Finite Group Databasebeta

← #25152752

Subgroups — #25152752

Order 11008.

1094 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19148398C211yesmaximal, characteristic
2#19139069C212yesmaximal, complemented
3#19135948C214yesmaximal, complemented
4#19135579C212yesmaximal, complemented
5#19134875C214yesmaximal
6C4×#13063240C211yesmaximal, characteristic
7C43×#2487C211yesmaximal, characteristic
8#14583608—24no—
9#14583606—22no—
10D4×Dic86—22no—
11#14583530—21no—
12#14583525—21no—
13#14583525C2212yes—
14#14583456—21no—
15C2×#13063304—21no—
16C2×#13063304C2212yes—
17#14583374C2212yes—
18C2×#13063240C2212yes—
19#14582830—24no—
20#14582804—22no—
21#14582772C2218yes—
22#14582768—22no—
23C4×C22⋊Dic43C2212yes—
24#14582666C2212yes—
25#14582665C2218yes—
26C2×C4×D4×C43C2211yescharacteristic
27C22×C4⋊C4×C43C2211yescharacteristic
28C43×#400—21no—
29C43×#399—21no—
30C43×#389—21no—
31C43×#386C2212yes—
32C43×#385C2212yes—
33C43×#380—21no—
34C43×#380C2211yescharacteristic
35C22×Dic86—21no—
36C22×Dic86D412yes—
37#13063325—21no—
38#13063325—21no—
39#13063324—24no—
40#13063317—22no—
41#13063316—24no—
42#13063304—22no—
43#13063304—22no—
44#13063304—22no—
45#13063304—22no—
46#13063304—22no—
47#13063304—22no—
48#13063304—22no—
49#13063304—22no—
50#13063302—24no—
51#13063302—24no—
52#13063302—24no—
53#13063302—24no—
54C4×Dic86—22no—
55C4×Dic86—22no—
56D4×Dic43—24no—
57D4×Dic43—24no—
58#13063284—22no—
59C2×C4⋊Dic43—21no—
60C2×C4⋊Dic43D412yes—
61C2×Dic43⋊C4—22no—
62C2×Dic43⋊C4—28no—
63C2×Dic43⋊C4D414yes—
64C2×C22⋊Dic43—22no—
65C2×C22⋊Dic43C2314yes—
66C2×C22⋊Dic43C2314yes—
67C2×C4×Dic43—22no—
68#13063240Q818yescomplemented
69C43⋊#95—22no—
70C43⋊#95—28no—
71C22×D4×C43C2311yescharacteristic
72C23×C4×C43D411yescharacteristic
73C23×C4×C43C2312yes—
74C23×C4×C43C2311yescharacteristic
75C4×D4×C43—22no—
76C4×D4×C43—22no—
77C2×C4⋊C4×C43—22no—
78C2×C4⋊C4×C43—22no—
79C2×C4⋊C4×C43—21no—
80C2×C4⋊C4×C43—21no—
81C2×C4⋊C4×C43—21no—
82C2×C4⋊C4×C43—22no—
83C2×C4⋊C4×C43D411yescharacteristic
84C2×C22⋊C4×C43—22no—
85C2×C22⋊C4×C43D411yescharacteristic
86C2×C22⋊C4×C43C2312yes—
87C2×C22⋊C4×C43C2311yescharacteristic
88C2×C42×C43D411yescharacteristic
89#95×C43—22no—
90#95×C43—21no—
91#95×C43—22no—
92C24×C43C2×D411yescharacteristic
93C24×C43C2411yescharacteristic
94C2×D4×C43—22no—
95C2×D4×C43—22no—
96C2×D4×C43C2×Q814yes—
97C22×C4×C43—21no—
98C22×C4×C43—21no—
99C22×C4×C43—22no—
100C22×C4×C43—22no—
101C22×C4×C43—24no—
102C22×C4×C43—42no—
103C22×C4×C43—42no—
104C22×C4×C43—82no—
105C22×C4×C43C2×D412yes—
106C22×C4×C43C2×D412yes—
107C22×C4×C43C2×D411yescharacteristic
108C22×C4×C43C2×D411yescharacteristic
109C22×C4×C43C2×D411yescharacteristic
110C22×C4×C43C2×D411yescharacteristic
111C22×C4×C43C2×D411yescharacteristic
112C22×Dic43—22no—
113C22×Dic43—22no—
114C22×Dic43—24no—
115C22×Dic43C2×D414yes—
116C22×Dic43C2×D414yes—
117C2×Dic86—22no—
118C2×Dic86—22no—
119C2×Dic86—22no—
120C2×Dic86—22no—
121C2×Dic86—44no—
122C2×Dic86—82no—
123C4⋊C4×C43—22no—
124C4⋊C4×C43—42no—
125C4⋊C4×C43—41no—
126C4⋊C4×C43—42no—
127C4⋊C4×C43—41no—
128C4⋊C4×C43—82no—
129C22⋊C4×C43—22no—
130C22⋊C4×C43—22no—
131C22⋊C4×C43—24no—
132C22⋊C4×C43—82no—
133C42×C43—21no—
134C42×C43—21no—
135C22⋊Dic43—24no—
136C22⋊Dic43—24no—
