Finite Group Databasebeta

← #29657041

Subgroups — #29657041

Order 15616.

876 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#21456826C211yesmaximal, characteristic
2#21456538C211yesmaximal, characteristic
3#21455850C211yesmaximal, characteristic
4#21445061C212yesmaximal, complemented
5#21445040C212yesmaximal, complemented
6#21444915C212yesmaximal, complemented
7#21444897C212yesmaximal, complemented
8#21444529C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3502C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840165C2211yescharacteristic
14#16840154—21no—
15#16840136C2211yescharacteristic
16#16840124—21no—
17#16840112C2211yescharacteristic
18#16840110C2211yescharacteristic
19#16840080—21no—
20#16840047C2212yes—
21#16840010—22no—
22#16839972C2212yes—
23#16839962C2212yes—
24#16839962C2212yes—
25#16839922—22no—
26#16839901—21no—
27#16839860—22no—
28C4×#11758201—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839511C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37Q8⋊#92418C2212yes—
38Q82×C61—21no—
39Q82×C61C2212yes—
40C61×#557—21no—
41C61×#557—22no—
42C61×#557—22no—
43C61×#557C2212yes—
44C61×#544—22no—
45C61×#544—21no—
46C61×#533—22no—
47C61×#533C2211yescharacteristic
48C61×#533C2211yescharacteristic
49C61×#519—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758222—21no—
52C2×#11758202—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758194—21no—
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—22no—
69#14519701—22no—
70#14519700—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C22×Q8×C61D411yescharacteristic
82C4⋊Q8×C61—24no—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—24no—
86C4⋊Q8×C61—22no—
87C4⋊Q8×C61—22no—
88C4⋊Q8×C61D411yescharacteristic
89C4⋊Q8×C61D411yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#125×C61—24no—
93#125×C61—24no—
94#125×C61—22no—
95#125×C61—24no—
96#125×C61—22no—
97#125×C61—21no—
98#125×C61—21no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61D412yes—
103C22⋊Q8×C61D412yes—
104C4×Q8×C61—24no—
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—22no—
108C4×Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C4×Q8×C61—21no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61D412yes—
116C4×Q8×C61D412yes—
117C4×Q8×C61C2312yes—
118C4×Q8×C61C2312yes—
119#117×C61—24no—
120#117×C61—21no—
121#117×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C2×Q8×C61—24no—
127C2×Q8×C61—24no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61C2×D412yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61—41no—
140#11758222—82no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×C4×D61—41no—
145C2×Dic122—21no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C42×C61—24no—
170C42×C61—24no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—22no—
174C42×C61—22no—
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2411yescharacteristic
179#11758202—82no—
180#11758201—44no—
181#11758201—44no—
182#11758201—82no—
183#11758198—41no—
184#11758198—41no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187#11758194—42no—
188C2×#92418—21no—
189C2×#92418—22no—
190C2×#92418—21no—
191C2×#92418—22no—
192C23×C61#12011yescharacteristic
193C22×D61—41no—
194Q8×C61—24no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—42no—
199Q8×C61—42no—
200Q8×C61—81no—
201Q8×C61—84no—
202Q8×C61#13714yes—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—81no—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C22⋊D61—82no—
221C2×Dic61—41no—
222D244—42no—
223D244—42no—
224D244—42no—
225C4×D61—82no—
226Dic122—42no—
227#92418—22no—
228#92418—41no—
229#92418—42no—
230#92418—42no—
231#18612—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61C2×#12011yescharacteristic
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—24no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61C2×#13712yes—
250C4×C61#57712yes—
251Dic61—81no—
252#3502D6111yescharacteristic, complemented
253#3216—611no—
254#3182—611no—
255#3103—611no—
256#1648—612no—
257#1645—612no—
258#1626—612no—
259#1623—612no—
260#1554—611no—
261#1491—611no—
262#1485—611no—
263C2×C61—41no—
264C2×C61#300411yescharacteristic
265C2×C61#301011yescharacteristic
266C2×C61#301811yescharacteristic
267D61—161no—
268Q82—21no—
269Q82C2×D6112yes—
270#557—21no—
271#557—22no—
272#557—22no—
273#557C2×D6112yes—
274#544—22no—
275#544—21no—
276#533—22no—
277#533C2×D6111yescharacteristic
