Finite Group Databasebeta

← #29577076

Subgroups — #29577076

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455071C211yesmaximal, characteristic
2#21445759C211yesmaximal, characteristic, complemented
3#21444388C211yesmaximal, characteristic, complemented
4#21444180C211yesmaximal, characteristic, complemented
5#21443726C211yesmaximal, characteristic, complemented
6#21443707C211yesmaximal, characteristic, complemented
7C61×#2880C211yesmaximal, characteristic, complemented
8C2×#14519802C411yescharacteristic, complemented
9C2×#14519802C411yescharacteristic, complemented
10C2×#14519751C2211yescharacteristic
11#16839907—21no—
12#16839907—21no—
13#16839901—21no—
14#16839896—21no—
15#16839763C411yescharacteristic, complemented
16#16839763C411yescharacteristic, complemented
17C2×#14519701—21no—
18C2×#14519698—21no—
19#16839726C2211yescharacteristic
20C2×C22⋊#92418C2211yescharacteristic
21#16839622C2211yescharacteristic
22#16839497—21no—
23#16839497—21no—
24#16839485—22no—
25C22×#41×C61C2211yescharacteristic
26C2×#131×C61—21no—
27C61×#408—21no—
28C61×#408—22no—
29C61×#407—21no—
30C61×#407C2211yescharacteristic
31C2×C22⋊C8×C61—21no—
32C2×C22⋊C8×C61C2211yescharacteristic
33C22×D244C2×C411yescharacteristic
34C22×C22⋊D61—21no—
35#14519802—42no—
36#14519802—42no—
37C2×#11758201—21no—
38C2×#11758201—21no—
39C2×#11758199—21no—
40C2×#11758199—21no—
41C2×#11758194C2×C411yescharacteristic
42C2×#11758194C2×C411yescharacteristic
43C22×#92418—21no—
44#14519751—21no—
45#14519751—21no—
46#14519751—21no—
47#14519751—21no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61C2×C411yescharacteristic
50C2×C22⋊Dic61—21no—
51C2×#11758198—21no—
52#14519701—21no—
53#14519701—21no—
54#14519701—42no—
55#14519698—21no—
56#14519698—21no—
57#14519698—42no—
58C22⋊#92418—21no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62#14519684—21no—
63C2×#41×C61—21no—
64C2×#41×C61—21no—
65C2×#41×C61—42no—
66C2×#41×C61—42no—
67C2×#41×C61C2×C411yescharacteristic
68C2×#41×C61C2×C411yescharacteristic
69C2×#41×C61D411yescharacteristic
70C2×#41×C61D411yescharacteristic
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61—21no—
73C22×D4×C61D411yescharacteristic
74C23×C4×C61—21no—
75C23×C4×C61C2311yescharacteristic
76#131×C61—42no—
77#131×C61—42no—
78C2×M16×C61—22no—
79C2×M16×C61—21no—
80C22×C8×C61—22no—
81C22×C8×C61—21no—
82C22⋊C8×C61—22no—
83C22⋊C8×C61—22no—
84C22⋊C8×C61—21no—
85C22⋊C8×C61—21no—
86C22⋊C8×C61D412yes—
87C22⋊C8×C61D411yescharacteristic
88C22⋊C8×C61D411yescharacteristic
89C24×C61—21no—
90C24×C61#4111yescharacteristic
91C23×D61—21no—
92#41×C61—22no—
93#41×C61—22no—
94#41×C61—22no—
95#41×C61—22no—
96#41×C61—41no—
97#41×C61—41no—
98#41×C61—41no—
99#41×C61—41no—
100#41×C61—44no—
101#41×C61—44no—
102C2×Q8×C61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—41no—
105C2×Q8×C61C22⋊C411yescharacteristic
106C2×Q8×C61C22⋊C411yescharacteristic
107C2×D4×C61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—41no—
114C2×D4×C61—44no—
115C2×D4×C61C22⋊C411yescharacteristic
116C2×D4×C61C22⋊C411yescharacteristic
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61—44no—
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C2×C22⋊D61—41no—
132C2×C22⋊D61—41no—
133C2×C22⋊D61—81no—
134C2×C22⋊D61—81no—
135C22×Dic61—21no—
136C22×Dic61—41no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—41no—
140C2×D244—41no—
141M16×C61—44no—
142M16×C61—41no—
143M16×C61—41no—
144C2×C8×C61—21no—
145C2×C8×C61—22no—
146C2×C8×C61—21no—
147C2×C8×C61—44no—
148C2×C8×C61—41no—
149C2×C8×C61—41no—
150C22⋊Dic61—81no—
