Finite Group Databasebeta

← #24989227

Subgroups — #24989227

Order 11008.

1134 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#13063303C211yesmaximal, characteristic
2#19134518C212yesmaximal, complemented
3#19132925C211yesmaximal, characteristic, complemented
4#19132921C212yesmaximal
5C43×#1463C211yesmaximal, characteristic, complemented
6C23×Dic86C2211yescharacteristic
7C23×C4×D43C411yescharacteristic
8C2×#13063303—24no—
9C2×#13063303—24no—
10C2×#13063303—22no—
11C2×#13063303—22no—
12C22×Dic43⋊C4C411yescharacteristic
13C22×Dic43⋊C4C411yescharacteristic
14C22×(C2×D43)⋊C4C411yescharacteristic
15C22×(C2×D43)⋊C4C2211yescharacteristic
16C2×#13063243—24no—
17#14582863C2212yes—
18#14582625—22no—
19#14582622—24no—
20C22×C4⋊C4×C43C2211yescharacteristic
21C43×#407C2212yes—
22C43×#324—22no—
23C24×D43C2×C411yescharacteristic
24C22×Dic86—21no—
25C22×Dic86—22no—
26C22×Dic86—21no—
27C22×Dic86—44no—
28C22×Dic86D414yes—
29C23×Dic43C2×C411yescharacteristic
30C23×Dic43C2×C411yescharacteristic
31C23×Dic43C2×C411yescharacteristic
32C22×C4×D43—22no—
33C22×C4×D43—22no—
34C22×C4×D43—22no—
35C22×C4×D43—44no—
36C2×#10621877—22no—
37#13063303—216no—
38#13063303—216no—
39C2×Dic43⋊C4—22no—
40C2×Dic43⋊C4—22no—
41C2×Dic43⋊C4—22no—
42C2×Dic43⋊C4—22no—
43C2×Dic43⋊C4—22no—
44C2×Dic43⋊C4—22no—
45C2×(C2×D43)⋊C4—22no—
46C2×(C2×D43)⋊C4—22no—
47C2×(C2×D43)⋊C4—21no—
48C2×(C2×D43)⋊C4—22no—
49C2×(C2×D43)⋊C4—21no—
50C2×(C2×D43)⋊C4—22no—
51C2×(C2×D43)⋊C4D414yes—
52#13063243—48no—
53C22⋊(C43⋊C8)—22no—
54C22⋊(C43⋊C8)—22no—
55C23×C4×C43C2×C411yescharacteristic
56C23×C4×C43C2×C411yescharacteristic
57C23×C4×C43C2311yescharacteristic
58C2×M16×C43—22no—
59C2×C4⋊C4×C43—22no—
60C2×C4⋊C4×C43—22no—
61C2×C4⋊C4×C43D412yes—
62C2×C4⋊C4×C43D412yes—
63C22⋊C8×C43—22no—
64C22⋊C8×C43—22no—
65C24×C43C22×C411yescharacteristic
66C23×D43—21no—
67C23×D43—21no—
68C23×D43—24no—
69C23×D43—42no—
70C23×D43—42no—
71C23×D43C22⋊C412yes—
72C22×C4×C43—21no—
73C22×C4×C43—21no—
74C22×C4×C43—21no—
75C22×C4×C43—21no—
76C22×C4×C43—42no—
77C22×C4×C43—42no—
78C22×C4×C43—42no—
79C22×C4×C43C22⋊C412yes—
80C22×C4×C43C22⋊C412yes—
81C22×C4×C43C2×D411yescharacteristic
82C22×C4×C43C2×D412yes—
83C22×C4×C43C2×D412yes—
84C22×C4×C43C2×D411yescharacteristic
85C22×Dic43—24no—
86C22×Dic43—21no—
87C22×Dic43—21no—
88C22×Dic43—21no—
89C22×Dic43—21no—
90C22×Dic43—21no—
91C22×Dic43—21no—
92C22×Dic43—42no—
93C22×Dic43—42no—
94C22×Dic43C22⋊C412yes—
95C22×Dic43C22⋊C412yes—
96C22×Dic43C22⋊C412yes—
97C2×C4×D43—28no—
98C2×C4×D43—48no—
99C2×C4×D43—44no—
100C2×C4×D43—48no—
101C2×C4×D43—44no—
102C2×Dic86—24no—
103C2×Dic86—24no—
104C2×Dic86—28no—
105C2×Dic86—44no—
106C2×Dic86—44no—
107C2×Dic86—48no—
108C2×Dic86—44no—
109M16×C43—44no—
110C2×C8×C43—42no—
111C4⋊C4×C43—24no—
112C4⋊C4×C43—24no—
113(C2×D43)⋊C4—28no—
114(C2×D43)⋊C4—24no—
115(C2×D43)⋊C4—24no—
116Dic43⋊C4—28no—
117Dic43⋊C4—28no—
