Finite Group Databasebeta

← #29656915

Subgroups — #29656915

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456826C211yesmaximal, characteristic, complemented
2#21456553C211yesmaximal, characteristic
3#21455850C211yesmaximal, characteristic, complemented
4#21445096C212yesmaximal, complemented
5#21445005C212yesmaximal, complemented
6#21444915C212yesmaximal, complemented
7#21444878C212yesmaximal, complemented
8#21444524C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3507C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840158C2211yescharacteristic
14#16840154—21no—
15#16840138C2211yescharacteristic
16#16840124—21no—
17#16840110C2211yescharacteristic
18#16840108C2211yescharacteristic
19#16840080—21no—
20#16840047C2212yes—
21#16840013—22no—
22#16839972C2212yes—
23#16839962C2212yes—
24#16839962C2212yes—
25#16839922—22no—
26#16839901—21no—
27#16839872—22no—
28C4×#11758201—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839508C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37Q8⋊#92418C2212yes—
38C61×#559C2212yes—
39C61×#557C2212yes—
40C61×#556—21no—
41C61×#556—22no—
42C61×#554—21no—
43C61×#552—22no—
44C61×#539—22no—
45C61×#536—21no—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#529—22no—
49C61×#520—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758222—21no—
52C2×#11758202—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758199—22no—
56C2×#11758194—21no—
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519708—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—22no—
69#14519700—22no—
70#14519698—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
81C2×#41×C61—21no—
82C22×Q8×C61D411yescharacteristic
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61D411yescharacteristic
85C4⋊Q8×C61D411yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87#127×C61C2311yescharacteristic
88#126×C61—24no—
89#126×C61—24no—
90#125×C61—24no—
91#125×C61—22no—
92#125×C61—21no—
93#124×C61—22no—
94#124×C61—22no—
95#124×C61—21no—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61D412yes—
103#121×C61—21no—
104#121×C61D412yes—
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—21no—
108C4×Q8×C61—21no—
109C4×Q8×C61—21no—
110C4×Q8×C61D412yes—
111C4×Q8×C61C2312yes—
112C4×Q8×C61C2312yes—
113C4×D4×C61—24no—
114C4×D4×C61—22no—
115C4×D4×C61—22no—
116C4×D4×C61—21no—
117C4×D4×C61D412yes—
118#117×C61—22no—
119#117×C61—24no—
120#117×C61—21no—
121C2×C4⋊C4×C61—22no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#41×C61—44no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—42no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—22no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—21no—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145#11758222—82no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×C4×D61—41no—
150C2×Dic122—21no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C2×D412yes—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C42×C61—24no—
173C42×C61—22no—
174C42×C61—22no—
175C42×C61—22no—
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2411yescharacteristic
180#11758202—82no—
181#11758201—44no—
182#11758201—82no—
183#11758199—44no—
184#11758198—41no—
185#11758198—41no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188#11758194—42no—
189C2×#92418—21no—
190C2×#92418—22no—
191C2×#92418—21no—
192C2×#92418—22no—
193C23×C61—22no—
194C23×C61#12011yescharacteristic
195C22×D61—41no—
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—22no—
199Q8×C61—22no—
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—81no—
203D4×C61—42no—
204D4×C61—44no—
205D4×C61—42no—
206D4×C61—42no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—41no—
215C2×C4×C61—44no—
216C2×C4×C61—42no—
217C2×C4×C61—81no—
218C2×C4×C61—84no—
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C22⋊D61—82no—
226C2×Dic61—41no—
227D244—42no—
228D244—42no—
229D244—42no—
230C4×D61—82no—
231Dic122—42no—
232#92418—22no—
233#92418—41no—
234#92418—42no—
235#92418—42no—
236#18602—611nomaximal, Sylow
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61C2×#12011yescharacteristic
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61#57512yes—
257C4×C61#57712yes—
258Dic61—81no—
259#3507D6111yescharacteristic, complemented
260#3216—611no—
261#3184—611no—
262#3103—611no—
263#1653—612no—
264#1640—612no—
265#1626—612no—
266#1620—612no—
267#1553—611no—
268#1491—611no—
269#1485—611no—
270C2×C61—41no—
271C2×C61—82no—
272C2×C61#300511yescharacteristic
273C2×C61#301011yescharacteristic
274C2×C61#301811yescharacteristic
