Finite Group Databasebeta

← #29653861

Subgroups — #29653861

Order 15616.

1142 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456572C212yesmaximal, complemented
2#21451553C211yesmaximal, characteristic
3#21444810C214yesmaximal, complemented
4#21444742C214yesmaximal, complemented
5#21444391C212yesmaximal, complemented
6C2×#16839486C211yesmaximal, characteristic, complemented
7C2×C61×#543C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519746—21no—
10#16840090C2214yescomplemented
11#16840080—22no—
12#16839993—24no—
13#16839912C2214yescomplemented
14#16839901—22no—
15C2×C22⋊#92418C2212yes—
16#16839687C2212yes—
17#16839678—22no—
18#16839671C2212yes—
19#16839641—21no—
20#16839622C2211yescharacteristic
21#16839499C2218yescomplemented
22#16839486C2214yes—
23C61×#543—24no—
24C61×#543—22no—
25C61×#543C2214yes—
26C2×C22⋊Q8×C61—22no—
27C2×C22⋊Q8×C61—21no—
28C2×C22⋊Q8×C61—22no—
29C2×C22⋊Q8×C61C2212yes—
30C22×C22⋊C4×C61—21no—
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C22⋊D61—21no—
34C22×C22⋊D61—22no—
35C2×#11758201—24no—
36C22×#92418—22no—
37#14519751—22no—
38#14519751—22no—
39#14519746—44no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2311yescharacteristic
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—21no—
44C2×#11758198—21no—
45C2×#11758198—22no—
46#14519708—24no—
47#14519701—24no—
48#14519697—24no—
49C22⋊#92418—22no—
50C22⋊#92418C2314yes—
51#14519684—22no—
52#14519684C2312yes—
53C25×C61D411yescharacteristic
54C22×Q8×C61—21no—
55C22×Q8×C61D412yescomplemented
56C23×C4×C61—21no—
57C23×C4×C61—21no—
58C23×C4×C61D411yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C22⋊Q8×C61—28no—
61C22⋊Q8×C61—28no—
62C22⋊Q8×C61—24no—
63C22⋊Q8×C61—28no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—24no—
67C22⋊Q8×C61—28no—
68C22⋊Q8×C61D414yescomplemented
69C22⋊Q8×C61C2314yes—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61D412yescomplemented
74C2×C22⋊C4×C61—24no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61D412yes—
81C2×C22⋊C4×C61C2312yes—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—22no—
85C24×C61—81no—
86C24×C61—81no—
87C24×C61SD1614yescomplemented
88C24×C61C2×D412yes—
89C24×C61C2×D411yescharacteristic
90C23×D61—21no—
91C2×Q8×C61—24no—
92C2×Q8×C61—22no—
93C2×Q8×C61—82no—
94C2×Q8×C61—82no—
95C2×Q8×C61C2×D414yes—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—24no—
100C22×C4×C61—24no—
101C22×C4×C61—21no—
102C22×C4×C61—81no—
103C22×C4×C61—81no—
104C22×C4×C61—82no—
105C22×C4×C61—82no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2411yescharacteristic
111C2×C22⋊D61—41no—
112C2×C22⋊D61—41no—
113C2×C22⋊D61—44no—
114C2×C22⋊D61—42no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—82no—
117C2×C22⋊D61—82no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C2×D244—22no—
121C2×D244—81no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—28no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61C2×D414yes—
128C22⋊C4×C61—28no—
129C22⋊C4×C61—24no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—24no—
132C22⋊C4×C61—24no—
133C22⋊Dic61—44no—
134C22⋊Dic61—42no—
135#11758201—84no—
136#11758198—42no—
137#11758198—82no—
138C4⋊Dic61—22no—
139C4⋊Dic61—81no—
140C2×#92418—24no—
141C2×#92418—82no—
142C23×C61—22no—
143C23×C61—24no—
144C23×C61—22no—
145C23×C61—24no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—42no—
149C23×C61—42no—
150C23×C61—42no—
151C23×C61—42no—
152C23×C61—81no—
153C23×C61—81no—
154C23×C61—84no—
155C23×C61—84no—
156C23×C61#12011yescharacteristic
157C23×C61#12011yescharacteristic
158C23×C61C2×SD1614yes—
159C23×C61C2×SD1612yes—
160C23×C61#13712yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167Q8×C61—44no—
168Q8×C61—82no—
169Q8×C61—84no—
170C2×C4×C61—24no—
171C2×C4×C61—22no—
172C2×C4×C61—24no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—44no—
179C2×C4×C61—44no—
180C2×C4×C61—42no—
181C2×C4×C61—81no—
182C2×C4×C61—84no—
183C2×C4×C61—84no—
184C2×C4×C61#12012yes—
185C2×C4×C61C22×D412yes—
186C22⋊D61—88no—
187C22⋊D61—82no—
188C22⋊D61—82no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195D244—44no—
196#92418—44no—
197#18286—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—24no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61—44no—
208C22×C61—48no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—82no—
212C22×C61—84no—
213C22×C61#44812yes—
214C22×C61#44812yes—
215C22×C61C2×#12011yescharacteristic
216C22×C61C22×SD1611yescharacteristic
217C22×C61C2×#13711yescharacteristic
218C2×D61—82no—
219C2×D61—82no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—22no—
224C4×C61—22no—
225C4×C61—42no—
226C4×C61—44no—
227C4×C61—44no—
228Dic61—82no—
229Dic61—81no—
