Finite Group Databasebeta

← #24997066

Subgroups — #24997066

Order 11008.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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#19145348C211yesmaximal, characteristic
2C2×#14583212C211yesmaximal, characteristic
3#19135431C211yesmaximal, characteristic
4#19134681C211yesmaximal, characteristic, complemented
5#19133167C211yesmaximal, characteristic, complemented
6#19132443C211yesmaximal, characteristic, complemented
7C43×#2503C211yesmaximal, characteristic, complemented
8C2×#13063299C2211yescharacteristic
9#14583227—21no—
10#14583217—21no—
11#14583212—22no—
12#14583212—22no—
13#14583130—22no—
14#14583049—21no—
15#14583038—21no—
16C2×#13063258—21no—
17C2×#13063245—21no—
18C2×#13063239—22no—
19C2×C22⋊(C43⋊C8)C2211yescharacteristic
20C2×C22⋊(C43⋊C8)C2211yescharacteristic
21#14582759C2211yescharacteristic
22#14582661—22no—
23#14582636—22no—
24C2×C22⋊Q8×C43C2211yescharacteristic
25C22×C4⋊C4×C43C2211yescharacteristic
26C43×#400—22no—
27C43×#391—21no—
28C43×#382—21no—
29C43×#381—22no—
30C4×C22⋊C4×C43C2211yescharacteristic
31C22×D172D411yescharacteristic
32C22×C22⋊D43—21no—
33C2×#10621884—22no—
34C22×C43⋊C8—21no—
35C22×C43⋊C8D412yes—
36#13063299—21no—
37#13063299—21no—
38#13063299—21no—
39#13063299—21no—
40C2×C4⋊Dic43—21no—
41C2×C4⋊Dic43D411yescharacteristic
42C2×C22⋊Dic43—21no—
43C2×(C2×D43)⋊C4—21no—
44#13063258—24no—
45#13063256—22no—
46#13063249—22no—
47#13063245—22no—
48#13063245—24no—
49#13063239—44no—
50C22⋊(C43⋊C8)—21no—
51C22⋊(C43⋊C8)—21no—
52C22⋊(C43⋊C8)—21no—
53C22⋊(C43⋊C8)D412yes—
54C43⋊#95—21no—
55C22×Q8×C43D411yescharacteristic
56C23×C4×C43—21no—
57C23×C4×C43C2311yescharacteristic
58C22⋊Q8×C43—21no—
59C22⋊Q8×C43—21no—
60C22⋊Q8×C43—21no—
61C22⋊Q8×C43—21no—
62C2×C4⋊C4×C43—21no—
63C2×C4⋊C4×C43—21no—
64C2×C4⋊C4×C43—21no—
65C2×C4⋊C4×C43—21no—
66C2×C4⋊C4×C43—21no—
67C2×C4⋊C4×C43—21no—
68C2×C4⋊C4×C43D412yes—
69C2×C4⋊C4×C43D411yescharacteristic
70C2×C22⋊C4×C43—21no—
71C2×C22⋊C4×C43D412yes—
72C2×C42×C43—21no—
73#95×C43—24no—
74#95×C43—24no—
75#95×C43—21no—
76C24×C43C2×D411yescharacteristic
77C23×D43—21no—
78C2×Q8×C43—21no—
79C2×Q8×C43—21no—
80C2×Q8×C43—42no—
81C22×C4×C43—22no—
82C22×C4×C43—21no—
83C22×C4×C43—21no—
84C22×C4×C43—22no—
85C22×C4×C43—42no—
86C22×C4×C43—41no—
87C22×C4×C43—41no—
88C22×C4×C43—42no—
89C22×C4×C43—42no—
90C22×C4×C43C2×D411yescharacteristic
91C22×C4×C43C2×D411yescharacteristic
92C22×C4×C43C2×D411yescharacteristic
93C22×C4×C43C2×D411yescharacteristic
94C22×C4×C43C2×D411yescharacteristic
95C22×C4×C43#4111yescharacteristic
96C2×C22⋊D43—41no—
97C2×C22⋊D43—41no—
98C2×C22⋊D43—42no—
99C2×C22⋊D43—42no—
100C22×Dic43—21no—
101C22×Dic43—41no—
102C2×D172—21no—
103C2×D172—21no—
104C2×D172—42no—
105C4⋊C4×C43—22no—
106C4⋊C4×C43—21no—
107C4⋊C4×C43—22no—
108C4⋊C4×C43—21no—
109C4⋊C4×C43—41no—
110C4⋊C4×C43—41no—
111C4⋊C4×C43—41no—
112C4⋊C4×C43—41no—
113C22⋊C4×C43—22no—
114C22⋊C4×C43—22no—
115C22⋊C4×C43—41no—
116C22⋊C4×C43—41no—
