Finite Group Databasebeta

← #44558846

Subgroups — #44558846

Order 32128.

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

normal subgroups only

This group has 704 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×#22548762C211yesmaximal, characteristic
2#30632453C212yesmaximal, complemented
3#30631671C211yesmaximal, characteristic, complemented
4#30631668C214yesmaximal, complemented
5C2×#22548633C212yesmaximal, complemented
6#101×D251C214yesmaximal, complemented
7#413×C251C211yesmaximal, characteristic, complemented
8#22548762—24no—
9#22548762—24no—
10C2×#16860810C412yes—
11C2×#16860810C2211yescharacteristic
12C2×#16860809C414yes—
13C2×#16860809C2212yes—
14C22×Dic502C412yes—
15C22×Dic502C2211yescharacteristic
16C2×Q8×D251C2212yes—
17#22548723—24no—
18#22548721—24no—
19C2×#16860794C2212yes—
20M16×D251C2218yes—
21#22548634C2212yes—
22#22548633C2218yes—
23#22548632C2212yes—
24C2×#41×C251C2211yescharacteristic
25C2×M16×C251C2212yes—
26#101×C251C2212yes—
27#100×C251C2212yes—
28#16860810—24no—
29#16860810—24no—
30#16860810D414yes—
31#16860810D414yes—
32#16860809—216no—
33#16860809—28no—
34C2×Dic502—24no—
35C2×Dic502—24no—
36C2×Dic502—22no—
37C2×Dic502—22no—
38C2×Dic502C2×C414yes—
39C2×Dic502C2×C412yes—
40C2×Dic502C2×C411yescharacteristic
41C2×Dic502C2312yes—
42Q8×D251—28no—
43C2×D1004C2×C411yescharacteristic
44C2×C22⋊D251C2×C414yes—
45C2×C22⋊D251C2×C412yes—
46C22×Dic251C2×C414yes—
47C22×Dic251C2×C412yes—
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×D251C22×C412yes—
66Dic502—22no—
67Dic502—22no—
68Dic502—24no—
69Dic502—28no—
70Dic502C22⋊C412yes—
71Dic502C22⋊C412yes—
72D1004C22⋊C412yes—
73D1004C22⋊C412yes—
74C22⋊D251—28no—
75C22⋊D251C22⋊C418yes—
76C2×Dic251—24no—
77C2×Dic251—28no—
78C2×Dic251C22×C414yes—
79C2×Dic251C22×C412yes—
80C4×D251—28no—
81C4×D251C2×D418yes—
82C251⋊C8—24no—
83C23×C251C22×C412yes—
84C23×C251C22×C411yescharacteristic
85Q8×C251—22no—
86D4×C251—24no—
87D4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251C22×C412yes—
91C2×C4×C251C22×C411yescharacteristic
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2×D412yes—
94C2×C4×C251C2411yescharacteristic
95C8×C251—24no—
96C22×C251—22no—
97C22×C251—41no—
98C22×C251—42no—
99C22×C251C2×C22⋊C412yes—
100C22×C251C23×C411yescharacteristic
101C2×D251—42no—
102C2×D251C2×C22⋊C414yes—
103C4×C251—22no—
104C4×C251C2×C22⋊C411yescharacteristic
105C4×C251C2×C22⋊C411yescharacteristic
106C4×C251C22×D412yes—
107Dic251—24no—
108Dic251C2×C22⋊C414yes—
109C2×C251—21no—
110C2×C251—22no—
111C2×C251—42no—
112C2×C251C22×C22⋊C411yescharacteristic
113D251—24no—
114C251#288911yescharacteristic, complemented, Sylow
115#2889—2511nomaximal, Sylow
116C2×2−1+4—2511no—
117#568—2512no—
118#413—2511no—
119#413—2514no—
120#413D25111yescharacteristic, complemented
121C2×#101—2512no—
122C2×#101—2514no—
1232−1+4—5024no—
1242−1+4—5024no—
125C2×#41—2514no—
126C2×#41—2512no—
127C2×#41—2512no—
128C2×#41—2511no—
129C2×#41C2×D25111yescharacteristic
130C22×Q8—2512no—
131C22×Q8—2512no—
132C22×Q8—2511no—
133#131—5024no—
134#131—5024no—
135C2×M16—2512no—
136C2×M16—2518no—
137C2×M16C2×D25112yes—
138#101—2512no—
139#101—2518no—
140#101C2×D25112yes—
141#100—2512no—
142#100C2×D25112yes—
143#41—22no—
144#41—22no—
145#41—5024no—
146#41—5024no—
147#41—50216no—
148#41—5028no—
149#41—2514no—
150#41—2514no—
151C2×Q8—5024no—
152C2×Q8—5024no—
153C2×Q8—5028no—
154C2×Q8—5022no—
155C2×Q8—5022no—
156C2×Q8—2512no—
157C2×Q8—2514no—
158C2×Q8—2511no—
159C2×Q8—2512no—
160C2×Q8C22×D25111yescharacteristic
161C2×D4—2514no—
162C2×D4—2512no—
163C2×D4—2511no—
164C2×D4C4×D25112yes—
165C2×D4C22×D25111yescharacteristic
166C22×C4—2512no—
167C22×C4—2514no—
168C22×C4—2514no—
169C22×C4—2512no—
170C22×C4C4×D25112yes—
171C22×C4C22×D25111yescharacteristic
172M16—22no—
173M16—5028no—
174M16—5022no—
175M16—2514no—
176M16C22×D25114yes—
177C2×C8—22no—
178C2×C8—5028no—
179C2×C8—5022no—
180C23—2512no—
181C23C2×C4×D25112yes—
182C23C2×C4×D25111yescharacteristic
183Q8—22no—
184Q8—5028no—
185Q8—5024no—
186Q8—5022no—
187Q8—5022no—
188Q8—2512no—
189Q8—2512no—
190D4—24no—
191D4—22no—
192D4—5028no—
193D4—2518no—
194D4—2512no—
195D4—2512no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—5028no—
199C2×C4—5024no—
200C2×C4—5028no—
201C2×C4—2514no—
202C2×C4—2512no—
203C2×C4—2518no—
204C2×C4C2×C4×D25112yes—
205C2×C4C2×C4×D25111yescharacteristic
206C2×C4D4×D25112yes—
207C2×C4D4×D25112yes—
208C2×C4C23×D25111yescharacteristic
209C8—24no—
210C8—5024no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—10042no—
215C22—2514no—
216C22C22⋊C4×D25112yes—
217C22C22×C4×D25111yescharacteristic
218C4—22no—
219C4—5024no—
220C4—2514no—
221C4C22⋊C4×D25111yescharacteristic
222C4C22⋊C4×D25111yescharacteristic
223C4C2×D4×D25112yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—5024no—
228C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC2×C4C23×D251
Fitting#413×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548762C2
cosocleC2×C4×C251C24
hypercenter#413D251
hypocenterC251#2889
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. #30631668C2semidirect
5. C2×#22548633C2semidirect
6. #101×D251C2semidirect
7. #413×C251C2semidirect
114. C251#2889semidirect
120. #413D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487622
#306324532
#306316712
#306316682
C2×#225486332
#101×D2512
#413×C2512
#2889251