Finite Group Databasebeta

← #24994399

Subgroups — #24994399

Order 11008.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 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#19137930C211yesmaximal, characteristic
2#19136952C211yesmaximal, characteristic, complemented
3#19134538C212yesmaximal
4#19132449C212yesmaximal, complemented
5C43×#1463C211yesmaximal, characteristic, complemented
6C22×#10621898C2211yescharacteristic
7C2×#13063308—21no—
8C2×#13063284—21no—
9C2×#13063251—24no—
10#14582819C412yescomplemented
11#14582816C412yescomplemented
12#14582778—21no—
13#14582770—22no—
14#14582762C2211yescharacteristic
15C4⋊C4×Dic43—24no—
16C4×C22⋊Dic43—21no—
17C4×C22⋊Dic43C2212yes—
18C22×C4⋊C4×C43C2211yescharacteristic
19C43×#407C2212yes—
20C43×#324—22no—
21C2×#10621898—22no—
22C2×#10621898—22no—
23C2×#10621898—44no—
24C22×Dic86C2×C411yescharacteristic
25C22×D172C2×C411yescharacteristic
26C22×C22⋊D43—21no—
27C22×C4×D43—21no—
28#13063308—44no—
29#13063284—28no—
30C2×C4⋊Dic43—22no—
31C2×Dic43⋊C4—22no—
32C2×C22⋊Dic43—22no—
33C2×C22⋊Dic43—21no—
34C2×(C2×D43)⋊C4—22no—
35C2×(C2×D43)⋊C4—22no—
36C2×C4×Dic43—22no—
37C2×C4×Dic43—22no—
38C2×C4×Dic43D414yes—
39#13063251—48no—
40C43⋊#95—22no—
41C43⋊#95—22no—
42C43⋊#95—22no—
43C43⋊#95—21no—
44C43⋊#95C2×C412yes—
45C23×C4×C43D411yescharacteristic
46C23×C4×C43D411yescharacteristic
47C23×C4×C43C2311yescharacteristic
48C2×M16×C43—22no—
49C2×C4⋊C4×C43—22no—
50C2×C4⋊C4×C43—22no—
51C2×C4⋊C4×C43C2×C412yes—
52C2×C4⋊C4×C43D412yes—
53C22⋊C8×C43—22no—
54C22⋊C8×C43—22no—
55C24×C43C2×D411yescharacteristic
56C23×D43—21no—
57C22×C4×C43—24no—
58C22×C4×C43—21no—
59C22×C4×C43—21no—
60C22×C4×C43—21no—
61C22×C4×C43—21no—
62C22×C4×C43—42no—
63C22×C4×C43—42no—
64C22×C4×C43C22⋊C412yes—
65C22×C4×C43C22⋊C412yes—
66C22×C4×C43C22×C411yescharacteristic
67C22×C4×C43C2×D411yescharacteristic
68C22×C4×C43C2×D412yes—
69C22×C4×C43#4112yes—
70C2×C22⋊D43—24no—
71C2×C22⋊D43—42no—
72C2×C22⋊D43—41no—
73C2×C22⋊D43—41no—
74C22×Dic43—21no—
75C22×Dic43—22no—
76C22×Dic43—24no—
77#10621898—28no—
78#10621898—44no—
79#10621898—44no—
80#10621898—44no—
81C2×D172—22no—
82C2×D172—42no—
83C2×C4×D43—42no—
84C2×C4×D43—42no—
85C2×C4×D43—42no—
86C2×Dic86—22no—
87C2×Dic86—42no—
88M16×C43—44no—
89C2×C8×C43—42no—
90C4⋊C4×C43—24no—
91C4⋊C4×C43—24no—
92C22⋊Dic43—28no—
93C22⋊Dic43—42no—
94(C2×D43)⋊C4—44no—
95(C2×D43)⋊C4—44no—
96C4⋊Dic43—28no—
97Dic43⋊C4—44no—
98C4×Dic43—28no—
99C4×Dic43—44no—
100C4×Dic43—82no—
101C23×C43—22no—
102C23×C43—22no—
103C23×C43—41no—
104C23×C43C2×C22⋊C411yescharacteristic
105C23×C43#12011yescharacteristic
106C23×C43#12311yescharacteristic
107C22×D43—22no—
108C22×D43—41no—
109C22×D43—42no—
110C22×D43—41no—
111C22×D43—42no—
112C2×C4×C43—22no—
113C2×C4×C43—22no—
114C2×C4×C43—24no—
115C2×C4×C43—22no—
116C2×C4×C43—22no—
117C2×C4×C43—24no—
118C2×C4×C43—22no—
119C2×C4×C43—41no—
120C2×C4×C43—41no—
121C2×C4×C43—42no—
122C2×C4×C43—42no—
123C2×C4×C43—42no—
124C2×C4×C43—81no—
125C2×C4×C43#10414yescomplemented
126C2×C4×C43C4×D412yes—
127C2×C4×C43#12112yes—
128C22⋊D43—44no—
129C22⋊D43—44no—
130C2×Dic43—28no—
131C2×Dic43—22no—
132C2×Dic43—41no—
133C2×Dic43—44no—
134C2×Dic43—41no—
135C2×Dic43—44no—
136C2×Dic43—42no—
137C2×Dic43—42no—
138C2×Dic43—42no—
139D172—24no—
140D172—42no—
141C4×D43—48no—
142Dic86—24no—
143Dic86—42no—
144C8×C43—82no—
145#9476—431nomaximal, Sylow
146C22×C43—21no—
147C22×C43—21no—
148C22×C43—21no—
149C22×C43—21no—
150C22×C43—21no—
151C22×C43—21no—
152C22×C43—21no—
153C22×C43—44no—
154C22×C43#38611yescharacteristic
155C22×C43C2×#10412yes—
156C22×C43B(4,𝔽2)12yes—
157C2×D43—44no—
158C2×D43—42no—
159C2×D43—82no—
160C2×D43—81no—
161C2×D43—81no—
162C4×C43—22no—
163C4×C43—22no—
164C4×C43—24no—
165C4×C43—42no—
166Dic43—48no—
167Dic43—42no—
168Dic43—82no—
