Finite Group Databasebeta

← #44550807

Subgroups — #44550807

Order 32128.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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#30632199C211yesmaximal, characteristic
2C2×C42×D251C211yesmaximal, characteristic, complemented
3C2×#22548643C211yesmaximal, characteristic, complemented
4C2×#22548641C211yesmaximal, characteristic, complemented
5#30631650C211yesmaximal, characteristic, complemented
6#30631446C211yesmaximal, characteristic, complemented
7#339×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×C4×D251—21no—
10C22×C4×D251C2211yescharacteristic
11C2×#16860794C2211yescharacteristic
12#22548699—21no—
13#22548695C412yescomplemented
14#22548690C412yescomplemented
15#22548685—21no—
16C4×D1004—22no—
17C2×C4×Dic251—21no—
18C2×C4×Dic251C2211yescharacteristic
19C42×D251—22no—
20C42×D251—22no—
21#22548643—22no—
22#22548641—22no—
23(C2×D251)⋊C8—22no—
24C2×M16×C251C2211yescharacteristic
25#117×C251C2211yescharacteristic
26C2×C42×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28#16860810—21no—
29#16860810—21no—
30#16860810—21no—
31#16860810—21no—
32C2×Dic502C2×C411yescharacteristic, complemented
33C2×D1004—22no—
34C2×D1004C2×C411yescharacteristic, complemented
35C2×C22⋊D251—21no—
36C22×Dic251—21no—
37C22×Dic251D411yescharacteristic
38C2×C4×D251—28no—
39C2×C4×D251—24no—
40C2×C4×D251—21no—
41C2×C4×D251—21no—
42C2×C4×D251—21no—
43C2×C4×D251—21no—
44C2×C4×D251—21no—
45C2×C4×D251C2×C412yes—
46C2×C4×D251D412yes—
47#16860794—21no—
48#16860794—21no—
49C2×C251⋊C8—21no—
50C4⋊Dic251—22no—
51(C2×D251)⋊C4—22no—
52(C2×D251)⋊C4—22no—
53C4×Dic251—24no—
54C4×Dic251—21no—
55C4×Dic251D412yescomplemented
56C22×C4×C251—21no—
57C22×C4×C251C2311yescharacteristic
58M16×C251—21no—
59M16×C251—21no—
60C2×C8×C251—21no—
61C4⋊C4×C251C2×C412yescomplemented
62C22⋊C4×C251—21no—
63C42×C251—21no—
64C42×C251—21no—
65C42×C251D412yescomplemented
66C22×D251—21no—
67C22×D251—21no—
68C22×D251—24no—
69C22×D251—21no—
70C22×D251C22⋊C412yes—
71Dic502—21no—
72Dic502—21no—
73D1004—21no—
74D1004—21no—
75D1004—44no—
76C22⋊D251—41no—
77C22⋊D251—41no—
78C2×Dic251—24no—
79C2×Dic251—21no—
80C2×Dic251—22no—
81C2×Dic251—21no—
82C2×Dic251—21no—
83C2×Dic251C22⋊C412yes—
84C4×D251—216no—
85C4×D251—22no—
86C4×D251—22no—
87C4×D251—22no—
88C4×D251—22no—
89C4×D251—41no—
90C4×D251—41no—
91C251⋊C8—41no—
92C23×C251C2×D411yescharacteristic
93C2×C4×C251—24no—
94C2×C4×C251—22no—
95C2×C4×C251—22no—
96C2×C4×C251C22×C411yescharacteristic
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251C2×D411yescharacteristic
99C2×C4×C251C2×D411yescharacteristic
100C2×C4×C251#4111yescharacteristic
101C2×C4×C251#4111yescharacteristic
102C8×C251—41no—
103C22×C251—22no—
104C22×C251C2×C22⋊C411yescharacteristic
105C22×C251#12011yescharacteristic
106C22×C251#12311yescharacteristic
107C2×D251—24no—
108C2×D251—24no—
109C2×D251—22no—
110C2×D251—22no—
111C2×D251—41no—
112C2×D251—42no—
113C2×D251#10414yescomplemented
114C4×C251—24no—
115C4×C251—42no—
116C4×C251C4×D411yescharacteristic
117C4×C251C4×D411yescharacteristic
118C4×C251#12111yescharacteristic
119C4×C251#12111yescharacteristic
120Dic251—24no—
121Dic251—22no—
122Dic251—41no—
123C2×C251—21no—
124C2×C251—21no—
125C2×C251#38611yescharacteristic
126C2×C251C2×#10412yes—
127D251—24no—
128D251—81no—
129C251#189611yescharacteristic, complemented, Sylow
130#1896—2511nomaximal, Sylow
131#525—2511no—
