Finite Group Databasebeta

← #25245324

Subgroups — #25245324

Order 11008.

3580 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 3580 conjugacy classes of subgroups — too many to draw. Only its 255 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×#13063373C211yesmaximal, characteristic
2C2×#14583081C214yesmaximal, complemented
3#19143853C212yesmaximal, complemented
4#19134553C214yesmaximal, complemented
5#19134540C212yesmaximal, complemented
6#19134519C211yesmaximal, characteristic, complemented
7C43×#2434C211yesmaximal, characteristic, complemented
8C2×#13063373—216no—
9C2×#13063373—24no—
10C2×#13063373—22no—
11C2×#13063373C2214yes—
12C22×#10621902—22no—
13C22×#10621898—22no—
14C22×#10621898C2212yes—
15C22×Q8×D43—22no—
16C2×#13063356C2214yescomplemented
17#14583081—216no—
18#14583059—24no—
19#14583054—22no—
20#14583049—22no—
21C2×#13063253C2218yescomplemented
22#14582898C2214yescomplemented
23C2×#13063244C2214yescomplemented
24#14582863C2212yes—
25C23×Q8×C43C2211yescharacteristic
26C2×C4⋊Q8×C43C2212yescomplemented
27C2×C22⋊Q8×C43C2211yescharacteristic
28C43×#395—22no—
29C43×#391C2212yescomplemented
30C43×#385—22no—
31C4×C22⋊C4×C43C2211yescharacteristic
32#13063373—232no—
33#13063373—232no—
34#13063373—232no—
35#13063373—28no—
36#13063373—216no—
37#13063373D4116yes—
38C2×#10621902—232no—
39C2×#10621902—28no—
40C2×#10621902—28no—
41C2×#10621898—28no—
42C2×#10621898—232no—
43C2×#10621898—28no—
44C2×#10621898—22no—
45C2×#10621898—22no—
46C2×#10621898—216no—
47C2×#10621898D414yes—
48C2×#10621898C2314yes—
49C22×Dic86—22no—
50C22×Dic86D412yes—
51C2×Q8×D43—28no—
52C2×Q8×D43—232no—
53C2×Q8×D43—28no—
54C22×D172—22no—
55C22×D172D412yes—
56C22×C22⋊D43—22no—
57C22×C4×D43—24no—
58C22×C4×D43—22no—
59#13063356—216no—
60C2×#10621885—24no—
61C2×#10621884—24no—
62C2×#10621877C2314yes—
63#13063253—216no—
64#13063248—24no—
65#13063245—24no—
66#13063244—28no—
67C22⋊(C43⋊C8)—22no—
68C22⋊(C43⋊C8)—22no—
69C22×Q8×C43—28no—
70C22×Q8×C43—22no—
71C22×Q8×C43—22no—
72C22×Q8×C43D412yes—
73C22×Q8×C43C2311yescharacteristic
74C22×Q8×C43C2312yes—
75C23×C4×C43D412yes—
76C23×C4×C43C2311yescharacteristic
77C4⋊Q8×C43—28no—
78C22⋊Q8×C43—22no—
79C22⋊Q8×C43—22no—
80C2×C4⋊C4×C43—21no—
81C2×C4⋊C4×C43—24no—
82C2×C4⋊C4×C43C2311yescharacteristic
83C2×C22⋊C4×C43—21no—
84C2×C22⋊C4×C43—21no—
85C2×C42×C43C2312yes—
86#95×C43—21no—
87#95×C43—24no—
88C24×C43C2×D411yescharacteristic
89C23×D43—22no—
90C2×Q8×C43—216no—
91C2×Q8×C43—216no—
92C2×Q8×C43—24no—
93C2×Q8×C43—28no—
94C2×Q8×C43—24no—
95C2×Q8×C43—24no—
96C2×Q8×C43—22no—
97C2×Q8×C43—24no—
98C2×Q8×C43C2×D418yes—
99C22×C4×C43—28no—
100C22×C4×C43—24no—
101C22×C4×C43—22no—
102C22×C4×C43—22no—
103C22×C4×C43—22no—
104C22×C4×C43—21no—
105C22×C4×C43—21no—
106C22×C4×C43C2×D412yes—
107C22×C4×C43C2×D412yes—
108C22×C4×C43C2×D414yes—
109C22×C4×C43C2×D411yescharacteristic
110C22×C4×C43C2411yescharacteristic
111C2×C22⋊D43—216no—
112C2×C22⋊D43—24no—
113C2×C22⋊D43—24no—
114C22×Dic43—22no—
115#10621902—264no—
116#10621902—232no—
117#10621902—232no—
118Q8×D43—264no—
119Q8×D43—232no—
120Q8×D43—232no—
121#10621898—264no—
122#10621898—232no—
123#10621898—216no—
124#10621898—216no—
125#10621898—232no—
126#10621898—28no—
127#10621898—28no—
128#10621898—28no—
129#10621898C2×D4116yes—
130C2×D172—216no—
131C2×D172—28no—
132C2×D172—22no—
133C2×D172—28no—
134C2×D172C2×D414yes—
