Finite Group Databasebeta

← #44560094

Subgroups — #44560094

Order 32128.

466 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 137 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#30632487C211yesmaximal, characteristic, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548743C211yesmaximal, characteristic, complemented
4#30632453C211yesmaximal, characteristic, complemented
5#30631958C214yesmaximal, complemented
6#30631951C214yesmaximal, complemented
7#30631669C211yesmaximal, characteristic, complemented
8C2×#22548631C211yesmaximal, characteristic, complemented
9#430×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C2×#16860810C2211yescharacteristic
12C22×D1004C2211yescharacteristic
13#22548748—21no—
14#22548748C2212yescomplemented
15#22548743—22no—
16#22548743—21no—
17#22548743—22no—
18#22548743—22no—
19#22548743C2214yes—
20#22548743C2212yescomplemented
21#22548737—22no—
22C2×#16860799—21no—
23C2×#16860799C2212yescomplemented
24C2×D2008—21no—
25C2×D2008C2212yescomplemented
26#22548723—22no—
27#22548721C2214yescomplemented
28C2×#16860794C2211yescharacteristic
29M16×D251—22no—
30#22548633C2212yescomplemented
31#22548631C2214yescomplemented
32#22548631C2212yescomplemented
33C2×M16×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic
35C2×M16×C251C2211yescharacteristic
36#108×C251C2214yescomplemented
37C23×D251—21no—
38#16860810—21no—
39#16860810—21no—
40#16860810—21no—
41#16860810—21no—
42#16860810—21no—
43#16860810—21no—
44#16860810D412yes—
45#16860810C2312yes—
46C2×Dic502D411yescharacteristic
47C2×Dic502C2311yescharacteristic
48C2×D1004—21no—
49C2×D1004—21no—
50C2×D1004—21no—
51C2×D1004—21no—
52C2×D1004—21no—
53C2×D1004D411yescharacteristic
54C2×D1004C2312yes—
55C2×D1004C2311yescharacteristic
56C2×C22⋊D251—21no—
57C2×C22⋊D251—21no—
58C2×C4×D251—21no—
59C2×C4×D251—21no—
60Dic1004—22no—
61#16860799—22no—
62#16860799—24no—
63#16860799—22no—
64#16860799—24no—
65D2008—22no—
66D2008—24no—
67D2008—24no—
68#16860794—21no—
69#16860794C2312yes—
70#16860793—24no—
71C2×C251⋊C8—21no—
72C8×D251—24no—
73C22×C4×C251C2311yescharacteristic
74M16×C251—21no—
75M16×C251—21no—
76M16×C251D412yes—
77M16×C251D412yes—
78M16×C251C2312yes—
79M16×C251C2312yes—
80C2×C8×C251—21no—
81C2×C8×C251—21no—
82C2×C8×C251C2312yes—
83C22×D251—21no—
84C22×D251—21no—
85C22×D251—22no—
86C22×D251—41no—
87C22×D251—41no—
88C22×D251—82no—
89Dic502—21no—
90Dic502—21no—
91Dic502—21no—
92Dic502C2×D412yes—
93D1004—21no—
94D1004—21no—
95D1004—22no—
96D1004—22no—
97D1004—21no—
98D1004—41no—
99D1004—41no—
100D1004C2×D412yes—
101C22⋊D251—22no—
102C22⋊D251—22no—
103C22⋊D251—41no—
104C22⋊D251—41no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C4×D251—22no—
108C4×D251—22no—
109C4×D251—41no—
110C4×D251—41no—
111C251⋊C8—22no—
112C23×C251#4111yescharacteristic
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251#4111yescharacteristic
118C2×C4×C251C2411yescharacteristic
119C8×C251—22no—
120C8×C251—22no—
121C8×C251C2×D414yes—
122C22×C251—41no—
123C22×C251#12111yescharacteristic
124C22×C251#12111yescharacteristic
125C22×C251C2×#4111yescharacteristic
126C2×D251—22no—
127C2×D251—41no—
128C2×D251—42no—
129C2×D251—81no—
130C2×D251—82no—
131C2×D251—82no—
132C2×D251—81no—
133C2×D251—82no—
134C4×C251#12111yescharacteristic
135C4×C251#12111yescharacteristic
136C4×C251C22×D411yescharacteristic
137C4×C251C22×D411yescharacteristic
138Dic251—22no—
139Dic251—41no—
140C2×C251—21no—
141C2×C251—21no—
142C2×C251—21no—
143C2×C251C2×#12111yescharacteristic
144D251—42no—
145D251—81no—
146D251—82no—
147C251#306811yescharacteristic, complemented, Sylow
148#3068—2511nomaximal, Sylow
149#575—2511no—
150C2×AGL(1,ℤ/8)—2511no—
151C2×AGL(1,ℤ/8)—2511no—
152#568—2511no—
153#472—2514no—
