Finite Group Databasebeta

← #29577060

Subgroups — #29577060

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455079C211yesmaximal, characteristic
2#21454268C211yesmaximal, characteristic
3#21444390C211yesmaximal, characteristic, complemented
4#21444182C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443707C211yesmaximal, characteristic, complemented
7C61×#1788C211yesmaximal, characteristic
8C2×#14519807C411yescharacteristic, complemented
9C2×#14519807C411yescharacteristic, complemented
10C2×#14519784—21no—
11C2×#14519756C2211yescharacteristic
12#16839924—21no—
13#16839924—21no—
14#16839918—21no—
15#16839913—21no—
16#16839763C411yescharacteristic, complemented
17#16839763C411yescharacteristic, complemented
18#16839729—22no—
19#16839729—21no—
20#16839726—21no—
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23C2×C22⋊#92418—21no—
24C2×C22⋊#92418C2211yescharacteristic
25#16839498—21no—
26#16839498—21no—
27#16839487—22no—
28C22×#41×C61C2211yescharacteristic
29C2×#103×C61—21no—
30C2×#102×C61—21no—
31C2×C22⋊C8×C61C2211yescharacteristic
32C61×#380C2211yescharacteristic
33C22×Dic122C2×C411yescharacteristic
34#14519807—42no—
35#14519807—42no—
36C2×#11758202—21no—
37C2×#11758202—21no—
38C2×#11758200—21no—
39C2×#11758200—21no—
40#14519784—42no—
41#14519784—42no—
42C2×#11758194—22no—
43C2×#11758194—21no—
44C2×#11758194C2×C411yescharacteristic
45C2×#11758194C2×C411yescharacteristic
46C22×#92418—22no—
47C22×#92418—21no—
48#14519756—21no—
49#14519756—21no—
50#14519756—21no—
51#14519756—21no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×#11758196—21no—
54C2×C22⋊Dic61—21no—
55#14519701—21no—
56#14519701—21no—
57#14519698—21no—
58#14519698—21no—
59C22⋊#92418—22no—
60C22⋊#92418—22no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63C22⋊#92418—21no—
64C22⋊#92418—21no—
65C22⋊#92418D411yescharacteristic
66C22⋊#92418D411yescharacteristic
67C22⋊#92418D412yes—
68C2×#41×C61—21no—
69C2×#41×C61—21no—
70C2×#41×C61—42no—
71C2×#41×C61—42no—
72C2×#41×C61C2×C411yescharacteristic
73C2×#41×C61C2×C411yescharacteristic
74C2×#41×C61D411yescharacteristic
75C2×#41×C61D411yescharacteristic
76C22×Q8×C61D411yescharacteristic
77C22×D4×C61—21no—
78C22×D4×C61D411yescharacteristic
79C23×C4×C61—21no—
80C23×C4×C61C2311yescharacteristic
81C22×C8×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C22⋊C4×C61—21no—
84#103×C61—42no—
85#102×C61—42no—
86C22⋊C8×C61—21no—
87C22⋊C8×C61—21no—
88#95×C61—21no—
89C24×C61—21no—
90C24×C61#4111yescharacteristic
91#41×C61—22no—
92#41×C61—22no—
93#41×C61—22no—
94#41×C61—22no—
95#41×C61—41no—
96#41×C61—41no—
97#41×C61—41no—
98#41×C61—41no—
99#41×C61—44no—
100#41×C61—44no—
101C2×Q8×C61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—41no—
104C2×Q8×C61C22⋊C411yescharacteristic
105C2×Q8×C61C22⋊C411yescharacteristic
106C2×D4×C61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—41no—
113C2×D4×C61—44no—
114C2×D4×C61C22⋊C411yescharacteristic
115C2×D4×C61C22⋊C411yescharacteristic
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—44no—
124C22×C4×C61—41no—
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—41no—
136C2×Dic122—41no—
137C2×C8×C61—41no—
138C2×C8×C61—41no—
139C2×C8×C61—41no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—41no—
142C22⋊C4×C61—81no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145#11758202—42no—
146#11758202—42no—
147#11758200—42no—
148#11758200—42no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151#11758196—41no—
152#11758196—41no—
153#11758194—44no—
154#11758194—41no—
155#11758194—41no—
156#11758194—41no—
157#11758194—41no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—22no—
162C2×#92418—21no—
163C2×#92418—44no—
164C2×#92418—41no—
165C2×#92418—41no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—42no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—81no—
176C23×C61C4×D411yescharacteristic
177C23×C61#12111yescharacteristic
178C23×C61#12311yescharacteristic
179Q8×C61—21no—
180Q8×C61—21no—
181Q8×C61—21no—
182Q8×C61—42no—
183Q8×C61#9912yes—
184D4×C61—24no—
185D4×C61—21no—
186D4×C61—21no—
187D4×C61—21no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61#9912yes—
193C2×C4×C61—24no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—81no—
210C2×C4×C61—81no—
211C2×C4×C61—81no—
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C4×D411yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12111yescharacteristic
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—81no—
220Dic122—41no—
221Dic122—41no—
222Dic122—81no—
223C8×C61—81no—
224#92418—41no—
225#92418—42no—
226#92418—41no—
227#92418—41no—
228#92418—41no—
229#9345—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—22no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61#38611yescharacteristic
249C22×C61#47111yescharacteristic
250C22×C61#47311yescharacteristic
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—22no—
255C4×C61—41no—
256C4×C61—161no—
257C4×C61C2×#9911yescharacteristic
258C4×C61C2×#9911yescharacteristic
259Dic61—81no—
260Dic61—81no—