135C2×C4×D43—232no—
136C2×C4×D43—216no—
137C2×C4×D43—216no—
138C2×C4×D43—24no—
139C2×C4×D43—24no—
140C2×Dic86—216no—
141C2×Dic86—28no—
142C2×Dic86—22no—
143C2×Dic86—28no—
144C2×Dic86C2×D414yes—
145C4⋊C4×C43—22no—
146C4⋊C4×C43—42no—
147C4⋊C4×C43—48no—
148C22⋊C4×C43—24no—
149C22⋊C4×C43—41no—
150C22⋊C4×C43—41no—
151C42×C43—24no—
152#10621885—48no—
153#10621884—48no—
154#10621877—28no—
155C2×C43⋊C8—24no—
156C23×C43—22no—
157C23×C43—22no—
158C23×C43—41no—
159C23×C43#12011yescharacteristic
160C23×C43#12011yescharacteristic
161C23×C43C22×D411yescharacteristic
162C22×D43—28no—
163C22×D43—24no—
164C22×D43—44no—
165C22×D43—44no—
166Q8×C43—216no—
167Q8×C43—216no—
168Q8×C43—44no—
169C2×C4×C43—28no—
170C2×C4×C43—28no—
171C2×C4×C43—24no—
172C2×C4×C43—24no—
173C2×C4×C43—22no—
174C2×C4×C43—24no—
175C2×C4×C43—24no—
176C2×C4×C43—24no—
177C2×C4×C43—42no—
178C2×C4×C43—42no—
179C2×C4×C43—42no—
180C2×C4×C43—41no—
181C2×C4×C43—41no—
182C2×C4×C43—42no—
183C2×C4×C43#12012yes—
184C2×C4×C43#12018yes—
185C2×C4×C43C22×D412yes—
186C2×C4×C43C22×D414yes—
187C22⋊D43—216no—
188C22⋊D43—216no—
189C2×Dic43—28no—
190C2×Dic43—24no—
191D172—232no—
192D172—28no—
193D172—24no—
194D172#12018yes—
195C4×D43—264no—
196C4×D43—216no—
197C4×D43—216no—
198Dic86—232no—
199Dic86—28no—
200Dic86—24no—
201Dic86#12018yes—
202C43⋊C8—44no—
203#32892—431nomaximal, Sylow
204C22×C43—21no—
205C22×C43—21no—
206C22×C43—21no—
207C22×C43—21no—
208C22×C43—21no—
209C22×C43—21no—
210C22×C43—21no—
211C22×C43—44no—
212C22×C43#45612yescomplemented
213C22×C43C2×#12012yes—
214C22×C43C2×#12011yescharacteristic
215C2×D43—28no—
216C2×D43—416no—
217C2×D43—44no—
218C2×D43—44no—
219C4×C43—28no—
220C4×C43—22no—
221C4×C43—44no—
222C4×C43—81no—
223C4×C43—81no—
224C4×C43C2×#12014yes—
225Dic43—28no—
226C22×2−1+4—431no—
227C2×#456—434no—
228#3007—432no—
229#2434D4311yescharacteristic, complemented
230#1793—434no—
231#1789—432no—
232#1784—431no—
233C2×C43—22no—
234C2×C43—22no—
235C2×C43—41no—
236C2×C43#284611yescharacteristic
237C2×C43C2×#45611yescharacteristic
238C2×C43#302011yescharacteristic
239D43—48no—
240C2×2−1+4—434no—
241C2×2−1+4—864no—
242C2×2−1+4—8616no—
243C2×2−1+4—862no—
244C22×#41—432no—
245C22×#41—862no—
246C22×#41—862no—
247C23×Q8—862no—
248C23×Q8C2×D4311yescharacteristic
249C2×#137—434no—
250C2×C4⋊Q8C2×D4312yescomplemented
251C2×C22⋊Q8C2×D4311yescharacteristic
252#456—8616no—
253#452—864no—
254#451—862no—
255#450—862no—
256C2×#104—438no—
257#417—434no—
258C2×#101—434no—
259#407—432no—
260#395—22no—
261#391C2×D4312yescomplemented
262#385—22no—
263C4×C22⋊C4C2×D4311yescharacteristic
264C43#3289211yescharacteristic, complemented, Sylow
2652−1+4—4316no—
2662−1+4—8616no—
2672−1+4—8632no—
2682−1+4—8632no—
2692−1+4—8632no—
2702−1+4—868no—
271C2×#41—434no—
272C2×#41—434no—
273C2×#41—8616no—
274C2×#41—868no—
275C2×#41—8632no—
276C2×#41—868no—
277C2×#41—8632no—
278C2×#41—868no—
279C2×#41—862no—
280C2×#41—862no—
281C2×#41—868no—
282C22×Q8—28no—
283C22×Q8—22no—
284C22×Q8—22no—
285C22×Q8—432no—
286C22×Q8—862no—
287C22×Q8—868no—
288C22×Q8—8632no—
289C22×Q8—868no—
290C22×Q8C22⋊D4312yes—
291C22×Q8C22×D4311yescharacteristic
292C22×Q8C22×D4312yes—
293C22×D4—432no—
294C22×D4—862no—
295C22×D4—862no—
296C23×C4—864no—
297C23×C4—862no—
298C23×C4C22⋊D4312yes—
299C23×C4C22×D4311yescharacteristic
300#137—8616no—
301C2×Q16—864no—
302C2×SD16—864no—
