Finite Group Databasebeta

← #29657995

Subgroups — #29657995

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457210C211yesmaximal, characteristic, complemented
2#21457206C211yesmaximal, characteristic
3#21456329C211yesmaximal, characteristic, complemented
4#21445089C212yesmaximal, complemented
5#21445082C212yesmaximal, complemented
6#21445075C212yesmaximal, complemented
7#21445068C212yesmaximal, complemented
8#21444557C211yesmaximal, characteristic, complemented
9#21444274C211yesmaximal, characteristic, complemented
10#21444270C211yesmaximal, characteristic
11C61×#3550C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840165C2211yescharacteristic
14#16840165C2211yescharacteristic
15#16840161—22no—
16#16840158C2211yescharacteristic
17#16840158C2211yescharacteristic
18#16840054—21no—
19#16840047C2212yes—
20#16840044—21no—
21#16840042C2212yes—
22#16840018—22no—
23#16840006—22no—
24#16839972C2212yes—
25#16839969—21no—
26#16839967C2212yes—
27#16839959—21no—
28#16839798—22no—
29#16839786C2211yescharacteristic
30#16839786C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839508C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#563—22no—
39C61×#559—21no—
40Q82×C61—21no—
41C61×#556—21no—
42C61×#556C2212yes—
43D4×Q8×C61—21no—
44D4×Q8×C61C2212yes—
45C61×#542—22no—
46C61×#542—22no—
47C61×#536—22no—
48C61×#533C2211yescharacteristic
49C61×#533C2211yescharacteristic
50C61×#532C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—22no—
65#14519710—22no—
66#14519710—22no—
67#14519708—22no—
68#14519708—22no—
69#14519708—22no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—22no—
75#14519700—22no—
76#14519700—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C2×#41×C61—21no—
82C22×Q8×C61—22no—
83C4⋊Q8×C61—24no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61D411yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#127×C61D411yescharacteristic
92#126×C61—24no—
93#126×C61—24no—
94#125×C61—21no—
95#125×C61D412yes—
96#124×C61—24no—
97#124×C61—22no—
98#124×C61—22no—
99#124×C61—21no—
100#123×C61—24no—
101#123×C61—24no—
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—24no—
109C22⋊Q8×C61D412yes—
110#121×C61—21no—
111#121×C61D412yes—
112C4×Q8×C61—22no—
113C4×Q8×C61—22no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61—21no—
117C4×Q8×C61C2312yes—
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—21no—
121C4×D4×C61C2312yes—
122#117×C61—21no—
123#117×C61—22no—
124#117×C61D412yes—
125C2×C42×C61C2311yescharacteristic
126#41×C61—81no—
127#41×C61—81no—
128C2×Q8×C61—24no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—82no—
134C2×Q8×C61—82no—
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—22no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C2×D412yes—
140C22×C4×C61—24no—
141C22×C4×C61—22no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×Dic122—21no—
146C2×Dic122—21no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61C2×D412yes—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C42×C61—24no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2411yescharacteristic
177C22⋊Dic61—41no—
178#11758202—82no—
179#11758200—82no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184#11758196—41no—
185#11758196—41no—
186C4×Dic61—41no—
187#11758194—81no—
188#11758194—81no—
189C2×#92418—24no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—22no—
193C2×#92418—22no—
194C23×C61—22no—
195C23×C61#12011yescharacteristic
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—41no—
199Q8×C61—41no—
200Q8×C61—41no—
201Q8×C61—41no—
202Q8×C61—42no—
203Q8×C61—81no—
204Q8×C61—84no—
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—42no—
208D4×C61—81no—
209D4×C61—82no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—84no—
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×Dic61—41no—
230C2×Dic61—41no—
231Dic122—41no—
232Dic122—41no—
233Dic122—41no—
234Dic122—41no—
235#92418—22no—
236#92418—22no—
237#92418—42no—
238#92418—42no—
239#18702—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61C2×#12011yescharacteristic
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61#45612yes—
256C4×C61#57611yescharacteristic
257C4×C61#57711yescharacteristic
258C4×C61#57711yescharacteristic
259C4×C61#57811yescharacteristic
260Dic61—81no—
261Dic61—81no—
262#3550D6111yescharacteristic
263#3255—611no—
264#3254—611no—
265#3160—611no—
266#1652—612no—
267#1651—612no—
268#1650—612no—
269#1649—612no—
270#1561—611no—
271#1493—611no—
272#1492—611no—
273C2×C61—41no—
274C2×C61—82no—