151#11758201—42no—
152#11758201—42no—
153#11758199—42no—
154#11758199—42no—
155#11758198—41no—
156#11758198—41no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161#11758194—41no—
162#11758194—41no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—41no—
166C2×#92418—41no—
167C2×#92418—41no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—42no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—81no—
178C23×C61C4×D411yescharacteristic
179C23×C61#12111yescharacteristic
180C23×C61#12311yescharacteristic
181C22×D61—41no—
182C22×D61—42no—
183C22×D61—81no—
184C22×D61—81no—
185Q8×C61—21no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—42no—
189Q8×C61#9912yes—
190D4×C61—24no—
191D4×C61—21no—
192D4×C61—21no—
193D4×C61—21no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61#9912yes—
199C2×C4×C61—24no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C22⋊C411yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C22⋊D61—82no—
220C22⋊D61—81no—
221C22⋊D61—81no—
222C2×Dic61—41no—
223C2×Dic61—81no—
224C2×Dic61—81no—
225C2×Dic61—81no—
226C2×Dic61—81no—
227D244—41no—
228D244—41no—
229D244—81no—
230C8×C61—41no—
231C8×C61—42no—
232C8×C61—41no—
233#92418—41no—
234#92418—41no—
235#92418—81no—
236#9347—611nomaximal, Sylow
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61#38611yescharacteristic
256C22×C61#47111yescharacteristic
257C22×C61#47211yescharacteristic
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—162no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—22no—
266C4×C61—41no—
267C4×C61C2×#9911yescharacteristic
268C4×C61C2×#9911yescharacteristic
269Dic61—81no—
270Dic61—161no—
271#3006—611no—
272#2880D6111yescharacteristic, complemented
273#1788—611no—
274#1519—611no—
275#1471—611no—
276#1305—611no—
277#1298—611no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—41no—
281C2×C61—41no—
282C2×C61—42no—
283C2×C61—81no—
284C2×C61#188111yescharacteristic
285C2×C61#189811yescharacteristic
286C2×C61#190711yescharacteristic
287D61—161no—
288C22×#41C2×D6111yescharacteristic
289C2×AGL(1,ℤ/8)—611no—
290C2×AGL(1,ℤ/8)—611no—
291C2×#131—21no—
292C2×#121—611no—
293#449—1221no—
294#449—1221no—
295#448—1221no—
296#447—1221no—
297#417—611no—
298#417—611no—
299C2×#103—1221no—
300C2×#102—1221no—
301#408—21no—
302#408—22no—
303#407—21no—
304#407—611no—
305#407C2×D6111yescharacteristic
306C2×C22⋊C8—21no—
307C2×C22⋊C8—611no—
308C2×C22⋊C8C2×D6111yescharacteristic
309#380—611no—
310#327—1221no—
311#327—1221no—
312#323—1222no—
313C61#934711yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—42no—
317C2×#41—42no—
318C2×#41C4×D6111yescharacteristic
319C2×#41C4×D6111yescharacteristic
320C2×#41C22⋊D6111yescharacteristic
321C2×#41C22⋊D6111yescharacteristic
322C22×Q8D24411yescharacteristic
323C22×D4—21no—
324C22×D4—611no—
325C22×D4—1221no—
326C22×D4D24411yescharacteristic
327C23×C4—21no—
328C23×C4C22×D6111yescharacteristic
329AGL(1,ℤ/8)—2442no—
330AGL(1,ℤ/8)—2442no—
331C2×SD16—1221no—
332C2×SD16—1221no—
333C2×D8—1221no—
334C2×D8—1221no—
335#131—42no—
336#131—42no—
337C2×M16—22no—
338C2×M16—21no—
339C2×M16—611no—
340C2×M16—611no—
341C22×C8—22no—
342C22×C8—21no—
343C22×C8—1221no—
344#121—1221no—
345#121—1221no—
346#121—1221no—
347#121—1221no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352#103—1221no—
353#103—1221no—
354#103—2442no—
355#102—1221no—
356#102—1221no—
357#102—2442no—
358C22⋊C8—22no—
359C22⋊C8—22no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C22⋊C8—1221no—
363C22⋊C8—1221no—
364C22⋊C8—1221no—
365C22⋊C8—1221no—
366C22⋊C8C22⋊D6112yes—
