Finite Group Databasebeta

← #29834497

Subgroups — #29834497

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840203C211yesmaximal, characteristic, complemented
2C22×#14519717C211yesmaximal, characteristic, complemented
3#21455400C211yesmaximal, characteristic
4#21447934C212yesmaximal, complemented
5#21447925C212yesmaximal, complemented
6#21447917C212yesmaximal, complemented
7#21447908C212yesmaximal, complemented
8#21447605C211yesmaximal, characteristic, complemented
9#21447539C211yesmaximal, characteristic, complemented
10#21447529C211yesmaximal, characteristic, complemented
11C2×C61×#467C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840273—21no—
14#16840203—24no—
15C2×#14519723C2212yescomplemented
16C2×#14519720C2212yescomplemented
17C2×#14519717—24no—
18C2×#14519717C2211yescharacteristic
19C2×#14519717C2212yes—
20C2×#14519717C2211yescharacteristic
21C22×#11758185C2212yes—
22#16840003C2211yescharacteristic
23#16840003C2211yescharacteristic
24#16839998C2211yescharacteristic
25#16839998C2211yescharacteristic
26#16839991—21no—
27#16839991—21no—
28#16839800C2211yescharacteristic
29#16839759—21no—
30#16839589C2212yescomplemented
31#16839587C2212yescomplemented
32#16839566C2212yescomplemented
33#16839566C2212yescomplemented
34#16839565C2212yescomplemented
35#16839562C2212yescomplemented
36C22×Q16×C61C2211yescharacteristic
37C23×C8×C61C2211yescharacteristic
38C2×C22⋊Q8×C61—21no—
39C61×#467—21no—
40C61×#467—21no—
41C61×#467C2212yes—
42C61×#467C2212yes—
43C2×#107×C61C2211yescharacteristic
44C2×#103×C61—21no—
45C2×#11758222—21no—
46#14519799—22no—
47#14519799—22no—
48C2×Dic244C2311yescharacteristic
49C2×#11758190—21no—
50C2×D488C2311yescharacteristic
51#14519760—41no—
52#14519756—21no—
53#14519756—21no—
54#14519751—21no—
55#14519751—21no—
56#14519746—41no—
57C4×C22⋊D61—41no—
58C4×C22⋊D61—41no—
59#14519736—21no—
60#14519723—24no—
61#14519720—24no—
62#14519717—22no—
63#14519717—24no—
64#14519717D414yescomplemented
65C2×#11758185—24no—
66C2×#11758185C2312yes—
67C2×#11758185C2312yes—
68#14519711C2311yescharacteristic
69#14519711C2311yescharacteristic
70#14519709—21no—
71#14519702—21no—
72#14519702—21no—
73#14519699—21no—
74#14519699—21no—
75C22×Q8×C61—21no—
76C23×C4×C61D411yescharacteristic
77C2×Q16×C61—22no—
78C2×Q16×C61—21no—
79C2×Q16×C61—22no—
80C2×Q16×C61C2312yes—
81C22×C8×C61—22no—
82C22×C8×C61—22no—
83C22×C8×C61D411yescharacteristic
84C22×C8×C61D411yescharacteristic
85C22×C8×C61D411yescharacteristic
86C22×C8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—41no—
92C22⋊Q8×C61—41no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—41no—
95C2×C22⋊C4×C61—41no—
96#107×C61—21no—
97#107×C61C2312yes—
98#103×C61—22no—
99#103×C61—22no—
100C24×C61D811yescharacteristic
101C2×Q8×C61—22no—
102C2×Q8×C61—41no—
103C2×Q8×C61—42no—
104C2×Q8×C61—42no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61D811yescharacteristic
110C22×C4×C61D811yescharacteristic
111C22×C4×C61D811yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C2×C22⋊D61—41no—
116C2×C22⋊D61—41no—
117C2×C22⋊D61—41no—
118#11758222—42no—
119#11758222—42no—
120C2×D244—21no—
121C2×C4×D61—41no—
122C2×Dic122—21no—
123Q16×C61—22no—
124Q16×C61—22no—
125Q16×C61—42no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×D412yes—
137C2×C8×C61#4112yes—