118#10621877—44no—
119C2×C43⋊C8—42no—
120C23×C43—22no—
121C23×C43—22no—
122C23×C43—22no—
123C23×C43C2×C22⋊C411yescharacteristic
124C23×C43C2×C22⋊C411yescharacteristic
125C23×C43C2×C22⋊C411yescharacteristic
126C22×D43—24no—
127C22×D43—24no—
128C22×D43—22no—
129C22×D43—22no—
130C22×D43—42no—
131C22×D43—416no—
132C22×D43—42no—
133C22×D43—42no—
134C22×D43—42no—
135C22×D43—42no—
136C22×D43—42no—
137C22×D43#10114yescomplemented
138C2×C4×C43—24no—
139C2×C4×C43—22no—
140C2×C4×C43—22no—
141C2×C4×C43—24no—
142C2×C4×C43—41no—
143C2×C4×C43—42no—
144C2×C4×C43—41no—
145C2×C4×C43—42no—
146C2×C4×C43—42no—
147C2×C4×C43—42no—
148C2×C4×C43—42no—
149C2×C4×C43—42no—
150C2×C4×C43#12012yes—
151C2×C4×C43#12012yes—
152C2×Dic43—24no—
153C2×Dic43—24no—
154C2×Dic43—22no—
155C2×Dic43—22no—
156C2×Dic43—22no—
157C2×Dic43—22no—
158C2×Dic43—24no—
159C2×Dic43—42no—
160C2×Dic43—42no—
161C2×Dic43—42no—
162C2×Dic43—42no—
163C2×Dic43—42no—
164C2×Dic43#9914yes—
165C4×D43—416no—
166Dic86—48no—
167Dic86—48no—
168C8×C43—82no—
169C43⋊C8—82no—
170#8550—431nomaximal, Sylow
171C22×C43—28no—
172C22×C43—21no—
173C22×C43—21no—
174C22×C43—21no—
175C22×C43—21no—
176C22×C43—21no—
177C22×C43—21no—
178C22×C43#37911yescharacteristic
179C22×C43C2×#9912yes—
180C22×C43C2×#10112yes—
181C22×C43#41312yes—
182C2×D43—24no—
183C2×D43—48no—
184C2×D43—44no—
185C2×D43—48no—
186C2×D43—42no—
187C2×D43—42no—
188C2×D43—44no—
189C4×C43—22no—
190C4×C43—22no—
191C4×C43—42no—
192C4×C43—42no—
193Dic43—24no—
194Dic43—42no—
195Dic43—42no—
196C22×C22⋊Q8—431no—
197#1784—432no—
198#1463—431no—
199#1463D4311yescharacteristic, complemented
200#1462—432no—
201C2×C43—22no—
202C2×C43—22no—
203C2×C43—22no—
204C2×C43#178111yescharacteristic
205C2×C43#178411yescharacteristic
206C2×C43#178511yescharacteristic
207D43—44no—
208C23×Q8—431no—
209C24×C4—431no—
210C2×C22⋊Q8—864no—
211C2×C22⋊Q8—864no—
212C2×C22⋊Q8—862no—
213C2×C22⋊Q8—862no—
214C22×C4⋊C4—431no—
215C22×C4⋊C4—431no—
216C22×C4⋊C4C2×D4311yescharacteristic
217C22×C22⋊C4—431no—
218C22×C22⋊C4—431no—
219C2×#101—864no—
220#407—432no—
221#407C2×D4312yes—
222#324—22no—
223#324—862no—
224#323—864no—
225C43#855011yescharacteristic, complemented, Sylow
226C25—431no—
227C22×Q8—434no—
228C22×Q8—861no—
229C22×Q8—862no—
230C22×Q8—861no—
231C22×Q8—1724no—
232C23×C4—431no—
233C23×C4—431no—
234C23×C4—431no—
235C23×C4—862no—
236C23×C4—862no—
237C23×C4—862no—
238C23×C4—1724no—
239C23×C4C4×D4311yescharacteristic
240C23×C4C4×D4311yescharacteristic
241C23×C4C22×D4311yescharacteristic
242C2×M16—22no—
243C2×M16—862no—
244C22⋊Q8—8616no—
245C22⋊Q8—8616no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—862no—
249C2×C4⋊C4—862no—
250C2×C4⋊C4—862no—
251C2×C4⋊C4—862no—
252C2×C4⋊C4—862no—
253C2×C4⋊C4—862no—
254C2×C4⋊C4D17212yes—
255C2×C4⋊C4C22⋊D4312yes—
256C2×C22⋊C4—434no—
257C2×C22⋊C4—862no—
258C2×C22⋊C4—862no—
259C2×C22⋊C4—862no—
260C2×C22⋊C4—861no—
