Finite Group Databasebeta

← #44548868

Subgroups — #44548868

Order 32128.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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#30632200C211yesmaximal, characteristic
2#30631649C211yesmaximal, characteristic, complemented
3#95×D251C211yesmaximal, characteristic, complemented
4#30631424C211yesmaximal, characteristic, complemented
5#30631422C211yesmaximal, characteristic, complemented
6#30631420C211yesmaximal, characteristic, complemented
7#328×C251C211yesmaximal, characteristic, complemented
8C2×#16860811C411yescharacteristic, complemented
9C22×C4×D251—21no—
10C22×C4×D251C2211yescharacteristic
11C2×#16860793—21no—
12#22548700—21no—
13#22548691C411yescharacteristic, complemented
14#22548690C411yescharacteristic, complemented
15#22548689C2211yescharacteristic
16#22548685—21no—
17#22548677—21no—
18(C22⋊D251)⋊C4—21no—
19#22548673C411yescharacteristic, complemented
20C22⋊(C251⋊C8)C2211yescharacteristic
21(C2×D251)⋊C8—21no—
22C251⋊#95—21no—
23C251⋊#95C2211yescharacteristic
24C22⋊Q8×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28#16860811—42no—
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004C2×C411yescharacteristic, complemented
32C2×C22⋊D251—21no—
33C22×Dic251—21no—
34C22×Dic251D411yescharacteristic
35C2×C4×D251—24no—
36C2×C4×D251—21no—
37C2×C4×D251—42no—
38C2×C4×D251—42no—
39C2×C4×D251—42no—
40C2×C4×D251C2×C411yescharacteristic
41C2×C4×D251C2×C411yescharacteristic
42C2×C4×D251D411yescharacteristic
43C2×C4×D251D411yescharacteristic
44#16860793—42no—
45C2×C251⋊C8—21no—
46C4⋊Dic251C2×C411yescharacteristic, complemented
47Dic251⋊C4—21no—
48(C2×D251)⋊C4—21no—
49(C2×D251)⋊C4—21no—
50(C2×D251)⋊C4—21no—
51(C2×D251)⋊C4—21no—
52C4×Dic251—21no—
53C2×Q8×C251C2×C411yescharacteristic, complemented
54C22×C4×C251—21no—
55C22×C4×C251C2311yescharacteristic
56C2×C8×C251—21no—
57C4⋊C4×C251—21no—
58C4⋊C4×C251C2×C411yescharacteristic, complemented
59C22⋊C4×C251—21no—
60C22×D251—21no—
61C22×D251—21no—
62C22×D251—22no—
63C22×D251—22no—
64C22×D251—21no—
65C22×D251C22⋊C411yescharacteristic
66C22×D251C22⋊C411yescharacteristic
67D1004—41no—
68D1004—42no—
69D1004—42no—
70C22⋊D251—41no—
71C22⋊D251—41no—
72C2×Dic251—22no—
73C2×Dic251—21no—
74C2×Dic251—41no—
75C2×Dic251—41no—
76C2×Dic251—41no—
77C2×Dic251C22⋊C411yescharacteristic
78C2×Dic251C22⋊C411yescharacteristic
79C4×D251—22no—
80C4×D251—22no—
81C4×D251—41no—
82C4×D251—41no—
83C4×D251—42no—
84C4×D251—48no—
85C251⋊C8—41no—
86C23×C251C2×D411yescharacteristic
87Q8×C251—41no—
88C2×C4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—21no—
91C2×C4×C251—41no—
92C2×C4×C251—41no—
93C2×C4×C251—41no—
94C2×C4×C251C22×C411yescharacteristic
95C2×C4×C251C2×D411yescharacteristic
96C8×C251—41no—
97C22×C251—21no—
98C22×C251—21no—
99C22×C251C2×C22⋊C411yescharacteristic
100C22×C251AGL(1,ℤ/8)11yescharacteristic
101C22×C251#13711yescharacteristic
102C2×D251—24no—
103C2×D251—24no—
104C2×D251—21no—
105C2×D251—21no—
106C2×D251—21no—
107C2×D251—21no—
108C2×D251—41no—
109C2×D251—41no—
110C2×D251—41no—
111C2×D251#9912yes—
112C2×D251#10412yes—
113C4×C251—21no—
114C4×C251—21no—
115C4×C251—42no—
116C4×C251—41no—
117C4×C251—41no—
118Dic251—21no—
119Dic251—21no—
120Dic251—42no—
121Dic251—41no—
122C2×C251—21no—
123C2×C251—21no—
124C2×C251C2×#9911yescharacteristic
125C2×C251#41711yescharacteristic