230C2×#543D6111yescharacteristic, complemented
231#3187—612no—
232#2617—611no—
233#1608—614no—
234#1596—614no—
235#1520—612no—
236C2×#323—611no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—22no—
240C2×C61—22no—
241C2×C61—44no—
242C2×C61C2×#44811yescharacteristic
243C2×C61#301911yescharacteristic
244C2×C61#302611yescharacteristic
245D61—161no—
246#543—24no—
247#543—22no—
248#543C2×D6114yes—
249C2×C22⋊Q8—22no—
250C2×C22⋊Q8—21no—
251C2×C22⋊Q8—22no—
252C2×C22⋊Q8C2×D6112yes—
253C2×#121—611no—
254C2×#120—1221no—
255C22×C22⋊C4—21no—
256C22×C22⋊C4C2×D6111yescharacteristic
257#483—614no—
258#481—1222no—
259#465—1224no—
260#450—614no—
261#448—1222no—
262C2×C22⋊C8—612no—
263#397—612no—
264#394—1222no—
265#392—612no—
266#386—1221no—
267#380—611no—
268#328—618no—
269#323—614no—
270C61#1828611yescharacteristic, complemented, Sylow
271C25C22⋊D6111yescharacteristic
272C22×Q8—21no—
273C22×Q8C22⋊D6112yescomplemented
274C22×D4—611no—
275C22×D4—1222no—
276C22×D4—1221no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4C22⋊D6111yescharacteristic
280C23×C4C22×D6111yescharacteristic
281C2×SD16—1224no—
282C22×C8—1222no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—28no—
287C22⋊Q8—24no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—28no—
291C22⋊Q8C22⋊D6114yescomplemented
292C22⋊Q8C22×D6114yes—
293#121—1222no—
294#121—1222no—
295#120—2444no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4C22⋊D6112yescomplemented
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4C22⋊D6112yes—
313C2×C22⋊C4C22×D6112yes—
314#106—1224no—
315#103—1224no—
316#102—1224no—
317C22⋊C8—614no—
318C22⋊C8—1222no—
319#95—612no—
320#95—1222no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—81no—
325C24—81no—
326C24—1221no—
327C24#1175820114yescomplemented
328C24C2×C22⋊D6112yes—
329C24C2×C22⋊D6111yescharacteristic
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8C2×C22⋊D6114yes—
335C2×D4—1222no—
336C2×D4—4881no—
337C2×D4—4882no—
338C2×D4—4882no—
339C2×D4—2442no—
340C2×D4—2441no—
341C2×D4—2441no—
342C2×D4—2444no—
343C2×D4—2442no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—1222no—
355C22×C4—1221no—
356C22×C4C2×C22⋊D6112yes—
357C22×C4C2×C22⋊D6112yes—
358C22×C4C2×C22⋊D6112yes—
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C23×D6111yescharacteristic
361SD16—4884no—
362C2×C8—1224no—
363C2×C8—4882no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—1222no—
370C4⋊C4—4881no—
371C4⋊C4C2×C22⋊D6114yes—
372C22⋊C4—28no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—4882no—
378C22⋊C4—2444no—
379C22⋊C4—2442no—
380C22⋊C4—2442no—
381C23—22no—
382C23—24no—
383C23—22no—
384C23—24no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—81no—
392C23—81no—
393C23—84no—
394C23—84no—
395C23—2442no—
396C23—2442no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23#1451974611yescharacteristic
401C23#1451974611yescharacteristic
402C23C2×#1175820114yes—
403C23C2×#1175820112yes—
404C23#1451980812yes—
405C23C22×C22⋊D6111yescharacteristic
406Q8—44no—
407Q8—82no—
408Q8—84no—
409D4—4888no—
410D4—4882no—
411D4—4882no—
412D4—2444no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2441no—
430C2×C4—2442no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4#1451974612yes—
434C2×C4C22×C22⋊D6112yes—
435C8—2444no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—84no—
451C22—4882no—
452C22—4882no—
453C22—4881no—
454C22—4881no—
455C22—4881no—
456C22#1683990112yes—
457C22#1683990112yes—
458C22C2×#1451974611yescharacteristic
459C22C22×#1175820111yescharacteristic
460C22C2×#1451980811yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—44no—
465C4—44no—
466C4—4882no—
467C4—4881no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—9761no—
474C2C2×#1683990111yescharacteristic
475C2#2145517611yescharacteristic
476C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#543C2
socleC22×C61C2×#120
prime radical#21451553C2
hypercenterC2×#543D61
hypocenterC61#18286
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #21456572C2semidirect
3. #21444810C2semidirect
4. #21444742C2semidirect
5. #21444391C2semidirect
6. C2×#16839486C2semidirect
7. C2×C61×#543C2semidirect
10. #16840090C22semidirect
13. #16839912C22semidirect
21. #16839499C22semidirect
55. C22×Q8×C61D4semidirect
68. C22⋊Q8×C61D4semidirect
73. C2×C4⋊C4×C61D4semidirect
87. C24×C61SD16semidirect
230. C2×#543D61semidirect
270. C61#18286semidirect
273. C22×Q8C22⋊D61semidirect
291. C22⋊Q8C22⋊D61semidirect
301. C2×C4⋊C4C22⋊D61semidirect
327. C24#11758201semidirect

Maximal subgroups

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

subgroupindex
#214565722
#214515532
#214448102
#214447422
#214443912
C2×#168394862
C2×C61×#5432
#1828661