Finite Group Databasebeta

← #25074742

Subgroups — #25074742

Order 11008.

1056 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 163 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#19149000C211yesmaximal, characteristic
2#19145723C212yesmaximal, complemented
3#19133738C214yesmaximal, complemented
4#19133709C214yesmaximal, complemented
5#19133257C211yesmaximal, characteristic, complemented
6#19133013C212yesmaximal, complemented
7C43×#3152C211yesmaximal, characteristic, complemented
8#14583641—22no—
9#14583615C2212yes—
10#14583614—22no—
11D4×D172—22no—
12#14583534C2212yes—
13#14583465—24no—
14#14583463—22no—
15#14583449C2212yes—
16#14583295—22no—
17#14583247—22no—
18#14583217—24no—
19#14583186C2218yescomplemented
20#14582952C2212yescomplemented
21#14582648C2212yescomplemented
22#14582643C2212yes—
23#14582628C2218yescomplemented
24C2×#137×C43—22no—
25C43×#532C2211yescharacteristic
26C2×C4⋊Q8×C43C2211yescharacteristic, complemented
27C43×#497—22no—
28C43×#494C2212yescomplemented
29C43×#492C2212yes—
30C4×M16×C43C2211yescharacteristic, complemented
31C2×#10621898—22no—
32C22×D172—22no—
33C2×D4×D43—24no—
34#13063325D412yes—
35#13063322D412yes—
36#13063321—24no—
37#13063314—24no—
38#13063313—22no—
39#13063311—28no—
40#13063300—24no—
41#13063299—24no—
42#13063299—22no—
43#13063299D414yes—
44#13063296—24no—
45#13063295—28no—
46#13063293—24no—
47#13063293—24no—
48C4×D172—22no—
49C4×D172C2314yes—
50#13063284—22no—
51#13063283—24no—
52#13063256—28no—
53C4⋊(C43⋊C8)C2314yes—
54#13063245—28no—
55C22⋊(C43⋊C8)—22no—
56C4×C43⋊C8—22no—
57C2×#41×C43D412yes—
58C22×Q8×C43—21no—
59#137×C43—48no—
60C2×Q16×C43—24no—
61C2×SD16×C43—24no—
62C2×M16×C43—21no—
63C4⋊Q8×C43—22no—
64C4⋊Q8×C43—21no—
65C4⋊Q8×C43C2312yes—
66C4⋊Q8×C43C2311yescharacteristic
67#127×C43C2311yescharacteristic
68#124×C43D412yes—
69#121×C43—22no—
70C4×D4×C43—22no—
71C2×C4⋊C4×C43—22no—
72C2×C42×C43C2311yescharacteristic
73#97×C43—21no—
74C4×C8×C43C2312yes—
75C23×D43—24no—
76#41×C43—44no—
77C2×Q8×C43—22no—
78C2×Q8×C43—44no—
79C2×Q8×C43—41no—
80C2×Q8×C43C2×D412yes—
81C2×D4×C43—22no—
82C2×D4×C43—22no—
83C2×D4×C43C2×D412yes—
84C22×C4×C43—22no—
85C22×C4×C43—41no—
86C22×C4×C43C2×D412yes—
87C22×C4×C43C2×D411yescharacteristic
88C2×C22⋊D43—28no—
89C2×C22⋊D43—24no—
90D4×D43—88no—
91#10621898—84no—
92C2×D172—22no—
93C2×D172—24no—
94C2×D172—22no—
95C2×D172—84no—
96C2×D172C2×D414yes—
97C2×C4×D43—24no—
98C2×Dic86—22no—
99Q16×C43—48no—
100SD16×C43—48no—
101M16×C43—24no—
102C2×C8×C43—22no—
103C4⋊C4×C43—22no—
104C4⋊C4×C43—24no—
105C4⋊C4×C43—42no—
106C22⋊C4×C43—24no—
107C42×C43C2×D412yes—
108C42×C43C2×D411yescharacteristic
109C42×C43C2411yescharacteristic
110C22⋊Dic43—24no—
111(C2×D43)⋊C4—28no—
112(C2×D43)⋊C4—24no—
113(C2×D43)⋊C4—24no—
114C4⋊Dic43—22no—
115C4⋊Dic43C2×D414yes—
116C4×Dic43—24no—
117C2×C43⋊C8—24no—
118C23×C43—22no—
119C23×C43#12011yescharacteristic
120C22×D43—24no—
121C22×D43—84no—
122C22×D43—88no—
123Q8×C43—42no—
124Q8×C43—44no—
125Q8×C43—82no—
126D4×C43—42no—
127D4×C43—44no—
128D4×C43—82no—
129C2×C4×C43—24no—
130C2×C4×C43—22no—
131C2×C4×C43—22no—
132C2×C4×C43—44no—
133C2×C4×C43—42no—
134C2×C4×C43—82no—
135C2×C4×C43#12012yes—
136C2×C4×C43#12011yescharacteristic
137C2×C4×C43C22×D412yes—
138C2×C4×C43C22×D411yescharacteristic
139C22⋊D43—84no—
140C22⋊D43—88no—
141C2×Dic43—24no—
142D172—28no—
143D172—44no—
144D172—82no—
145D172—84no—
146C4×D43—48no—
147Dic86—82no—
148C8×C43—24no—
149C43⋊C8—44no—
150#18715—431nomaximal, Sylow
151C22×C43—22no—
152C22×C43—21no—
153C22×C43—44no—
154C22×C43—42no—
155C22×C43C2×#12011yescharacteristic
156C2×D43—44no—
157C2×D43—44no—
158C2×D43—44no—
159C2×D43—168no—
160C4×C43—22no—
161C4×C43—21no—
162C4×C43—42no—
163C4×C43—82no—
164C4×C43#45312yes—
165C4×C43#57714yes—
166Dic43—44no—
167#3549—431no—
168#3227—432no—
169#3152D4311yescharacteristic, complemented
170#1634—434no—
171#1629—434no—
172#1543—431no—
