Finite Group Databasebeta

← #44550666

Subgroups — #44550666

Order 32128.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×M16×D251C211yesmaximal, characteristic, complemented
2#30632199C211yesmaximal, characteristic
3C2×#22548643C211yesmaximal, characteristic, complemented
4C2×#22548641C211yesmaximal, characteristic, complemented
5C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
6#30631446C211yesmaximal, characteristic, complemented
7#339×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×C4×D251C2211yescharacteristic
10C2×#16860794C2211yescharacteristic
11C2×#16860793—21no—
12M16×D251—22no—
13M16×D251—22no—
14C2×C8×D251—21no—
15#22548699—21no—
16#22548695C411yescharacteristic, complemented
17#22548695C411yescharacteristic, complemented
18#22548690C411yescharacteristic, complemented
19#22548690C411yescharacteristic, complemented
20#22548685—21no—
21C4×D1004—21no—
22C4×D1004—21no—
23C2×(C2×D251)⋊C4—21no—
24C2×C4×Dic251C2211yescharacteristic
25#22548643—22no—
26#22548641—22no—
27C251⋊#95—21no—
28C2×M16×C251C2211yescharacteristic
29#117×C251C2211yescharacteristic
30C2×C42×C251C2211yescharacteristic
31C23×D251D411yescharacteristic
32#16860810—21no—
33#16860810—21no—
34#16860810—21no—
35#16860810—21no—
36C2×Dic502C2×C411yescharacteristic
37C2×D1004—21no—
38C2×D1004—21no—
39C2×D1004C2×C411yescharacteristic
40C2×C22⋊D251—21no—
41C22×Dic251—21no—
42C22×Dic251D411yescharacteristic
43C2×C4×D251—21no—
44C2×C4×D251—21no—
45C2×C4×D251—21no—
46C2×C4×D251—21no—
47C2×C4×D251—21no—
48C2×C4×D251C2×C411yescharacteristic
49C2×C4×D251C2×C411yescharacteristic
50C2×C4×D251D411yescharacteristic
51C2×C4×D251D411yescharacteristic
52#16860794—21no—
53#16860794D412yescomplemented
54#16860793—24no—
55C2×C251⋊C8—21no—
56C8×D251—24no—
57C4⋊Dic251—21no—
58C4⋊Dic251—21no—
59(C2×D251)⋊C4—24no—
60(C2×D251)⋊C4—21no—
61(C2×D251)⋊C4—21no—
62(C2×D251)⋊C4—21no—
63(C2×D251)⋊C4—21no—
64C4×Dic251—21no—
65C4×Dic251—21no—
66C22×C4×C251—21no—
67C22×C4×C251C2311yescharacteristic
68M16×C251—21no—
69M16×C251D412yescomplemented
70C2×C8×C251—21no—
71C4⋊C4×C251C2×C411yescharacteristic
72C4⋊C4×C251C2×C411yescharacteristic
73C22⋊C4×C251—21no—
74C42×C251—21no—
75C42×C251—21no—
76C42×C251—21no—
77C22×D251—21no—
78C22×D251—21no—
79C22×D251—22no—
80C22×D251—21no—
81C22×D251—41no—
82C22×D251C22⋊C411yescharacteristic
83C22×D251C22⋊C411yescharacteristic
84Dic502—21no—
85Dic502—21no—
86D1004—21no—
87D1004—21no—
88D1004—42no—
89D1004—42no—
90C22⋊D251—41no—
91C22⋊D251—41no—
92C2×Dic251—21no—
93C2×Dic251—21no—
94C2×Dic251—22no—
95C2×Dic251—21no—
96C2×Dic251—41no—
97C2×Dic251—41no—
98C2×Dic251C22⋊C411yescharacteristic
99C2×Dic251C22⋊C411yescharacteristic
100C4×D251—21no—
101C4×D251—21no—
102C4×D251—21no—
103C4×D251—21no—
104C4×D251—21no—
105C4×D251—21no—
106C4×D251—21no—
107C4×D251—21no—
108C4×D251—41no—
109C4×D251—41no—
110C251⋊C8—22no—
111C23×C251C2×D411yescharacteristic
112C2×C4×C251—22no—
113C2×C4×C251—21no—
114C2×C4×C251—21no—
115C2×C4×C251—41no—
116C2×C4×C251—41no—
117C2×C4×C251C22×C411yescharacteristic
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2×D411yescharacteristic
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251#4111yescharacteristic
122C2×C4×C251#4111yescharacteristic
123C8×C251—22no—
124C22×C251—21no—
125C22×C251—21no—
126C22×C251C2×C22⋊C411yescharacteristic
127C22×C251#12111yescharacteristic
128C22×C251#12111yescharacteristic
129C2×D251—21no—
130C2×D251—21no—
131C2×D251—21no—
132C2×D251—21no—
133C2×D251—21no—
134C2×D251—42no—
135C2×D251—42no—
136C2×D251—41no—
137C2×D251—41no—
138C2×D251—41no—
139C2×D251#10412yescomplemented
140C4×C251—42no—
141C4×C251—41no—
142C4×C251—41no—
143C4×C251C4×D411yescharacteristic
144C4×C251C4×D411yescharacteristic
145C4×C251#12011yescharacteristic
146C4×C251#12311yescharacteristic
147Dic251—21no—
148Dic251—21no—
149Dic251—42no—
150Dic251—41no—
151C2×C251—21no—
152C2×C251—21no—
153C2×C251#38611yescharacteristic
