Finite Group Databasebeta

← #44559985

Subgroups — #44559985

Order 32128.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#22548743C211yesmaximal, characteristic, complemented
2C2×M16×D251C211yesmaximal, characteristic, complemented
3#30632199C211yesmaximal, characteristic, complemented
4C2×#22548690C211yesmaximal, characteristic, complemented
5#30631941C212yesmaximal, complemented
6#30631934C212yesmaximal, complemented
7#30631924C212yesmaximal, complemented
8#30631917C212yesmaximal, complemented
9#30631691C211yesmaximal, characteristic, complemented
10#30631686C211yesmaximal, characteristic, complemented
11#428×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×D1004—21no—
14C22×D1004C2211yescharacteristic
15C22×C4×D251C2211yescharacteristic
16#22548743—24no—
17C2×#16860799C2212yescomplemented
18C2×D2008C2212yescomplemented
19C2×#16860794C2211yescharacteristic
20C2×#16860793C2212yescomplemented
21M16×D251—24no—
22C2×C8×D251C2212yescomplemented
23#22548699—21no—
24#22548695C2211yescharacteristic, complemented
25#22548695C2211yescharacteristic, complemented
26#22548690—21no—
27#22548690—21no—
28#22548690C2212yes—
29#22548690C2212yes—
30#22548690C2211yescharacteristic, complemented
31#22548690C2211yescharacteristic, complemented
32#22548685—21no—
33C4×D1004—21no—
34C4×D1004—21no—
35C2×(C2×D251)⋊C4—21no—
36#22548638C2212yescomplemented
37#22548635C2212yescomplemented
38#22548635C2212yescomplemented
39#22548635C2212yescomplemented
40C2×M16×C251C2211yescharacteristic
41#117×C251C2211yescharacteristic
42C2×C4⋊C4×C251C2211yescharacteristic
43#107×C251C2212yescomplemented
44#106×C251C2212yescomplemented
45C23×D251—21no—
46C23×D251D411yescharacteristic
47#16860810—22no—
48#16860810—22no—
49C2×Dic502C2311yescharacteristic
50C2×D1004—22no—
51C2×D1004—22no—
52C2×D1004—21no—
53C2×D1004—21no—
54C2×D1004—22no—
55C2×D1004—21no—
56C2×D1004—22no—
57C2×D1004—44no—
58C2×D1004C2312yes—
59C2×D1004C2311yescharacteristic
60C2×C22⋊D251—21no—
61C22×Dic251D411yescharacteristic
62C2×C4×D251—22no—
63C2×C4×D251—21no—
64C2×C4×D251—22no—
65C2×C4×D251D411yescharacteristic
66C2×C4×D251D411yescharacteristic
67C2×C4×D251C2311yescharacteristic
68C2×C4×D251C2311yescharacteristic
69#16860799—24no—
70D2008—24no—
71#16860794—22no—
72#16860793—24no—
73C2×C251⋊C8C2312yes—
74C8×D251—24no—
75C4⋊Dic251—21no—
76C4⋊Dic251—21no—
77(C2×D251)⋊C4—21no—
78(C2×D251)⋊C4—21no—
79(C2×D251)⋊C4—22no—
80(C2×D251)⋊C4—22no—
81(C2×D251)⋊C4—21no—
82(C2×D251)⋊C4—21no—
83C22×C4×C251—21no—
84C22×C4×C251C2311yescharacteristic
85M16×C251D414yescomplemented
86C2×C8×C251C2312yes—
87C4⋊C4×C251—21no—
88C4⋊C4×C251C2311yescharacteristic
89C4⋊C4×C251C2312yes—
90C4⋊C4×C251C2311yescharacteristic
91C22⋊C4×C251—21no—
92C42×C251—21no—
93C22×D251—22no—
94C22×D251—22no—
95C22×D251—21no—
96C22×D251—22no—
97C22×D251—41no—
98C22×D251—42no—
99C22×D251—42no—
100C22×D251—42no—
101C22×D251C2×D411yescharacteristic
102C22×D251C2×D411yescharacteristic
103Dic502—21no—