303C2×M16—434no—
304C4⋊Q8—28no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4C22×D4311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C42C22×D4312yes—
313#104—8616no—
314#103—864no—
315#102—864no—
316#101—868no—
317C22⋊C8—862no—
318C22⋊C8—862no—
319#95—21no—
320#95—24no—
321C24—862no—
322C24C2×C22⋊D4311yescharacteristic
323#41—4316no—
324#41—868no—
325#41—868no—
326#41—8664no—
327#41—8616no—
328#41—8632no—
329#41—8616no—
330#41—8632no—
331#41—8664no—
332#41—8632no—
333#41—8632no—
334#41—868no—
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—24no—
338C2×Q8—28no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—434no—
344C2×Q8—868no—
345C2×Q8—8616no—
346C2×Q8—868no—
347C2×Q8—8632no—
348C2×Q8—8664no—
349C2×Q8—8632no—
350C2×Q8—862no—
351C2×Q8D4×D4318yes—
352C2×D4—434no—
353C2×D4—868no—
354C2×D4—8616no—
355C2×D4—868no—
356C2×D4—8616no—
357C2×D4—864no—
358C2×D4—864no—
359C2×D4—862no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—8632no—
368C22×C4—8616no—
369C22×C4—8616no—
370C22×C4—862no—
371C22×C4—864no—
372C22×C4—864no—
373C22×C4D4×D4312yes—
374C22×C4D4×D4312yes—
375C22×C4C2×C22⋊D4311yescharacteristic
376C22×C4C2×C22⋊D4314yes—
377C22×C4C23×D4311yescharacteristic
378Q16—1728no—
379SD16—1728no—
380M16—868no—
381C2×C8—864no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C22⋊C4—24no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C42—24no—
389C23—22no—
390C23—22no—
391C23—41no—
392C23—868no—
393C23—864no—
394C23—1724no—
395C23—1724no—
396C23#1306329311yescharacteristic
397C23#1306329311yescharacteristic
398C23C22×C22⋊D4311yescharacteristic
399Q8—216no—
400Q8—216no—
401Q8—44no—
402Q8—438no—
403Q8—868no—
404Q8—8632no—
405Q8—864no—
406D4—438no—
407D4—868no—
408D4—8632no—
409D4—8616no—
410D4—8616no—
411D4—864no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—8664no—
427C2×C4—8616no—
428C2×C4—868no—
429C2×C4—864no—
430C2×C4—8616no—
431C2×C4#1306329318yes—
432C2×C4#1306329312yes—
433C2×C4C2×D4×D4314yes—
434C2×C4C2×D4×D4312yes—
435C8—1724no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—44no—
444C22—868no—
445C22—17216no—
446C22—1724no—
447C22—1724no—
448C22#1458308112yescomplemented
449C22C2×#1306329312yes—
450C22C2×#1306329311yescharacteristic
451C4—28no—
452C4—22no—
453C4—44no—
454C4—81no—
455C4—81no—
456C4—868no—
457C4#120×D4314yes—
458C2—22no—
459C2—22no—
460C2—41no—
461C2—1728no—
462C2#1914287111yescharacteristic
463C2C2×#1458308111yescharacteristic
464C2#1914395811yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2434C2
socleC22×C43C2×#120
prime radicalC22×#13063373C2
hypercenter#2434D43
hypocenterC43#32892
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C2×#14583081C2semidirect
3. #19143853C2semidirect
4. #19134553C2semidirect
5. #19134540C2semidirect
6. #19134519C2semidirect
7. C43×#2434C2semidirect
16. C2×#13063356C22semidirect
21. C2×#13063253C22semidirect
22. #14582898C22semidirect
23. C2×#13063244C22semidirect
26. C2×C4⋊Q8×C43C22semidirect
29. C43×#391C22semidirect
212. C22×C43#456semidirect
229. #2434D43semidirect
250. C2×C4⋊Q8C2×D43semidirect
261. #391C2×D43semidirect
264. C43#32892semidirect
448. C22#14583081semidirect

Maximal subgroups

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

subgroupindex
C22×#130633732
C2×#145830812
#191438532
#191345532
#191345402
#191345192
C43×#24342
#3289243