Finite Group Databasebeta

← #29687396

Subgroups — #29687396

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455734C211yesmaximal, characteristic
2#21454509C211yesmaximal, characteristic
3C2×#16839720C211yesmaximal, characteristic, complemented
4#21446466C212yesmaximal, complemented
5#21446417C212yesmaximal, complemented
6#21446121C211yesmaximal, characteristic
7#21445916C211yesmaximal, characteristic, complemented
8#21445706C212yesmaximal, complemented
9#21445662C212yesmaximal, complemented
10#21445598C211yesmaximal, characteristic, complemented
11C61×#1855C211yesmaximal, characteristic, complemented
12C2×#14519785C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840273C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16840104C412yescomplemented
17#16840098C411yescharacteristic, complemented
18#16840093C411yescharacteristic, complemented
19#16840087C412yescomplemented
20#16840081C411yescharacteristic, complemented
21#16840076C411yescharacteristic, complemented
22#16839830—21no—
23#16839830—21no—
24#16839825—21no—
25#16839825—21no—
26#16839800C2212yes—
27#16839790C2211yescharacteristic
28C2×#14519703C2212yes—
29#16839759C2212yes—
30#16839730C2211yescharacteristic
31#16839720—21no—
32#16839720—21no—
33#16839720C2212yes—
34#16839720C2212yes—
35#16839720C2211yescharacteristic
36C2×#14519687—21no—
37C2×#14519687C2212yes—
38C2×C8×Dic61—21no—
39C4×C22⋊Dic61C2211yescharacteristic
40#16839523C2212yes—
41#16839523C2212yes—
42#16839523C2212yes—
43#16839516C2212yes—
44C22×M16×C61C2211yescharacteristic
45C2×#108×C61C2212yes—
46C61×#407C2211yescharacteristic
47C2×C22⋊C8×C61C2211yescharacteristic
48C61×#341C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519785—24no—
51C2×#11758189—21no—
52C22×#92418—24no—
53C22×#92418D411yescharacteristic
54C22×#92418D411yescharacteristic
55C22×#92418D411yescharacteristic
56C22×#92418D411yescharacteristic
57C22×#92418C2311yescharacteristic
58C22×#92418C2311yescharacteristic
59C2×C8×D61—21no—
60#14519760—21no—
61#14519756C2×C411yescharacteristic
62#14519756C2×C411yescharacteristic
63#14519751C2×C411yescharacteristic
64#14519751C2×C411yescharacteristic
65#14519746—21no—
66C4×C22⋊D61—21no—
67C4×C22⋊D61—21no—
68#14519736—21no—
69#14519736—21no—
70#14519736C2×C412yes—
71#14519736C2312yes—
72#14519736C2311yescharacteristic
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61C2312yes—
79#14519711C2×C412yes—
80#14519711C2×C412yes—
81#14519709C2×C412yes—
82#14519709C2×C412yes—
83#14519706—21no—
84#14519706—21no—
85#14519703—24no—
86#14519702C2×C412yes—
87#14519702C2×C412yes—
88#14519699C2×C412yes—
89#14519699C2×C412yes—
90#14519689—21no—
91#14519689—21no—
92#14519687—22no—
93#14519687—22no—
94#14519687—21no—
95#14519687—24no—
96C8×Dic61—22no—
97C8×Dic61—22no—
98C8×Dic61—21no—
99#14519684—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—21no—
102C2×M16×C61—24no—
103C2×M16×C61C2311yescharacteristic
104C2×M16×C61C2311yescharacteristic
105C2×M16×C61C2312yes—
106C22×C8×C61—21no—
107C22×C8×C61C2312yes—
108#108×C61—24no—
109C22⋊C8×C61—21no—
110C22⋊C8×C61—21no—
111C22⋊C8×C61C2×C411yescharacteristic
112C22⋊C8×C61C2×C411yescharacteristic
113C22⋊C8×C61C2×C412yes—
114C24×C61#4111yescharacteristic
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C2×C22⋊D61—21no—
125C2×C22⋊D61—21no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129#11758222—22no—
130#11758222—22no—
131C2×D244C22×C411yescharacteristic
132C2×C4×D61—21no—
133C2×Dic122C22×C411yescharacteristic
134M16×C61—24no—
135M16×C61—22no—
136M16×C61—22no—
137M16×C61—22no—
138M16×C61#4114yes—