275D61—161no—
276#559C2×D6112yes—
277#557C2×D6112yes—
278#556—21no—
279#556—22no—
280#554—21no—
281#552—22no—
282#539—22no—
283#536—21no—
284#533C2×D6111yescharacteristic
285#532—611no—
286#532C2×D6111yescharacteristic
287#529—22no—
288#520—21no—
289C2×C4×Q8C2×D6111yescharacteristic
290#498—611no—
291#497—1221no—
292#493—611no—
293#489—1221no—
294#486—611no—
295#486—611no—
296#481—1221no—
297#475—612no—
298#469—1222no—
299#462—612no—
300#460—612no—
301#460—612no—
302#452—1222no—
303#448—1221no—
304#442—1222no—
305C4×SD16—1222no—
306#432—611no—
307#431—611no—
308C4×M16—611no—
309#332—612no—
310#331—612no—
311Q8⋊C8—612no—
312Q8⋊C8—612no—
313Q8⋊C8—612no—
314Q8⋊C8—612no—
315C61#1860211yescharacteristic, complemented, Sylow
316C2×#41—21no—
317C2×#41—1221no—
318C22×Q8C22⋊D6111yescharacteristic
319C2×Q16—1222no—
320C2×SD16—1222no—
321C2×SD16—1222no—
322C2×D8—1222no—
323C2×M16—1221no—
324C4⋊Q8—22no—
325C4⋊Q8—611no—
326C4⋊Q8C22⋊D6111yescharacteristic
327C4⋊Q8C22⋊D6111yescharacteristic
328C4⋊Q8C22×D6111yescharacteristic
329#127—611no—
330#127C22×D6111yescharacteristic
331#126—24no—
332#126—24no—
333#125—22no—
334#125—24no—
335#125—21no—
336#124—22no—
337#124—22no—
338#124—21no—
339#124—1221no—
340#123—24no—
341#123—24no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—24no—
345C22⋊Q8—21no—
346C22⋊Q8C22⋊D6112yes—
347#121—21no—
348#121—1221no—
349#121—1221no—
350#121C22⋊D6112yes—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8C22⋊D6112yes—
357C4×Q8C22×D6112yes—
358C4×Q8C22×D6112yes—
359C4×D4—24no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—21no—
363C4×D4—1221no—
364C4×D4—1221no—
365C4×D4C22⋊D6112yes—
366#117—22no—
367#117—24no—
368#117—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4C22⋊D6111yescharacteristic
372C2×C42—21no—
373C2×C42C22×D6111yescharacteristic
374#106—1222no—
375#106—1222no—
376C4⋊C8—612no—
377C4⋊C8—612no—
378#103—1222no—
379#103—1222no—
380#102—1222no—
381#102—1222no—
382#102—1222no—
383#102—1222no—
384C22⋊C8—1221no—
385C22⋊C8—1221no—
386#97—1221no—
387#97—1221no—
388C4×C8—611no—
389C4×C8—611no—
390C4×C8—1221no—
391#41—44no—
392#41—4882no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—1221no—
398C2×Q8C2×C22⋊D6112yes—
399C2×Q8C2×C22⋊D6112yes—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—2441no—
407C2×D4C2×C22⋊D6112yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—2441no—
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416Q16—4882no—
417SD16—4882no—
418SD16—2444no—
419D8—2444no—
420M16—2442no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1222no—
424C2×C8—1222no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4C2×C22⋊D6112yes—
440C4⋊C4C2×C22⋊D6112yes—
441C4⋊C4C2×C22⋊D6112yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42C2×C22⋊D6111yescharacteristic
455C42C2×C22⋊D6111yescharacteristic
456C42C2×C22⋊D6111yescharacteristic
457C42C23×D6111yescharacteristic
458C23—22no—
459C23—2441no—
460C23#1451974611yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—42no—
467Q8—81no—
468Q8—2442no—
469D4—42no—
470D4—44no—
471D4—42no—
472D4—42no—
473D4—4882no—
474D4—2442no—
475D4—2442no—
476D4—2442no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—81no—
488C2×C4—84no—
489C2×C4—4882no—
490C2×C4—2441no—
491C2×C4#1451974611yescharacteristic
492C2×C4#1451974611yescharacteristic
493C2×C4#1451974611yescharacteristic
494C2×C4C22×C22⋊D6111yescharacteristic
495C2×C4C22×C22⋊D6111yescharacteristic
496C2×C4C22×C22⋊D6111yescharacteristic
497C8—1222no—
498C8—2441no—
499C8—2442no—
500C8—2442no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—4881no—
508C22—4881no—
509C22—4881no—
510C22C2×#1451974611yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—4881no—
521C4#1684056012yes—
522C4#1684057012yes—
523C2—41no—
524C2—82no—
525C2—9761no—
526C2#2145506111yescharacteristic
527C2#2145510511yescharacteristic
528C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radical#21456553C2
hypercenter#3507D61
hypocenterC61#18602
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. #21456826C2semidirect
3. #21455850C2semidirect
4. #21445096C2semidirect
5. #21445005C2semidirect
6. #21444915C2semidirect
7. #21444878C2semidirect
8. #21444524C2semidirect
9. #21444266C2semidirect
10. #21444240C2semidirect
11. C61×#3507C2semidirect
259. #3507D61semidirect
315. C61#18602semidirect

Maximal subgroups

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

subgroupindex
#214568262
#214565532
#214558502
#214450962
#214450052
#214449152
#214448782
#214445242
#214442662
#214442402
C61×#35072
#1860261