Finite Group Databasebeta

← #29654440

Subgroups — #29654440

Order 15616.

996 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456621C212yesmaximal, complemented
2#21451703C211yesmaximal, characteristic
3#21444861C212yesmaximal, complemented
4#21444844C212yesmaximal, complemented
5#21444759C212yesmaximal, complemented
6#21444742C212yesmaximal, complemented
7#21444352C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#3501C211yesmaximal, characteristic, complemented
10C2×#14519751C2211yescharacteristic
11#16840095C2214yes—
12#16840075—22no—
13#16840010—24no—
14#16839912C2214yes—
15#16839901—22no—
16#16839726C2212yes—
17#16839701C2211yescharacteristic
18#16839698C2211yescharacteristic
19#16839679C2211yescharacteristic
20#16839677—21no—
21#16839671C2211yescharacteristic
22#16839670—21no—
23#16839653—21no—
24#16839641—21no—
25#16839625C2211yescharacteristic
26#16839499C2214yes—
27#16839499C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C61×#554—24no—
31C61×#554—22no—
32D4×Q8×C61—22no—
33C61×#544C2212yes—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×D244C2311yescharacteristic
42C22×C22⋊D61—21no—
43C22×C22⋊D61—21no—
44C2×#11758201—24no—
45C2×#11758194—22no—
46#14519751—22no—
47#14519751—22no—
48C2×C4⋊Dic61C2311yescharacteristic
49C2×#11758196—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56#14519710—24no—
57#14519701—24no—
58#14519697—24no—
59C22⋊#92418—22no—
60C22⋊#92418C2314yes—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C22×Q8×C61—21no—
67C22×Q8×C61D412yescomplemented
68C22×D4×C61D411yescharacteristic
69C23×C4×C61D411yescharacteristic
70C23×C4×C61C2311yescharacteristic
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—24no—
74#125×C61—22no—
75#125×C61—24no—
76C22⋊Q8×C61—24no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61D414yescomplemented
87C22⋊Q8×C61C2314yes—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61D412yescomplemented
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61C2311yescharacteristic
104C2×C42×C61D411yescharacteristic
105C24×C61C2×D411yescharacteristic
106C24×C61C2×D411yescharacteristic
107C23×D61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—82no—
112C2×Q8×C61—82no—
113C2×Q8×C61C2×D414yes—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—81no—
124C22×C4×C61—81no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—81no—
135C2×C22⋊D61—81no—
136C2×C22⋊D61—81no—
137C2×C22⋊D61—81no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C2×D244—22no—
142C2×D244—81no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D414yes—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—22no—
159C42×C61—24no—
160C42×C61—21no—
161C42×C61—21no—
162C22⋊Dic61—81no—
163C22⋊Dic61—81no—
164#11758201—84no—
165#11758198—41no—
166#11758198—41no—
167#11758198—42no—
168#11758198—81no—
169#11758198—81no—
170C4⋊Dic61—22no—
171#11758196—81no—
172#11758194—82no—
173C2×#92418—24no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61#12012yes—
183C23×C61#13712yes—
184C23×C61C22×D411yescharacteristic
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192Q8×C61—44no—
193Q8×C61—44no—
194Q8×C61—82no—
195Q8×C61—82no—
196D4×C61—28no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—21no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—81no—
211C2×C4×C61—82no—
212C2×C4×C61—82no—
213C2×C4×C61—82no—
214C2×C4×C61—82no—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#13712yes—
218C2×C4×C61C22×D412yes—
219C22⋊D61—84no—
220C22⋊D61—84no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230D244—44no—
231#92418—44no—
232#18352—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—42no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61C2×#12011yescharacteristic
246C22×C61C2×#13711yescharacteristic
247C22×C61C2×#13711yescharacteristic
248C2×D61—81no—
249C2×D61—81no—
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C4×C61—22no—
256C4×C61—22no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—44no—
261C4×C61—41no—
262C4×C61—44no—
263Dic61—81no—
264Dic61—81no—
265Dic61—81no—
266#3501D6111yescharacteristic, complemented
267#3193—612no—
268#2635—611no—
269#1617—612no—
270#1614—612no—
271#1599—612no—
272#1596—612no—
273#1511—612no—
274#1465—611no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—42no—
278C2×C61—42no—