117C42×C43—42no—
118C22⋊Dic43—42no—
119#10621884—82no—
120(C2×D43)⋊C4—41no—
121(C2×D43)⋊C4—41no—
122C4⋊Dic43—21no—
123C4⋊Dic43—21no—
124C4⋊Dic43—42no—
125C2×C43⋊C8—22no—
126C2×C43⋊C8—22no—
127C2×C43⋊C8—42no—
128C2×C43⋊C8—42no—
129C23×C43—41no—
130C23×C43—41no—
131C23×C43#12011yescharacteristic
132C23×C43#12111yescharacteristic
133C23×C43#12711yescharacteristic
134C23×C43#13512yes—
135C23×C43AGL(1,ℤ/8)12yes—
136C22×D43—41no—
137C22×D43—42no—
138C22×D43—81no—
139C22×D43—81no—
140Q8×C43—42no—
141Q8×C43—81no—
142C2×C4×C43—22no—
143C2×C4×C43—22no—
144C2×C4×C43—22no—
145C2×C4×C43—41no—
146C2×C4×C43—41no—
147C2×C4×C43—41no—
148C2×C4×C43—42no—
149C2×C4×C43—42no—
150C2×C4×C43—42no—
151C2×C4×C43—42no—
152C2×C4×C43—41no—
153C2×C4×C43—41no—
154C2×C4×C43—42no—
155C2×C4×C43—42no—
156C2×C4×C43—44no—
157C2×C4×C43—82no—
158C2×C4×C43—81no—
159C2×C4×C43—81no—
160C2×C4×C43#12011yescharacteristic
161C2×C4×C43#12011yescharacteristic
162C2×C4×C43#12111yescharacteristic
163C2×C4×C43#12111yescharacteristic
164C22⋊D43—82no—
165C22⋊D43—81no—
166C22⋊D43—81no—
167C2×Dic43—41no—
168C2×Dic43—42no—
169C2×Dic43—81no—
170C2×Dic43—81no—
171C2×Dic43—81no—
172D172—42no—
173D172—42no—
174C43⋊C8—42no—
175C43⋊C8—42no—
176#9953—431nomaximal, Sylow
177C22×C43—21no—
178C22×C43—21no—
179C22×C43—21no—
180C22×C43—21no—
181C22×C43—21no—
182C22×C43—21no—
183C22×C43—42no—
184C22×C43—42no—
185C22×C43#39211yescharacteristic
186C22×C43#44912yes—
187C22×C43#46312yes—
188C22×C43#49511yescharacteristic
189C22×C43#49611yescharacteristic
190C2×D43—81no—
191C2×D43—81no—
192C2×D43—81no—
193C2×D43—162no—
194C4×C43—21no—
195C4×C43—21no—
196C4×C43—21no—
197C4×C43—82no—
198C4×C43—81no—
199C4×C43—82no—
200C4×C43—81no—
201C4×C43#45812yes—
202Dic43—81no—
203Dic43—161no—
204#3187—431no—
205C2×#480—431no—
206#2503D4311yescharacteristic, complemented
207#1964—431no—
208#1823—431no—
209#1520—431no—
210#1282—431no—
211C2×C43—21no—
212C2×C43—22no—
213C2×C43—22no—
214C2×C43—21no—
215C2×C43#200911yescharacteristic
216C2×C43#201211yescharacteristic
217C2×C43#202211yescharacteristic
218D43—161no—
219C2×C22⋊Q8C2×D4311yescharacteristic
220C2×#121—431no—
221C22×C4⋊C4C2×D4311yescharacteristic
222#483—861no—
223#481—861no—
224#480—862no—
225#480—862no—
226#465—862no—
227#450—861no—
228#448—861no—
229C2×#107—861no—
230C2×#102—861no—
231C2×#99—862no—
232C2×C22⋊C8—431no—
233C2×C22⋊C8—431no—
234#400—22no—
235#391—21no—
236#382—21no—
237#381—22no—
238#380—431no—
239C4×C22⋊C4C2×D4311yescharacteristic
240#340—862no—
241#328—862no—
242C43#995311yescharacteristic, complemented, Sylow
243C22×Q8C22⋊D4311yescharacteristic
244C22×D4—431no—
245C22×D4—861no—
246C23×C4—21no—
247C23×C4C22×D4311yescharacteristic
248C2×SD16—862no—
249C22×C8—432no—
250C22×C8—861no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255#121—861no—
256#121—861no—
257#121—861no—
258#121—861no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—431no—
266C2×C4⋊C4—861no—
