Finite Group Databasebeta

← #44550906

Subgroups — #44550906

Order 32128.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#30632249C211yesmaximal, characteristic
2#30632160C211yesmaximal, characteristic, complemented
3C2×#22548629C211yesmaximal, characteristic, complemented
4C2×#22548629C211yesmaximal, characteristic, complemented
5C2×#22548629C211yesmaximal, characteristic, complemented
6#30631453C211yesmaximal, characteristic, complemented
7#342×C251C211yesmaximal, characteristic, complemented
8C22×C22⋊D251C2211yescharacteristic
9C22×C22⋊D251C2211yescharacteristic
10#22548699—21no—
11#22548699C412yescomplemented
12#22548689—22no—
13#22548685—21no—
14#22548685C412yescomplemented
15C4×C22⋊D251—22no—
16(C22⋊D251)⋊C4—22no—
17#22548671—21no—
18C2×C22⋊Dic251C2211yescharacteristic
19C2×C22⋊Dic251C2211yescharacteristic
20C22⋊C4×D251—21no—
21#22548629—22no—
22#22548629—22no—
23#22548629—22no—
24C2×C22⋊C4×C251C2211yescharacteristic
25C2×C22⋊C4×C251C2211yescharacteristic
26C2×C22⋊C4×C251C2211yescharacteristic
27C23×D251C2×C411yescharacteristic, complemented
28C23×D251D411yescharacteristic
29C2×D1004—22no—
30C2×C22⋊D251—21no—
31C2×C22⋊D251—21no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251—21no—
34C2×C22⋊D251—21no—
35C2×C22⋊D251—21no—
36C2×C22⋊D251—21no—
37C2×C22⋊D251—21no—
38C2×C22⋊D251—21no—
39C2×C22⋊D251—21no—
40C2×C22⋊D251C2×C412yes—
41C2×C22⋊D251D412yes—
42C22×Dic251—21no—
43C22×Dic251—21no—
44C22×Dic251C2×C411yescharacteristic, complemented
45C22×Dic251D411yescharacteristic
46C2×C4×D251—22no—
47C4⋊Dic251—22no—
48Dic251⋊C4—22no—
49C22⋊Dic251—21no—
50C22⋊Dic251—21no—
51C22⋊Dic251—21no—
52C22⋊Dic251D412yescomplemented
53(C2×D251)⋊C4—22no—
54(C2×D251)⋊C4—22no—
55(C2×D251)⋊C4—22no—
56C4×Dic251—22no—
57C24×C251C2311yescharacteristic
58C22×C4×C251—21no—
59C22×C4×C251—21no—
60C22×C4×C251—21no—
61C22⋊C4×C251—21no—
62C22⋊C4×C251—21no—
63C22⋊C4×C251—21no—
64C22⋊C4×C251—21no—
65C22⋊C4×C251C2×C412yescomplemented
66C22⋊C4×C251D412yescomplemented
67C22×D251—21no—
68C22×D251—21no—
69C22×D251—21no—
70C22×D251—21no—
71C22×D251—21no—
72C22×D251—81no—
73C22×D251—81no—
74C22×D251C22⋊C412yes—
75D1004—82no—
76C22⋊D251—22no—
77C22⋊D251—22no—
78C22⋊D251—22no—
79C22⋊D251—22no—
80C22⋊D251—41no—
81C22⋊D251—41no—
82C22⋊D251—41no—
83C22⋊D251—41no—
84C2×Dic251—21no—
85C2×Dic251—21no—
86C2×Dic251—21no—
87C2×Dic251—21no—
88C2×Dic251—21no—
89C2×Dic251—22no—
90C2×Dic251—41no—
91C2×Dic251—81no—
92C2×Dic251—81no—
93C2×Dic251C22⋊C412yes—
94C4×D251—82no—
95C23×C251—81no—
96C23×C251C22×C411yescharacteristic
97C23×C251C2×D411yescharacteristic
98C23×C251C2×D411yescharacteristic
99C23×C251C2×D411yescharacteristic
100C23×C251C2×D411yescharacteristic
101C23×C251#4111yescharacteristic
102C23×C251#4111yescharacteristic
103C2×C4×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251—41no—
106C2×C4×C251—81no—
107C2×C4×C251—81no—
108C2×C4×C251—81no—
109C22×C251—41no—
110C22×C251—41no—
111C22×C251—41no—
112C22×C251—41no—
113C22×C251—41no—
114C22×C251—41no—
115C22×C251—41no—
116C22×C251C2×C22⋊C411yescharacteristic
117C22×C251C4×D411yescharacteristic
118C22×C251C4×D411yescharacteristic
119C22×C251#12011yescharacteristic
120C22×C251#12111yescharacteristic
121C22×C251#12111yescharacteristic
122C22×C251#12311yescharacteristic
123C2×D251—22no—
124C2×D251—42no—
125C2×D251—42no—
126C2×D251—42no—
127C2×D251—41no—
128C2×D251—81no—
129C2×D251—81no—
130C2×D251—81no—
131C4×C251—42no—
132C4×C251—42no—
133C4×C251—81no—
134Dic251—22no—
135Dic251—42no—
136Dic251—41no—
137Dic251—81no—
138C2×C251—21no—
139C2×C251—21no—
140C2×C251—21no—
141C2×C251—21no—
142C2×C251—21no—
143C2×C251—21no—
144C2×C251—21no—