275C2×C61#301211yescharacteristic
276C2×C61#301311yescharacteristic
277C2×C61C2×#45611yescharacteristic
278#563—22no—
279#559—21no—
280Q82—21no—
281#556—21no—
282#556C2×D6112yes—
283D4×Q8—21no—
284D4×Q8C2×D6112yes—
285#542—22no—
286#542—22no—
287#536—22no—
288#533—611no—
289#533C2×D6111yescharacteristic
290#533C2×D6111yescharacteristic
291#532C2×D6111yescharacteristic
292#500—611no—
293#500—611no—
294#499—1222no—
295#498—611no—
296#498—611no—
297#477—1221no—
298#475—612no—
299#475—1221no—
300#474—612no—
301#470—1222no—
302#468—1222no—
303#462—612no—
304#462—1221no—
305#461—612no—
306#460—1221no—
307#427—1222no—
308#423—611no—
309#423—611no—
310C2×C4×C8—611no—
311#332—612no—
312#332—612no—
313Q8⋊C8—612no—
314Q8⋊C8—612no—
315D4⋊C8—612no—
316D4⋊C8—612no—
317C61#1870211yescharacteristic, complemented, Sylow
318C2×#41—21no—
319C22×Q8—22no—
320C2×Q16—1222no—
321C2×SD16—1222no—
322C2×M16—1221no—
323C2×M16—1221no—
324C22×C8—1221no—
325C4⋊Q8—24no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—611no—
329C4⋊Q8—611no—
330C4⋊Q8C22⋊D6111yescharacteristic
331C4⋊Q8C22⋊D6111yescharacteristic
332C4⋊Q8C22⋊D6111yescharacteristic
333C4⋊Q8C22×D6111yescharacteristic
334C4⋊Q8C22×D6111yescharacteristic
335#127C22⋊D6111yescharacteristic
336#126—24no—
337#126—24no—
338#125—21no—
339#125—1221no—
340#125C22⋊D6112yes—
341#124—24no—
342#124—22no—
343#124—22no—
344#124—21no—
345#123—24no—
346#123—24no—
347C22⋊Q8—24no—
348C22⋊Q8—22no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—1221no—
355C22⋊Q8—1221no—
356C22⋊Q8C22⋊D6112yes—
357#121—21no—
358#121C22⋊D6112yes—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—1221no—
365C4×Q8—1221no—
366C4×Q8C22×D6112yes—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4C22×D6112yes—
371#117—21no—
372#117—22no—
373#117—1221no—
374#117C22⋊D6112yes—
375C2×C42C22×D6111yescharacteristic
376#107—1222no—
377#107—1222no—
378#107—1222no—
379#106—1222no—
380#106—1222no—
381#106—1222no—
382C4⋊C8—612no—
383C4⋊C8—612no—
384C4⋊C8—1221no—
385C4⋊C8—1221no—
386#103—1222no—
387#103—1222no—
388#103—1222no—
389#102—1222no—
390C4×C8—611no—
391C4×C8—611no—
392C4×C8—1221no—
393#41—81no—
394#41—81no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8C2×C22⋊D6112yes—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4C2×C22⋊D6112yes—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414Q16—4882no—
415SD16—4882no—
416M16—4881no—
417M16—4881no—
418C2×C8—1224no—
419C2×C8—1221no—
420C2×C8—1221no—
421C2×C8—1222no—
422C2×C8—1222no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C4⋊C4C2×C22⋊D6112yes—
442C4⋊C4C2×C22⋊D6112yes—
443C4⋊C4C2×C22⋊D6112yes—
444C4⋊C4C2×C22⋊D6112yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—2441no—
453C42—24no—
454C42—22no—
455C42—22no—
456C42—2441no—
457C42C2×C22⋊D6111yescharacteristic
458C42C2×C22⋊D6111yescharacteristic
459C42C2×C22⋊D6111yescharacteristic
460C42C23×D6111yescharacteristic
461C23—22no—
462C23#1451974611yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469Q8—41no—
470Q8—81no—
471Q8—84no—
472Q8—2441no—
473Q8—2441no—
474Q8—2441no—
475Q8—2441no—
476D4—22no—
477D4—22no—
478D4—42no—
479D4—81no—
480D4—82no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4#1451974611yescharacteristic
497C2×C4#1451974611yescharacteristic
498C2×C4#1451974611yescharacteristic
499C2×C4C22×C22⋊D6111yescharacteristic
500C2×C4C22×C22⋊D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C8—1222no—
503C8—1222no—
504C8—2442no—
505C8—2442no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22C2×#1451974611yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—4881no—
522C4—4881no—
523C4#1683994512yes—
524C4#1684056511yescharacteristic
525C4#1684057011yescharacteristic
526C4#1684057311yescharacteristic
527C4#1684057711yescharacteristic
528C2—41no—
529C2—82no—
530C2#2145512311yescharacteristic
531C2#2145513411yescharacteristic
532C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3550C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3550D61
hypocenterC61#18702
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #21457210C2semidirect
3. #21456329C2semidirect
4. #21445089C2semidirect
5. #21445082C2semidirect
6. #21445075C2semidirect
7. #21445068C2semidirect
8. #21444557C2semidirect
9. #21444274C2semidirect
317. C61#18702semidirect

Maximal subgroups

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

subgroupindex
#214572102
#214572062
#214563292
#214450892
#214450822
#214450752
#214450682
#214445572
#214442742
#214442702
C61×#35502
#1870261