Finite Group Databasebeta

← #44558842

Subgroups — #44558842

Order 32128.

756 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22548761C211yesmaximal, characteristic
2#30632453C212yesmaximal, complemented
3#30631671C211yesmaximal, characteristic, complemented
4#30631670C214yesmaximal, complemented
5C2×#22548631C212yesmaximal, complemented
6#100×D251C214yesmaximal, complemented
7#413×C251C211yesmaximal, characteristic, complemented
8#22548761—24no—
9#22548761—24no—
10C2×#16860811C2212yes—
11C2×#16860810C412yes—
12C2×#16860810C2211yescharacteristic
13C22×D1004C412yes—
14C22×D1004C2211yescharacteristic
15C2×D4×D251C414yes—
16C2×D4×D251C2212yes—
17#22548723—24no—
18#22548721—24no—
19C2×#16860794C2212yescomplemented
20M16×D251C2218yescomplemented
21#22548634C2212yescomplemented
22#22548632C2212yescomplemented
23#22548631C2218yescomplemented
24C2×#41×C251C2211yescharacteristic
25C2×M16×C251C2212yescomplemented
26#101×C251C2212yescomplemented
27#100×C251C2212yescomplemented
28C23×D251C2×C412yes—
29C23×D251C2×C414yes—
30#16860811—28no—
31#16860810—24no—
32#16860810—24no—
33#16860810D414yes—
34#16860810D414yes—
35C2×Dic502C2×C411yescharacteristic
36C2×D1004—24no—
37C2×D1004—24no—
38C2×D1004—22no—
39C2×D1004—22no—
40C2×D1004C2×C414yes—
41C2×D1004C2×C412yes—
42C2×D1004C2×C411yescharacteristic
43C2×D1004C2312yes—
44C2×C22⋊D251C2×C414yes—
45C2×C22⋊D251C2×C412yes—
46D4×D251—216no—
47D4×D251—28no—
48C2×C4×D251C2×C414yes—
49C2×C4×D251C2312yes—
50#16860794—22no—
51#16860794C2314yes—
52#16860793—28no—
53C2×C251⋊C8—22no—
54C8×D251—28no—
55#41×C251—22no—
56#41×C251—22no—
57C2×Q8×C251C2311yescharacteristic
58C2×D4×C251C2×C412yes—
59C2×D4×C251C2311yescharacteristic
60C22×C4×C251C2×C412yes—
61C22×C4×C251C2311yescharacteristic
62M16×C251—22no—
63M16×C251C2314yes—
64C2×C8×C251—22no—
65C22×D251—24no—
66C22×D251—28no—
67C22×D251—84no—
68C22×D251—82no—
69C22×D251C22×C414yes—
70C22×D251C22×C412yes—
71Dic502C22⋊C412yes—
72Dic502C22⋊C412yes—
73D1004—28no—
74D1004—22no—
75D1004—22no—
76D1004—24no—
77D1004C22⋊C412yes—
78D1004C22⋊C412yes—
79C22⋊D251—28no—
80C22⋊D251C22⋊C418yes—
81C2×Dic251C22×C412yes—
82C4×D251—28no—
83C4×D251C2×D418yes—
84C251⋊C8—24no—
85C23×C251C22×C412yes—
86C23×C251C22×C411yescharacteristic
87Q8×C251—22no—
88D4×C251—24no—
89D4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251C22×C412yes—
93C2×C4×C251C22×C411yescharacteristic
94C2×C4×C251C2×D412yes—
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2411yescharacteristic
97C8×C251—24no—
98C22×C251—22no—
99C22×C251—41no—
100C22×C251—42no—
101C22×C251C2×C22⋊C412yes—
102C22×C251C23×C411yescharacteristic
103C2×D251—24no—
104C2×D251—42no—
105C2×D251—44no—
106C2×D251—88no—
107C2×D251—84no—
108C2×D251C2×C22⋊C414yes—
109C4×C251—22no—
110C4×C251C2×C22⋊C411yescharacteristic
111C4×C251C2×C22⋊C411yescharacteristic
112C4×C251C22×D412yes—
113Dic251C2×C22⋊C414yes—
114C2×C251—21no—
115C2×C251—22no—
116C2×C251—42no—
117C2×C251C22×C22⋊C411yescharacteristic
118D251—24no—
119D251—44no—
120C251#288811yescharacteristic, complemented, Sylow
121#2888—2511nomaximal, Sylow
