Finite Group Databasebeta

← #29687394

Subgroups — #29687394

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#21455732C211yesmaximal, characteristic
2#21454509C211yesmaximal, characteristic
3C2×#16839718C211yesmaximal, characteristic, complemented
4#21446465C212yesmaximal, complemented
5#21446416C212yesmaximal, complemented
6#21446119C211yesmaximal, characteristic
7#21446023C211yesmaximal, characteristic, complemented
8#21445702C212yesmaximal, complemented
9#21445660C212yesmaximal, complemented
10#21445597C211yesmaximal, characteristic, complemented
11C61×#1826C211yesmaximal, characteristic, complemented
12C2×#14519785C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840273C2211yescharacteristic
16#16840103C412yescomplemented
17#16840097C411yescharacteristic, complemented
18#16840092C411yescharacteristic, complemented
19#16840086C412yescomplemented
20#16840080C411yescharacteristic, complemented
21#16840075C411yescharacteristic, complemented
22#16839830—21no—
23#16839830—21no—
24#16839825—21no—
25#16839825—21no—
26C2×#14519712C2212yes—
27#16839798C2212yes—
28#16839790C2211yescharacteristic
29C2×#14519705C2212yes—
30#16839756C2212yes—
31#16839730C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33#16839720C2211yescharacteristic
34#16839718—21no—
35#16839718—21no—
36#16839718C2212yes—
37#16839718C2212yes—
38C2×#14519688—21no—
39C2×#14519688C2212yes—
40C2×C4×#92418—21no—
41#16839526C2212yes—
42#16839522C2212yes—
43#16839517C2212yes—
44C2×#117×C61C2211yescharacteristic
45C61×#407C2211yescharacteristic
46C4×C22⋊C4×C61C2211yescharacteristic
47C61×#339C2212yes—
48C61×#339C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519785—24no—
51C2×#11758194—21no—
52C2×#11758194—24no—
53C2×#11758194C2312yes—
54C2×#11758189—21no—
55C22×#92418—21no—
56C22×#92418—24no—
57C22×#92418D411yescharacteristic
58C22×#92418D411yescharacteristic
59C22×#92418D411yescharacteristic
60C22×#92418D411yescharacteristic
61C22×#92418C2312yes—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C2×C8×D61—21no—
65#14519760—21no—
66#14519756C2×C411yescharacteristic
67#14519756C2×C411yescharacteristic
68#14519751C2×C411yescharacteristic
69#14519751C2×C411yescharacteristic
70#14519746—21no—
71C4×C22⋊D61—21no—
72C4×C22⋊D61—21no—
73#14519736C2311yescharacteristic
74#14519712—24no—
75#14519710C2×C412yes—
76#14519710C2×C412yes—
77#14519708C2×C412yes—
78#14519708C2×C412yes—
79#14519706—21no—
80#14519706—21no—
81#14519705—24no—
82#14519700C2×C412yes—
83#14519700C2×C412yes—
84#14519697C2×C412yes—
85#14519697C2×C412yes—
86C22⋊#92418C2×C411yescharacteristic
87C22⋊#92418C2×C411yescharacteristic
88C22⋊#92418C2×C412yes—
89#14519689—21no—
90#14519689—21no—
91#14519688—22no—
92#14519688—22no—
93#14519688—24no—
94#14519687—21no—
95C4×#92418—22no—
96C4×#92418—22no—
97C8×Dic61—21no—
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61C2311yescharacteristic
100C2×M16×C61C2311yescharacteristic
101#117×C61—21no—
102#117×C61—21no—
103#117×C61C2×C412yes—
104#117×C61C2312yes—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110C2×C42×C61C2312yes—
111C22⋊C8×C61—21no—
112C22⋊C8×C61—21no—
113#95×C61—21no—
114C24×C61#4111yescharacteristic
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—21no—
127C2×C22⋊D61—21no—
128C2×C22⋊D61—21no—
129#11758222—22no—
130#11758222—22no—
131C2×D244C22×C411yescharacteristic
132C2×C4×D61—21no—
133C2×Dic122C22×C411yescharacteristic
134M16×C61—22no—
135M16×C61—22no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61C22×C412yes—
141C4⋊C4×C61C22×C412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—42no—
145C42×C61—22no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61—22no—
149C22⋊Dic61—21no—
150C22⋊Dic61—21no—
151C22⋊Dic61—21no—
152#11758198—21no—
153#11758198—21no—
154C4⋊Dic61C22×C411yescharacteristic
155C4⋊Dic61C22×C411yescharacteristic
156#11758196—21no—
157#11758196—21no—
158C4×Dic61—21no—
159#11758194—24no—
160#11758194—22no—
161#11758194#4114yes—
162C2×#92418—24no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418C22×C412yes—
172C2×#92418C22×C412yes—
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×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C4×D411yescharacteristic
202C2×C4×C61C4×D411yescharacteristic
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C61C2×#4112yes—
207C22⋊D61—42no—
208C22⋊D61—42no—
209C22⋊D61—42no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213D244—22no—
214C4×D61—41no—
215C4×D61—41no—
216Dic122—22no—
217C8×C61—41no—
218C8×C61—41no—
219#92418—22no—
220#92418—22no—
221#92418—21no—
222#92418—21no—
223#92418—24no—
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×C61—42no—
244C4×C61—42no—
245C4×C61—82no—
246C4×C61C4×#4112yes—
247C4×C61C2×#12311yescharacteristic
248C4×C61#52511yescharacteristic
249Dic61—41no—
250Dic61—41no—
