Finite Group Databasebeta

← #29661346

Subgroups — #29661346

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457025C211yesmaximal, characteristic, complemented
2#21456822C211yesmaximal, characteristic
3#21456368C211yesmaximal, characteristic, complemented
4#21445153C214yesmaximal, complemented
5#21445130C214yesmaximal, complemented
6#21444446C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444417C211yesmaximal, characteristic
9C2×C61×#563C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840172C2212yescomplemented
12#16840172C2212yescomplemented
13#16840172C2212yescomplemented
14#16840161—22no—
15#16840157C2212yes—
16#16840132C2212yescomplemented
17#16840121C2212yes—
18#16840097—21no—
19#16840092—21no—
20#16840066—22no—
21#16839981—22no—
22#16839918—21no—
23#16839913—21no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839802—21no—
27#16839802—21no—
28C4×#11758194C2211yescharacteristic
29#16839504C2214yescomplemented
30#16839504C2214yescomplemented
31#16839502C2214yes—
32C61×#563—22no—
33C61×#563—22no—
34C61×#563—22no—
35C61×#563C2214yes—
36C2×C4⋊Q8×C61—21no—
37C2×#126×C61—21no—
38C2×#126×C61—21no—
39C2×#125×C61C2211yescharacteristic
40C2×#125×C61C2211yescharacteristic
41C2×#124×C61C2211yescharacteristic
42C2×#123×C61—21no—
43C2×C22⋊Q8×C61—21no—
44C2×C22⋊Q8×C61—21no—
45C2×#11758202—22no—
46C2×#11758200—22no—
47C2×#11758194—21no—
48#14519777C2311yescharacteristic
49#14519777C2311yescharacteristic
50#14519769—21no—
51#14519756—21no—
52#14519756—21no—
53C4×Dic122—21no—
54C4×Dic122—21no—
55#14519736—21no—
56#14519710—22no—
57#14519710—22no—
58#14519710—22no—
59#14519708—22no—
60#14519708—22no—
61#14519708—22no—
62C4⋊#92418—21no—
63C4⋊#92418—21no—
64#14519701—22no—
65#14519700—22no—
66#14519700—22no—
67#14519698—22no—
68C22⋊#92418—21no—
69C22⋊#92418—21no—
70#14519688C2312yes—
71#14519688C2312yes—
72#14519688C2312yes—
73C4×#92418—21no—
74C22×Q8×C61—21no—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—21no—
78C4⋊Q8×C61—24no—
79C4⋊Q8×C61—24no—
80#126×C61—24no—
81#126×C61—24no—
82#126×C61—24no—
83#126×C61—24no—
84#125×C61—21no—
85#125×C61—21no—
86#125×C61—21no—
87#125×C61—21no—
88#125×C61D412yes—
89#125×C61D412yes—
90#125×C61C2312yes—
91#125×C61C2312yes—
92#124×C61—21no—
93#124×C61—21no—
94#124×C61D412yes—
95#124×C61C2312yes—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—24no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C24×C61—21no—
120C2×Q8×C61—24no—
121C2×Q8×C61—21no—
122C2×Q8×C61—81no—
123C2×Q8×C61—82no—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—21no—
126C2×D4×C61—81no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—82no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C2×Dic122—21no—
139C2×Dic122—21no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C2×D412yes—
155C4⋊C4×C61C2×D412yes—
156C4⋊C4×C61C2×D412yes—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C42×C61—24no—
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C42×C61C2411yescharacteristic
170C22⋊Dic61—41no—
171#11758202—82no—
172#11758200—82no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177#11758196—41no—
178#11758196—41no—
179C4×Dic61—41no—
180#11758194—24no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C23×C61—22no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61#12011yescharacteristic
189Q8×C61—42no—
190Q8×C61—81no—
191Q8×C61—84no—
192D4×C61—42no—
193D4×C61—81no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—84no—
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×Dic61—41no—
215C2×Dic61—41no—
216Dic122—42no—
217Dic122—42no—
218#92418—24no—
219#92418—42no—
220#92418—42no—
221#19007—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—84no—
228C22×C61#45612yes—
229C22×C61C2×#12011yescharacteristic
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—42no—
240C4×C61—42no—
241Dic61—81no—
242Dic61—81no—
243C2×#563D6111yescharacteristic
244#3236—611no—
245#3215—611no—
246#3166—611no—
247#1664—614no—
248#1660—614no—
249#1534—611no—
250#1528—611no—
251#1527—611no—
252C2×C61—22no—
253C2×C61—82no—
254C2×C61C2×#45611yescharacteristic
255C2×C61#301911yescharacteristic
256C2×C61#302011yescharacteristic
257#563—22no—
