Finite Group Databasebeta

← #44560093

Subgroups — #44560093

Order 32128.

466 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 145 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#30631957C214yesmaximal, complemented
5#30631950C214yesmaximal, complemented
6#30631736C211yesmaximal, characteristic, complemented
7#30631669C211yesmaximal, characteristic, complemented
8C2×#22548631C211yesmaximal, characteristic, complemented
9#568×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12#22548746—21no—
13#22548745C2212yescomplemented
14#22548743—22no—
15#22548743—22no—
16#22548741—21no—
17#22548740—22no—
18#22548740C2214yes—
19#22548740C2212yescomplemented
20#22548735—22no—
21C2×#16860799C2212yescomplemented
22C2×#16860797—21no—
23C2×D2008C2212yescomplemented
24C2×#16860795—21no—
25C2×#16860794C2211yescharacteristic
26C2×#16860794C2211yescharacteristic
27#22548651C2214yescomplemented
28#22548633C2212yescomplemented
29#22548631C2214yescomplemented
30#22548631C2212yescomplemented
31C2×#41×C251C2211yescharacteristic
32#131×C251—22no—
33#131×C251C2214yescomplemented
34C2×M16×C251—22no—
35C2×M16×C251C2211yescharacteristic
36C2×M16×C251C2211yescharacteristic
37C23×D251—21no—
38#16860810—21no—
39#16860810—21no—
40#16860810—21no—
41#16860810—21no—
42C2×Dic502C2311yescharacteristic
43C2×D1004—21no—
44C2×D1004—21no—
45C2×D1004—21no—
46C2×D1004—21no—
47C2×D1004—21no—
48C2×D1004C2312yes—
49C2×D1004C2311yescharacteristic
50C2×C22⋊D251—21no—
51C2×C4×D251—21no—
52#16860800—22no—
53#16860799—24no—
54#16860798—22no—
55#16860797—24no—
56#16860797—22no—
57D2008—24no—
58#16860795—24no—
59#16860795—22no—
60#16860794—21no—
61#16860794—21no—
62#16860794C2312yes—
63#16860794C2312yes—
64C2×C251⋊C8—21no—
65C2×C251⋊C8—21no—
66#41×C251—21no—
67#41×C251—21no—
68#41×C251D412yes—
69#41×C251C2312yes—
70C2×Q8×C251D411yescharacteristic
71C2×D4×C251—21no—
72C2×D4×C251D411yescharacteristic
73C22×C4×C251—21no—
74C22×C4×C251C2311yescharacteristic
75M16×C251—21no—
76M16×C251—24no—
77M16×C251D412yes—
78M16×C251D412yes—
79M16×C251C2312yes—
80C2×C8×C251—21no—
81C2×C8×C251—24no—
82C2×C8×C251C2312yes—
83C22×D251—21no—
84C22×D251—22no—
85C22×D251—41no—
86C22×D251—41no—
87C22×D251—82no—
88Dic502—21no—
89Dic502—21no—
90D1004—22no—
91D1004—21no—
92D1004—21no—
93D1004—22no—
94D1004—41no—
95D1004—41no—
96C22⋊D251—41no—
97C22⋊D251—41no—
98C2×Dic251—21no—
99C4×D251—41no—
100C4×D251—41no—
101C251⋊C8—22no—
102C251⋊C8—22no—
103C23×C251—21no—
104C23×C251#4111yescharacteristic
105Q8×C251—21no—
106Q8×C251C2×D412yes—
107D4×C251—22no—
108D4×C251—22no—
109D4×C251—21no—
110D4×C251C2×D412yes—
111C2×C4×C251—21no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
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×C251C2×D414yes—
122C22×C251—22no—
123C22×C251—41no—
124C22×C251—81no—
125C22×C251#12111yescharacteristic
126C22×C251#12111yescharacteristic
127C22×C251C2×#4111yescharacteristic
128C2×D251—41no—
129C2×D251—42no—
130C2×D251—82no—
131C2×D251—82no—
132C2×D251—81no—
133C2×D251—82no—
134C4×C251—22no—
135C4×C251#12111yescharacteristic
136C4×C251#12111yescharacteristic
137C4×C251C22×D411yescharacteristic
138C4×C251C22×D411yescharacteristic
139Dic251—41no—
140C2×C251—21no—
141C2×C251—21no—
142C2×C251—21no—
143C2×C251—42no—
144C2×C251C2×#12111yescharacteristic
