Finite Group Databasebeta

← #29834490

Subgroups — #29834490

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840200C211yesmaximal, characteristic, complemented
2#21455400C211yesmaximal, characteristic, complemented
3C2×#16839998C211yesmaximal, characteristic, complemented
4#21447932C212yesmaximal, complemented
5#21447922C212yesmaximal, complemented
6#21447915C212yesmaximal, complemented
7#21447905C212yesmaximal, complemented
8#21447607C211yesmaximal, characteristic, complemented
9#21447536C211yesmaximal, characteristic, complemented
10#21447532C211yesmaximal, characteristic, complemented
11C22×M32×C61C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×D488C2211yescharacteristic
14#16840273—21no—
15C2×#14519751—21no—
16#16840200—24no—
17C2×#14519722C2212yescomplemented
18C2×D976C2212yescomplemented
19#16840003C2211yescharacteristic, complemented
20#16840003C2211yescharacteristic, complemented
21#16839998—21no—
22#16839998—21no—
23#16839998C2212yes—
24#16839998C2212yes—
25#16839998C2211yescharacteristic, complemented
26#16839998C2211yescharacteristic, complemented
27#16839991—21no—
28#16839991—21no—
29#16839800C2211yescharacteristic
30C2×#14519711C2211yescharacteristic
31#16839759—21no—
32C2×#14519699—21no—
33#16839590C2212yescomplemented
34#16839588C2212yescomplemented
35#16839567C2212yescomplemented
36#16839564C2212yescomplemented
37#16839564C2212yescomplemented
38#16839564C2212yescomplemented
39C23×C8×C61C2211yescharacteristic
40C2×M32×C61—24no—
41C2×M32×C61C2211yescharacteristic
42C2×M32×C61C2211yescharacteristic
43C2×M32×C61C2212yescomplemented
44C22×C16×C61C2212yescomplemented
45C2×#11758222—21no—
46C22×D244—21no—
47C22×C22⋊D61—41no—
48#14519799—22no—
49#14519799—22no—
50C2×Dic244C2311yescharacteristic
51C2×#11758190—21no—
52C2×D488—22no—
53C2×D488—22no—
54C2×D488—21no—
55C2×D488C2312yes—
56C2×D488C2311yescharacteristic
57#14519760—41no—
58#14519756—21no—
59#14519756—21no—
60#14519751—21no—
61#14519751—21no—
62#14519751—22no—
63#14519751—22no—
64#14519751—41no—
65#14519751—41no—
66#14519746—41no—
67C4×C22⋊D61—41no—
68C4×C22⋊D61—41no—
69#14519736—21no—
70C2×C4⋊Dic61—21no—
71C2×#11758198—41no—
72#14519722—24no—
73D976—24no—
74#14519711—21no—
75#14519711C2312yes—
76#14519711C2311yescharacteristic
77#14519711C2311yescharacteristic
78#14519709—21no—
79#14519702—21no—
80#14519702—21no—
81#14519699—21no—
82#14519699—21no—
83#14519699—22no—
84#14519699—22no—
85C23×C4×C61D411yescharacteristic
86C22×C8×C61—22no—
87C22×C8×C61—22no—
88C22×C8×C61D411yescharacteristic
89C22×C8×C61D411yescharacteristic
90C22×C8×C61D411yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94M32×C61—22no—
95M32×C61—24no—
96M32×C61D414yescomplemented
97C2×C16×C61—24no—
98C2×C16×C61C2312yes—
99C2×C16×C61C2312yes—
100C24×C61D811yescharacteristic
101C23×D61—41no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61D811yescharacteristic
105C22×C4×C61D811yescharacteristic
106C22×C4×C61D811yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C2×C22⋊D61—42no—
111C2×C22⋊D61—42no—
112C2×C22⋊D61—41no—
113C2×C22⋊D61—41no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—84no—
116C22×Dic61—41no—
117#11758222—42no—
118#11758222—42no—
119C2×D244—22no—
120C2×D244—21no—
121C2×D244—42no—
122C2×D244—42no—
123C2×D244—41no—
124C2×C4×D61—41no—
125C2×Dic122—21no—
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
139C22⋊Dic61—41no—
140C22⋊Dic61—41no—
141C22⋊Dic61—41no—
142#11758198—42no—
143#11758198—42no—
144#11758198—41no—
145#11758198—41no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—41no—
150#11758196—41no—
151#11758196—41no—
152C4×Dic61—41no—
153Dic244—21no—
154Dic244—21no—
155D488—21no—
156D488—22no—
157D488—22no—
158D488—21no—
159D488—42no—
