Finite Group Databasebeta

← #44560488

Subgroups — #44560488

Order 32128.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#30632237C211yesmaximal, characteristic
2C2×C4×Dic502C211yesmaximal, characteristic, complemented
3#30632001C212yesmaximal, complemented
4#30631983C212yesmaximal, complemented
5#30631888C212yesmaximal, complemented
6#30631828C212yesmaximal, complemented
7#30631789C214yesmaximal, complemented
8#432×C251C211yesmaximal, characteristic, complemented
9C2×#16860810—21no—
10C22×Dic502C2211yescharacteristic
11C2×#16860799C2214yes—
12#22548715C2211yescharacteristic
13#22548712C2211yescharacteristic
14#22548703—21no—
15#22548689—21no—
16#22548689C2212yes—
17C4×Dic502—21no—
18C4×Dic502—21no—
19C4×Dic502C2212yes—
20C4×Dic502C2212yes—
21C4×D1004—21no—
22C4×D1004C2212yes—
23C2×C4⋊Dic251C2211yescharacteristic
24C2×Dic251⋊C4—21no—
25C2×C4×Dic251—21no—
26#22548649C2214yes—
27#22548640C2214yes—
28#22548637C2214yes—
29C2×C42×C251C2211yescharacteristic
30C4⋊C8×C251C2212yes—
31C22⋊C8×C251C2212yes—
32#97×C251C2212yes—
33#16860810—44no—
34C2×Dic502—24no—
35C2×Dic502—21no—
36C2×Dic502—21no—
37C2×Dic502C2312yes—
38C2×D1004—21no—
39C2×D1004—21no—
40C2×D1004C2312yes—
41C2×C22⋊D251—22no—
42C22×Dic251—21no—
43C22×Dic251—21no—
44C2×C4×D251—22no—
45#16860799—28no—
46C4⋊Dic251—21no—
47C4⋊Dic251—21no—
48C4⋊Dic251C2312yes—
49C4⋊Dic251C2312yes—
50Dic251⋊C4—22no—
51Dic251⋊C4—22no—
52(C2×D251)⋊C4—22no—
53(C2×D251)⋊C4—22no—
54C4×Dic251—22no—
55C4×Dic251—22no—
56C22×C4×C251D411yescharacteristic
57C22×C4×C251D411yescharacteristic
58C22×C4×C251C2311yescharacteristic
59C2×C8×C251C2314yes—
60C42×C251D411yescharacteristic
61C42×C251D411yescharacteristic
62C42×C251C2311yescharacteristic
63C42×C251C2311yescharacteristic
64C22×D251—22no—
65Dic502—24no—
66Dic502—22no—
67Dic502—42no—
68Dic502#4114yescomplemented
69D1004—24no—
70D1004—42no—
71D1004—42no—
72C22⋊D251—44no—
73C2×Dic251—24no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—42no—
78C4×D251—44no—
79C23×C251C2×D411yescharacteristic
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251C2411yescharacteristic
89C8×C251—24no—
90C22×C251—21no—
91C22×C251—21no—
92C22×C251AGL(1,ℤ/8)12yes—
93C22×C251C22×D411yescharacteristic
94C2×D251—42no—
95C2×D251—42no—
96C2×D251—42no—
97C4×C251—21no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—21no—
101C4×C251#13712yes—
102C4×C251C2×#4112yes—
103Dic251—24no—
104Dic251—42no—
105Dic251—42no—
106C2×C251—22no—
107C2×C251#52511yescharacteristic
108C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C251C2×#13711yescharacteristic
110D251—82no—
111C251#311211yescharacteristic, complemented, Sylow
112#3112—2511nomaximal, Sylow
113#532—2511no—
114C2×C4×Q8—2511no—
115#480—2512no—
116#477—2512no—
117#460—2512no—
118#448—2512no—
119#440—2514no—
120#432D25111yescharacteristic, complemented
121C2×#41—5021no—
122C22×Q8—2511no—
123C2×SD16—2514no—
124C4⋊Q8—2511no—
125#127—2511no—
126#124—5021no—
127#121—5021no—
128#121—2512no—
129C4×Q8—5021no—
130C4×Q8—5021no—
131C4×Q8—2512no—
132C4×Q8—2512no—
133C4×D4—5021no—
134C4×D4—2512no—
135C2×C4⋊C4—5021no—
136C2×C4⋊C4—2511no—
137C2×C42—5021no—
138C2×C42C2×D25111yescharacteristic
139#107—2514no—
140C4⋊C8C2×D25112yes—
141#103—2514no—
142#102—2514no—
143C22⋊C8C2×D25112yes—
144#97C2×D25112yes—
145#41—10044no—
146C2×Q8—5024no—
147C2×Q8—5021no—
148C2×Q8—5021no—
149C2×Q8—2512no—
150C2×D4—5022no—
151C2×D4—5021no—
152C2×D4—5021no—
153C2×D4—2512no—
154C22×C4—5021no—
155C22×C4—5022no—
156C22×C4—5021no—
157C22×C4D100411yescharacteristic
158C22×C4D100411yescharacteristic
159C22×C4C22×D25111yescharacteristic
160SD16—5028no—
161C2×C8C22×D25114yes—
162C4⋊C4—5022no—
163C4⋊C4—5022no—
164C4⋊C4—5021no—
165C4⋊C4—5021no—
166C4⋊C4—2512no—
167C4⋊C4—2512no—
168C22⋊C4—5022no—
169C22⋊C4—5022no—
170C42—5022no—
171C42—5022no—
172C42D100411yescharacteristic
173C42D100411yescharacteristic
174C42C22×D25111yescharacteristic
175C42C22×D25111yescharacteristic
176C23—5022no—
177C23C2×D100411yescharacteristic
178Q8—10042no—
179Q8—5024no—
180Q8—5022no—
181Q8—2514no—
182D4—10042no—
183D4—10044no—
184D4—10042no—
185D4—5024no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—10044no—
190C2×C4—10042no—
191C2×C4—5022no—
192C2×C4—5022no—
193C2×C4—5024no—
194C2×C4—5022no—
195C2×C4C2×D100411yescharacteristic
196C2×C4C2×D100411yescharacteristic
197C2×C4C2×D100411yescharacteristic
198C2×C4C2×D100411yescharacteristic
199C2×C4C2×D100411yescharacteristic
200C2×C4C23×D25111yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—10042no—
205C22—10042no—
206C22—10042no—
207C22#2254874312yes—
208C22C22×D100411yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—10042no—
214C4—10042no—
215C4—5024no—
216C4#2254874812yes—
217C4#41×D25112yes—
218C2—22no—
219C2—20082no—
220C2#3063219911yescharacteristic
221C2C2×#2254874311yescharacteristic
222C2C2×#2254874811yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#432×C251C2
socleC22×C251C22×D4
prime radical#30632237C2
hypercenter#432D251
hypocenterC251#3112
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic502C2semidirect
3. #30632001C2semidirect
4. #30631983C2semidirect
5. #30631888C2semidirect
6. #30631828C2semidirect
7. #30631789C2semidirect
8. #432×C251C2semidirect
68. Dic502#41semidirect
111. C251#3112semidirect
120. #432D251semidirect

Maximal subgroups

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

subgroupindex
#306322372
C2×C4×Dic5022
#306320012
#306319832
#306318882
#306318282
#306317892
#432×C2512
#3112251