Finite Group Databasebeta

← #44548931

Subgroups — #44548931

Order 32128.

340 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 71 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#121×D251C211yesmaximal, characteristic
2C22⋊C8×D251C211yesmaximal, characteristic, complemented
3#30631546C211yesmaximal, characteristic, complemented
4#30631422C211yesmaximal, characteristic, complemented
5#30631421C211yesmaximal, characteristic, complemented
6#30631420C211yesmaximal, characteristic, complemented
7#327×C251C211yesmaximal, characteristic, complemented
8C2×D4×D251—21no—
9C2×D4×D251C411yescharacteristic, complemented
10C22×C4×D251C2211yescharacteristic
11C2×C8×D251—21no—
12#22548690C411yescharacteristic, complemented
13#22548689C2211yescharacteristic
14#22548688—21no—
15#22548687—21no—
16#22548686C411yescharacteristic, complemented
17C2×Dic251⋊C4—21no—
18C4⋊C4×D251C411yescharacteristic, complemented
19C2×(C2×D251)⋊C4—21no—
20C22⋊C4×D251—21no—
21C22⋊(C251⋊C8)C2211yescharacteristic, complemented
22(C2×D251)⋊C8—21no—
23C251⋊#95C2211yescharacteristic, complemented
24#121×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic, complemented
26#95×C251C2211yescharacteristic, complemented
27C23×D251—21no—
28C23×D251D411yescharacteristic
29C2×D1004—21no—
30C2×D1004C2×C411yescharacteristic
31C2×C22⋊D251—21no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251—21no—
34D4×D251—42no—
35D4×D251—42no—
36D4×D251—42no—
37C22×Dic251—21no—
38C22×Dic251D411yescharacteristic
39C2×C4×D251—21no—
40C2×C4×D251—42no—
41C2×C4×D251C2×C411yescharacteristic
42C2×C4×D251C2×C411yescharacteristic
43C2×C4×D251D411yescharacteristic
44C2×C4×D251D411yescharacteristic
45C2×C251⋊C8—21no—
46C8×D251—42no—
47C4⋊Dic251C2×C411yescharacteristic
48Dic251⋊C4—21no—
49Dic251⋊C4—42no—
50C22⋊Dic251—21no—
51(C2×D251)⋊C4—21no—
52(C2×D251)⋊C4—21no—
53(C2×D251)⋊C4—42no—
54C2×D4×C251—21no—
55C2×D4×C251C2×C411yescharacteristic
56C22×C4×C251—21no—
57C22×C4×C251C2311yescharacteristic
58C2×C8×C251—21no—
59C4⋊C4×C251C2×C411yescharacteristic
60C22⋊C4×C251—21no—
61C22×D251—21no—
62C22×D251—41no—
63C22×D251—41no—
64C22×D251—42no—
65C22×D251—42no—
66C22×D251—42no—
67C22×D251C22⋊C411yescharacteristic
68C22×D251C22⋊C411yescharacteristic
69C22×D251D812yes—
70C22×D251SD1612yes—
71D1004—41no—
72D1004—42no—
73C22⋊D251—41no—
74C22⋊D251—41no—
75C22⋊D251—42no—
76C22⋊D251—42no—
77C2×Dic251—22no—
78C2×Dic251—21no—
79C2×Dic251—41no—
80C2×Dic251—41no—
81C2×Dic251—41no—
82C2×Dic251C22⋊C411yescharacteristic
83C2×Dic251C22⋊C411yescharacteristic
84C4×D251—22no—
85C4×D251—22no—
86C4×D251—41no—
87C4×D251—41no—
88C4×D251—42no—
89C251⋊C8—41no—
90C23×C251—21no—
91C23×C251C2×D411yescharacteristic
92D4×C251—41no—
93D4×C251—41no—
94D4×C251—41no—
95C2×C4×C251—21no—
96C2×C4×C251—41no—
97C2×C4×C251—41no—
98C2×C4×C251—41no—
99C2×C4×C251—41no—
100C2×C4×C251C22×C411yescharacteristic
101C2×C4×C251C2×D411yescharacteristic
102C8×C251—41no—
103C22×C251—21no—
104C22×C251—21no—
105C22×C251—41no—
106C22×C251—41no—
107C22×C251—41no—
108C22×C251C2×C22⋊C411yescharacteristic
109C22×C251C2×D811yescharacteristic
110C22×C251C2×SD1611yescharacteristic
111C2×D251—21no—
112C2×D251—21no—
113C2×D251—21no—
114C2×D251—21no—
115C2×D251—42no—
116C2×D251—42no—
117C2×D251—41no—
118C2×D251—41no—
119C2×D251—41no—
120C2×D251—84no—
121C2×D251#10212yes—
122C2×D251#10212yes—
123C4×C251—21no—
124C4×C251—21no—
125C4×C251—41no—
126C4×C251—81no—
127Dic251—21no—
128Dic251—41no—
129Dic251—81no—
130Dic251#9912yes—
131C2×C251—21no—
132C2×C251—21no—
133C2×C251—81no—
134C2×C251C2×#9911yescharacteristic