160#11758190—42no—
161C16×C61—22no—
162C16×C61C2×D414yescomplemented
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—41no—
166C23×C61C2×D811yescharacteristic
167C23×C61C2×D811yescharacteristic
168C23×C61C2×D811yescharacteristic
169C22×D61—41no—
170C22×D61—42no—
171C22×D61—82no—
172C22×D61—82no—
173C22×D61—82no—
174C2×C4×C61—22no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61#12112yes—
182C2×C4×C61C2×D811yescharacteristic
183C2×C4×C61C2×D811yescharacteristic
184C2×C4×C61C2×D811yescharacteristic
185C2×C4×C61#13512yes—
186C2×C4×C61C22×D411yescharacteristic
187C22⋊D61—82no—
188C22⋊D61—84no—
189C22⋊D61—82no—
190C22⋊D61—82no—
191C22⋊D61—84no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—82no—
197D244—41no—
198D244—42no—
199D244—42no—
200D244—41no—
201D244—82no—
202C4×D61—82no—
203Dic122—41no—
204Dic122—41no—
205C8×C61—21no—
206C8×C61—21no—
207C8×C61#12112yes—
208C8×C61C22×D411yescharacteristic
209C8×C61C2×#4111yescharacteristic
210#33605—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—44no—
219C22×C61#46612yes—
220C22×C61#50912yescomplemented
221C22×C61C22×D811yescharacteristic
222C2×D61—82no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—82no—
226C2×D61—81no—
227C2×D61—82no—
228C2×D61—164no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61#46612yes—
232C4×C61C2×#12111yescharacteristic
233C4×C61C2×#13511yescharacteristic
234Dic61—82no—
235Dic61—81no—
236Dic61—81no—
237Dic61—81no—
238C2×#509—611no—
239C22×M32D6111yescharacteristic, complemented
240#3050—611no—
241C2×#466—611no—
242#2220—612no—
243#2218—612no—
244#2217—612no—
245#2215—612no—
246#2159—611no—
247#2140—611no—
248#2139—611no—
249C2×C61—22no—
250C2×C61—22no—
251C2×C61—41no—
252C2×C61C2×#46611yescharacteristic
253C2×C61C2×#50911yescharacteristic
254C2×C61#341411yescharacteristic
255D61—161no—
256D61—162no—
257C2×#135—611no—
258C22×D8—611no—
259C23×C8C2×D6111yescharacteristic
260#525—1221no—
261C2×#121—1221no—
262#509—1224no—
263C2×SD32—612no—
264C2×D16—612no—
265C2×M32—24no—
266C2×M32C2×D6111yescharacteristic
267C2×M32C2×D6111yescharacteristic
268C2×M32C2×D6112yescomplemented
269C22×C16C2×D6112yescomplemented
270#467—611no—
271#467—611no—
272#466—612no—
273#466—612no—
274#466—611no—
275#466—611no—
276#466—1221no—
277#466—1221no—
278#465—1221no—
279#465—1221no—
280#427—611no—
281C2×#107—611no—
282#416—1221no—
283C2×#102—1221no—
284#367—612no—
285#366—612no—
286#358—612no—
287#357—612no—
288#357—612no—
289#357—612no—
290C61#3360511yescharacteristic, complemented, Sylow
291C2×#41—1221no—
292C22×D4—1221no—
293C22×D4—2441no—
294C23×C4D24411yescharacteristic
295#135—1222no—
296#135—1222no—
297C2×Q16—611no—
298C2×SD16—1221no—
299C2×D8—612no—
300C2×D8—611no—
301C2×D8—1221no—
302C2×D8—1222no—
303C2×D8—1222no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8D24411yescharacteristic
307C22×C8D24411yescharacteristic
308C22×C8D24411yescharacteristic
309C22×C8C22×D6111yescharacteristic
310C22×C8C22×D6111yescharacteristic
311C22×C8C22×D6111yescharacteristic
312#123—2441no—
313C22⋊Q8—1221no—
314C22⋊Q8—1221no—
315#121—1222no—
316#121—1222no—
317#121—1221no—
318#121—1221no—
319#121—2441no—
320#121—2441no—
321#120—2441no—
322C4×D4—2441no—
323C4×D4—2441no—
324#117—1221no—
325C2×C4⋊C4—1221no—
326C2×C22⋊C4—2441no—
327SD32—1224no—
328D16—1224no—
329M32—22no—
330M32—24no—
331M32C22⋊D6114yescomplemented
332C2×C16—24no—
333C2×C16C22×D6112yes—
334C2×C16C22×D6112yes—
335#107—611no—
336#107—611no—
337#107—612no—
338#107—1221no—
339#106—1221no—
340#103—1221no—
341#103—1221no—
342#102—1222no—
343#102—1222no—
344#102—1221no—
345#102—1221no—
346C24—2441no—
347C24D48811yescharacteristic
348#41—2442no—
349#41—2442no—
350C2×Q8—1221no—
351C2×D4—1222no—
352C2×D4—1221no—