104Dic502—21no—
105D1004—21no—
106D1004—22no—
107D1004—22no—
108D1004—21no—
109D1004—42no—
110D1004—44no—
111D1004—42no—
112D1004—44no—
113D1004—42no—
114C22⋊D251—42no—
115C2×Dic251—22no—
116C2×Dic251—21no—
117C2×Dic251C2×D411yescharacteristic
118C2×Dic251C2×D411yescharacteristic
119C4×D251—22no—
120C4×D251—22no—
121C4×D251—21no—
122C4×D251—21no—
123C4×D251—21no—
124C4×D251—21no—
125C4×D251—42no—
126C251⋊C8—22no—
127C23×C251C2×D411yescharacteristic
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—22no—
131C2×C4×C251—42no—
132C2×C4×C251C2×D411yescharacteristic
133C2×C4×C251C2×D412yes—
134C2×C4×C251#4112yes—
135C2×C4×C251C2411yescharacteristic
136C8×C251C2×D414yescomplemented
137C22×C251—21no—
138C22×C251—21no—
139C22×C251#12112yes—
140C22×C251C22×D411yescharacteristic
141C2×D251—21no—
142C2×D251—21no—
143C2×D251—21no—
144C2×D251—21no—
145C2×D251—21no—
146C2×D251—44no—
147C2×D251—41no—
148C2×D251—42no—
149C2×D251—42no—
150C2×D251—41no—
151C2×D251—41no—
152C2×D251—42no—
153C2×D251—84no—
154C2×D251AGL(1,ℤ/8)12yescomplemented
155C4×C251—42no—
156C4×C251—41no—
157C4×C251—41no—
158C4×C251#12112yes—
159C4×C251C22×D411yescharacteristic
160C4×C251C2×#4111yescharacteristic
161Dic251—21no—
162Dic251—21no—
163Dic251—41no—
164C2×C251—22no—
165C2×C251C2×#12111yescharacteristic
166C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C251#57511yescharacteristic
168D251—22no—
169D251—41no—
170D251—81no—
171D251—82no—
172C251#305511yescharacteristic, complemented, Sylow
173#3055—2511nomaximal, Sylow
174C2×AGL(1,ℤ/8)—2511no—
175C22×M16—2511no—
176#525—2511no—
177C2×#121—2511no—
178#469—2512no—
179#468—2512no—
180#466—2512no—
181#465—2512no—
182#428D25111yescharacteristic, complemented
183#418—2511no—
184#417—2511no—
185C2×#41—2511no—
186C22×D4—5021no—
187C22×D4—2511no—
188C23×C4—2511no—
189AGL(1,ℤ/8)—5024no—
190C2×SD16—2512no—
191C2×D8—2512no—
192C2×M16—5024no—
193C2×M16—2511no—
194C2×M16—2512no—
195C2×M16C2×D25111yescharacteristic
196C22×C8—2512no—
197#123—5021no—
198C22⋊Q8—2511no—
199C22⋊Q8—2511no—
200#121—5021no—
201#121—5021no—
202#121—2512no—
203#121—2512no—
204#121—2511no—
205#121—2511no—
206#120—5021no—
207C4×D4—5021no—
208C4×D4—5021no—
209#117C2×D25111yescharacteristic
210C2×C4⋊C4C2×D25111yescharacteristic
211C2×C22⋊C4—5021no—
212#107C2×D25112yescomplemented
213#106C2×D25112yescomplemented
214#103—2512no—
215#102—2512no—
216#102—2512no—
217#102—2512no—
218C24—5021no—
219C24—2511no—
220#41—5022no—
221#41—5022no—
222C2×Q8—2511no—
223C2×D4—10044no—
224C2×D4—5021no—
225C2×D4—5021no—
226C2×D4—5022no—
227C2×D4—5022no—
228C2×D4—5021no—
229C2×D4—5021no—
230C2×D4—5022no—
231C2×D4—5022no—
232C2×D4—2511no—
233C2×D4—2512no—
234C22×C4—21no—
235C22×C4—5021no—
236C22×C4—5022no—
237C22×C4—5022no—
238C22×C4—2511no—
239C22×C4—2511no—
240C22×C4—2511no—
241C22×C4—2511no—
242C22×C4—2511no—
243C22×C4C22×D25111yescharacteristic
244SD16—5024no—
245D8—5024no—
246M16—5022no—
247M16—5024no—
248M16D100414yescomplemented