154C2×C251C2×#10411yescharacteristic
155C2×C251#42111yescharacteristic
156D251—22no—
157D251—41no—
158D251—81no—
159C251#187111yescharacteristic, complemented, Sylow
160#1871—2511nomaximal, Sylow
161C22×M16—2511no—
162#525—2511no—
163C2×#104—2511no—
164C2×#104—2511no—
165C4×C22⋊C4—2511no—
166#339—2511no—
167#339D25111yescharacteristic, complemented
168C2×#41—2511no—
169C23×C4—2511no—
170C2×M16—5021no—
171C2×M16—5022no—
172C2×M16—5022no—
173C2×M16—2511no—
174C2×M16C2×D25111yescharacteristic
175C22×C8—5021no—
176#123—5021no—
177C22⋊Q8—2511no—
178C22⋊Q8—2511no—
179#121—2511no—
180#121—2511no—
181#120—5021no—
182C4×D4—5021no—
183C4×D4—5021no—
184#117C2×D25111yescharacteristic
185C2×C22⋊C4—5021no—
186C2×C42—2511no—
187C2×C42C2×D25111yescharacteristic
188#104—5022no—
189#104—5022no—
190#95—5021no—
191C24—2511no—
192#41—5021no—
193#41—5021no—
194#41—5021no—
195#41—5021no—
196C2×Q8—2511no—
197C2×D4—5021no—
198C2×D4—5021no—
199C2×D4—5021no—
200C2×D4—2511no—
201C22×C4—21no—
202C22×C4—5021no—
203C22×C4—5021no—
204C22×C4—5021no—
205C22×C4—5021no—
206C22×C4—5021no—
207C22×C4—5021no—
208C22×C4—2511no—
209C22×C4—2511no—
210C22×C4—2511no—
211C22×C4—2511no—
212C22×C4—2511no—
213C22×C4C22×D25111yescharacteristic
214M16—21no—
215M16—5024no—
216M16—5021no—
217M16—2512no—
218M16D100412yescomplemented
219C2×C8—21no—
220C2×C8—5024no—
221C2×C8—5021no—
222C4⋊C4—5021no—
223C4⋊C4—5021no—
224C4⋊C4C4×D25111yescharacteristic
225C4⋊C4C4×D25111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—5024no—
228C22⋊C4—5021no—
229C22⋊C4—5021no—
230C22⋊C4—5021no—
231C22⋊C4—5021no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—5021no—
236C42—5021no—
237C23—10041no—
238C23—5022no—
239C23—5021no—
240C23—5021no—
241C23—5021no—
242C23—2511no—
243C23—2511no—
244C23D4×D25111yescharacteristic
245Q8—5021no—
246Q8—5021no—
247D4—10042no—
248D4—10042no—
249D4—10041no—
250D4—10041no—
251D4—5021no—
252D4—5021no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—10041no—
259C2×C4—10041no—
260C2×C4—10041no—
261C2×C4—10041no—
262C2×C4—5022no—
263C2×C4—5021no—
264C2×C4—5021no—
265C2×C4—5021no—
266C2×C4—5021no—
267C2×C4—5021no—
268C2×C4—5021no—
269C2×C4—5021no—
270C2×C4—5021no—
271C2×C4—5021no—
272C2×C4—5021no—
273C2×C4—5021no—
274C2×C4—2511no—
275C2×C4—2511no—
276C2×C4C2×C4×D25111yescharacteristic
277C2×C4D4×D25111yescharacteristic
278C2×C4D4×D25111yescharacteristic
279C2×C4C2×D100411yescharacteristic
280C2×C4#1686081011yescharacteristic
281C2×C4#1686081111yescharacteristic
282C8—22no—
283C8—5022no—
284C22—21no—
285C22—21no—
286C22—10042no—
287C22—10042no—
288C22—10041no—
289C22—10041no—
290C22—10041no—
291C22—5021no—
292C22—5021no—
293C22—5021no—
294C22—5021no—
295C22—5021no—
296C22—2512no—
297C22C22⋊C4×D25111yescharacteristic
298C22#2254868811yescharacteristic
299C22#2254869011yescharacteristic
300C4—42no—
301C4—41no—
302C4—41no—
303C4—10042no—
304C4—10041no—
305C4—5021no—
306C4—5021no—
307C4#2254867711yescharacteristic
308C4C4×D100411yescharacteristic
309C4#2254868511yescharacteristic
310C4#2254870011yescharacteristic
311C2—21no—
312C2—21no—
313C2—20081no—
314C2—10041no—
315C2—5022no—
316C2#3063156511yescharacteristic
317C2#104×D25111yescharacteristic
318C2#3063170211yescharacteristic
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#1871
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
1. C2×M16×D251C2semidirect
3. C2×#22548643C2semidirect
4. C2×#22548641C2semidirect
5. C4×(C2×D251)⋊C4C2semidirect
6. #30631446C2semidirect
7. #339×C251C2semidirect
16. #22548695C4semidirect
17. #22548695C4semidirect
18. #22548690C4semidirect
19. #22548690C4semidirect
53. #16860794D4semidirect
69. M16×C251D4semidirect
139. C2×D251#104semidirect
159. C251#1871semidirect
167. #339D251semidirect
218. M16D1004semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2512
#306321992
C2×#225486432
C2×#225486412
C4×(C2×D251)⋊C42
#306314462
#339×C2512
#1871251