258#563—22no—
259#563—22no—
260#563C2×D6114yes—
261#533—611no—
262C2×C4⋊Q8—21no—
263C2×#126—21no—
264C2×#126—21no—
265C2×#125C2×D6111yescharacteristic
266C2×#125C2×D6111yescharacteristic
267C2×#124C2×D6111yescharacteristic
268C2×#123—21no—
269C2×C22⋊Q8—21no—
270C2×C22⋊Q8—21no—
271#501—612no—
272#501—612no—
273#501—612no—
274#499—1222no—
275#497—612no—
276#491—612no—
277#489—612no—
278#484—1221no—
279#483—1221no—
280#479—1222no—
281#464—1222no—
282#451—1221no—
283#450—1221no—
284#432—611no—
285#432—611no—
286#428—1221no—
287#428—1221no—
288C4×M16—611no—
289#330—614no—
290#330—614no—
291#329—614no—
292C61#1900711yescharacteristic, complemented, Sylow
293C22×Q8—21no—
294C22×Q8C22⋊D6111yescharacteristic
295C22×D4C22⋊D6111yescharacteristic
296C23×C4—21no—
297C2×Q16—1222no—
298C2×SD16—1222no—
299C2×M16—1221no—
300C4⋊Q8—24no—
301C4⋊Q8—24no—
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304#126—24no—
305#126—24no—
306#126—24no—
307#126—24no—
308#125—21no—
309#125—21no—
310#125—21no—
311#125—21no—
312#125—1221no—
313#125C22⋊D6112yes—
314#125C22⋊D6112yes—
315#125C22×D6112yes—
316#125C22×D6112yes—
317#124—21no—
318#124—21no—
319#124C22⋊D6112yes—
320#124C22×D6112yes—
321#123—24no—
322#123—24no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—1221no—
328C22⋊Q8—1221no—
329C4×Q8—1221no—
330C4×Q8—1221no—
331#117—1221no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C22⋊D6111yescharacteristic
339C2×C4⋊C4C22⋊D6111yescharacteristic
340C2×C4⋊C4C22⋊D6111yescharacteristic
341C2×C4⋊C4C22⋊D6111yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42C22×D6111yescharacteristic
349#107—1222no—
350#107—1222no—
351#107—1222no—
352#106—1222no—
353#106—1222no—
354#106—1222no—
355C4⋊C8—1221no—
356C4⋊C8—1221no—
357#103—1222no—
358#103—1222no—
359#103—1222no—
360#102—1222no—
361C22⋊C8—1221no—
362C22⋊C8—1221no—
363#97—612no—
364#97—612no—
365#97—612no—
366C4×C8—1221no—
367C24—21no—
368C2×Q8—24no—
369C2×Q8—21no—
370C2×Q8—81no—
371C2×Q8—82no—
372C2×Q8—1221no—
373C2×Q8—1221no—
374C2×Q8C2×C22⋊D6112yes—
375C2×D4—21no—
376C2×D4—81no—
377C2×D4C2×C22⋊D6112yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—82no—
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388Q16—4882no—
389SD16—4882no—
390M16—1224no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4C2×C22⋊D6112yes—
415C4⋊C4C2×C22⋊D6112yes—
416C4⋊C4C2×C22⋊D6112yes—
417C4⋊C4C2×C22⋊D6112yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—2441no—
426C42—24no—
427C42—2441no—
428C42C2×C22⋊D6111yescharacteristic
429C42C2×C22⋊D6111yescharacteristic
430C42C2×C22⋊D6111yescharacteristic
431C42C23×D6111yescharacteristic
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23#1451974611yescharacteristic
437Q8—42no—
438Q8—81no—
439Q8—84no—
440Q8—2442no—
441Q8—2442no—
442D4—42no—
443D4—81no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—84no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4#1451974611yescharacteristic
461C2×C4#1451974611yescharacteristic
462C2×C4#1451974611yescharacteristic
463C2×C4C22×C22⋊D6111yescharacteristic
464C2×C4C22×C22⋊D6111yescharacteristic
465C2×C4C22×C22⋊D6111yescharacteristic
466C8—1224no—
467C8—2442no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—84no—
475C22#1683994512yes—
476C22C2×#1451974611yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—4881no—
489C4—4881no—
490C2—22no—
491C2—82no—
492C2C2×#1683994511yescharacteristic
493C2#2145517611yescharacteristic
494C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#563C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenterC2×#563D61
hypocenterC61#19007
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #21457025C2semidirect
3. #21456368C2semidirect
4. #21445153C2semidirect
5. #21445130C2semidirect
6. #21444446C2semidirect
7. #21444421C2semidirect
11. #16840172C22semidirect
12. #16840172C22semidirect
13. #16840172C22semidirect
16. #16840132C22semidirect
29. #16839504C22semidirect
30. #16839504C22semidirect
292. C61#19007semidirect

Maximal subgroups

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

subgroupindex
#214570252
#214568222
#214563682
#214451532
#214451302
#214444462
#214444212
#214444172
C2×C61×#5632
#1900761