145C2×C251#38611yescharacteristic
146D251—42no—
147D251—81no—
148C251#191311yescharacteristic, complemented, Sylow
149#1913—2511nomaximal, Sylow
150#534—2511no—
151#518—2511no—
152C2×#99—2511no—
153C2×#99—2511no—
154C2×#99—2511no—
155#342—2511no—
156#342D25111yescharacteristic, complemented
157C22×D4—2511no—
158C22×D4—2511no—
159#123—5021no—
160#123—2512no—
161#121—5022no—
162#120—5021no—
163#120—2512no—
164C4×D4—5022no—
165C4×D4—5022no—
166#117—5021no—
167C2×C22⋊C4—5021no—
168C2×C22⋊C4—2511no—
169C2×C22⋊C4—2511no—
170C2×C22⋊C4C2×D25111yescharacteristic
171C2×C22⋊C4C2×D25111yescharacteristic
172C2×C22⋊C4C2×D25111yescharacteristic
173#99—5022no—
174#99—5022no—
175#99—5022no—
176C24—2511no—
177C24—2511no—
178C24C22×D25111yescharacteristic
179C2×D4—5022no—
180C2×D4—5021no—
181C2×D4—5021no—
182C2×D4—5021no—
183C2×D4—5021no—
184C2×D4—5021no—
185C2×D4—5021no—
186C2×D4—5021no—
187C2×D4—5021no—
188C2×D4—5021no—
189C2×D4—5021no—
190C2×D4—2512no—
191C2×D4—2512no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—5022no—
196C22×C4—5021no—
197C22×C4—5021no—
198C22×C4—2511no—
199C22×C4—2511no—
200C4⋊C4—5022no—
201C4⋊C4—5022no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—5022no—
207C22⋊C4—5022no—
208C22⋊C4—5022no—
209C22⋊C4—5021no—
210C22⋊C4—5021no—
211C22⋊C4—5021no—
212C22⋊C4—2512no—
213C22⋊C4C4×D25112yescomplemented
214C22⋊C4D100412yescomplemented
215C42—5022no—
216C23—81no—
217C23—20081no—
218C23—20081no—
219C23—5021no—
220C23—5021no—
221C23—5021no—
222C23—5021no—
223C23—5021no—
224C23—2512no—
225C23C2×C4×D25111yescharacteristic
226C23D4×D25111yescharacteristic
227C23D4×D25111yescharacteristic
228C23D4×D25111yescharacteristic
229C23C2×D100411yescharacteristic
230C23#1686081011yescharacteristic
231C23#1686081111yescharacteristic
232D4—20082no—
233D4—10041no—
234D4—10041no—
235D4—10041no—
236D4—10041no—
237D4—5022no—
238D4—5022no—
239D4—5022no—
240D4—5022no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—20082no—
248C2×C4—20081no—
249C2×C4—20081no—
250C2×C4—10041no—
251C2×C4—5021no—
252C2×C4—5021no—
253C2×C4—5021no—
254C2×C4—5022no—
255C2×C4—5021no—
256C2×C4—5021no—
257C2×C4—2512no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—20081no—
266C22—20081no—
267C22—20081no—
268C22—10042no—
269C22—10042no—
270C22—10042no—
271C22—10041no—
272C22—5022no—
273C22C22⋊C4×D25111yescharacteristic
274C22#2254867711yescharacteristic
275C22C4×D100411yescharacteristic
276C22#2254868511yescharacteristic
277C22#2254868811yescharacteristic
278C22#2254869011yescharacteristic
279C22#2254870011yescharacteristic
280C4—42no—
281C4—42no—
282C4—81no—
283C4—20081no—
284C4—10042no—
285C4—10041no—
286C4—5022no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—20081no—
295C2—10042no—
296C2#3063156511yescharacteristic
subgroupG/H
centerC2#30631565
commutatorC23×C251C22×C4
FrattiniC24C22×D251
Fitting#342×C251C2
socleC2×C251#386
prime radical#30632249C2
cosocleC24×C251C23
hypercenter#342D251
hypocenterC251#1913
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #30632160C2semidirect
3. C2×#22548629C2semidirect
4. C2×#22548629C2semidirect
5. C2×#22548629C2semidirect
6. #30631453C2semidirect
7. #342×C251C2semidirect
11. #22548699C4semidirect
14. #22548685C4semidirect
27. C23×D251C2×C4semidirect
44. C22×Dic251C2×C4semidirect
52. C22⋊Dic251D4semidirect
65. C22⋊C4×C251C2×C4semidirect
66. C22⋊C4×C251D4semidirect
148. C251#1913semidirect
156. #342D251semidirect
213. C22⋊C4C4×D251semidirect
214. C22⋊C4D1004semidirect

Maximal subgroups

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

subgroupindex
#306322492
#306321602
C2×#225486292
#306314532
#342×C2512
#1913251