279C2×C61#300411yescharacteristic
280C2×C61#302111yescharacteristic
281C2×C61#302611yescharacteristic
282D61—161no—
283#554—24no—
284#554—22no—
285D4×Q8—22no—
286#544C2×D6112yes—
287#543—21no—
288#543C2×D6112yes—
289#533—21no—
290C2×C22⋊Q8—21no—
291C2×C22⋊Q8C2×D6112yes—
292C2×#121—611no—
293#518—21no—
294C2×C4×D4C2×D6111yescharacteristic
295#484—614no—
296#480—1222no—
297#468—1224no—
298#450—614no—
299#448—1222no—
300#407—612no—
301#400—611no—
302#399—611no—
303#394—611no—
304#394—1221no—
305#392—611no—
306#392—1221no—
307#388—1221no—
308#386—1221no—
309#381—611no—
310#328—614no—
311#328—614no—
312#324—612no—
313#323—612no—
314C61#1835211yescharacteristic, complemented, Sylow
315C22×Q8—21no—
316C22×Q8C22⋊D6112yescomplemented
317C22×D4—611no—
318C22×D4—1221no—
319C22×D4—1221no—
320C22×D4C22⋊D6111yescharacteristic
321C23×C4C22⋊D6111yescharacteristic
322C23×C4C22×D6111yescharacteristic
323C2×SD16—1224no—
324C2×M16—1222no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—24no—
328#125—22no—
329#125—24no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—24no—
340C22⋊Q8C22⋊D6114yescomplemented
341C22⋊Q8C22×D6114yes—
342#121—1222no—
343#121—1222no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—24no—
347C4×D4—24no—
348C4×D4—22no—
349C4×D4—22no—
350#117—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4C22⋊D6112yescomplemented
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4C22⋊D6111yescharacteristic
367C2×C22⋊C4C22×D6111yescharacteristic
368C2×C42C22⋊D6111yescharacteristic
369#107—1224no—
370#103—1224no—
371#102—1224no—
372C22⋊C8—614no—
373C22⋊C8—1222no—
374#95—611no—
375#95—611no—
376#95—1221no—
377#95—1221no—
378#95—1221no—
379C24—1221no—
380C24C2×C22⋊D6111yescharacteristic
381C24C2×C22⋊D6111yescharacteristic
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8C2×C22⋊D6114yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—1222no—
392C2×D4—4881no—
393C2×D4—4881no—
394C2×D4—4881no—
395C2×D4—4881no—
396C2×D4—4881no—
397C2×D4—2442no—
398C2×D4—2442no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6112yes—
414C22×C4C2×C22⋊D6112yes—
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417SD16—4884no—
418M16—4882no—
419C2×C8—1224no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—1222no—
431C4⋊C4—4881no—
432C4⋊C4C2×C22⋊D6114yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—4881no—
439C22⋊C4—4881no—
440C22⋊C4—4881no—
441C22⋊C4—4881no—
442C22⋊C4—2442no—
443C22⋊C4—2441no—
444C22⋊C4—2441no—
445C42—24no—
446C42—21no—
447C42—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—2441no—
457C23—2441no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23—2441no—
463C23#1451974612yes—
464C23#1451980812yes—
465C23C22×C22⋊D6111yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—82no—
469Q8—82no—
470D4—28no—
471D4—4884no—
472D4—4884no—
473D4—2444no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4#1451974611yescharacteristic
502C2×C4#1451974611yescharacteristic
503C2×C4#1451980812yes—
504C2×C4C22×C22⋊D6112yes—
505C8—2444no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—4881no—
519C22—4881no—
520C22—4881no—
521C22—4881no—
522C22—4881no—
523C22—4881no—
524C22—4881no—
525C22C2×#1451974611yescharacteristic
526C22C2×#1451980811yescharacteristic
527C22C2×#1451980811yescharacteristic
528C4—22no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—44no—
534C4—41no—
535C4—44no—
536C4—4881no—
537C4—4881no—
538C4—4881no—
539C2—21no—
540C2—21no—
541C2—42no—
542C2—42no—
543C2—9761no—
544C2#2145505311yescharacteristic
545C2#2145519111yescharacteristic
546C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radical#21451703C2
hypercenter#3501D61
hypocenterC61#18352
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21456621C2semidirect
3. #21444861C2semidirect
4. #21444844C2semidirect
5. #21444759C2semidirect
6. #21444742C2semidirect
7. #21444352C2semidirect
8. #21444157C2semidirect
9. C61×#3501C2semidirect
67. C22×Q8×C61D4semidirect
86. C22⋊Q8×C61D4semidirect
98. C2×C4⋊C4×C61D4semidirect
266. #3501D61semidirect
314. C61#18352semidirect
316. C22×Q8C22⋊D61semidirect
340. C22⋊Q8C22⋊D61semidirect
356. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214566212
#214517032
#214448612
#214448442
#214447592
#214447422
#214443522
#214441572
C61×#35012
#1835261