261#3007—611no—
262#2880—611no—
263#1788D6111yescharacteristic
264#1520—611no—
265#1472—611no—
266#1305—611no—
267#1298—611no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—41no—
271C2×C61—41no—
272C2×C61—42no—
273C2×C61—81no—
274C2×C61#188111yescharacteristic
275C2×C61#189811yescharacteristic
276C2×C61#190811yescharacteristic
277C22×#41C2×D6111yescharacteristic
278C2×#137—611no—
279C2×#137—611no—
280C2×#131—1221no—
281C2×C22⋊Q8—611no—
282#452—1221no—
283#452—1221no—
284#451—1221no—
285#450—1221no—
286#417—611no—
287#417—611no—
288C2×#103—21no—
289C2×#102—21no—
290#408—1222no—
291#408—1221no—
292#407—611no—
293#407—611no—
294#407—1221no—
295C2×C22⋊C8—611no—
296C2×C22⋊C8—1221no—
297C2×C22⋊C8C2×D6111yescharacteristic
298#380C2×D6111yescharacteristic
299#328—1221no—
300#328—1221no—
301#324—1222no—
302C61#934511yescharacteristic, complemented, Sylow
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—42no—
306C2×#41—42no—
307C2×#41C4×D6111yescharacteristic
308C2×#41C4×D6111yescharacteristic
309C2×#41D24411yescharacteristic
310C2×#41D24411yescharacteristic
311C22×Q8—611no—
312C22×Q8C22⋊D6111yescharacteristic
313C22×D4—21no—
314C22×D4C22⋊D6111yescharacteristic
315C23×C4—21no—
316C23×C4C22×D6111yescharacteristic
317#137—2442no—
318#137—2442no—
319C2×Q16—1221no—
320C2×Q16—1221no—
321C2×SD16—1221no—
322C2×SD16—1221no—
323#131—2442no—
324#131—2442no—
325C2×M16—611no—
326C2×M16—611no—
327C2×M16—1222no—
328C2×M16—1221no—
329C22×C8—21no—
330C22×C8—1222no—
331C22×C8—1221no—
332C22⋊Q8—1221no—
333C22⋊Q8—1221no—
334C22⋊Q8—1221no—
335C22⋊Q8—1221no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—1221no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—1221no—
341#103—42no—
342#103—1221no—
343#103—1221no—
344#102—42no—
345#102—1221no—
346#102—1221no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—612no—
350C22⋊C8—611no—
351C22⋊C8—611no—
352C22⋊C8—1222no—
353C22⋊C8—1222no—
354C22⋊C8—1221no—
355C22⋊C8—1221no—
356C22⋊C8—1221no—
357C22⋊C8—1221no—
358#95—21no—
359C24—21no—
360C24#1175822411yescharacteristic
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—44no—
370#41—44no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—41no—
374C2×Q8—1221no—
375C2×Q8—1221no—
376C2×Q8—2441no—
377C2×Q8—2441no—
378C2×Q8#1175819811yescharacteristic
379C2×Q8#1175819811yescharacteristic
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—44no—
388C2×D4#1175819811yescharacteristic
389C2×D4#1175819811yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—44no—
398C22×C4—41no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4C2×C4×D6111yescharacteristic
402C22×C4C2×D24411yescharacteristic
403C22×C4D4×D6111yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4#1175822411yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407Q16—2442no—
408Q16—2442no—
409SD16—2442no—
410SD16—2442no—
411M16—2444no—
412M16—2441no—
413M16—2441no—
414M16—2441no—
415M16—2441no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—1221no—
420C2×C8—1221no—
421C2×C8—1221no—
422C2×C8—1222no—
423C2×C8—1221no—
424C2×C8—2444no—
425C2×C8—2441no—
426C2×C8—2441no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C22⋊C4—81no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—81no—
446C23#1451973711yescharacteristic
447C23#1451974911yescharacteristic
448C23#1451976111yescharacteristic
449Q8—21no—
450Q8—21no—
451Q8—21no—
452Q8—42no—
453Q8—4881no—
454Q8—2441no—
455Q8—2441no—
456Q8#1451969112yes—
457D4—24no—
458D4—21no—
459D4—21no—
460D4—21no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4#1451969112yes—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—4881no—
486C2×C4—2442no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4C2×#1175819811yescharacteristic
490C2×C4#1451973711yescharacteristic
491C2×C4#1451974711yescharacteristic
492C2×C4#1451974911yescharacteristic
493C8—81no—
494C8—2441no—
495C8—2442no—
496C8—2441no—
497C8—2441no—
498C8—2441no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22#1683963911yescharacteristic
518C22#1684002311yescharacteristic
519C22#1684003511yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—21no—
524C4—41no—
525C4—161no—
526C4—4881no—
527C4—4881no—
528C4C2×#1451969111yescharacteristic
529C4C2×#1451969111yescharacteristic
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—42no—
535C2—81no—
536C2#2144617911yescharacteristic
537C2#2144627011yescharacteristic
538C2#2144633211yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1788C2
socleC22×C61#386
prime radicalC22×#41×C61C22
cosocleC23×C4×C61C23
hypercenter#1788D61
hypocenterC61#9345
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
3. #21444390C2semidirect
4. #21444182C2semidirect
5. #21443725C2semidirect
6. #21443707C2semidirect
8. C2×#14519807C4semidirect
9. C2×#14519807C4semidirect
16. #16839763C4semidirect
17. #16839763C4semidirect
302. C61#9345semidirect

Maximal subgroups

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

subgroupindex
#214550792
#214542682
#214443902
#214441822
#214437252
#214437072
C61×#17882
#934561