Finite Group Databasebeta

← #24997438

Subgroups — #24997438

Order 11008.

734 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 105 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#19144595C212yesmaximal, complemented
2#19135427C211yesmaximal, characteristic, complemented
3#19133020C212yesmaximal, complemented
4#19132424C211yesmaximal, characteristic, complemented
5C43×#2397C211yesmaximal, characteristic, complemented
6#14583449C2211yescharacteristic
7#14583212—22no—
8#14583179—22no—
9#14583094—22no—
10#14583033—22no—
11C4×#10621882—24no—
12#14582952C2212yes—
13#14582923C2211yescharacteristic
14C2×#13063253—22no—
15D4⋊(C43⋊C8)—24no—
16C2×C4×D4×C43—22no—
17C2×C4×D4×C43C2212yes—
18C43×#399—22no—
19C43×#392—22no—
20C43×#376C2211yescharacteristic
21C2×#10621898—21no—
22C2×#10621882—24no—
23C2×#10621877—21no—
24#13063325D411yescharacteristic, complemented
25#13063322D411yescharacteristic, complemented
26#13063313—21no—
27#13063299—22no—
28C4×D172—22no—
29#13063258—24no—
30C4⋊(C43⋊C8)—22no—
31C4⋊(C43⋊C8)—22no—
32#13063253—44no—
33#13063246—24no—
34#13063245—24no—
35C22⋊(C43⋊C8)—22no—
36C4×C43⋊C8D414yescomplemented
37C22×D4×C43—22no—
38C22×D4×C43D412yes—
39C23×C4×C43—22no—
40C4×D4×C43—22no—
41C4×D4×C43—22no—
42C4×D4×C43—22no—
43C4×D4×C43—22no—
44C4×D4×C43—44no—
45C4×D4×C43—44no—
46C2×C4⋊C4×C43—22no—
47C2×C4⋊C4×C43D412yes—
48C2×C22⋊C4×C43—24no—
49C2×C22⋊C4×C43—22no—
50C2×C42×C43D412yes—
51C2×C42×C43C2311yescharacteristic
52#95×C43—22no—
53C24×C43—22no—
54C2×D4×C43—22no—
55C2×D4×C43—22no—
56C2×D4×C43—44no—
57C2×D4×C43—48no—
58C2×D4×C43—44no—
59C22×C4×C43—22no—
60C22×C4×C43—22no—
61C22×C4×C43—42no—
62C22×C4×C43—42no—
63C22×C4×C43—44no—
64C22×C4×C43—44no—
65C22×C4×C43C2×D412yes—
66C22×C4×C43C2×D411yescharacteristic
67C2×C22⋊D43—41no—
68#10621898—42no—
69#10621898—42no—
70C2×D172—22no—
71C2×D172—21no—
72C2×C4×D43—41no—
73C2×Dic86—21no—
74C4⋊C4×C43—22no—
75C4⋊C4×C43—22no—
76C4⋊C4×C43—44no—
77C22⋊C4×C43—48no—
78C22⋊C4×C43—42no—
79C22⋊C4×C43—42no—
80C42×C43—22no—
81C42×C43D814yescomplemented
82C42×C43C2×D411yescharacteristic
83C42×C43C2×D412yes—
84C42×C43#4111yescharacteristic
85#10621882—48no—
86(C2×D43)⋊C4—42no—
87C4⋊Dic43—22no—
88#10621877—42no—
89C2×C43⋊C8—24no—
90C2×C43⋊C8—41no—
91C23×C43—24no—
92C23×C43—44no—
93C23×C43—42no—
94C23×C43—42no—
95C23×C43—44no—
96C23×C43#12011yescharacteristic
97C22×D43—41no—
98D4×C43—24no—
99D4×C43—24no—
100D4×C43—42no—
101D4×C43—42no—
102D4×C43—416no—
103C2×C4×C43—24no—
104C2×C4×C43—42no—
105C2×C4×C43—44no—
106C2×C4×C43—42no—
107C2×C4×C43—44no—
108C2×C4×C43—84no—
109C2×C4×C43—82no—
110C2×C4×C43—82no—
111C2×C4×C43#12012yes—
112C2×C4×C43#12111yescharacteristic
113C2×C4×C43#12112yes—
114C2×C4×C43#12711yescharacteristic
115C2×C4×C43C2×D812yes—
116C2×C4×C43AGL(1,ℤ/8)12yes—
117C22⋊D43—82no—
118C2×Dic43—41no—
119D172—44no—
120D172—42no—
121C4×D43—81no—
122C4×D43—81no—
123Dic86—42no—
124C43⋊C8—44no—
125C43⋊C8—81no—
126#10012—431nomaximal, Sylow
127C22×C43—22no—
128C22×C43—48no—
129C22×C43—44no—
130C22×C43—84no—
131C22×C43—82no—
132C22×C43—82no—
133C22×C43#39211yescharacteristic
134C22×C43#44712yes—
135C2×D43—82no—
136C2×D43—81no—
137C4×C43—22no—
138C4×C43—22no—
139C4×C43—44no—
140C4×C43—82no—
141C4×C43#45912yes—
142C4×C43#49311yescharacteristic
143C4×C43#49611yescharacteristic
144Dic43—81no—
145#3100—432no—