132C22×C42—2511no—
133C2×#104—2511no—
134C2×#104—2511no—
135#407—2511no—
136#339—2511no—
137#339D25111yescharacteristic, complemented
138C2×#41—2511no—
139C23×C4—5021no—
140C23×C4—2511no—
141C2×M16—2511no—
142C2×M16C2×D25111yescharacteristic
143#123—5021no—
144C22⋊Q8—2512no—
145#121—2512no—
146#120—5021no—
147C4×D4—5022no—
148#117C2×D25111yescharacteristic
149C2×C42—5022no—
150C2×C42—5021no—
151C2×C42—5022no—
152C2×C42—2511no—
153C2×C42C2×D25111yescharacteristic
154#104—5022no—
155#104—5022no—
156C22⋊C8—5022no—
157C24—2511no—
158#41—5021no—
159#41—5021no—
160#41—5021no—
161#41—5021no—
162C2×Q8—2511no—
163C2×D4—5022no—
164C2×D4—5021no—
165C2×D4—2511no—
166C22×C4—21no—
167C22×C4—5024no—
168C22×C4—5028no—
169C22×C4—5021no—
170C22×C4—5021no—
171C22×C4—5021no—
172C22×C4—5021no—
173C22×C4—5021no—
174C22×C4—5021no—
175C22×C4—2511no—
176C22×C4—2512no—
177C22×C4—2512no—
178C22×C4C22×D25111yescharacteristic
179M16—21no—
180M16—21no—
181M16—5021no—
182M16—5021no—
183C2×C8—21no—
184C2×C8—5021no—
185C4⋊C4—5022no—
186C4⋊C4C4×D25112yescomplemented
187C22⋊C4—21no—
188C22⋊C4—5022no—
189C22⋊C4—5022no—
190C42—21no—
191C42—21no—
192C42—5024no—
193C42—5021no—
194C42—2512no—
195C42D100412yescomplemented
196C23—5024no—
197C23—5021no—
198C23—5021no—
199C23—5021no—
200C23—2512no—
201C23D4×D25111yescharacteristic
202Q8—5021no—
203Q8—5021no—
204D4—10044no—
205D4—10041no—
206D4—10041no—
207D4—5021no—
208D4—5021no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—10041no—
213C2×C4—10041no—
214C2×C4—50216no—
215C2×C4—5021no—
216C2×C4—5024no—
217C2×C4—5021no—
218C2×C4—5022no—
219C2×C4—5021no—
220C2×C4—5022no—
221C2×C4—5022no—
222C2×C4—5022no—
223C2×C4—5022no—
224C2×C4—2512no—
225C2×C4C2×C4×D25111yescharacteristic
226C2×C4D4×D25111yescharacteristic
227C2×C4D4×D25111yescharacteristic
228C2×C4C2×D100411yescharacteristic
229C2×C4#1686081011yescharacteristic
230C2×C4#1686081111yescharacteristic
231C8—41no—
232C8—10041no—
233C22—22no—
234C22—10042no—
235C22—10041no—
236C22—5024no—
237C22—5024no—
238C22—5022no—
239C22—5022no—
240C22—2514no—
241C22C22⋊C4×D25111yescharacteristic
242C22#2254868511yescharacteristic
243C22#2254870011yescharacteristic
244C4—24no—
245C4—42no—
246C4—10041no—
247C4—5024no—
248C4—5022no—
249C4#2254867711yescharacteristic
250C4C4×D100411yescharacteristic
251C4#2254868811yescharacteristic
252C4#2254869011yescharacteristic
253C2—21no—
254C2—21no—
255C2—20081no—
256C2—5024no—
257C2#3063156511yescharacteristic
258C2#104×D25112yes—
subgroupG/H
centerC2×C4D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#339×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632199C2
cosocleC22×C4×C251C23
hypercenter#339D251
hypocenterC251#1896
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×C42×D251C2semidirect
3. C2×#22548643C2semidirect
4. C2×#22548641C2semidirect
5. #30631650C2semidirect
6. #30631446C2semidirect
7. #339×C251C2semidirect
13. #22548695C4semidirect
14. #22548690C4semidirect
32. C2×Dic502C2×C4semidirect
34. C2×D1004C2×C4semidirect
55. C4×Dic251D4semidirect
61. C4⋊C4×C251C2×C4semidirect
65. C42×C251D4semidirect
113. C2×D251#104semidirect
129. C251#1896semidirect
137. #339D251semidirect
186. C4⋊C4C4×D251semidirect
195. C42D1004semidirect

Maximal subgroups

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

subgroupindex
#306321992
C2×C42×D2512
C2×#225486432
C2×#225486412
#306316502
#306314462
#339×C2512
#1896251