Finite Group Databasebeta

← #29654265

Subgroups — #29654265

Order 15616.

950 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456818C212yesmaximal, complemented
2#21451506C211yesmaximal, characteristic
3#21444793C212yesmaximal, complemented
4#21444776C212yesmaximal, complemented
5#21444759C212yesmaximal, complemented
6#21444742C212yesmaximal
7#21444331C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic, complemented
9C61×#3458C211yesmaximal, characteristic, complemented
10C2×#14519765C2211yescharacteristic
11#16840154—22no—
12#16840124—22no—
13#16839984—24no—
14#16839917C2214yes—
15#16839912C2214yes—
16#16839819C2212yes—
17#16839682C2211yescharacteristic
18#16839679C2211yescharacteristic
19#16839674C2211yescharacteristic
20#16839671C2211yescharacteristic
21#16839662—21no—
22#16839653—21no—
23#16839652—21no—
24#16839641—21no—
25#16839608C2211yescharacteristic
26#16839499C2214yes—
27#16839499C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C61×#544—22no—
31C61×#544—22no—
32C61×#544—21no—
33C61×#544C2212yes—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—24no—
37C61×#533—21no—
38C61×#533C2212yes—
39C2×#117×C61—21no—
40C2×#117×C61C2211yescharacteristic
41C22×Dic122C2311yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C22⋊D61—21no—
44C2×#11758202—24no—
45C2×#11758201—24no—
46#14519765—22no—
47#14519765—22no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×C4×Dic61—21no—
56C4⋊#92418—22no—
57#14519700—24no—
58#14519697—24no—
59C22⋊#92418C2314yes—
60#14519688—22no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C23×C4×C61D411yescharacteristic
67C4⋊Q8×C61—24no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—24no—
71C4⋊Q8×C61D412yescomplemented
72C4⋊Q8×C61D412yescomplemented
73#125×C61—24no—
74#125×C61—22no—
75#125×C61—24no—
76#125×C61—22no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61C2314yes—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yescomplemented
89#117×C61—22no—
90#117×C61—22no—
91#117×C61—22no—
92#117×C61—22no—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61C2311yescharacteristic
102C2×C42×C61—22no—
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61C2×D414yes—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—81no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C2×D244—22no—
130C2×D244—81no—
131C2×Dic122—22no—
132C2×Dic122—81no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61C2×D414yes—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C42×C61—24no—
152C42×C61—24no—
153C42×C61—21no—
154C42×C61—21no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—81no—
158#11758202—84no—
159#11758201—84no—
160#11758198—42no—
161#11758198—41no—
162#11758198—41no—
163#11758198—42no—
164#11758198—42no—
165#11758198—81no—
166#11758196—81no—
167C4×Dic61—42no—
168C2×#92418—24no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61#13712yes—
174C23×C61C22×D411yescharacteristic
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182Q8×C61—44no—
183Q8×C61—82no—
184Q8×C61—82no—
185Q8×C61—82no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—24no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—24no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—82no—
201C2×C4×C61—82no—
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12012yes—
205C2×C4×C61#13712yes—
206C2×C4×C61C22×D412yes—
207C22⋊D61—84no—
208C2×Dic61—22no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217D244—44no—
218Dic122—44no—
219#92418—44no—
220#18333—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—22no—
224C22×C61—42no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—81no—
228C22×C61C2×#12011yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C22×C61C2×#13711yescharacteristic
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—22no—
241C4×C61—22no—
242C4×C61—22no—
243C4×C61—41no—
244C4×C61—44no—
245C4×C61—41no—
246C4×C61—44no—
247Dic61—42no—
248Dic61—81no—
249Dic61—81no—
250#3458D6111yescharacteristic, complemented
251#3215—612no—
252#2610—611no—
253#1605—612no—
254#1602—612no—
255#1599—612no—
256#1596—612no—
257#1506—612no—
258#1464—611no—
259C2×C61—21no—
260C2×C61—21no—
261C2×C61—42no—
262C2×C61#300411yescharacteristic
263C2×C61#301811yescharacteristic
264C2×C61#302711yescharacteristic
265D61—161no—
266#544—22no—