146#2397D4311yescharacteristic, complemented
147#1963—431no—
148#1486—432no—
149#1274—431no—
150C2×C43—22no—
151C2×C43—84no—
152C2×C43#201012yes—
153C2×C43#201111yescharacteristic
154D43—161no—
155#532—431no—
156C2×C4×D4—22no—
157C2×C4×D4C2×D4312yes—
158#480—862no—
159#474—862no—
160#459—862no—
161#447—862no—
162C4×D8—864no—
163#431—432no—
164#423—431no—
165C2×#104—862no—
166#399—22no—
167#392—22no—
168#376C2×D4311yescharacteristic
169D4⋊C8—864no—
170C43#1001211yescharacteristic, complemented, Sylow
171C2×#41—861no—
172C22×D4—22no—
173C22×D4C22⋊D4312yes—
174C23×C4—22no—
175C2×D8—864no—
176C2×M16—861no—
177C4⋊Q8—431no—
178#127—431no—
179#124—861no—
180#121—862no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—862no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊D4312yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C42C22⋊D4312yes—
193C2×C42C22×D4311yescharacteristic
194#107—864no—
195C4⋊C8—862no—
196C4⋊C8—862no—
197#104—1724no—
198#102—864no—
199#102—864no—
200C22⋊C8—862no—
201C4×C8—434no—
202#95—22no—
203C24—22no—
204#41—1722no—
205#41—1722no—
206C2×Q8—861no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—48no—
212C2×D4—862no—
213C2×D4—861no—
214C2×D4—1721no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—1721no—
222C22×C4C2×C22⋊D4312yes—
223C22×C4C2×C22⋊D4311yescharacteristic
224D8—1728no—
225M16—1722no—
226C2×C8—864no—
227C2×C8—1721no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—44no—
231C4⋊C4—862no—
232C22⋊C4—48no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—1722no—
236C42—22no—
237C42#1062188214yescomplemented
238C42D4×D4311yescharacteristic
239C42D4×D4312yes—
240C42#1062190011yescharacteristic
241C23—24no—
242C23—44no—
243C23—42no—
244C23—42no—
245C23—44no—
246C23—1721no—
247C23#1306329411yescharacteristic
248Q8—1722no—
249D4—24no—
250D4—24no—
251D4—42no—
252D4—42no—
253D4—416no—
254D4—3442no—
255D4—1724no—
256D4—1722no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—3441no—
266C2×C4—3441no—
267C2×C4—1721no—
268C2×C4#1306329312yes—
269C2×C4#1306329611yescharacteristic
270C2×C4#1306329712yes—
271C2×C4#1306332111yescharacteristic
272C2×C4C2×#1062188212yes—
273C2×C4#1306335112yes—
274C8—3441no—
275C8—1724no—
276C22—22no—
277C22—48no—
278C22—44no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—3442no—
283C22—3441no—
284C22#1458280711yescharacteristic
285C22#1458303312yes—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—82no—
290C4—3441no—
291C4#1458309112yes—
292C4#1458327511yescharacteristic
293C4#1458328611yescharacteristic
294C2—22no—
295C2—84no—
296C2—6881no—
297C2#1913566012yes—
298C2#1913566611yescharacteristic
subgroupG/H
centerC2×C4#13063321
commutator, cosocleC2×C42×C43C23
FrattiniC2×C42C22×D43
FittingC43×#2397C2
socleC22×C43#392
hypercenter#2397D43
hypocenterC43#10012
LCSG ▷ C2×C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #19144595C2semidirect
2. #19135427C2semidirect
3. #19133020C2semidirect
4. #19132424C2semidirect
5. C43×#2397C2semidirect
24. #13063325D4semidirect
25. #13063322D4semidirect
36. C4×C43⋊C8D4semidirect
81. C42×C43D8semidirect
146. #2397D43semidirect
170. C43#10012semidirect
237. C42#10621882semidirect

Maximal subgroups

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

subgroupindex
#191445952
#191354272
#191330202
#191324242
C43×#23972
#1001243