251Dic61—41no—
252#3091—611no—
253#2929—611no—
254C2×#405—611no—
255#1929—612no—
256#1922—612no—
257#1871—611no—
258#1855—611no—
259#1826D6111yescharacteristic, complemented
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×M16—611no—
272C23×C8—611no—
273#525—611no—
274C2×#117C2×D6111yescharacteristic
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×#108—612no—
286#427—612no—
287#424—611no—
288C2×#104—612no—
289#416—612no—
290#408—611no—
291#407C2×D6111yescharacteristic
292C2×C22⋊C8—611no—
293#405—611no—
294#405—612no—
295#405—612no—
296#405—1221no—
297#405—1221no—
298C2×#97—612no—
299C2×#97—1221no—
300C2×C4×C8—1221no—
301C4×C22⋊C4C2×D6111yescharacteristic
302#341—612no—
303#339—612no—
304#339C2×D6112yes—
305#339C2×D6112yes—
306#336—612no—
307C61#2159311yescharacteristic, complemented, Sylow
308C2×#41—611no—
309C23×C4C22×D6111yescharacteristic
310#131—1224no—
311C2×M16—612no—
312C2×M16—1221no—
313C2×M16—1224no—
314C2×M16—1221no—
315C2×M16C22×D6111yescharacteristic
316C2×M16C22×D6111yescharacteristic
317C22×C8—611no—
318C22×C8—611no—
319C22×C8—611no—
320C22×C8—611no—
321C22×C8—611no—
322C22×C8—611no—
323C22×C8—612no—
324C22×C8—1221no—
325C22×C8—1224no—
326C22×C8—1221no—
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—21no—
336#117—21no—
337#117—611no—
338#117C4×D6112yes—
339#117C22×D6112yes—
340C2×C4⋊C4C4×D6111yescharacteristic
341C2×C4⋊C4C4×D6111yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C42—21no—
345C2×C42C22×D6112yes—
346#108—1224no—
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—611no—
361C22⋊C8—611no—
362C22⋊C8—612no—
363C22⋊C8—1221no—
364C22⋊C8—1221no—
365#97—1224no—
366#97—1222no—
367#97—1222no—
368#97—1221no—
369C4×C8—1222no—
370C4×C8—1222no—
371C4×C8—1221no—
372#95—21no—
373C24#1175822411yescharacteristic
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—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—1221no—
387C22×C4C2×C4×D6111yescharacteristic
388C22×C4C2×C4×D6111yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4#1175822411yescharacteristic
392C22×C4C23×D6111yescharacteristic
393M16—22no—
394M16—22no—
395M16—614no—
396M16—1224no—
397M16—1222no—
398M16—2442no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—611no—
402C2×C8—611no—
403C2×C8—614no—
404C2×C8—612no—
405C2×C8—612no—
406C2×C8—611no—
407C2×C8—611no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1224no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—2442no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—611no—
421C4⋊C4—611no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4C2×C4×D6112yes—
425C4⋊C4C2×C4×D6112yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—42no—
429C22⋊C4—1221no—
430C22⋊C4—1221no—
431C22⋊C4—1221no—
432C22⋊C4—1221no—
433C22⋊C4—1221no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—1221no—
439C23—22no—
440C23—22no—
441C23—41no—
442C23—1221no—
443C23#1451973711yescharacteristic
444C23#1451973711yescharacteristic
445C23C2×#1175822411yescharacteristic
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—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4#1451973711yescharacteristic
471C2×C4#1451973711yescharacteristic
472C2×C4C22×C4×D6111yescharacteristic
473C2×C4C2×D4×D6111yescharacteristic
474C2×C4#41×D6112yes—
475C2×C4#41×D6112yes—
476C8—41no—
477C8—41no—
478C8—612no—
479C8—612no—
480C8—1222no—
481C8—1222no—
482C8—1221no—
483C8—1221no—
484C8—1224no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—44no—
493C22—2441no—
494C22—2441no—
495C22—2441no—
496C22#1683988112yes—
497C22C2×#1451973711yescharacteristic
498C22#1684028912yes—
499C4—21no—
500C4—21no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—2441no—
505C4—2441no—
506C4—2441no—
507C4#1684023012yes—
508C4#123×D6111yescharacteristic
509C4#1684027011yescharacteristic
510C2—22no—
511C2—22no—
512C2—41no—
513C2—4881no—
514C2#2144833611yescharacteristic
515C2C2×#1683988111yescharacteristic
516C2#2145471811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1826C2
socleC22×C61C2×C4×D4
prime radical#16840273C22
cosocleC22×C4×C61C24
hypercenter#1826D61
hypocenterC61#21593
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
3. C2×#16839718C2semidirect
4. #21446465C2semidirect
5. #21446416C2semidirect
7. #21446023C2semidirect
8. #21445702C2semidirect
9. #21445660C2semidirect
10. #21445597C2semidirect
11. C61×#1826C2semidirect
16. #16840103C4semidirect
17. #16840097C4semidirect
18. #16840092C4semidirect
19. #16840086C4semidirect
20. #16840080C4semidirect
21. #16840075C4semidirect
259. #1826D61semidirect
307. C61#21593semidirect

Maximal subgroups

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

subgroupindex
#214557322
#214545092
C2×#168397182
#214464652
#214464162
#214461192
#214460232
#214457022
#214456602
#214455972
C61×#18262
#2159361