353C2×D4—4884no—
354C2×D4—2441no—
355C2×D4—2441no—
356C2×D4—2441no—
357C2×D4—2442no—
358C2×D4—2442no—
359C2×D4—2442no—
360C2×D4—2441no—
361C2×D4—2442no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—2441no—
365C22×C4—2441no—
366C22×C4D48811yescharacteristic
367C22×C4D48811yescharacteristic
368C22×C4D48811yescharacteristic
369C22×C4C2×D24411yescharacteristic
370C22×C4C2×D24411yescharacteristic
371C22×C4C2×D24411yescharacteristic
372Q16—1221no—
373Q16—1221no—
374SD16—2442no—
375D8—1221no—
376D8—1222no—
377D8—1222no—
378D8—1221no—
379D8—2442no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8C2×D24411yescharacteristic
388C2×C8C2×D24411yescharacteristic
389C2×C8C2×D24411yescharacteristic
390C2×C8#1175822212yes—
391C2×C8C2×C22⋊D6112yes—
392C2×C8C23×D6111yescharacteristic
393C4⋊C4—1222no—
394C4⋊C4—1221no—
395C4⋊C4—1221no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C22⋊C4—2442no—
403C22⋊C4—2442no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42—2441no—
407C16—22no—
408C16C2×C22⋊D6114yescomplemented
409C23—22no—
410C23—22no—
411C23—41no—
412C23—4882no—
413C23—4882no—
414C23—4882no—
415C23—2441no—
416C23—2442no—
417C23C2×D48811yescharacteristic
418C23C2×D48811yescharacteristic
419C23C2×D48811yescharacteristic
420Q8—2441no—
421Q8—2441no—
422D4—4884no—
423D4—4882no—
424D4—4884no—
425D4—4882no—
426D4—4882no—
427D4—4882no—
428D4—2442no—
429D4—2441no—
430D4—2442no—
431D4—2441no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—4882no—
440C2×C4—4882no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2442no—
444C2×C4—2441no—
445C2×C4#1451975112yes—
446C2×C4C2×D48811yescharacteristic
447C2×C4C2×D48811yescharacteristic
448C2×C4C2×D48811yescharacteristic
449C2×C4#1451979912yes—
450C2×C4C22×D24411yescharacteristic
451C8—21no—
452C8—21no—
453C8#1451975112yes—
454C8C22×C22⋊D6111yescharacteristic
455C8C2×#1175822211yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—9764no—
465C22—4882no—
466C22—4881no—
467C22—4881no—
468C22—4882no—
469C22—4881no—
470C22—4882no—
471C22#1683999812yes—
472C22#1684020012yescomplemented
473C22C22×D48811yescharacteristic
474C4—21no—
475C4—21no—
476C4—4882no—
477C4—4881no—
478C4—4881no—
479C4—4881no—
480C4#1683999812yes—
481C4C2×#1451975111yescharacteristic
482C4C2×#1451979911yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—9762no—
487C2—9761no—
488C2C2×#1683999811yescharacteristic
489C2C2×#1684020011yescharacteristic
490C2#2145895511yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC22×M32×C61C2
socleC22×C61C22×D8
hypercenterC22×M32D61
hypocenterC61#33605
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. #21455400C2semidirect
3. C2×#16839998C2semidirect
4. #21447932C2semidirect
5. #21447922C2semidirect
6. #21447915C2semidirect
7. #21447905C2semidirect
8. #21447607C2semidirect
9. #21447536C2semidirect
10. #21447532C2semidirect
11. C22×M32×C61C2semidirect
17. C2×#14519722C22semidirect
18. C2×D976C22semidirect
19. #16840003C22semidirect
20. #16840003C22semidirect
25. #16839998C22semidirect
26. #16839998C22semidirect
33. #16839590C22semidirect
34. #16839588C22semidirect
35. #16839567C22semidirect
36. #16839564C22semidirect
37. #16839564C22semidirect
38. #16839564C22semidirect
43. C2×M32×C61C22semidirect
44. C22×C16×C61C22semidirect
96. M32×C61D4semidirect
162. C16×C61C2×D4semidirect
220. C22×C61#509semidirect
239. C22×M32D61semidirect
268. C2×M32C2×D61semidirect
269. C22×C16C2×D61semidirect
290. C61#33605semidirect
331. M32C22⋊D61semidirect
408. C16C2×C22⋊D61semidirect
472. C22#16840200semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
#214554002
C2×#168399982
#214479322
#214479222
#214479152
#214479052
#214476072
#214475362
#214475322
C22×M32×C612
#3360561