135C2×C251C2×#10211yescharacteristic
136C2×C251#42111yescharacteristic
137D251—22no—
138D251—22no—
139D251—81no—
140C251#151911yescharacteristic, complemented, Sylow
141#1519—2511nomaximal, Sylow
142C2×#121—2511no—
143C2×C22⋊C8—2511no—
144#380—2511no—
145#327—2511no—
146#327—2511no—
147#327—2511no—
148#327D25111yescharacteristic, complemented
149C22×D4—5021no—
150C22×D4—2511no—
151C23×C4—2511no—
152C22×C8—5021no—
153#121—5021no—
154#121—5021no—
155#121—2511no—
156#121—2511no—
157#121—2511no—
158#121C2×D25111yescharacteristic
159C2×C4⋊C4—5021no—
160C2×C4⋊C4—2511no—
161C2×C22⋊C4—5021no—
162C2×C22⋊C4—5021no—
163C22⋊C8—5021no—
164C22⋊C8—2511no—
165C22⋊C8C2×D25111yescharacteristic, complemented
166#95—2511no—
167#95C2×D25111yescharacteristic, complemented
168C24—5021no—
169C24—2511no—
170C2×D4—21no—
171C2×D4—10042no—
172C2×D4—10042no—
173C2×D4—10042no—
174C2×D4—5021no—
175C2×D4—5021no—
176C2×D4—5021no—
177C2×D4—5021no—
178C2×D4—2511no—
179C2×D4C4×D25111yescharacteristic
180C22×C4—21no—
181C22×C4—10042no—
182C22×C4—5021no—
183C22×C4—5021no—
184C22×C4—2511no—
185C22×C4—2511no—
186C22×C4—2511no—
187C22×C4—2511no—
188C22×C4—2511no—
189C22×C4C22×D25111yescharacteristic
190C2×C8—21no—
191C2×C8—10042no—
192C2×C8—5021no—
193C4⋊C4—10042no—
194C4⋊C4—5021no—
195C4⋊C4—2511no—
196C4⋊C4C4×D25111yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—10042no—
199C22⋊C4—5021no—
200C22⋊C4—5021no—
201C22⋊C4—5021no—
202C23—21no—
203C23—10041no—
204C23—10041no—
205C23—10042no—
206C23—10042no—
207C23—10042no—
208C23—5021no—
209C23—2512no—
210C23—2512no—
211C23—2511no—
212C23—2511no—
213C23D4×D25111yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—10041no—
218D4—10042no—
219D4—10042no—
220D4—10041no—
221D4—10041no—
222D4—10042no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—10041no—
229C2×C4—10041no—
230C2×C4—10042no—
231C2×C4—10041no—
232C2×C4—10041no—
233C2×C4—10041no—
234C2×C4—5022no—
235C2×C4—5022no—
236C2×C4—5021no—
237C2×C4—5022no—
238C2×C4—2511no—
239C2×C4—2511no—
240C2×C4C2×C4×D25111yescharacteristic
241C2×C4D4×D25111yescharacteristic
242C8—41no—
243C8—10041no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—20084no—
250C22—10042no—
251C22—10042no—
252C22—10041no—
253C22—10041no—
254C22—10041no—
255C22—5021no—
256C22—5021no—
257C22—5021no—
258C22—5021no—
259C22—2512no—
260C22—2512no—
261C22C22⋊C4×D25111yescharacteristic
262C22D8×D25111yescharacteristic
263C22SD16×D25111yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—81no—
268C4—20081no—
269C4—10041no—
270C4—5021no—
271C4—2512no—
272C2—21no—
273C2—21no—
274C2—81no—
275C2—20081no—
276C2—5022no—
277C2—5022no—
278C2#99×D25111yescharacteristic
279C2#102×D25111yescharacteristic
280C2#3063170211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#327×C251C2
socleC22×C251C2×C22⋊C4
prime radical#121×D251C2
cosocleC22×C4×C251C23
hypercenter#327D251
hypocenterC251#1519
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D251C2semidirect
3. #30631546C2semidirect
4. #30631422C2semidirect
5. #30631421C2semidirect
6. #30631420C2semidirect
7. #327×C251C2semidirect
9. C2×D4×D251C4semidirect
12. #22548690C4semidirect
16. #22548686C4semidirect
18. C4⋊C4×D251C4semidirect
21. C22⋊(C251⋊C8)C22semidirect
23. C251⋊#95C22semidirect
25. C22⋊C8×C251C22semidirect
26. #95×C251C22semidirect
140. C251#1519semidirect
148. #327D251semidirect
165. C22⋊C8C2×D251semidirect
167. #95C2×D251semidirect

Maximal subgroups

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

subgroupindex
#121×D2512
C22⋊C8×D2512
#306315462
#306314222
#306314212
#306314202
#327×C2512
#1519251