267#544—22no—
268#544—21no—
269#544C2×D6112yes—
270#543—21no—
271#543C2×D6112yes—
272#533—24no—
273#533—21no—
274#533C2×D6112yes—
275C2×#124—611no—
276C2×#117—21no—
277C2×#117C2×D6111yescharacteristic
278#497—1222no—
279#489—1222no—
280#464—1224no—
281#451—614no—
282#450—614no—
283#432—612no—
284#395—611no—
285#394—611no—
286#393—611no—
287#392—611no—
288#390—1221no—
289#388—1221no—
290#388—1221no—
291#386—1221no—
292#376—611no—
293#328—614no—
294#328—614no—
295#324—612no—
296#323—612no—
297C61#1833311yescharacteristic, complemented, Sylow
298C22×Q8—611no—
299C22×D4—611no—
300C22×D4—1221no—
301C23×C4C22⋊D6111yescharacteristic
302C2×Q16—1224no—
303C2×SD16—1224no—
304C4⋊Q8—24no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—24no—
308C4⋊Q8C22⋊D6112yescomplemented
309C4⋊Q8C22⋊D6112yescomplemented
310#125—24no—
311#125—22no—
312#125—24no—
313#125—22no—
314#124—1222no—
315#124—1222no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—24no—
323C22⋊Q8C22×D6114yes—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8C22⋊D6114yescomplemented
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4C22⋊D6111yescharacteristic
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C22⋊C4—22no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1221no—
345C2×C22⋊C4—1221no—
346C2×C22⋊C4C22⋊D6111yescharacteristic
347C2×C22⋊C4C22×D6111yescharacteristic
348C2×C42—22no—
349C2×C42—1221no—
350C2×C42C22⋊D6111yescharacteristic
351C2×C42C22×D6111yescharacteristic
352C4⋊C8—1222no—
353#103—1224no—
354#102—1224no—
355C22⋊C8—614no—
356#97—1222no—
357#95—611no—
358#95—611no—
359#95—1221no—
360#95—1221no—
361#95—1221no—
362C24—1221no—
363C24C2×C22⋊D6111yescharacteristic
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—1222no—
368C2×Q8—4881no—
369C2×Q8C2×C22⋊D6114yes—
370C2×D4—1222no—
371C2×D4—4881no—
372C2×D4—4881no—
373C2×D4—2442no—
374C2×D4—2442no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4C2×C22⋊D6111yescharacteristic
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C2×C22⋊D6111yescharacteristic
389C22×C4C23×D6111yescharacteristic
390Q16—4884no—
391SD16—4884no—
392C2×C8—1224no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—4881no—
407C4⋊C4C2×C22⋊D6114yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—4881no—
413C22⋊C4—4881no—
414C22⋊C4—2442no—
415C22⋊C4—2442no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C22⋊C4—2442no—
419C42—24no—
420C42—24no—
421C42—21no—
422C42—21no—
423C42—2442no—
424C42C2×C22⋊D6112yes—
425C42C2×C22⋊D6112yes—
426C23—21no—
427C23—21no—
428C23—41no—
429C23—41no—
430C23—2441no—
431C23—2441no—
432C23—2441no—
433C23—2441no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23#1451980812yes—
438C23C22×C22⋊D6111yescharacteristic
439Q8—44no—
440Q8—82no—
441Q8—82no—
442Q8—82no—
443Q8—2444no—
444D4—4884no—
445D4—2444no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—1222no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4#1451974611yescharacteristic
472C2×C4#1451974611yescharacteristic
473C2×C4#1451974612yes—
474C2×C4#1451980812yes—
475C2×C4C22×C22⋊D6112yes—
476C8—2444no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22—81no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22C2×#1451974611yescharacteristic
492C22C2×#1451980811yescharacteristic
493C22C2×#1451980811yescharacteristic
494C4—22no—
495C4—22no—
496C4—22no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—44no—
501C4—41no—
502C4—44no—
503C4—4881no—
504C4—4881no—
505C4—2442no—
506C2—21no—
507C2—21no—
508C2—42no—
509C2—9761no—
510C2#2145505311yescharacteristic
511C2#2145516811yescharacteristic
512C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radical#16839671C22
hypercenter#3458D61
hypocenterC61#18333
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #21456818C2semidirect
3. #21444793C2semidirect
4. #21444776C2semidirect
5. #21444759C2semidirect
7. #21444331C2semidirect
8. #21444153C2semidirect
9. C61×#3458C2semidirect
71. C4⋊Q8×C61D4semidirect
72. C4⋊Q8×C61D4semidirect
88. C4×Q8×C61D4semidirect
250. #3458D61semidirect
297. C61#18333semidirect
308. C4⋊Q8C22⋊D61semidirect
309. C4⋊Q8C22⋊D61semidirect
327. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214568182
#214515062
#214447932
#214447762
#214447592
#214447422
#214443312
#214441532
C61×#34582
#1833361