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#568D25111yescharacteristic, complemented
153#472—2514no—
154#471—2514no—
155#430—2511no—
156#413—2511no—
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)—2514no—
168AGL(1,ℤ/8)—2512no—
169#135—5022no—
170C2×SD16—5021no—
171C2×SD16—2512no—
172C2×D8—5021no—
173C2×D8—2512no—
174#131—22no—
175#131C2×D25114yescomplemented
176C2×M16—22no—
177C2×M16—2511no—
178C2×M16—2511no—
179C2×M16C2×D25111yescharacteristic
180C2×M16C2×D25111yescharacteristic
181#108—2514no—
182#101—2512no—
183#100—2514no—
184#100—2512no—
185C24—5021no—
186#41—21no—
187#41—21no—
188#41—5021no—
189#41—5021no—
190#41—5021no—
191#41—5021no—
192#41D100412yes—
193#41C22×D25112yes—
194C2×Q8—2511no—
195C2×Q8D100411yescharacteristic
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×D4D100411yescharacteristic
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—24no—
219M16—5021no—
220M16—5021no—
221M16—2512no—
222M16—2512no—
223M16C22⋊D25112yes—
224M16C22⋊D25112yes—
225M16C22×D25112yes—
226C2×C8—21no—
227C2×C8—24no—
228C2×C8—5021no—
229C2×C8—5021no—
230C2×C8C22×D25112yes—
231C23—21no—
232C23—20082no—
233C23—10041no—
234C23—10041no—
235C23—5021no—
236C23—5022no—
237C23#1686081011yescharacteristic
238Q8—21no—
239Q8—5021no—
240Q8—5021no—
241Q8C2×D100412yes—
242D4—22no—
243D4—22no—
244D4—21no—
245D4—10041no—
246D4—10041no—
247D4—10041no—
248D4—10041no—
249D4—5021no—
250D4—5022no—
251D4—5022no—
252D4—5021no—
253D4C2×D100412yes—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—10041no—
258C2×C4—10041no—
259C2×C4—5021no—
260C2×C4C2×C22⋊D25111yescharacteristic
261C2×C4C2×C22⋊D25111yescharacteristic
262C2×C4C2×D100411yescharacteristic
263C2×C4C2×D100411yescharacteristic
264C2×C4#1686081011yescharacteristic
265C2×C4C23×D25111yescharacteristic
266C8—22no—
267C8—5022no—
268C8—5022no—
269C8C2×C22⋊D25114yes—
270C22—22no—
271C22—41no—
272C22—81no—
273C22—20082no—
274C22—20081no—
275C22—20082no—
276C22—20082no—
277C22—10041no—
278C22—10042no—
279C22#2254868911yescharacteristic
280C22#2254868911yescharacteristic
281C22C2×#1686081011yescharacteristic
282C4—22no—
283C4—10041no—
284C4#2254868911yescharacteristic
285C4#2254868911yescharacteristic
286C4C22×C22⋊D25111yescharacteristic
287C4C22×D100411yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—42no—
292C2—20082no—
293C2—20081no—
294C2C2×#2254868911yescharacteristic
subgroupG/H
centerC2C2×#22548689
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#568×C251C2
socleC2×C251C2×#121
hypercenter#568D251
hypocenterC251#3068
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #30632485C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548740C2semidirect
4. #30631957C2semidirect
5. #30631950C2semidirect
6. #30631736C2semidirect
7. #30631669C2semidirect
8. C2×#22548631C2semidirect
9. #568×C251C2semidirect
13. #22548745C22semidirect
19. #22548740C22semidirect
21. C2×#16860799C22semidirect
23. C2×D2008C22semidirect
27. #22548651C22semidirect
28. #22548633C22semidirect
29. #22548631C22semidirect
30. #22548631C22semidirect
33. #131×C251C22semidirect
147. C251#3068semidirect
152. #568D251semidirect
175. #131C2×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
#306319572
#306319502
#306317362
#306316692
C2×#225486312
#568×C2512
#3068251