Finite Group Databasebeta

← #44560096

Subgroups — #44560096

Order 32128.

466 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 129 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#30632485C211yesmaximal, characteristic, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548740C211yesmaximal, characteristic, complemented
4#30632453C211yesmaximal, characteristic, complemented
5#30631956C214yesmaximal, complemented
6#30631953C214yesmaximal, complemented
7#30631735C211yesmaximal, characteristic, complemented
8C2×#22548631C211yesmaximal, characteristic, complemented
9#413×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12#22548746C2212yescomplemented
13#22548745—21no—
14#22548743—22no—
15#22548743C2214yes—
16#22548741C2212yescomplemented
17#22548740—21no—
18#22548740—22no—
19#22548740—22no—
20#22548735—22no—
21C2×#16860799—21no—
22C2×#16860797C2212yescomplemented
23C2×D2008—21no—
24C2×#16860795C2212yescomplemented
25#22548723C2214yescomplemented
26#22548721—22no—
27C2×#16860794C2211yescharacteristic
28C2×#16860794C2211yescharacteristic
29M16×D251—22no—
30#22548650C2214yescomplemented
31#22548631C2214yescomplemented
32C2×#41×C251C2211yescharacteristic
33C2×M16×C251C2211yescharacteristic
34C2×M16×C251C2211yescharacteristic
35#101×C251C2212yescomplemented
36#100×C251C2212yescomplemented
37C23×D251—21no—
38#16860810—21no—
39#16860810—21no—
40#16860810D412yes—
41#16860810C2312yes—
42C2×Dic502D411yescharacteristic
43C2×D1004—21no—
44C2×D1004—21no—
45C2×D1004—21no—
46C2×D1004—21no—
47C2×D1004—21no—
48C2×D1004D411yescharacteristic
49C2×D1004C2312yes—
50C2×C22⋊D251—21no—
51C2×C4×D251—21no—
52#16860800—22no—
53#16860799—24no—
54#16860798—22no—
55#16860797—22no—
56#16860797—24no—
57D2008—24no—
58#16860795—22no—
59#16860795—24no—
60#16860794—21no—
61#16860794D412yes—
62#16860794D412yes—
63#16860794C2312yes—
64#16860793—24no—
65C2×C251⋊C8—21no—
66C2×C251⋊C8C2312yes—
67C8×D251—24no—
68#41×C251—21no—
69#41×C251—21no—
70#41×C251—21no—
71#41×C251—21no—
72C2×Q8×C251C2311yescharacteristic
73C2×D4×C251—21no—
74C2×D4×C251C2311yescharacteristic
75C22×C4×C251—21no—
76C22×C4×C251C2311yescharacteristic
77M16×C251—21no—
78M16×C251—21no—
79M16×C251C2312yes—
80M16×C251C2312yes—
81C2×C8×C251—21no—
82C2×C8×C251—21no—
83C22×D251—21no—
84C22×D251—22no—
85C22×D251—41no—
86C22×D251—41no—
87C22×D251—82no—
88Dic502—21no—
89Dic502C2×D412yes—
90D1004—21no—
91D1004—22no—
92D1004—22no—
93D1004—41no—
94D1004—41no—
95D1004C2×D412yes—
96C22⋊D251—22no—
97C22⋊D251—22no—
98C2×Dic251—21no—
99C4×D251—22no—
100C4×D251—22no—
101C251⋊C8—22no—
102C251⋊C8C2×D414yes—
103C23×C251—21no—
104C23×C251#4111yescharacteristic
105Q8×C251—21no—
106Q8×C251—21no—
107D4×C251—21no—
108D4×C251—21no—
109D4×C251—41no—
110D4×C251—41no—
111C2×C4×C251—21no—
112C2×C4×C251—41no—
113C2×C4×C251—41no—
114C2×C4×C251C2×D411yescharacteristic
115C2×C4×C251C2×D411yescharacteristic
116C2×C4×C251C2×D411yescharacteristic
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251#4111yescharacteristic
119C2×C4×C251C2411yescharacteristic
120C8×C251—22no—
121C8×C251—22no—
122C22×C251—41no—
123C22×C251—41no—
124C22×C251—81no—
125C22×C251#12111yescharacteristic
126C22×C251#12111yescharacteristic
127C22×C251C2×#4111yescharacteristic
128C2×D251—22no—
129C2×D251—42no—
130C2×D251—81no—
131C2×D251—82no—
132C2×D251—82no—
133C2×D251—82no—
134C4×C251—41no—
135C4×C251#12111yescharacteristic
136C4×C251#12111yescharacteristic
137C4×C251C22×D411yescharacteristic
138C4×C251C22×D411yescharacteristic
