Finite Group Databasebeta

← #44550665

Subgroups — #44550665

Order 32128.

376 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#16860794C211yesmaximal, characteristic, complemented
2#30632198C211yesmaximal, characteristic
3C2×#22548643C212yesmaximal, complemented
4#30631447C212yesmaximal, complemented
5C4×C22⋊C4×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2211yescharacteristic
7C2×#16860794—24no—
8C2×#16860794—21no—
9C2×#16860794C2212yes—
10C22×C251⋊C8—21no—
11#22548698—21no—
12#22548694C411yescharacteristic, complemented
13#22548694C411yescharacteristic, complemented
14#22548689C411yescharacteristic, complemented
15#22548689C411yescharacteristic, complemented
16#22548684—21no—
17C4×C22⋊D251—21no—
18C4×C22⋊D251—21no—
19#22548674C2211yescharacteristic
20#22548643—24no—
21C23×C4×C251C2211yescharacteristic
22C2×C22⋊C4×C251—21no—
23C2×C42×C251C2212yes—
24#95×C251—21no—
25#16860810—22no—
26#16860810—22no—
27C2×Dic502C2×C411yescharacteristic
28C2×D1004C2×C411yescharacteristic
29C2×C22⋊D251—21no—
30C2×C22⋊D251—21no—
31C2×C22⋊D251—21no—
32C2×C4×D251—21no—
33#16860794—24no—
34#16860794—22no—
35#16860794D414yescomplemented
36C2×C251⋊C8—24no—
37C2×C251⋊C8—22no—
38C4⋊Dic251C2×C411yescharacteristic
39C4⋊Dic251C2×C411yescharacteristic
40Dic251⋊C4—21no—
41Dic251⋊C4—21no—
42C22⋊Dic251—21no—
43C22⋊Dic251—21no—
44C22⋊Dic251—21no—
45(C2×D251)⋊C4—21no—
46(C2×D251)⋊C4—21no—
47C4×Dic251—21no—
48C24×C251D411yescharacteristic
49C22×C4×C251—22no—
50C22×C4×C251—22no—
51C22×C4×C251—22no—
52C22×C4×C251C2×C411yescharacteristic
53C22×C4×C251C2×C411yescharacteristic
54C22×C4×C251D411yescharacteristic
55C22×C4×C251D411yescharacteristic
56C22×C4×C251D411yescharacteristic
57C22×C4×C251C2311yescharacteristic
58C22⋊C4×C251—24no—
59C42×C251—22no—
60C42×C251—22no—
61C22×D251—21no—
62Dic502—22no—
63D1004—22no—
64C22⋊D251—42no—
65C22⋊D251—42no—
66C22⋊D251—42no—
67C2×Dic251—21no—
68C2×Dic251—21no—
69C2×Dic251—21no—
70C4×D251—41no—
71C4×D251—41no—
72C251⋊C8—24no—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251—41no—
76C23×C251C22⋊C411yescharacteristic
77C23×C251C22⋊C411yescharacteristic
78C23×C251C2×D411yescharacteristic
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—21no—
83C2×C4×C251—21no—
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—24no—
87C2×C4×C251—42no—
88C2×C4×C251—42no—
89C2×C4×C251C22⋊C411yescharacteristic
90C2×C4×C251C22⋊C411yescharacteristic
91C2×C4×C251C22×C411yescharacteristic
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2×D411yescharacteristic
94C2×C4×C251#4112yes—
95C22×C251—21no—
96C22×C251—21no—
97C22×C251—21no—
98C22×C251—21no—
99C22×C251—21no—
100C22×C251—21no—
101C22×C251—21no—
102C22×C251—44no—
103C22×C251#10412yescomplemented
104C22×C251C2×C22⋊C411yescharacteristic
105C22×C251#12112yes—
106C2×D251—41no—
107C2×D251—41no—
108C2×D251—41no—
109C4×C251—21no—
110C4×C251—21no—
111C4×C251—44no—
112C4×C251C4×D412yes—
113C4×C251#12011yescharacteristic
114C4×C251#12311yescharacteristic
115Dic251—41no—
116Dic251—41no—
117Dic251—41no—
118C2×C251—22no—
119C2×C251—22no—
120C2×C251—41no—
121C2×C251#38611yescharacteristic
122C2×C251C2×#10411yescharacteristic
123C2×C251#42111yescharacteristic