139C2×C8×C61—24no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C22×C412yes—
146C2×C8×C61C22×C412yes—
147C22⋊Dic61—22no—
148C22⋊Dic61—21no—
149C22⋊Dic61—21no—
150C22⋊Dic61—22no—
151C22⋊Dic61—21no—
152C22⋊Dic61—42no—
153#11758198—21no—
154#11758198—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61C22×C412yes—
158C4⋊Dic61C22×C412yes—
159C4⋊Dic61C22×C411yescharacteristic
160C4⋊Dic61C22×C411yescharacteristic
161#11758196—21no—
162#11758196—21no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C4×Dic61—21no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175C2×#92418C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177C2×#92418#4114yes—
178#11758189—42no—
179C8×D61—42no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61C4×D411yescharacteristic
184C23×C61C4×D411yescharacteristic
185C23×C61C2×#4111yescharacteristic
186C22×D61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61C4×D411yescharacteristic
195C2×C4×C61C4×D411yescharacteristic
196C2×C4×C61C23×C411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4112yes—
199C2×C4×C61C2×#4112yes—
200C22⋊D61—42no—
201C22⋊D61—42no—
202C22⋊D61—42no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213D244—22no—
214C4×D61—41no—
215C4×D61—41no—
216Dic122—22no—
217C8×C61—22no—
218C8×C61—22no—
219C8×C61—24no—
220C8×C61—41no—
221C8×C61—41no—
222#92418—21no—
223#92418—21no—
224#92418#12312yes—
225#92418#12312yes—
226#21593—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—44no—
235C22×C61#44412yes—
236C22×C61C2×C4×D411yescharacteristic
237C22×C61#52912yes—
238C2×D61—41no—
239C2×D61—41no—
240C2×D61—41no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61C4×#4112yes—
244C4×C61C2×#12311yescharacteristic
245C4×C61#52511yescharacteristic
246Dic61—42no—
247Dic61—41no—
248Dic61—42no—
249Dic61—41no—
250Dic61—41no—
251Dic61—82no—
252#3091—611no—
253#2929—611no—
254C2×#405—611no—
255#1929—612no—
256#1922—612no—
257#1871—611no—
258#1855D6111yescharacteristic, complemented
259#1826—611no—
260#1773—612no—
261#1765—612no—
262#1752—611no—
263C2×C61—22no—
264C2×C61—22no—
265C2×C61—41no—
266C2×C61C4×#12311yescharacteristic
267C2×C61C2×#44411yescharacteristic
268C2×C61#295511yescharacteristic
269D61—81no—
270C2×#131—611no—
271C22×M16C2×D6111yescharacteristic
272C23×C8—611no—
273#525—611no—
274C2×#117—611no—
275#485—612no—
276#484—611no—
277#483—611no—
278#482—612no—
279#481—611no—
280#480—611no—
281#435—1221no—
282#435—1221no—
283C8×D4—1221no—
284C8×D4—1221no—
285C2×#108C2×D6112yes—
286#427—612no—
287#424—611no—
288C2×#104—612no—
289#416—612no—
290#408—611no—
291#407C2×D6111yescharacteristic
292C2×C22⋊C8C2×D6111yescharacteristic
293#405—612no—
294#405—612no—
295#405—611no—
296#405—1221no—
297#405—1221no—
298C2×#97—612no—
299C2×#97—1221no—
300C2×C4×C8—1221no—
301C4×C22⋊C4—611no—
302#341C2×D6112yes—
303#339—612no—
304#339—612no—
305#339—612no—
306#336—612no—
307C61#2159311yescharacteristic, complemented, Sylow
308C2×#41—611no—
309C23×C4C22×D6111yescharacteristic
310#131—1224no—
311C2×M16—21no—
312C2×M16—24no—
313C2×M16—1221no—
314C2×M16C22×D6111yescharacteristic
315C2×M16C22×D6111yescharacteristic
316C2×M16C22×D6112yes—
317C22×C8—21no—
318C22×C8—611no—
319C22×C8—611no—
320C22×C8—611no—
321C22×C8—611no—
322C22×C8—611no—
323C22×C8—611no—
324C22×C8—1221no—
325C22×C8—1224no—
326C22×C8C22×D6112yes—
327#123—1221no—
328C22⋊Q8—611no—
329C22⋊Q8—611no—
330#121—611no—
331#121—611no—
332#120—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335#117—612no—
336#117—612no—
337#117—611no—
338#117—1221no—