122C2×H(5,𝔽2)—2511no—
123#568—2512no—
124#413—2511no—
125#413—2514no—
126#413D25111yescharacteristic, complemented
127C2×#100—2512no—
128C2×#100—2514no—
129H(5,𝔽2)—5024no—
130H(5,𝔽2)—5024no—
131C2×#41—2512no—
132C2×#41—2512no—
133C2×#41—2511no—
134C2×#41C2×D25111yescharacteristic
135C22×D4—2514no—
136C22×D4—2512no—
137C22×D4—2512no—
138C22×D4—2511no—
139#131—5024no—
140#131—5024no—
141C2×M16—2512no—
142C2×M16—2518no—
143C2×M16C2×D25112yescomplemented
144#101—2512no—
145#101C2×D25112yescomplemented
146#100—2512no—
147#100—2518no—
148#100C2×D25112yescomplemented
149C24—2514no—
150C24—2512no—
151#41—22no—
152#41—22no—
153#41—5024no—
154#41—5024no—
155#41—5028no—
156#41—2514no—
157#41—2514no—
158C2×Q8—2511no—
159C2×Q8C22×D25111yescharacteristic
160C2×D4—5024no—
161C2×D4—5024no—
162C2×D4—50216no—
163C2×D4—5028no—
164C2×D4—5022no—
165C2×D4—5022no—
166C2×D4—2514no—
167C2×D4—2512no—
168C2×D4—2514no—
169C2×D4—2512no—
170C2×D4—2511no—
171C2×D4—2512no—
172C2×D4C4×D25112yes—
173C2×D4C22×D25111yescharacteristic
174C22×C4—2514no—
175C22×C4—2512no—
176C22×C4C4×D25112yes—
177C22×C4C22×D25111yescharacteristic
178M16—22no—
179M16—5028no—
180M16—5022no—
181M16—2514no—
182M16C22×D25114yes—
183C2×C8—22no—
184C2×C8—5028no—
185C2×C8—5022no—
186C23—20082no—
187C23—20084no—
188C23—5028no—
189C23—5024no—
190C23—2514no—
191C23—2512no—
192C23C2×C4×D25112yes—
193C23C2×C4×D25111yescharacteristic
194Q8—22no—
195Q8—2512no—
196Q8—2512no—
197D4—24no—
198D4—22no—
199D4—5028no—
200D4—5028no—
201D4—5024no—
202D4—5022no—
203D4—5022no—
204D4—2518no—
205D4—2512no—
206D4—2512no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—5028no—
210C2×C4—2512no—
211C2×C4—2518no—
212C2×C4C2×C4×D25112yes—
213C2×C4C2×C4×D25111yescharacteristic
214C2×C4D4×D25112yes—
215C2×C4D4×D25112yes—
216C2×C4C23×D25111yescharacteristic
217C8—24no—
218C8—5024no—
219C22—22no—
220C22—41no—
221C22—42no—
222C22—20084no—
223C22—20088no—
224C22—10044no—
225C22—10042no—
226C22—5024no—
227C22—2514no—
228C22C22⋊C4×D25112yes—
229C22C22×C4×D25111yescharacteristic
230C4—22no—
231C4—2514no—
232C4C22⋊C4×D25111yescharacteristic
233C4C22⋊C4×D25111yescharacteristic
234C4C2×D4×D25112yes—
235C2—21no—
236C2—22no—
237C2—42no—
238C2—10044no—
239C2—5024no—
240C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC2×C4C23×D251
Fitting#413×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548761C2
cosocleC2×C4×C251C24
hypercenter#413D251
hypocenterC251#2888
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #30632453C2semidirect
3. #30631671C2semidirect
4. #30631670C2semidirect
5. C2×#22548631C2semidirect
6. #100×D251C2semidirect
7. #413×C251C2semidirect
19. C2×#16860794C22semidirect
20. M16×D251C22semidirect
21. #22548634C22semidirect
22. #22548632C22semidirect
23. #22548631C22semidirect
25. C2×M16×C251C22semidirect
26. #101×C251C22semidirect
27. #100×C251C22semidirect
120. C251#2888semidirect
126. #413D251semidirect
143. C2×M16C2×D251semidirect
145. #101C2×D251semidirect
148. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
#306324532
#306316712
#306316702
C2×#225486312
#100×D2512
#413×C2512
#2888251