126C2×C251C2×#10411yescharacteristic
127D251—22no—
128D251—22no—
129D251—81no—
130C251#150411yescharacteristic, complemented, Sylow
131#1504—2511nomaximal, Sylow
132#525—2511no—
133#407—2511no—
134C2×#95—2511no—
135#328—2511no—
136#328D25111yescharacteristic, complemented
137#327—2511no—
138#327—2511no—
139C2×#41—2511no—
140C23×C4—5021no—
141C23×C4—2511no—
142C2×M16—5021no—
143#123—5021no—
144C22⋊Q8—2511no—
145C22⋊Q8C2×D25111yescharacteristic
146#121—2511no—
147#121—2511no—
148#120—5021no—
149C4×D4—5021no—
150C4×D4—5021no—
151#117—2511no—
152C22⋊C8—5021no—
153C22⋊C8—2511no—
154C22⋊C8C2×D25111yescharacteristic
155#95—5021no—
156#95—2511no—
157#95C2×D25111yescharacteristic
158C24—2511no—
159#41—10042no—
160C2×Q8C4×D25111yescharacteristic, complemented
161C2×D4—5021no—
162C2×D4—5021no—
163C2×D4—5021no—
164C2×D4—2511no—
165C22×C4—21no—
166C22×C4—10042no—
167C22×C4—10042no—
168C22×C4—10042no—
169C22×C4—5024no—
170C22×C4—5021no—
171C22×C4—5021no—
172C22×C4—2511no—
173C22×C4—2511no—
174C22×C4—2511no—
175C22×C4—2511no—
176C22×C4—2511no—
177C22×C4C22×D25111yescharacteristic
178M16—10042no—
179C2×C8—21no—
180C2×C8—5021no—
181C4⋊C4—21no—
182C4⋊C4—5021no—
183C4⋊C4—2511no—
184C4⋊C4C4×D25111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—5021no—
187C22⋊C4—5021no—
188C22⋊C4—5021no—
189C22⋊C4—5021no—
190C42—5021no—
191C23—5022no—
192C23—5022no—
193C23—5021no—
194C23—5021no—
195C23—5021no—
196C23—2511no—
197C23—2511no—
198C23D4×D25111yescharacteristic
199Q8—41no—
200D4—10041no—
201D4—10042no—
202D4—10042no—
203D4—10041no—
204D4—10041no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—10041no—
212C2×C4—10041no—
213C2×C4—10048no—
214C2×C4—10042no—
215C2×C4—10041no—
216C2×C4—10041no—
217C2×C4—10041no—
218C2×C4—5022no—
219C2×C4—5022no—
220C2×C4—5022no—
221C2×C4—5021no—
222C2×C4—2511no—
223C2×C4—2511no—
224C2×C4C2×C4×D25111yescharacteristic
225C2×C4D4×D25111yescharacteristic
226C8—41no—
227C8—10041no—
228C22—21no—
229C22—21no—
230C22—10041no—
231C22—10041no—
232C22—10041no—
233C22—5024no—
234C22—5024no—
235C22—5021no—
236C22—5021no—
237C22—5021no—
238C22—5021no—
239C22—2512no—
240C22—2512no—
241C22C22⋊C4×D25111yescharacteristic
242C22#2254874211yescharacteristic
243C22#2254874711yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—10042no—
250C4—10041no—
251C4—5021no—
252C4—5021no—
253C2—21no—
254C2—21no—
255C2—20081no—
256C2—5022no—
257C2—5022no—
258C2#99×D25111yescharacteristic
259C2#3063168711yescharacteristic
260C2#104×D25111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#328×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632200C2
cosocleC22×C4×C251C23
hypercenter#328D251
hypocenterC251#1504
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #30631649C2semidirect
3. #95×D251C2semidirect
4. #30631424C2semidirect
5. #30631422C2semidirect
6. #30631420C2semidirect
7. #328×C251C2semidirect
8. C2×#16860811C4semidirect
13. #22548691C4semidirect
14. #22548690C4semidirect
19. #22548673C4semidirect
31. C2×D1004C2×C4semidirect
46. C4⋊Dic251C2×C4semidirect
53. C2×Q8×C251C2×C4semidirect
58. C4⋊C4×C251C2×C4semidirect
130. C251#1504semidirect
136. #328D251semidirect
160. C2×Q8C4×D251semidirect
184. C4⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
#306322002
#306316492
#95×D2512
#306314242
#306314222
#306314202
#328×C2512
#1504251