173#1484—432no—
174C2×C43—41no—
175C2×C43—82no—
176C2×C43#300912yes—
177C2×C43C2×#45311yescharacteristic
178D43—84no—
179C2×#137—22no—
180H(3,𝔽4)—862no—
181#555—432no—
182#555—862no—
183D42—432no—
184D42—862no—
185#535—864no—
186#535—862no—
187#532—432no—
188#532C2×D4311yescharacteristic
189C2×C4⋊Q8C2×D4311yescharacteristic, complemented
190#498—862no—
191#497—22no—
192#494C2×D4312yescomplemented
193#492C2×D4312yes—
194#486—862no—
195#481—864no—
196#476—438no—
197#431—432no—
198C4×M16C2×D4311yescharacteristic, complemented
199#333—432no—
200#331—432no—
201D4⋊C8—438no—
202C43#1871511yescharacteristic, complemented, Sylow
203C2×#41—862no—
204C2×#41C22⋊D4312yes—
205C22×Q8—21no—
206C22×D4—862no—
207C22×D4—864no—
208#137—48no—
209C2×Q16—24no—
210C2×SD16—24no—
211C2×M16—21no—
212C4⋊Q8—22no—
213C4⋊Q8—21no—
214C4⋊Q8—432no—
215C4⋊Q8C22×D4312yes—
216C4⋊Q8C22×D4311yescharacteristic
217#127—432no—
218#127—864no—
219#127C22×D4311yescharacteristic
220#124—868no—
221#124—864no—
222#124—862no—
223#124C22⋊D4312yes—
224#121—22no—
225#121—434no—
226#121—864no—
227#121—864no—
228#121—864no—
229#121—862no—
230#120—868no—
231#120—864no—
232#120—864no—
233C4×D4—22no—
234C4×D4—434no—
235C4×D4—862no—
236#117—864no—
237#117—862no—
238C2×C4⋊C4—22no—
239C2×C42C22×D4311yescharacteristic
240#106—868no—
241C4⋊C8—434no—
242#102—868no—
243C22⋊C8—862no—
244#97—21no—
245C4×C8—862no—
246C4×C8C22×D4312yes—
247C24—864no—
248#41—44no—
249#41—3444no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—862no—
254C2×Q8C2×C22⋊D4312yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—434no—
258C2×D4—864no—
259C2×D4—868no—
260C2×D4—864no—
261C2×D4—862no—
262C2×D4—862no—
263C2×D4—3444no—
264C2×D4—3448no—
265C2×D4C2×C22⋊D4312yes—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—864no—
269C22×C4D4×D4312yes—
270C22×C4C2×C22⋊D4311yescharacteristic
271Q16—48no—
272SD16—48no—
273M16—24no—
274C2×C8—22no—
275C2×C8—864no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—434no—
280C4⋊C4—862no—
281C22⋊C4—24no—
282C22⋊C4—868no—
283C22⋊C4—864no—
284C22⋊C4—864no—
285C22⋊C4—864no—
286C42—864no—
287C42D4×D4312yes—
288C42C2×C22⋊D4311yescharacteristic
289C42C23×D4311yescharacteristic
290C23—22no—
291C23—864no—
292C23—3448no—
293C23—3444no—
294C23#1306329311yescharacteristic
295Q8—42no—
296Q8—44no—
297Q8—82no—
298Q8—3442no—
299D4—42no—
300D4—44no—
301D4—82no—
302D4—868no—
303D4—3442no—
304D4—3444no—
305D4—3444no—
306D4—3448no—
307D4—1724no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—864no—
315C2×C4—1728no—
316C2×C4#1306329312yes—
317C2×C4#1306329311yescharacteristic
318C2×C4C2×D4×D4312yes—
319C2×C4C22×C22⋊D4311yescharacteristic
320C8—24no—
321C8—1724no—
322C22—22no—
323C22—21no—
324C22—44no—
325C22—42no—
326C22—1724no—
327C22—1724no—
328C22—1724no—
329C22—6888no—
330C22C2×#1306329311yescharacteristic
331C4—22no—
332C4—21no—
333C4—42no—
334C4—82no—
335C4—1724no—
336C4#1458306612yes—
337C4#1458370814yes—
338C2—41no—
339C2—82no—
340C2—3444no—
341C2#1914387112yes—
342C2C2×#1458306611yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3152C2
socleC22×C43C2×#120
prime radical#19149000C2
hypercenter#3152D43
hypocenterC43#18715
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #19145723C2semidirect
3. #19133738C2semidirect
4. #19133709C2semidirect
5. #19133257C2semidirect
6. #19133013C2semidirect
7. C43×#3152C2semidirect
19. #14583186C22semidirect
20. #14582952C22semidirect
21. #14582648C22semidirect
23. #14582628C22semidirect
26. C2×C4⋊Q8×C43C22semidirect
28. C43×#494C22semidirect
30. C4×M16×C43C22semidirect
169. #3152D43semidirect
189. C2×C4⋊Q8C2×D43semidirect
192. #494C2×D43semidirect
198. C4×M16C2×D43semidirect
202. C43#18715semidirect

Maximal subgroups

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

subgroupindex
#191490002
#191457232
#191337382
#191337092
#191332572
#191330132
C43×#31522
#1871543