137C22⋊Dic43—24no—
138C22⋊Dic43—24no—
139C22⋊Dic43C2×Q818yescomplemented
140C4⋊Dic43—22no—
141C4⋊Dic43—22no—
142C4⋊Dic43—22no—
143C4⋊Dic43—22no—
144C4⋊Dic43—22no—
145C4⋊Dic43—22no—
146Dic43⋊C4—24no—
147Dic43⋊C4—24no—
148Dic43⋊C4—24no—
149Dic43⋊C4—24no—
150Dic43⋊C4—88no—
151C4×Dic43—24no—
152C4×Dic43—44no—
153C23×C43—21no—
154C23×C43—21no—
155C23×C43—41no—
156C23×C43—41no—
157C23×C43—41no—
158C23×C43—41no—
159C23×C43C4⋊Q812yes—
160C23×C43C22×D412yes—
161C23×C43C22×D411yescharacteristic
162C23×C43C22×Q812yes—
163C23×C432−1+412yes—
164D4×C43—28no—
165C2×C4×C43—21no—
166C2×C4×C43—21no—
167C2×C4×C43—24no—
168C2×C4×C43—22no—
169C2×C4×C43—21no—
170C2×C4×C43—21no—
171C2×C4×C43—42no—
172C2×C4×C43—41no—
173C2×C4×C43—42no—
174C2×C4×C43—41no—
175C2×C4×C43—41no—
176C2×C4×C43—48no—
177C2×C4×C43—42no—
178C2×C4×C43—42no—
179C2×C4×C43—42no—
180C2×C4×C43—42no—
181C2×C4×C43—42no—
182C2×C4×C43—42no—
183C2×C4×C43—81no—
184C2×C4×C43#12011yescharacteristic
185C2×C4×C43#12012yes—
186C2×C4×C43#12011yescharacteristic
187C2×C4×C43C4⋊Q812yes—
188C2×Dic43—24no—
189C2×Dic43—24no—
190C2×Dic43—28no—
191C2×Dic43—44no—
192C2×Dic43—44no—
193C2×Dic43—44no—
194C2×Dic43—44no—
195C2×Dic43—82no—
196C2×Dic43—82no—
197Dic86—44no—
198Dic86—82no—
199Dic86—82no—
200Dic86—82no—
201#25853—431nomaximal, Sylow
202C22×C43—21no—
203C22×C43—22no—
204C22×C43—21no—
205C22×C43—22no—
206C22×C43—22no—
207C22×C43—42no—
208C22×C43—42no—
209C22×C43—42no—
210C22×C43—41no—
211C22×C43—41no—
212C22×C43—81no—
213C22×C43C2×#12011yescharacteristic
214C22×C43C2×C4⋊Q811yescharacteristic
215C22×C43#53611yescharacteristic
216C22×C43D4×Q814yes—
217C4×C43—22no—
218C4×C43—22no—
219C4×C43—21no—
220C4×C43—41no—
221C4×C43—41no—
222C4×C43—82no—
223C4×C43—82no—
224C4×C43#45812yes—
225Dic43—48no—
226Dic43—44no—
227Dic43—44no—
228#3501—431no—
229#2502—432no—
230#2487D4311yescharacteristic
231#2062—434no—
232#1996—432no—
233#1867—434no—
234C4×#99—431no—
235C2×C43—24no—
236C2×C43—21no—
237C2×C43—21no—
238C2×C43—42no—
239C2×C43#243411yescharacteristic
240C2×C43C2×#45811yescharacteristic
241C2×C43#302711yescharacteristic
242#554—862no—
243#554—864no—
244D4×Q8—862no—
245#544—861no—
246#543—432no—
247#543—861no—
248#533—861no—
249C2×C22⋊Q8—432no—
250C2×C22⋊Q8—861no—
251#518—432no—
252C2×C4×D4C2×D4311yescharacteristic
253C22×C4⋊C4C2×D4311yescharacteristic
254C2×#99—432no—
255#400—21no—
256#399—21no—
257#398—864no—
258#391—862no—
259#389—21no—
260#386C2×D4312yes—
261#385—438no—
262#385C2×D4312yes—
263#384—862no—
264#380—21no—
265#380C2×D4311yescharacteristic
266C4×C22⋊C4—432no—
267#342—432no—
268#342—438no—
269C43#2585311yescharacteristic, complemented, Sylow
270C22×Q8—432no—
271C22×Q8—861no—
272C22×D4C22×D4311yescharacteristic
273C23×C4C22⋊D4311yescharacteristic
274C23×C4C22×D4312yes—
275C23×C4C22×D4311yescharacteristic
276C4⋊Q8—861no—
277C4⋊Q8—861no—
278C4⋊Q8—864no—
279#125—862no—
280#125—864no—
281C22⋊Q8—864no—
282C22⋊Q8—864no—
283C22⋊Q8—864no—
284C22⋊Q8—864no—
285C22⋊Q8—862no—
286C22⋊Q8—862no—
287C22⋊Q8—862no—
288C22⋊Q8—862no—
289C22⋊Q8—862no—
290C22⋊Q8—862no—
291C22⋊Q8—862no—
292C22⋊Q8—862no—
293C4×Q8—862no—
294C4×Q8—862no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—864no—
298C4×D4—864no—