261C2×C22⋊C4—862no—
262C2×C22⋊C4—861no—
263#101—1728no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—862no—
267C22⋊C8—862no—
268C24—432no—
269C24—864no—
270C24—861no—
271C24—861no—
272C24—1722no—
273C24—1722no—
274C24C2×C4×D4311yescharacteristic
275C2×Q8—868no—
276C2×Q8—864no—
277C2×Q8—864no—
278C2×Q8—1724no—
279C2×Q8—1724no—
280C2×Q8—1728no—
281C2×Q8—1724no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—432no—
290C22×C4—432no—
291C22×C4—432no—
292C22×C4—868no—
293C22×C4—864no—
294C22×C4—861no—
295C22×C4—861no—
296C22×C4—861no—
297C22×C4—861no—
298C22×C4—861no—
299C22×C4—861no—
300C22×C4—1724no—
301C22×C4—1728no—
302C22×C4—1724no—
303C22×C4—1728no—
304C22×C4—1722no—
305C22×C4—1722no—
306C22×C4(C2×D43)⋊C412yes—
307C22×C4(C2×D43)⋊C412yes—
308C22×C4C2×D17211yescharacteristic
309C22×C4D4×D4312yes—
310C22×C4D4×D4312yes—
311C22×C4C2×C22⋊D4311yescharacteristic
312M16—44no—
313M16—1724no—
314C2×C8—42no—
315C2×C8—1722no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—868no—
319C4⋊C4—868no—
320C22⋊C4—868no—
321C22⋊C4—864no—
322C22⋊C4—864no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—434no—
327C23—864no—
328C23—864no—
329C23—862no—
330C23—862no—
331C23—1722no—
332C23—1722no—
333C23—1722no—
334C23—17216no—
335C23—1722no—
336C23—1722no—
337C23—1722no—
338C23C22⋊C4×D4311yescharacteristic
339C23C22⋊C4×D4311yescharacteristic
340C23C2×(C2×D43)⋊C411yescharacteristic
341Q8—1728no—
342Q8—1728no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—434no—
356C2×C4—864no—
357C2×C4—864no—
358C2×C4—862no—
359C2×C4—862no—
360C2×C4—864no—
361C2×C4—862no—
362C2×C4—862no—
363C2×C4—17216no—
364C2×C4—1722no—
365C2×C4—1722no—
366C2×C4—1722no—
367C2×C4—1722no—
368C2×C4—1722no—
369C2×C4#1306329312yes—
370C2×C4#1306329512yes—
371C8—82no—
372C8—3442no—
373C22—28no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—864no—
381C22—1724no—
382C22—1728no—
383C22—1724no—
384C22—1728no—
385C22—1722no—
386C22—1722no—
387C22#1458275411yescharacteristic
388C22#99×D4312yes—
389C22#101×D4312yes—
390C22#1458288112yes—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—42no—
395C4—864no—
396C4—1722no—
397C4—1722no—
398C2—22no—
399C2—22no—
400C2—22no—
401C2—1724no—
402C2#1913450811yescharacteristic
403C2#1913451711yescharacteristic
404C2#1913452111yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC24×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1463C2
socleC22×C43#379
prime radicalC24×D43C2×C4
cosocleC23×C4×C43C23
hypercenter#1463D43
hypocenterC43#8550
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #19134518C2semidirect
3. #19132925C2semidirect
5. C43×#1463C2semidirect
137. C22×D43#101semidirect
199. #1463D43semidirect
225. C43#8550semidirect

Maximal subgroups

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

subgroupindex
C22×#130633032
#191345182
#191329252
#191329212
C43×#14632
#855043