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—22no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—42no—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—42no—
145C22⋊Dic61—41no—
146C22⋊Dic61—41no—
147C22⋊Dic61—41no—
148#11758198—41no—
149#11758198—41no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152#11758196—41no—
153#11758196—41no—
154C4×Dic61—41no—
155Dic244—21no—
156Dic244—21no—
157D488—21no—
158D488—21no—
159#11758190—42no—
160#11758185—22no—
161#11758185C2×D414yes—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—41no—
165C23×C61C2×D811yescharacteristic
166C23×C61C2×D811yescharacteristic
167C23×C61C2×D811yescharacteristic
168C22×D61—41no—
169Q8×C61—42no—
170Q8×C61—42no—
171Q8×C61—82no—
172C2×C4×C61—22no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—82no—
182C2×C4×C61—82no—
183C2×C4×C61#12112yes—
184C2×C4×C61C2×D811yescharacteristic
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61C2×D811yescharacteristic
187C2×C4×C61#13512yes—
188C2×C4×C61C22×D411yescharacteristic
189C22⋊D61—82no—
190C22⋊D61—82no—
191C22⋊D61—82no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195D244—41no—
196D244—41no—
197C4×D61—82no—
198Dic122—41no—
199Dic122—41no—
200C8×C61—21no—
201C8×C61—21no—
202C8×C61#12112yes—
203C8×C61C22×D411yescharacteristic
204C8×C61C2×#4111yescharacteristic
205#33606—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—44no—
214C22×C61#46612yes—
215C22×C61#51012yescomplemented
216C22×C61C22×D811yescharacteristic
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—82no—
223C4×C61—82no—
224C4×C61#46612yes—
225C4×C61C2×#12111yescharacteristic
226C4×C61C2×#13511yescharacteristic
227Dic61—81no—
228Dic61—81no—
229Dic61—81no—
230C2×#510—611no—
231C22×M32—611no—
232#3050—611no—
233C2×#467D6111yescharacteristic, complemented
234#2220—612no—
235#2219—612no—
236#2217—612no—
237#2216—612no—
238#2159—611no—
239#2141—611no—
240#2139—611no—
241C2×C61—22no—
242C2×C61—22no—
243C2×C61—41no—
244C2×C61C2×#46611yescharacteristic
245C2×C61C2×#51011yescharacteristic
246C2×C61#341411yescharacteristic
247D61—161no—
248C2×#135—611no—
249C22×Q16C2×D6111yescharacteristic
250C23×C8C2×D6111yescharacteristic
251#525—1221no—
252C2×C22⋊Q8—21no—
253#510—1224no—
254C2×Q32—612no—
255C2×SD32—612no—
256C2×M32—611no—
257C2×M32—611no—
258C2×M32—612no—
259C2×M32—1224no—
260C22×C16—612no—
261#467—21no—
262#467—21no—
263#467—611no—
264#467—611no—
265#467C2×D6112yes—
266#467C2×D6112yes—
267#466—611no—
268#466—611no—
269#465—1221no—
270#465—1221no—
271#427—611no—
272C2×#107C2×D6111yescharacteristic
273#416—1221no—
274C2×#103—21no—
275#367—612no—
276#366—612no—
277#358—612no—
278#358—612no—
279#358—612no—
280#357—612no—
281C61#3360611yescharacteristic, complemented, Sylow
282C2×#41—1221no—
283C22×Q8—21no—
284C23×C4C22⋊D6111yescharacteristic
285#135—1222no—
286#135—1222no—
287C2×Q16—22no—
288C2×Q16—22no—
289C2×Q16—21no—
290C2×Q16—611no—
291C2×Q16C22×D6112yes—
292C2×SD16—1221no—
293C2×D8—611no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C22⋊D6111yescharacteristic
297C22×C8C22⋊D6111yescharacteristic
298C22×C8C22⋊D6111yescharacteristic
299C22×C8C22×D6111yescharacteristic
300C22×C8C22×D6111yescharacteristic
301C22×C8C22×D6111yescharacteristic
302#123—2441no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—41no—
306C22⋊Q8—41no—
307C22⋊Q8—1221no—
308C22⋊Q8—1221no—
309#121—1221no—
310#121—1221no—
311#120—2441no—
312C4×D4—2441no—
313C4×D4—2441no—