267C2×C4⋊C4C22⋊D4312yes—
268C2×C4⋊C4C22⋊D4311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—861no—
271C2×C22⋊C4—861no—
272C2×C22⋊C4C22⋊D4312yes—
273C2×C42—21no—
274#107—864no—
275#106—862no—
276#103—862no—
277#102—864no—
278#102—862no—
279#99—1724no—
280C22⋊C8—432no—
281C22⋊C8—861no—
282C22⋊C8—861no—
283C22⋊C8—861no—
284#95—24no—
285#95—24no—
286#95—21no—
287#95—861no—
288C24—861no—
289C24C2×C22⋊D4311yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—42no—
293C2×D4—861no—
294C2×D4—861no—
295C2×D4—1722no—
296C2×D4—1721no—
297C2×D4—1721no—
298C2×D4—1722no—
299C2×D4—1722no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—861no—
310C22×C4—1721no—
311C22×C4D4×D4311yescharacteristic
312C22×C4D4×D4311yescharacteristic
313C22×C4D4×D4311yescharacteristic
314C22×C4#1062190011yescharacteristic
315C22×C4C2×C22⋊D4311yescharacteristic
316C22×C4C2×C22⋊D4311yescharacteristic
317SD16—3442no—
318C2×C8—862no—
319C2×C8—862no—
320C2×C8—1722no—
321C2×C8—1722no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—861no—
331C4⋊C4—861no—
332C4⋊C4—1722no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—1722no—
338C22⋊C4—1721no—
339C22⋊C4—1721no—
340C42—42no—
341C23—41no—
342C23—41no—
343C23—3441no—
344C23—3441no—
345C23—1721no—
346C23—1722no—
347C23#1306329411yescharacteristic
348C23#1306329611yescharacteristic
349C23#1306332111yescharacteristic
350C23#1306334512yes—
351C23#1306335012yes—
352Q8—42no—
353Q8—81no—
354D4—3442no—
355D4—3441no—
356D4—3441no—
357D4—1722no—
358D4—1722no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—3441no—
378C2×C4—3441no—
379C2×C4—3441no—
380C2×C4—1721no—
381C2×C4—1722no—
382C2×C4#1306329311yescharacteristic
383C2×C4#1306329311yescharacteristic
384C2×C4#1306329711yescharacteristic
385C2×C4#1306329711yescharacteristic
386C8—1722no—
387C8—1722no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—3441no—
397C22—3441no—
398C22—3441no—
399C22—6882no—
400C22#1458280711yescharacteristic
401C22#1458304412yes—
402C22#1458311112yes—
403C22#1458328111yescharacteristic
404C22#1458328911yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—82no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4—3441no—
413C4—6881no—
414C4#1458308812yes—
415C2—22no—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—6881no—
420C2#1913565511yescharacteristic
421C2#1913567211yescharacteristic
422C2#1913572311yescharacteristic
subgroupG/H
centerC22#14582807
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC43×#2503C2
socleC22×C43#392
prime radicalC2×#13063299C22
hypercenter#2503D43
hypocenterC43#9953
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
4. #19134681C2semidirect
5. #19133167C2semidirect
6. #19132443C2semidirect
7. C43×#2503C2semidirect
206. #2503D43semidirect
242. C43#9953semidirect

Maximal subgroups

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

subgroupindex
#191453482
C2×#145832122
#191354312
#191346812
#191331672
#191324432
C43×#25032
#995343