299#117—862no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—434no—
307C2×C4⋊C4—432no—
308C2×C4⋊C4—862no—
309C2×C4⋊C4—861no—
310C2×C4⋊C4—868no—
311C2×C4⋊C4C22⋊D4311yescharacteristic
312C2×C22⋊C4—22no—
313C2×C22⋊C4—434no—
314C2×C22⋊C4—434no—
315C2×C22⋊C4—862no—
316C2×C22⋊C4C22⋊D4311yescharacteristic
317C2×C22⋊C4C22×D4312yes—
318C2×C22⋊C4C22×D4311yescharacteristic
319C2×C42—862no—
320C2×C42C22⋊D4311yescharacteristic
321#99—438no—
322#95—22no—
323#95—21no—
324#95—22no—
325#95—868no—
326#95—862no—
327C24C2×C22⋊D4311yescharacteristic
328C24C23×D4311yescharacteristic
329C2×Q8—862no—
330C2×Q8—862no—
331C2×Q8—862no—
332C2×Q8—862no—
333C2×Q8—3442no—
334C2×Q8—1724no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4Q8×D4314yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—82no—
346C22×C4—434no—
347C22×C4—434no—
348C22×C4—862no—
349C22×C4—862no—
350C22×C4—864no—
351C22×C4D4×D4312yes—
352C22×C4D4×D4312yes—
353C22×C4C2×C22⋊D4311yescharacteristic
354C22×C4C2×C22⋊D4311yescharacteristic
355C22×C4C2×C22⋊D4311yescharacteristic
356C22×C4C2×C22⋊D4311yescharacteristic
357C22×C4C2×C22⋊D4311yescharacteristic
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—82no—
364C4⋊C4—862no—
365C4⋊C4—864no—
366C4⋊C4—862no—
367C4⋊C4—862no—
368C4⋊C4—864no—
369C4⋊C4—862no—
370C4⋊C4—864no—
371C4⋊C4—862no—
372C4⋊C4—862no—
373C4⋊C4—864no—
374C4⋊C4—3448no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—24no—
378C22⋊C4—82no—
379C22⋊C4—438no—
380C22⋊C4—864no—
381C22⋊C4—864no—
382C22⋊C4—864no—
383C22⋊C4—864no—
384C42—21no—
385C42—21no—
386C42—864no—
387C42—1724no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#1306332312yes—
395C23C2×D4×D4312yes—
396C23C22×C22⋊D4311yescharacteristic
397C23C2×Q8×D4312yes—
398C23#1306337212yes—
399Q8—3442no—
400Q8—3442no—
401Q8—3442no—
402Q8—1724no—
403D4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—868no—
424C2×C4—864no—
425C2×C4—864no—
426C2×C4—3442no—
427C2×C4—3442no—
428C2×C4—1724no—
429C2×C4—1724no—
430C2×C4—1724no—
431C2×C4—1724no—
432C2×C4#1306329311yescharacteristic
433C2×C4#1306329312yes—
434C2×C4#1306329311yescharacteristic
435C2×C4#1306332312yes—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—22no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22C2×#1306329311yescharacteristic
448C22C2×#1306332311yescharacteristic
449C22#1458347011yescharacteristic
450C22#1458356214yes—
451C4—21no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—82no—
458C4—1728no—
459C4—1724no—
460C4—1724no—
461C4#1458308812yes—
462C2—21no—
463C2—24no—
464C2—21no—
465C2—42no—
466C2#1913824011yescharacteristic
467C2C2×#1458308811yescharacteristic
468C2#1914399611yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC24×C43C24
FrattiniC24C23×D43
FittingC43×#2487C2
socleC22×C43C2×#120
prime radicalC22×D4×C43C23
hypercenter#2487D43
hypocenterC43#25853
LCSG ▷ C24×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #19139069C2semidirect
3. #19135948C2semidirect
4. #19135579C2semidirect
68. #13063240Q8semidirect
139. C22⋊Dic43C2×Q8semidirect
269. C43#25853semidirect

Maximal subgroups

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

subgroupindex
#191483982
#191390692
#191359482
#191355792
#191348752
C4×#130632402
C43×#24872
#2585343