314#117—1221no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—41no—
317C2×C22⋊C4—41no—
318Q32—1224no—
319SD32—1224no—
320M32—614no—
321M32—1224no—
322M32—1222no—
323C2×C16—612no—
324C2×C16—612no—
325C2×C16—1224no—
326#107—21no—
327#107—611no—
328#107—611no—
329#107C22×D6112yes—
330#106—1221no—
331#103—22no—
332#103—22no—
333#103—1221no—
334#103—1221no—
335#102—1221no—
336#102—1221no—
337C24#1175819911yescharacteristic
338#41—2442no—
339#41—2442no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—1221no—
345C2×D4—1221no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—2441no—
354C22×C4#1175819911yescharacteristic
355C22×C4#1175819911yescharacteristic
356C22×C4#1175819911yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360Q16—22no—
361Q16—22no—
362Q16—42no—
363Q16—1221no—
364Q16—1221no—
365SD16—2442no—
366D8—1221no—
367D8—1221no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8D4×D6112yes—
376C2×C8#1175822412yes—
377C2×C8C2×C22⋊D6111yescharacteristic
378C2×C8C2×C22⋊D6111yescharacteristic
379C2×C8C2×C22⋊D6111yescharacteristic
380C2×C8C23×D6111yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C22⋊C4—2441no—
396C42—2441no—
397C16—614no—
398C16—1222no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—2441no—
403C23C2×#1175819911yescharacteristic
404C23C2×#1175819911yescharacteristic
405C23C2×#1175819911yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—82no—
409Q8—2441no—
410Q8—2441no—
411D4—4882no—
412D4—4882no—
413D4—4882no—
414D4—2441no—
415D4—2441no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—4882no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4#1451974912yes—
432C2×C4C2×#1175819911yescharacteristic
433C2×C4C2×#1175819911yescharacteristic
434C2×C4C2×#1175819911yescharacteristic
435C2×C4#1451979712yes—
436C2×C4C22×C22⋊D6111yescharacteristic
437C8—21no—
438C8—21no—
439C8#1451974912yes—
440C8C2×D4×D6111yescharacteristic
441C8C2×#1175822411yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—4881no—
451C22—4881no—
452C22—4881no—
453C22#1683999612yes—
454C22#1684020312yescomplemented
455C22C22×#1175819911yescharacteristic
456C4—21no—
457C4—21no—
458C4—82no—
459C4—82no—
460C4—4881no—
461C4—4881no—
462C4—4881no—
463C4#1683999612yes—
464C4C2×#1451974911yescharacteristic
465C4C2×#1451979711yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—9761no—
470C2C2×#1683999611yescharacteristic
471C2C2×#1684020311yescharacteristic
472C2#2145895311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#467C2
socleC22×C61C22×D8
prime radical#21455400C2
hypercenterC2×#467D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C22×#14519717C2semidirect
4. #21447934C2semidirect
5. #21447925C2semidirect
6. #21447917C2semidirect
7. #21447908C2semidirect
8. #21447605C2semidirect
9. #21447539C2semidirect
10. #21447529C2semidirect
11. C2×C61×#467C2semidirect
15. C2×#14519723C22semidirect
16. C2×#14519720C22semidirect
30. #16839589C22semidirect
31. #16839587C22semidirect
32. #16839566C22semidirect
33. #16839566C22semidirect
34. #16839565C22semidirect
35. #16839562C22semidirect
64. #14519717D4semidirect
215. C22×C61#510semidirect
233. C2×#467D61semidirect
281. C61#33606semidirect
454. C22#16840203semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C22×#145197172
#214554002
#214479342
#214479252
#214479172
#214479082
#214476052
#214475392
#214475292
C2×C61×#4672
#3360661