Finite Group Databasebeta

← #44559998

Subgroups — #44559998

Order 32128.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×#22548745C211yesmaximal, characteristic, complemented
2C2×M16×D251C211yesmaximal, characteristic, complemented
3#30632194C211yesmaximal, characteristic
4C2×#22548695C211yesmaximal, characteristic, complemented
5#30631944C212yesmaximal, complemented
6#30631932C212yesmaximal, complemented
7#30631927C212yesmaximal, complemented
8#30631915C212yesmaximal, complemented
9#30631727C211yesmaximal, characteristic, complemented
10#30631697C211yesmaximal, characteristic, complemented
11#417×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×C4×D251C2211yescharacteristic
14#22548745—24no—
15C2×#16860800C2212yescomplemented
16C2×#16860798C2212yescomplemented
17C2×#16860794C2211yescharacteristic
18C2×#16860793C2212yes—
19M16×D251—24no—
20C2×C8×D251C2212yes—
21#22548696—21no—
22#22548695—21no—
23#22548695—21no—
24#22548695C2212yes—
25#22548695C2212yes—
26#22548691C2211yescharacteristic
27#22548691C2211yescharacteristic
28#22548686C2211yescharacteristic
29#22548686C2211yescharacteristic
30#22548683—21no—
31C4×C22⋊D251—21no—
32C4×C22⋊D251—21no—
33#22548674C2211yescharacteristic
34C2×C4⋊Dic251—21no—
35C2×(C2×D251)⋊C4—21no—
36#22548648C2212yescomplemented
37#22548646C2212yescomplemented
38#22548640C2212yescomplemented
39#22548640C2212yescomplemented
40C2×#41×C251C2211yescharacteristic
41C2×M16×C251C2211yescharacteristic
42C2×C4⋊C4×C251C2211yescharacteristic
43#103×C251C2212yescomplemented
44#102×C251C2212yescomplemented
45C23×D251D411yescharacteristic
46C2×Dic502—21no—
47C2×Dic502—22no—
48C2×Dic502—22no—
49C2×Dic502C2312yes—
50C2×C22⋊D251—21no—
51C2×C22⋊D251—21no—
52C22×Dic251—21no—
53C22×Dic251D411yescharacteristic
54C2×C4×D251—22no—
55C2×C4×D251—22no—
56C2×C4×D251D411yescharacteristic
57C2×C4×D251D411yescharacteristic
58C2×C4×D251C2311yescharacteristic
59C2×C4×D251C2311yescharacteristic
60#16860800—24no—
61#16860798—24no—
62#16860794D414yescomplemented
63#16860793—24no—
64C2×C251⋊C8C2312yes—
65C8×D251—24no—
66C4⋊Dic251—22no—
67C4⋊Dic251—22no—
68C4⋊Dic251C2311yescharacteristic
69C4⋊Dic251C2311yescharacteristic
70Dic251⋊C4—21no—
71Dic251⋊C4—21no—
72C22⋊Dic251—21no—
73C22⋊Dic251—21no—
74C22⋊Dic251—21no—
75(C2×D251)⋊C4—22no—
76(C2×D251)⋊C4—22no—
77(C2×D251)⋊C4—21no—
78(C2×D251)⋊C4—21no—
79C4×Dic251—21no—
80#41×C251—22no—
81#41×C251—22no—
82C2×Q8×C251C2311yescharacteristic
83C2×D4×C251—21no—
84C2×D4×C251C2311yescharacteristic
85C22×C4×C251—21no—
86C22×C4×C251—21no—
87C22×C4×C251C2311yescharacteristic
88M16×C251—22no—
89C2×C8×C251C2312yes—
90C4⋊C4×C251—21no—
91C4⋊C4×C251C2312yes—
92C22×D251—22no—
93C22×D251—22no—
94C22×D251—41no—
95C22×D251C2×D411yescharacteristic
96C22×D251C2×D411yescharacteristic
97Dic502—22no—
98Dic502—22no—
99Dic502—42no—
100C22⋊D251—42no—
101C22⋊D251—42no—
102C2×Dic251—22no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—22no—
106C2×Dic251—42no—
107C2×Dic251C2×D411yescharacteristic
108C2×Dic251C2×D411yescharacteristic
109C4×D251—22no—
110C4×D251—22no—
111C4×D251—21no—
112C4×D251—21no—
113C4×D251—21no—
114C4×D251—21no—
115C251⋊C8C2×D414yes—
116C23×C251—21no—
117C23×C251C2×D411yescharacteristic
118Q8×C251—21no—
119Q8×C251—21no—
120D4×C251—21no—
121D4×C251—21no—
122D4×C251—42no—
123C2×C4×C251—22no—
124C2×C4×C251—21no—
125C2×C4×C251—42no—
126C2×C4×C251—42no—
127C2×C4×C251C2×D411yescharacteristic
128C2×C4×C251C2×D412yes—
129C2×C4×C251#4112yes—
130C2×C4×C251C2411yescharacteristic
131C8×C251—22no—
132C22×C251—21no—
133C22×C251—21no—
134C22×C251—41no—
135C22×C251—41no—
136C22×C251—41no—
137C22×C251#12112yes—
138C22×C251C22×D411yescharacteristic
139C2×D251—21no—
140C2×D251—21no—
141C2×D251—21no—
142C2×D251—21no—
143C2×D251—21no—
144C2×D251—44no—
145C2×D251#13712yescomplemented
146C4×C251—41no—
147C4×C251—42no—
148C4×C251#12112yes—
149C4×C251C22×D411yescharacteristic
150C4×C251C2×#4111yescharacteristic
151Dic251—21no—
152Dic251—21no—
153Dic251—42no—
154Dic251—41no—
155Dic251—41no—
156C2×C251—22no—
157C2×C251—81no—
158C2×C251C2×#12111yescharacteristic
159C2×C251C2×#13711yescharacteristic
