Finite Group Databasebeta

← #44559996

Subgroups — #44559996

Order 32128.

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

normal subgroups only

This group has 496 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×#22548748C211yesmaximal, characteristic, complemented
2C2×M16×D251C211yesmaximal, characteristic, complemented
3#30632199C211yesmaximal, characteristic
4C2×#22548695C211yesmaximal, characteristic, complemented
5#30631946C212yesmaximal, complemented
6#30631934C212yesmaximal, complemented
7#30631929C212yesmaximal, complemented
8#30631917C212yesmaximal, complemented
9#30631695C211yesmaximal, characteristic, complemented
10#30631686C211yesmaximal, characteristic, complemented
11#428×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C22×Dic502C2211yescharacteristic
14C22×C4×D251C2211yescharacteristic
15#22548748—24no—
16C2×Dic1004C2212yescomplemented
17C2×#16860799C2212yescomplemented
18C2×#16860794C2211yescharacteristic
19C2×#16860793C2212yes—
20M16×D251—24no—
21C2×C8×D251C2212yes—
22#22548699—21no—
23#22548695—21no—
24#22548695—21no—
25#22548695C2211yescharacteristic
26#22548695C2211yescharacteristic
27#22548695C2212yes—
28#22548695C2212yes—
29#22548690C2211yescharacteristic
30#22548690C2211yescharacteristic
31#22548685—21no—
32C4×D1004—21no—
33C4×D1004—21no—
34C2×C4⋊Dic251—21no—
35C2×(C2×D251)⋊C4—21no—
36#22548638C2212yescomplemented
37#22548638C2212yescomplemented
38#22548638C2212yescomplemented
39#22548635C2212yescomplemented
40C2×M16×C251C2211yescharacteristic
41#117×C251C2211yescharacteristic
42C2×C4⋊C4×C251C2211yescharacteristic
43#107×C251C2212yescomplemented
44#106×C251C2212yescomplemented
45C23×D251D411yescharacteristic
46#16860810—22no—
47#16860810—22no—
48C2×Dic502—21no—
49C2×Dic502—22no—
50C2×Dic502—22no—
51C2×Dic502C2312yes—
52C2×Dic502C2311yescharacteristic
53C2×D1004—21no—
54C2×D1004—21no—
55C2×D1004C2311yescharacteristic
56C2×C22⋊D251—21no—
57C22×Dic251—21no—
58C22×Dic251D411yescharacteristic
59C2×C4×D251—22no—
60C2×C4×D251—21no—
61C2×C4×D251—22no—
62C2×C4×D251D411yescharacteristic
63C2×C4×D251D411yescharacteristic
64C2×C4×D251C2311yescharacteristic
65C2×C4×D251C2311yescharacteristic
66Dic1004—24no—
67#16860799—24no—
68#16860794—22no—
69#16860793—24no—
70C2×C251⋊C8C2312yes—
71C8×D251—24no—
72C4⋊Dic251—22no—
73C4⋊Dic251—22no—
74C4⋊Dic251—21no—
75C4⋊Dic251—21no—
76(C2×D251)⋊C4—22no—
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—21no—
79(C2×D251)⋊C4—21no—
80(C2×D251)⋊C4—21no—
81(C2×D251)⋊C4—21no—
82C22×C4×C251—21no—
83C22×C4×C251C2311yescharacteristic
84M16×C251D414yescomplemented
85C2×C8×C251C2312yes—
86C4⋊C4×C251—21no—
87C4⋊C4×C251C2311yescharacteristic
88C4⋊C4×C251C2312yes—
89C4⋊C4×C251C2311yescharacteristic
90C22⋊C4×C251—21no—
91C42×C251—21no—
92C22×D251—22no—
93C22×D251—22no—
94C22×D251—21no—
95C22×D251—41no—
96C22×D251C2×D411yescharacteristic
97C22×D251C2×D411yescharacteristic
98Dic502—22no—
99Dic502—21no—
100Dic502—22no—
101Dic502—21no—
102Dic502—42no—
103D1004—21no—
104D1004—21no—
105D1004—42no—
106D1004—42no—
107C22⋊D251—42no—
108C2×Dic251—22no—
109C2×Dic251—21no—
110C2×Dic251—22no—
111C2×Dic251—42no—
112C2×Dic251C2×D411yescharacteristic
113C2×Dic251C2×D411yescharacteristic
114C4×D251—22no—
115C4×D251—22no—
116C4×D251—21no—
117C4×D251—21no—
118C4×D251—21no—
119C4×D251—21no—
120C4×D251—42no—
121C251⋊C8—22no—
122C23×C251C2×D411yescharacteristic
123C2×C4×C251—21no—
124C2×C4×C251—21no—
125C2×C4×C251—22no—
126C2×C4×C251—42no—
127C2×C4×C251C2×D411yescharacteristic
128C2×C4×C251C2×D412yes—
129C2×C4×C251#4112yes—
130C2×C4×C251C2411yescharacteristic
131C8×C251C2×D414yes—
132C22×C251—21no—
133C22×C251—21no—
134C22×C251#12112yes—
135C22×C251C22×D411yescharacteristic
136C2×D251—21no—
137C2×D251—21no—
138C2×D251—21no—
139C2×D251—21no—
140C2×D251—21no—