339#117—1221no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—611no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C42—612no—
345C2×C42—1221no—
346#108—24no—
347#107—612no—
348#107—612no—
349#106—612no—
350#106—612no—
351C4⋊C8—1221no—
352C4⋊C8—1221no—
353#104—1224no—
354#103—612no—
355#103—612no—
356#102—612no—
357#102—612no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—1221no—
361C22⋊C8—1221no—
362C22⋊C8C4×D6111yescharacteristic
363C22⋊C8C4×D6111yescharacteristic
364C22⋊C8C4×D6112yes—
365#97—1224no—
366#97—1222no—
367#97—1222no—
368#97—1221no—
369C4×C8—1222no—
370C4×C8—1222no—
371C4×C8—1221no—
372#95—1221no—
373C24#1175822611yescharacteristic
374#41—1222no—
375#41—1222no—
376C2×Q8—611no—
377C2×D4—611no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1222no—
387C22×C4C2×C4×D6111yescharacteristic
388C22×C4C2×C4×D6111yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4#1175822611yescharacteristic
392C22×C4C23×D6111yescharacteristic
393M16—24no—
394M16—22no—
395M16—22no—
396M16—22no—
397M16—2442no—
398M16#1175822414yes—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—611no—
406C2×C8—611no—
407C2×C8—614no—
408C2×C8—611no—
409C2×C8—611no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—2442no—
416C2×C8C2×C4×D6112yes—
417C2×C8C2×C4×D6112yes—
418C4⋊C4—612no—
419C4⋊C4—612no—
420C4⋊C4—611no—
421C4⋊C4—611no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C22⋊C4—1222no—
427C22⋊C4—1221no—
428C22⋊C4—1222no—
429C22⋊C4—1221no—
430C22⋊C4—1221no—
431C22⋊C4—1221no—
432C22⋊C4—1221no—
433C22⋊C4—2442no—
434C42—1222no—
435C42—1221no—
436C42—1222no—
437C42—1222no—
438C42—1222no—
439C23—22no—
440C23—22no—
441C23—41no—
442C23—1221no—
443C23#1451973911yescharacteristic
444C23#1451973911yescharacteristic
445C23C2×#1175822611yescharacteristic
446Q8—1222no—
447D4—1222no—
448D4—2442no—
449D4—2442no—
450D4—2442no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4#1451973911yescharacteristic
471C2×C4#1451973911yescharacteristic
472C2×C4C22×C4×D6111yescharacteristic
473C2×C4C2×D4×D6111yescharacteristic
474C2×C4#41×D6112yes—
475C2×C4C2×#1175822412yes—
476C8—22no—
477C8—22no—
478C8—24no—
479C8—41no—
480C8—41no—
481C8—612no—
482C8—612no—
483C8—1221no—
484C8—1221no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—44no—
493C22—2441no—
494C22—2441no—
495C22—2441no—
496C22#1683988312yes—
497C22C2×#1451973911yescharacteristic
498C22#1684029812yes—
499C4—21no—
500C4—21no—
501C4—4882no—
502C4—2442no—
503C4—2441no—
504C4—2442no—
505C4—2441no—
506C4—2441no—
507C4C4×#1175822412yes—
508C4#123×D6111yescharacteristic
509C4#1684026811yescharacteristic
510C2—22no—
511C2—22no—
512C2—41no—
513C2—4881no—
514C2#2144833811yescharacteristic
515C2C2×#1683988311yescharacteristic
516C2#2145472111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1855C2
socleC22×C61C2×C4×D4
prime radical#16840273C22
cosocleC22×C4×C61C24
hypercenter#1855D61
hypocenterC61#21593
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
3. C2×#16839720C2semidirect
4. #21446466C2semidirect
5. #21446417C2semidirect
7. #21445916C2semidirect
8. #21445706C2semidirect
9. #21445662C2semidirect
10. #21445598C2semidirect
11. C61×#1855C2semidirect
16. #16840104C4semidirect
17. #16840098C4semidirect
18. #16840093C4semidirect
19. #16840087C4semidirect
20. #16840081C4semidirect
21. #16840076C4semidirect
258. #1855D61semidirect
307. C61#21593semidirect

Maximal subgroups

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

subgroupindex
#214557342
#214545092
C2×#168397202
#214464662
#214464172
#214461212
#214459162
#214457062
#214456622
#214455982
C61×#18552
#2159361