249C2×C8—5024no—
250C2×C8—2512no—
251C2×C8C22×D25112yes—
252C4⋊C4—21no—
253C4⋊C4—5021no—
254C4⋊C4—5021no—
255C4⋊C4C22×D25111yescharacteristic
256C4⋊C4C22×D25111yescharacteristic
257C4⋊C4C22×D25112yes—
258C22⋊C4—21no—
259C22⋊C4—5021no—
260C22⋊C4—5021no—
261C22⋊C4—5022no—
262C22⋊C4—5022no—
263C22⋊C4—5021no—
264C22⋊C4—5021no—
265C42—21no—
266C23—10042no—
267C23—10042no—
268C23—10041no—
269C23—10042no—
270C23—5022no—
271C23—5021no—
272C23—5022no—
273C23—5022no—
274C23—2511no—
275C23—2511no—
276C23D4×D25111yescharacteristic
277Q8—5021no—
278Q8—5021no—
279D4—10044no—
280D4—10042no—
281D4—10044no—
282D4—10042no—
283D4—10042no—
284D4—10042no—
285D4—5021no—
286D4—5022no—
287D4—5022no—
288D4—5021no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—10042no—
294C2×C4—5021no—
295C2×C4—5022no—
296C2×C4—5021no—
297C2×C4—5021no—
298C2×C4—5021no—
299C2×C4—5021no—
300C2×C4—5022no—
301C2×C4—5022no—
302C2×C4—2511no—
303C2×C4—2511no—
304C2×C4D4×D25111yescharacteristic
305C2×C4C2×D100412yes—
306C2×C4#1686081112yes—
307C2×C4C23×D25111yescharacteristic
308C8—5022no—
309C8C2×D100414yescomplemented
310C22—21no—
311C22—21no—
312C22—20084no—
313C22—10044no—
314C22—10042no—
315C22—10041no—
316C22—10041no—
317C22—10042no—
318C22—10041no—
319C22—10042no—
320C22—5021no—
321C22—5021no—
322C22—5021no—
323C22—5021no—
324C22—5021no—
325C22—2512no—
326C22#2254869012yes—
327C22C2×D4×D25111yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—10041no—
332C4—5021no—
333C4—5021no—
334C4#2254869012yes—
335C4C22×D100411yescharacteristic
336C4C2×#1686081111yescharacteristic
337C2—22no—
338C2—20082no—
339C2—20081no—
340C2—10041no—
341C2—5022no—
342C2C2×#2254869011yescharacteristic
343C2AGL(1,ℤ/8)×D25111yescharacteristic
344C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
hypercenter#428D251
hypocenterC251#3055
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#22548743C2semidirect
2. C2×M16×D251C2semidirect
3. #30632199C2semidirect
4. C2×#22548690C2semidirect
5. #30631941C2semidirect
6. #30631934C2semidirect
7. #30631924C2semidirect
8. #30631917C2semidirect
9. #30631691C2semidirect
10. #30631686C2semidirect
11. #428×C251C2semidirect
17. C2×#16860799C22semidirect
18. C2×D2008C22semidirect
20. C2×#16860793C22semidirect
22. C2×C8×D251C22semidirect
24. #22548695C22semidirect
25. #22548695C22semidirect
30. #22548690C22semidirect
31. #22548690C22semidirect
36. #22548638C22semidirect
37. #22548635C22semidirect
38. #22548635C22semidirect
39. #22548635C22semidirect
43. #107×C251C22semidirect
44. #106×C251C22semidirect
85. M16×C251D4semidirect
136. C8×C251C2×D4semidirect
154. C2×D251AGL(1,ℤ/8)semidirect
172. C251#3055semidirect
182. #428D251semidirect
212. #107C2×D251semidirect
213. #106C2×D251semidirect
248. M16D1004semidirect
309. C8C2×D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
C2×M16×D2512
#306321992
C2×#225486902
#306319412
#306319342
#306319242
#306319172
#306316912
#306316862
#428×C2512
#3055251