154#471—2514no—
155#430D25111yescharacteristic, complemented
156#413—2511no—
157C2×#100—2511no—
158C2×#41—2511no—
159C2×#41—2511no—
160C22×D4—2511no—
161#137—5021no—
162#137—2512no—
163AGL(1,ℤ/8)—5022no—
164AGL(1,ℤ/8)—5021no—
165AGL(1,ℤ/8)—5022no—
166AGL(1,ℤ/8)—5022no—
167AGL(1,ℤ/8)—2514no—
168AGL(1,ℤ/8)—2512no—
169#135—5022no—
170C2×SD16—5021no—
171C2×SD16—2512no—
172C2×D8—5021no—
173C2×D8—2512no—
174#131—5022no—
175#131—2514no—
176C2×M16—5022no—
177C2×M16—2511no—
178C2×M16C2×D25111yescharacteristic
179C2×M16C2×D25111yescharacteristic
180C2×M16C2×D25111yescharacteristic
181#108C2×D25114yescomplemented
182#101—2512no—
183#100—2514no—
184#100—2512no—
185C24—5021no—
186#41—5021no—
187#41—5021no—
188#41—5021no—
189#41—5021no—
190#41—5021no—
191#41—5021no—
192#41—2512no—
193#41—2512no—
194C2×Q8—2511no—
195C2×Q8—2511no—
196C2×D4—5021no—
197C2×D4—5021no—
198C2×D4—5021no—
199C2×D4—5021no—
200C2×D4—5021no—
201C2×D4—5021no—
202C2×D4—5021no—
203C2×D4—2511no—
204C2×D4—2511no—
205C2×D4—2512no—
206C22×C4—5021no—
207C22×C4—5021no—
208C22×C4C22×D25111yescharacteristic
209Q16—5022no—
210SD16—5024no—
211SD16—5022no—
212SD16—5022no—
213SD16—5024no—
214D8—5024no—
215D8—5022no—
216D8—5024no—
217M16—21no—
218M16—21no—
219M16—5021no—
220M16—5024no—
221M16—2512no—
222M16D100412yes—
223M16D100412yes—
224M16C22×D25112yes—
225M16C22×D25112yes—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—5021no—
229C2×C8—5024no—
230C2×C8C22×D25112yes—
231C23—20082no—
232C23—10041no—
233C23—10041no—
234C23—5021no—
235C23—5022no—
236C23—5021no—
237C23#1686081111yescharacteristic
238Q8—5021no—
239Q8—5021no—
240Q8—5021no—
241Q8—2512no—
242D4—10041no—
243D4—10041no—
244D4—10041no—
245D4—10041no—
246D4—5022no—
247D4—5022no—
248D4—5022no—
249D4—5021no—
250D4—5021no—
251D4—5022no—
252D4—5021no—
253D4—2512no—
254C2×C4—10041no—
255C2×C4—10041no—
256C2×C4—5021no—
257C2×C4—5021no—
258C2×C4—5022no—
259C2×C4—5022no—
260C2×C4D4×D25111yescharacteristic
261C2×C4D4×D25111yescharacteristic
262C2×C4C2×D100411yescharacteristic
263C2×C4C2×D100411yescharacteristic
264C2×C4#1686081111yescharacteristic
265C2×C4C23×D25111yescharacteristic
266C8—22no—
267C8—22no—
268C8—5022no—
269C8C2×D100414yes—
270C22—41no—
271C22—20082no—
272C22—20081no—
273C22—20082no—
274C22—20082no—
275C22—20081no—
276C22—10041no—
277C22—10042no—
278C22—5022no—
279C22#2254869011yescharacteristic
280C22#2254869011yescharacteristic
281C22C2×#1686081111yescharacteristic
282C4—10041no—
283C4—5022no—
284C4#2254869011yescharacteristic
285C4#2254869011yescharacteristic
286C4C2×D4×D25111yescharacteristic
287C4C22×D100411yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—20082no—
292C2—20081no—
293C2—10042no—
294C2C2×#2254869011yescharacteristic
subgroupG/H
centerC2C2×#22548690
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#430×C251C2
socleC2×C251C2×#121
hypercenter#430D251
hypocenterC251#3068
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #430

Complements

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

NH ≅ G/Ntype
1. #30632487C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548743C2semidirect
4. #30632453C2semidirect
5. #30631958C2semidirect
6. #30631951C2semidirect
7. #30631669C2semidirect
8. C2×#22548631C2semidirect
9. #430×C251C2semidirect
14. #22548748C22semidirect
20. #22548743C22semidirect
23. C2×#16860799C22semidirect
25. C2×D2008C22semidirect
27. #22548721C22semidirect
30. #22548633C22semidirect
31. #22548631C22semidirect
32. #22548631C22semidirect
36. #108×C251C22semidirect
147. C251#3068semidirect
155. #430D251semidirect
181. #108C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324872
C2×#225487432
#306324532
#306319582
#306319512
#306316692
C2×#225486312
#430×C2512
#3068251