139Dic251—22no—
140C2×C251—21no—
141C2×C251—21no—
142C2×C251—21no—
143C2×C251—81no—
144C2×C251C2×#12111yescharacteristic
145D251—42no—
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#430—2511no—
156#413D25111yescharacteristic, complemented
157C2×#100—2511no—
158C2×#41—2511no—
159C2×#41C2×D25111yescharacteristic
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)—2512no—
168AGL(1,ℤ/8)—2514no—
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×M16—2511no—
179C2×M16C2×D25111yescharacteristic
180C2×M16C2×D25111yescharacteristic
181#108—2514no—
182#101C2×D25112yescomplemented
183#100—2514no—
184#100C2×D25112yescomplemented
185C24—5021no—
186#41—21no—
187#41—21no—
188#41—21no—
189#41—21no—
190#41—5021no—
191#41—5021no—
192#41—2512no—
193#41—2512no—
194C2×Q8—2511no—
195C2×Q8C22×D25111yescharacteristic
196C2×D4—21no—
197C2×D4—5021no—
198C2×D4—5021no—
199C2×D4—5021no—
200C2×D4—5021no—
201C2×D4—5021no—
202C2×D4—5021no—
203C2×D4—2511no—
204C2×D4—2512no—
205C2×D4C22×D25111yescharacteristic
206C22×C4—21no—
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—5024no—
220M16—5021no—
221M16—2512no—
222M16—2512no—
223M16—2512no—
224M16C22×D25112yes—
225M16C22×D25112yes—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—5024no—
229C2×C8—5021no—
230C2×C8—2512no—
231C23—21no—
232C23—20082no—
233C23—10041no—
234C23—10041no—
235C23—5022no—
236C23—5021no—
237C23#1686081011yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—5021no—
241Q8—2512no—
242D4—21no—
243D4—21no—
244D4—41no—
245D4—41no—
246D4—10041no—
247D4—10041no—
248D4—5022no—
249D4—5022no—
250D4—5022no—
251D4—5022no—
252D4—5021no—
253D4—2512no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—5021no—
258C2×C4—5022no—
259C2×C4—5022no—
260C2×C4D4×D25111yescharacteristic
261C2×C4D4×D25111yescharacteristic
262C2×C4D4×D25111yescharacteristic
263C2×C4D4×D25111yescharacteristic
264C2×C4#1686081011yescharacteristic
265C2×C4C23×D25111yescharacteristic
266C8—22no—
267C8—22no—
268C8—5022no—
269C8—2514no—
270C22—41no—
271C22—41no—
272C22—81no—
273C22—20082no—
274C22—20082no—
275C22—20082no—
276C22—20081no—
277C22—10042no—
278C22—5022no—
279C22#2254868811yescharacteristic
280C22#2254868811yescharacteristic
281C22C2×#1686081011yescharacteristic
282C4—41no—
283C4—5022no—
284C4#2254868811yescharacteristic
285C4#2254868811yescharacteristic
286C4C2×D4×D25111yescharacteristic
287C4C2×D4×D25111yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—81no—
292C2—20082no—
293C2—10042no—
294C2C2×#2254868811yescharacteristic
subgroupG/H
centerC2C2×#22548688
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#413×C251C2
socleC2×C251C2×#121
hypercenter#413D251
hypocenterC251#3068
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #30632485C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548740C2semidirect
4. #30632453C2semidirect
5. #30631956C2semidirect
6. #30631953C2semidirect
7. #30631735C2semidirect
8. C2×#22548631C2semidirect
9. #413×C251C2semidirect
12. #22548746C22semidirect
16. #22548741C22semidirect
22. C2×#16860797C22semidirect
24. C2×#16860795C22semidirect
25. #22548723C22semidirect
30. #22548650C22semidirect
31. #22548631C22semidirect
35. #101×C251C22semidirect
36. #100×C251C22semidirect
147. C251#3068semidirect
156. #413D251semidirect
182. #101C2×D251semidirect
184. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324852
C2×#225487432
C2×#225487402
#306324532
#306319562
#306319532
#306317352
C2×#225486312
#413×C2512
#3068251