367C22⋊C8C22⋊D6111yescharacteristic
368C22⋊C8C22⋊D6111yescharacteristic
369#95—1221no—
370C24—21no—
371C24—1221no—
372C24#1175822211yescharacteristic
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—44no—
382#41—44no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—41no—
386C2×Q8#1175819811yescharacteristic
387C2×Q8#1175819811yescharacteristic
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—41no—
395C2×D4—44no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—4881no—
399C2×D4—4881no—
400C2×D4—2441no—
401C2×D4—2441no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4#1175819811yescharacteristic
405C2×D4#1175819811yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—44no—
414C22×C4—1221no—
415C22×C4—2441no—
416C22×C4C2×C4×D6111yescharacteristic
417C22×C4C2×D24411yescharacteristic
418C22×C4#1175822211yescharacteristic
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422SD16—2442no—
423SD16—2442no—
424D8—2442no—
425D8—2442no—
426M16—44no—
427M16—41no—
428M16—41no—
429M16—2441no—
430M16—2441no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—21no—
434C2×C8—44no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—2441no—
440C2×C8—2441no—
441C2×C8—2441no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—2441no—
445C4⋊C4—2441no—
446C22⋊C4—4881no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—81no—
459C23—4881no—
460C23—4881no—
461C23—2442no—
462C23—2441no—
463C23C4×C22⋊D6111yescharacteristic
464C23#1451975111yescharacteristic
465C23#1451976011yescharacteristic
466Q8—21no—
467Q8—21no—
468Q8—21no—
469Q8—42no—
470Q8#1451969112yes—
471D4—24no—
472D4—21no—
473D4—21no—
474D4—21no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—4881no—
480D4—4882no—
481D4—4881no—
482D4—4881no—
483D4—2441no—
484D4—2441no—
485D4#1451969112yes—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—4881no—
503C2×C4—4881no—
504C2×C4—4881no—
505C2×C4—4881no—
506C2×C4—2441no—
507C2×C4C2×#1175819811yescharacteristic
508C2×C4C4×C22⋊D6111yescharacteristic
509C2×C4#1451974611yescharacteristic
510C2×C4#1451975111yescharacteristic
511C8—41no—
512C8—42no—
513C8—41no—
514C8—4881no—
515C8—2441no—
516C8—2441no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—81no—
532C22—81no—
533C22—82no—
534C22—82no—
535C22—9762no—
536C22—4881no—
537C22—4881no—
538C22—4881no—
539C22#1683964111yescharacteristic
540C22#1684002511yescharacteristic
541C22#1684003211yescharacteristic
542C4—21no—
543C4—21no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—9761no—
548C4—4881no—
549C4C2×#1451969111yescharacteristic
550C4C2×#1451969111yescharacteristic
551C2—21no—
552C2—21no—
553C2—41no—
554C2—41no—
555C2—42no—
556C2—81no—
557C2—9761no—
558C2#2144618111yescharacteristic
559C2#2144627211yescharacteristic
560C2#2144632711yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2880C2
socleC22×C61#386
prime radical#21455071C2
cosocleC23×C4×C61C23
hypercenter#2880D61
hypocenterC61#9347
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2880

Complements

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

NH ≅ G/Ntype
2. #21445759C2semidirect
3. #21444388C2semidirect
4. #21444180C2semidirect
5. #21443726C2semidirect
6. #21443707C2semidirect
7. C61×#2880C2semidirect
8. C2×#14519802C4semidirect
9. C2×#14519802C4semidirect
15. #16839763C4semidirect
16. #16839763C4semidirect
272. #2880D61semidirect
313. C61#9347semidirect

Maximal subgroups

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

subgroupindex
#214550712
#214457592
#214443882
#214441802
#214437262
#214437072
C61×#28802
#934761