128.

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

normal subgroups only

This group has 430 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×#22548709C211yesmaximal, characteristic
2#30631625C212yesmaximal, complemented
3#30631622C212yesmaximal
4#30631618C212yesmaximal, complemented
5#30631613C212yesmaximal
6#30631606C212yesmaximal, complemented
7#30631598C212yesmaximal
8#30631555C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C251C211yesmaximal, characteristic, complemented
10#22548709—24no—
11C2×Dic251⋊C4C2211yescharacteristic
12C2×Dic251⋊C4C2211yescharacteristic
13C2×Dic251⋊C4C2211yescharacteristic
14C2×Dic251⋊C4C2214yes—
15C2×(C2×D251)⋊C4C2211yescharacteristic
16C2×(C2×D251)⋊C4C2211yescharacteristic
17C2×(C2×D251)⋊C4C2211yescharacteristic
18C2×(C2×D251)⋊C4C2214yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2214yes—
23C251⋊#95C2214yes—
24C22×Q8×C251C2212yes—
25C4⋊Q8×C251—24no—
26C2×C4⋊C4×C251C2212yes—
27C2×C4⋊C4×C251C2212yes—
28C2×C42×C251C2211yescharacteristic
29C23×D251C2311yescharacteristic
30C22×Dic251C2314yes—
31C22×Dic251C2311yescharacteristic
32C22×Dic251C2311yescharacteristic
33C22×Dic251C2311yescharacteristic
34Dic251⋊C4—22no—
35Dic251⋊C4—22no—
36Dic251⋊C4—22no—
37Dic251⋊C4—44no—
38(C2×D251)⋊C4—22no—
39(C2×D251)⋊C4—22no—
40(C2×D251)⋊C4—22no—
41(C2×D251)⋊C4—44no—
42C2×Q8×C251—24no—
43C2×Q8×C251—44no—
44C22×C4×C251C2314yes—
45C22×C4×C251C2311yescharacteristic
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2311yescharacteristic
48C4⋊C4×C251—24no—
49C4⋊C4×C251—24no—
50C42×C251D414yes—
51C22×D251—21no—
52C22×D251—22no—
53C22×D251—22no—
54C22×D251—22no—
55C2×Dic251—21no—
56C2×Dic251—21no—
57C2×Dic251—21no—
58C2×Dic251—24no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—24no—
63C2×Dic251—24no—
64C23×C251C2411yescharacteristic
65Q8×C251—48no—
66C2×C4×C251—22no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—24no—
70C2×C4×C251—24no—
71C2×C4×C251—24no—
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D412yes—
74C2×C4×C251C2×D412yes—
75C22×C251C22×D411yescharacteristic
76C22×C251H(5,𝔽2)12yes—
77C22×C2512−1+412yes—
78C22×C2512−1+412yes—
79C2×D251—41no—
80C2×D251—41no—
81C2×D251—41no—
82C2×D251—44no—
83C4×C251—22no—
84C4×C251—22no—
85C4×C251—22no—
86C4×C251—44no—
87Dic251—41no—
88Dic251—41no—
89Dic251—41no—
90Dic251—44no—
91C2×C251#53511yescharacteristic
92C2×C251#53611yescharacteristic
93C2×C251#53611yescharacteristic
94C2×C251#56314yes—
95D251—81no—
96C251#282111yescharacteristic, complemented, Sylow
97#2821—2511nomaximal, Sylow
98C2×C4⋊Q8D25111yescharacteristic, complemented
99C2×#126—2511no—
100#400—2512no—
101#399—2512no—
102#398—2512no—
103#397—2512no—
104#395—2512no—
105#393—2512no—
106#383—2511no—
107C22×Q8C2×D25112yes—
108C4⋊Q8—24no—
109#126—5024no—
110C2×C4⋊C4—2514no—
111C2×C4⋊C4—2511no—
112C2×C4⋊C4—2511no—
113C2×C4⋊C4—2511no—
114C2×C4⋊C4C2×D25112yes—
115C2×C4⋊C4C2×D25112yes—
116C2×C22⋊C4—2514no—
117C2×C22⋊C4—2511no—
118C2×C22⋊C4—2511no—
119C2×C22⋊C4—2511no—
120C2×C42C2×D25111yescharacteristic
121#95—2512no—
122#95—2512no—
123#95—2512no—
124#95—2514no—
125#95—2514no—
126C24—2511no—
127C2×Q8—24no—
128C2×Q8—44no—
129C22×C4—2514no—
130C22×C4—2511no—
131C22×C4—2511no—
132C22×C4—2511no—
133C22×C4C22×D25114yes—
134C22×C4C22×D25111yescharacteristic
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25111yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—10044no—
140C4⋊C4—5022no—
141C4⋊C4—5022no—
142C4⋊C4—5022no—
143C22⋊C4—10044no—
144C22⋊C4—5022no—
145C22⋊C4—5022no—
146C22⋊C4—5022no—
147C42C22⋊D25114yes—
148C23—5021no—
149C23—5022no—
150C23—5022no—
151C23—5022no—
152C23C23×D25111yescharacteristic
153Q8—48no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—5021no—
161C2×C4—5021no—
162C2×C4—5021no—
163C2×C4—5024no—
164C2×C4—5022no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5024no—
168C2×C4—5024no—
169C2×C4C2×C22⋊D25112yes—
170C2×C4C2×C22⋊D25112yes—
171C2×C4C2×C22⋊D25112yes—
172C22—10041no—
173C22—10041no—
174C22—10041no—
175C22—10044no—
176C22C22×C22⋊D25111yescharacteristic
177C22#2254876012yes—
178C22#2254876312yes—
179C22#2254876312yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—10041no—
185C4—10041no—
186C4—10041no—
187C4—10044no—
188C2—20081no—
189C2#3063225411yescharacteristic
190C2#3063226111yescharacteristic
191C2#3063226111yescharacteristic
192C2#3063243614yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×C4⋊Q8×C251C2
prime radicalC23×D251C23
hypercenterC2×C4⋊Q8D251
hypocenterC251#2821
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #30631625C2semidirect
4. #30631618C2semidirect
6. #30631606C2semidirect
8. #30631555C2semidirect
9. C2×C4⋊Q8×C251C2semidirect
96. C251#2821semidirect
98. C2×C4⋊Q8D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487092
#306316252
#306316222
#306316182
#306316132
#306316062
#306315982
#306315552
C2×C4⋊Q8×C2512
#2821251