278#533C2×D6111yescharacteristic
279#532—611no—
280#519—21no—
281C2×C4×Q8C2×D6111yescharacteristic
282#500—611no—
283#497—1221no—
284#492—611no—
285#489—1221no—
286#487—611no—
287#486—611no—
288#481—1221no—
289#475—612no—
290#468—1222no—
291#462—612no—
292#460—612no—
293#460—612no—
294#452—1222no—
295#448—1221no—
296#440—1222no—
297C4×SD16—1222no—
298#432—611no—
299#431—611no—
300C4×M16—611no—
301#333—612no—
302#332—612no—
303Q8⋊C8—612no—
304Q8⋊C8—612no—
305Q8⋊C8—612no—
306Q8⋊C8—612no—
307C61#1861211yescharacteristic, complemented, Sylow
308C2×#41—1221no—
309C22×Q8C22⋊D6111yescharacteristic
310C2×Q16—1222no—
311C2×SD16—1222no—
312C2×SD16—1222no—
313C2×SD16—1222no—
314C2×M16—1221no—
315C4⋊Q8—24no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—24no—
319C4⋊Q8—22no—
320C4⋊Q8—22no—
321C4⋊Q8—611no—
322C4⋊Q8C22⋊D6111yescharacteristic
323C4⋊Q8C22⋊D6111yescharacteristic
324C4⋊Q8C22×D6111yescharacteristic
325C4⋊Q8C22×D6111yescharacteristic
326#127—611no—
327#125—24no—
328#125—24no—
329#125—22no—
330#125—24no—
331#125—22no—
332#125—21no—
333#125—21no—
334#124—1221no—
335C22⋊Q8—24no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C22⋊Q8C22⋊D6112yes—
339C22⋊Q8C22⋊D6112yes—
340#121—1221no—
341#121—1221no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—24no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8C22⋊D6112yes—
354C4×Q8C22⋊D6112yes—
355C4×Q8C22×D6112yes—
356C4×Q8C22×D6112yes—
357C4×D4—1221no—
358C4×D4—1221no—
359#117—24no—
360#117—21no—
361#117—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C22⋊D6111yescharacteristic
364C2×C42—21no—
365C2×C42C22×D6111yescharacteristic
366#107—1222no—
367#106—1222no—
368C4⋊C8—612no—
369C4⋊C8—612no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#102—1222no—
374#102—1222no—
375#102—1222no—
376C22⋊C8—1221no—
377C22⋊C8—1221no—
378#97—1221no—
379#97—1221no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8—1221no—
383#41—4882no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—1221no—
390C2×Q8C2×C22⋊D6112yes—
391C2×Q8C2×C22⋊D6112yes—
392C2×Q8C2×C22⋊D6112yes—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—2441no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—2441no—
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403Q16—4882no—
404SD16—4882no—
405SD16—2444no—
406SD16—2444no—
407M16—2442no—
408C2×C8—1221no—
409C2×C8—1221no—
410C2×C8—1222no—
411C2×C8—1222no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4C2×C22⋊D6112yes—
433C4⋊C4C2×C22⋊D6112yes—
434C4⋊C4C2×C22⋊D6112yes—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C42—24no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42C2×C22⋊D6111yescharacteristic
446C42C2×C22⋊D6111yescharacteristic
447C42C2×C22⋊D6111yescharacteristic
448C42C23×D6111yescharacteristic
449C23—2441no—
450C23#1451974611yescharacteristic
451Q8—24no—
452Q8—22no—
453Q8—22no—
454Q8—22no—
455Q8—42no—
456Q8—42no—
457Q8—81no—
458Q8—84no—
459Q8—2442no—
460Q8#1451980814yes—
461D4—4882no—
462D4—2442no—
463D4—2442no—
464D4—2442no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—81no—
476C2×C4—4882no—
477C2×C4—2441no—
478C2×C4#1451974611yescharacteristic
479C2×C4#1451974611yescharacteristic
480C2×C4#1451974611yescharacteristic
481C2×C4C22×C22⋊D6111yescharacteristic
482C2×C4C22×C22⋊D6111yescharacteristic
483C2×C4C22×C22⋊D6111yescharacteristic
484C8—1222no—
485C8—2441no—
486C8—2442no—
487C8—2442no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—4881no—
492C22—4881no—
493C22—4881no—
494C22C2×#1451974611yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—24no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—4881no—
506C4C2×#1451980812yes—
507C4#1684057012yes—
508C2—41no—
509C2—9761no—
510C2#2145505311yescharacteristic
511C2#2145510511yescharacteristic
512C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenter#3502D61
hypocenterC61#18612
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
4. #21445061C2semidirect
5. #21445040C2semidirect
6. #21444915C2semidirect
7. #21444897C2semidirect
8. #21444529C2semidirect
9. #21444266C2semidirect
10. #21444240C2semidirect
11. C61×#3502C2semidirect
252. #3502D61semidirect
307. C61#18612semidirect

Maximal subgroups

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

subgroupindex
#214568262
#214565382
#214558502
#214450612
#214450402
#214449152
#214448972
#214445292
#214442662
#214442402
C61×#35022
#1861261