141C2×D251—44no—
142C2×D251—41no—
143C2×D251—41no—
144C2×D251—41no—
145C2×D251#13712yescomplemented
146C4×C251—41no—
147C4×C251—42no—
148C4×C251—41no—
149C4×C251#12112yes—
150C4×C251C22×D411yescharacteristic
151C4×C251C2×#4111yescharacteristic
152Dic251—21no—
153Dic251—21no—
154Dic251—41no—
155Dic251—42no—
156C2×C251—22no—
157C2×C251C2×#12111yescharacteristic
158C2×C251C2×#13711yescharacteristic
159C2×C251#57511yescharacteristic
160D251—22no—
161D251—41no—
162D251—81no—
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#428D25111yescharacteristic, complemented
174#419—2511no—
175#417—2511no—
176C2×#41—2511no—
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—2512no—
191C22⋊Q8—2512no—
192C22⋊Q8—2511no—
193C22⋊Q8—2511no—
194#121—2511no—
195#121—2511no—
196#120—5021no—
197C4×D4—5021no—
198C4×D4—5021no—
199#117C2×D25111yescharacteristic
200C2×C4⋊C4—5021no—
201C2×C4⋊C4C2×D25111yescharacteristic
202C2×C22⋊C4—5021no—
203#107C2×D25112yescomplemented
204#106C2×D25112yescomplemented
205#103—2512no—
206#103—2512no—
207#103—2512no—
208#102—2512no—
209C24—2511no—
210#41—5022no—
211#41—5022no—
212C2×Q8—5022no—
213C2×Q8—5022no—
214C2×Q8—5021no—
215C2×Q8—2512no—
216C2×Q8—2511no—
217C2×D4—5021no—
218C2×D4—5021no—
219C2×D4—5021no—
220C2×D4—2511no—
221C22×C4—21no—
222C22×C4—5021no—
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—5022no—
235M16—5024no—
236M16D100414yescomplemented
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⋊C4C22×D25112yes—
246C4⋊C4C22×D25111yescharacteristic
247C4⋊C4C22×D25111yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—5022no—
250C22⋊C4—5022no—
251C22⋊C4—5021no—
252C22⋊C4—5021no—
253C22⋊C4—5021no—
254C22⋊C4—5021no—
255C42—21no—
256C23—10041no—
257C23—5022no—
258C23—5021no—
259C23—5022no—
260C23—2511no—
261C23—2511no—
262C23D4×D25111yescharacteristic
263Q8—10042no—
264Q8—5022no—
265Q8—5021no—
266Q8—5022no—
267Q8—5021no—
268D4—10042no—
269D4—10042no—
270D4—10042no—
271D4—5021no—
272D4—5021no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—10042no—
278C2×C4—10042no—
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×D25111yescharacteristic
291C2×C4C2×D100412yes—
292C2×C4#1686081112yes—
293C2×C4C23×D25111yescharacteristic
294C8—5022no—
295C8C2×D100414yes—
296C22—21no—
297C22—21no—
298C22—10044no—
299C22—10041no—
300C22—10041no—
301C22—10041no—
302C22—5021no—
303C22—5021no—
304C22—5021no—
305C22—5021no—
306C22—5021no—
307C22—2512no—
308C22#2254869012yes—
309C22C2×D4×D25111yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—10041no—
314C4—10042no—
315C4—5021no—
316C4—5021no—
317C4#2254869012yes—
318C4C22×D100411yescharacteristic
319C4C2×#1686081111yescharacteristic
320C2—22no—
321C2—20081no—
322C2—10041no—
323C2—5022no—
324C2C2×#2254869011yescharacteristic
325C2#137×D25111yescharacteristic
326C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radical#30632199C2
hypercenter#428D251
hypocenterC251#3056
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×#22548748C2semidirect
2. C2×M16×D251C2semidirect
4. C2×#22548695C2semidirect
5. #30631946C2semidirect
6. #30631934C2semidirect
7. #30631929C2semidirect
8. #30631917C2semidirect
9. #30631695C2semidirect
10. #30631686C2semidirect
11. #428×C251C2semidirect
16. C2×Dic1004C22semidirect
17. C2×#16860799C22semidirect
36. #22548638C22semidirect
37. #22548638C22semidirect
38. #22548638C22semidirect
39. #22548635C22semidirect
43. #107×C251C22semidirect
44. #106×C251C22semidirect
84. M16×C251D4semidirect
145. C2×D251#137semidirect
163. C251#3056semidirect
173. #428D251semidirect
203. #107C2×D251semidirect
204. #106C2×D251semidirect
236. M16D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C2×M16×D2512
#306321992
C2×#225486952
#306319462
#306319342
#306319292
#306319172
#306316952
#306316862
#428×C2512
#3056251