160C2×C251#57511yescharacteristic
161D251—22no—
162D251—41no—
163C251#305611yescharacteristic, complemented, Sylow
164#3056—2511nomaximal, Sylow
165C2×#137—2511no—
166C22×M16—2511no—
167#525—2511no—
168C2×C22⋊Q8—2511no—
169#470—2512no—
170#468—2512no—
171#467—2512no—
172#465—2512no—
173#428—2511no—
174#419—2511no—
175#417D25111yescharacteristic, complemented
176C2×#41C2×D25111yescharacteristic
177C22×Q8—2511no—
178C23×C4—2511no—
179#137—5024no—
180C2×Q16—2512no—
181C2×SD16—2512no—
182C2×M16—5024no—
183C2×M16—2511no—
184C2×M16—2512no—
185C2×M16C2×D25111yescharacteristic
186C22×C8—2512no—
187#123—5021no—
188C22⋊Q8—5021no—
189C22⋊Q8—5021no—
190C22⋊Q8—2511no—
191C22⋊Q8—2511no—
192C22⋊Q8—2512no—
193C22⋊Q8—2512no—
194#121—2511no—
195#121—2511no—
196#120—5021no—
197C4×D4—5021no—
198C4×D4—5021no—
199#117—2511no—
200C2×C4⋊C4—5021no—
201C2×C4⋊C4C2×D25111yescharacteristic
202C2×C22⋊C4—5021no—
203#107—2512no—
204#106—2512no—
205#103—2512no—
206#103—2512no—
207#103C2×D25112yescomplemented
208#102C2×D25112yescomplemented
209C24—2511no—
210#41—22no—
211#41—22no—
212C2×Q8—5022no—
213C2×Q8—5022no—
214C2×Q8—5021no—
215C2×Q8—2512no—
216C2×Q8C22×D25111yescharacteristic
217C2×D4—21no—
218C2×D4—5021no—
219C2×D4—5021no—
220C2×D4C22×D25111yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—5021no—
224C22×C4—5022no—
225C22×C4—5022no—
226C22×C4—2511no—
227C22×C4—2511no—
228C22×C4—2511no—
229C22×C4—2511no—
230C22×C4—2511no—
231C22×C4C22×D25111yescharacteristic
232Q16—5024no—
233SD16—5024no—
234M16—22no—
235M16—5024no—
236M16—2514no—
237C2×C8—5024no—
238C2×C8—2512no—
239C2×C8C22×D25112yes—
240C4⋊C4—21no—
241C4⋊C4—5022no—
242C4⋊C4—5022no—
243C4⋊C4—5021no—
244C4⋊C4—5021no—
245C4⋊C4—2511no—
246C4⋊C4—2511no—
247C4⋊C4C22×D25112yes—
248C22⋊C4—5022no—
249C22⋊C4—5022no—
250C22⋊C4—5021no—
251C22⋊C4—5021no—
252C22⋊C4—5021no—
253C22⋊C4—5021no—
254C22⋊C4—5021no—
255C42—5021no—
256C23—21no—
257C23—10041no—
258C23—5022no—
259C23—5022no—
260C23—2511no—
261C23—2511no—
262C23D4×D25111yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—10042no—
266Q8—5022no—
267Q8—5022no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—10042no—
272D4—10042no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—10042no—
278C2×C4—5021no—
279C2×C4—5021no—
280C2×C4—5022no—
281C2×C4—5022no—
282C2×C4—5021no—
283C2×C4—5021no—
284C2×C4—5021no—
285C2×C4—5021no—
286C2×C4—5022no—
287C2×C4—5022no—
288C2×C4—2511no—
289C2×C4—2511no—
290C2×C4D4×D25112yes—
291C2×C4D4×D25111yescharacteristic
292C2×C4#1686081012yes—
293C2×C4C23×D25111yescharacteristic
294C8—22no—
295C8—2514no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—10044no—
302C22—5021no—
303C22—5021no—
304C22—5021no—
305C22—5021no—
306C22—5021no—
307C22—2512no—
308C22#2254868812yes—
309C22C2×D4×D25111yescharacteristic
310C4—42no—
311C4—41no—
312C4—10041no—
313C4—10041no—
314C4—10042no—
315C4—5021no—
316C4—5021no—
317C4#2254868812yes—
318C4C2×D4×D25111yescharacteristic
319C4C2×#1686081011yescharacteristic
320C2—22no—
321C2—81no—
322C2—10041no—
323C2—5022no—
324C2C2×#2254868811yescharacteristic
325C2#137×D25111yescharacteristic
326C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#417×C251C2
socleC22×C251C22×D4
prime radical#30632194C2
hypercenter#417D251
hypocenterC251#3056
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22548745C2semidirect
2. C2×M16×D251C2semidirect
4. C2×#22548695C2semidirect
5. #30631944C2semidirect
6. #30631932C2semidirect
7. #30631927C2semidirect
8. #30631915C2semidirect
9. #30631727C2semidirect
10. #30631697C2semidirect
11. #417×C251C2semidirect
15. C2×#16860800C22semidirect
16. C2×#16860798C22semidirect
36. #22548648C22semidirect
37. #22548646C22semidirect
38. #22548640C22semidirect
39. #22548640C22semidirect
43. #103×C251C22semidirect
44. #102×C251C22semidirect
62. #16860794D4semidirect
145. C2×D251#137semidirect
163. C251#3056semidirect
175. #417D251semidirect
207. #103C2×D251semidirect
208. #102C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487452
C2×M16×D2512
#306321942
C2×#225486952
#306319442
#306319322
#306319272
#306319152
#306317272
#306316972
#417×C2512
#3056251