169#2406—431no—
170#2274—431no—
171#1789—432no—
172#1463D4311yescharacteristic, complemented
173#1284—432no—
174C2×C43—22no—
175C2×C43—22no—
176C2×C43—41no—
177C2×C43#188911yescharacteristic
178C2×C43#189611yescharacteristic
179C2×C43#191211yescharacteristic
180D43—82no—
181C22×#41—431no—
182C2×#123—861no—
183C2×#117—861no—
184C22×C4⋊C4C2×D4311yescharacteristic
185C2×#104—864no—
186#407C2×D4312yes—
187#395—432no—
188#394—432no—
189#386—861no—
190#385—862no—
191#381—431no—
192C4×C4⋊C4—864no—
193C4×C22⋊C4—432no—
194C4×C22⋊C4—861no—
195#324—22no—
196C43#947611yescharacteristic, complemented, Sylow
197C2×#41—862no—
198C2×#41—862no—
199C2×#41—1724no—
200C22×Q8—431no—
201C22×D4—431no—
202C22×D4—861no—
203C23×C4—861no—
204C23×C4D17211yescharacteristic
205C23×C4D17211yescharacteristic
206C23×C4C22×D4311yescharacteristic
207C2×M16—22no—
208#123—1724no—
209#117—868no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—862no—
213C2×C4⋊C4—862no—
214C2×C4⋊C4C4×D4312yes—
215C2×C4⋊C4C22⋊D4312yes—
216C2×C22⋊C4—862no—
217C2×C22⋊C4—861no—
218C2×C22⋊C4—862no—
219C2×C22⋊C4—862no—
220C2×C42—434no—
221C2×C42—862no—
222C2×C42—862no—
223#104—1728no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226#95—432no—
227#95—862no—
228#95—862no—
229#95—862no—
230#95—861no—
231C24—861no—
232C24C2×D17211yescharacteristic
233#41—868no—
234#41—1724no—
235#41—1724no—
236#41—1724no—
237C2×Q8—862no—
238C2×Q8—1722no—
239C2×D4—864no—
240C2×D4—862no—
241C2×D4—1722no—
242C2×D4—1722no—
243C2×D4—1721no—
244C2×D4—1721no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—861no—
253C22×C4—864no—
254C22×C4—862no—
255C22×C4—1722no—
256C22×C4—1722no—
257C22×C4—1722no—
258C22×C4(C2×D43)⋊C412yes—
259C22×C4(C2×D43)⋊C412yes—
260C22×C4C2×C4×D4311yescharacteristic
261C22×C4D4×D4312yes—
262C22×C4#1062190012yes—
263C22×C4C2×C22⋊D4311yescharacteristic
264M16—44no—
265C2×C8—42no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—868no—
269C4⋊C4—1724no—
270C22⋊C4—868no—
271C22⋊C4—1724no—
272C22⋊C4—1722no—
273C22⋊C4—1724no—
274C42—868no—
275C42—3442no—
276C42—1724no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—862no—
281C23—1721no—
282C23—1722no—
283C23—1722no—
284C23—1721no—
285C23C2×(C2×D43)⋊C411yescharacteristic
286C23#1306329511yescharacteristic
287C23#1306330911yescharacteristic
288Q8—864no—
289Q8—1722no—
290D4—864no—
291D4—1722no—
292D4—1724no—
293D4—1724no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—868no—
308C2×C4—862no—
309C2×C4—1728no—
310C2×C4—1722no—
311C2×C4—1722no—
312C2×C4—1722no—
313C2×C4—1721no—
314C2×C4—1724no—
315C2×C4—1724no—
316C2×C4—1721no—
317C2×C4#1306325114yescomplemented
318C2×C4(C22⋊D43)⋊C412yes—
319C2×C4#1306329712yes—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—3442no—
330C22—3441no—
331C22—3441no—
332C22—1724no—
333C22—1722no—
334C22#1458277611yescharacteristic
335C22C2×#1306325112yes—
336C22#1458308712yes—
337C4—22no—
338C4—22no—
339C4—24no—
340C4—42no—
341C4—3442no—
342C4—1728no—
343C4—1722no—
344C2—22no—
345C2—22no—
346C2—41no—
347C2—3442no—
348C2#1913500111yescharacteristic
349C2#1913503611yescharacteristic
350C2#1913513011yescharacteristic
subgroupG/H
centerC22#14582776
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1463C2
socleC22×C43#386
prime radicalC22×#10621898C22
cosocleC23×C4×C43C23
hypercenter#1463D43
hypocenterC43#9476
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #19136952C2semidirect
4. #19132449C2semidirect
5. C43×#1463C2semidirect
10. #14582819C4semidirect
11. #14582816C4semidirect
125. C2×C4×C43#104semidirect
172. #1463D43semidirect
196. C43#9476semidirect
317. C2×C4#13063251semidirect

Maximal subgroups

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

subgroupindex
#191379302
#191369522
#191345382
#191324492
C43×#14632
#947643