124D251—81no—
125C251#187111yescharacteristic, complemented, Sylow
126#1871—2511nomaximal, Sylow
127C22×M16—2511no—
128#525—2511no—
129C2×#104—2512no—
130C4×C22⋊C4D25111yescharacteristic, complemented
131#339—2512no—
132C2×#41—2511no—
133C23×C4C2×D25111yescharacteristic
134C2×M16—5021no—
135C2×M16—5024no—
136C2×M16—2512no—
137C22×C8—5021no—
138#123—5021no—
139C22⋊Q8—2511no—
140C22⋊Q8—2511no—
141#121—2511no—
142#121—2511no—
143#120—5021no—
144C4×D4—5021no—
145C4×D4—5021no—
146#117—2511no—
147C2×C22⋊C4—21no—
148C2×C42C2×D25112yes—
149#104—5024no—
150#95—21no—
151C24C22⋊D25111yescharacteristic
152#41—5022no—
153#41—5022no—
154C2×Q8—2511no—
155C2×D4—5021no—
156C2×D4—5021no—
157C2×D4—5021no—
158C2×D4—2511no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—5021no—
163C22×C4C4×D25111yescharacteristic
164C22×C4C4×D25111yescharacteristic
165C22×C4C22⋊D25111yescharacteristic
166C22×C4D100411yescharacteristic
167C22×C4D100411yescharacteristic
168C22×C4C22×D25111yescharacteristic
169M16—5024no—
170M16—5022no—
171M16—2514no—
172C2×C8—5024no—
173C2×C8—5022no—
174C4⋊C4—5021no—
175C4⋊C4—5021no—
176C4⋊C4—2511no—
177C4⋊C4—2511no—
178C22⋊C4—24no—
179C22⋊C4—5021no—
180C22⋊C4—5021no—
181C22⋊C4—5021no—
182C22⋊C4—5021no—
183C22⋊C4—5021no—
184C42—22no—
185C42—22no—
186C42—5021no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—5021no—
191C23(C2×D251)⋊C411yescharacteristic
192C23(C2×D251)⋊C411yescharacteristic
193C23C2×C22⋊D25111yescharacteristic
194Q8—5022no—
195D4—10042no—
196D4—10042no—
197D4—10042no—
198D4—5022no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—10041no—
210C2×C4—10041no—
211C2×C4—5021no—
212C2×C4—5021no—
213C2×C4—5021no—
214C2×C4(C2×D251)⋊C411yescharacteristic
215C2×C4(C2×D251)⋊C411yescharacteristic
216C2×C4C2×C4×D25111yescharacteristic
217C2×C4D4×D25112yes—
218C2×C4C2×D100411yescharacteristic
219C2×C4#1686080912yes—
220C8—5024no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—10041no—
230C22—10041no—
231C22—10041no—
232C22#2254864312yescomplemented
233C22C2×(C2×D251)⋊C411yescharacteristic
234C22#2254868712yes—
235C4—21no—
236C4—21no—
237C4—44no—
238C4—10041no—
239C4—10041no—
240C4—10041no—
241C4(C22⋊D251)⋊C412yes—
242C4#2254868511yescharacteristic
243C4#2254869911yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—20081no—
248C2#3063156411yescharacteristic
249C2C2×#2254864311yescharacteristic
250C2#3063170511yescharacteristic
subgroupG/H
centerC2×C4C2×D1004
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC4×C22⋊C4×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenterC4×C22⋊C4D251
hypocenterC251#1871
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#16860794C2semidirect
3. C2×#22548643C2semidirect
4. #30631447C2semidirect
5. C4×C22⋊C4×C251C2semidirect
12. #22548694C4semidirect
13. #22548694C4semidirect
14. #22548689C4semidirect
15. #22548689C4semidirect
35. #16860794D4semidirect
103. C22×C251#104semidirect
125. C251#1871semidirect
130. C4×C22⋊C4D251semidirect
232. C22#22548643semidirect

Maximal subgroups

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

subgroupindex
C22×#168607942
#306321982
C2×#225486432
#306314472
C4×C22⋊C4×C2512
#1871251