Finite Group Databasebeta

← #29576587

Subgroups — #29576587

Order 15616.

648 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456570C211yesmaximal, characteristic
2#21448866C211yesmaximal, characteristic, complemented
3#21446654C211yesmaximal, characteristic, complemented
4#21444386C211yesmaximal, characteristic, complemented
5C2×#16839495C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C61×#1294C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519751C2211yescharacteristic
10C2×C4×C22⋊D61—21no—
11#16840076—21no—
12#16840076C412yescomplemented
13#16839998—22no—
14#16839898—21no—
15#16839898C412yescomplemented
16C2×#14519691—21no—
17C2×#14519691—21no—
18C2×C22⋊#92418C2211yescharacteristic
19#16839651—21no—
20#16839645—21no—
21#16839641—21no—
22#16839640—21no—
23C4×C22⋊Dic61C2211yescharacteristic
24#16839609—21no—
25#16839495—22no—
26#16839495—21no—
27#16839495—21no—
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#380C2211yescharacteristic
30C2×#95×C61C2211yescharacteristic
31C22×D244C2×C411yescharacteristic, complemented
32C22×D244D411yescharacteristic
33C22×C22⋊D61—21no—
34C22×C22⋊D61—21no—
35C22×C4×D61—21no—
36C2×D488—22no—
37C22×#92418—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42#14519751—21no—
43#14519751—21no—
44#14519751C2×C412yes—
45#14519751D412yes—
46C4×C22⋊D61—42no—
47C4×C22⋊D61—42no—
48C2×C4⋊Dic61C2×C411yescharacteristic, complemented
49C2×C4⋊Dic61D411yescharacteristic
50C2×#11758196—21no—
51C2×C22⋊Dic61D411yescharacteristic
52C2×C22⋊Dic61D411yescharacteristic
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×C4×Dic61—21no—
59C2×C4×Dic61—21no—
60#14519711—22no—
61#14519699—22no—
62#14519699—22no—
63#14519691—42no—
64#14519691—42no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67#14519684—21no—
68#14519684—21no—
69#14519684—21no—
70C23×C4×C61—21no—
71C23×C4×C61C2311yescharacteristic
72C22×C8×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61C2×C412yescomplemented
77#95×C61—21no—
78#95×C61—21no—
79#95×C61D411yescharacteristic
80#95×C61D411yescharacteristic
81C24×C61C2×D411yescharacteristic
82C23×D61—21no—
83C23×D61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—42no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61—81no—
91C22×C4×C61—81no—
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—22no—
99C2×C22⋊D61—22no—
100C2×C22⋊D61—41no—
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C2×C22⋊D61—41no—
105C2×C22⋊D61—41no—
106C2×C22⋊D61—41no—
107C2×C22⋊D61—41no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—41no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—81no—
118C2×D244—81no—
119C2×D244C22⋊C412yes—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C2×C4×D61—81no—
123C2×C8×C61—22no—
124C2×C8×C61—81no—
125C4⋊C4×C61—81no—
126C22⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C22⋊Dic61—21no—
129C22⋊Dic61—21no—
130C22⋊Dic61—21no—
131C22⋊Dic61D812yescomplemented
132#11758198—22no—
133#11758198—22no—
134#11758198—42no—
135#11758198—42no—
136#11758198—41no—
137#11758198—41no—
138#11758198—81no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61C22⋊C412yes—
143#11758196—42no—
144C4×Dic61—42no—
145C4×Dic61—81no—
146C2×#92418—41no—
147C2×#92418—81no—
148D488—82no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61C4×D411yescharacteristic
152C23×C61#12011yescharacteristic
153C23×C61#12111yescharacteristic
154C23×C61C2×D811yescharacteristic
155C23×C61#13511yescharacteristic
156C23×C61AGL(1,ℤ/8)11yescharacteristic
157C23×C61AGL(1,ℤ/8)11yescharacteristic
158C22×D61—22no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—81no—
169C22×D61—81no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—82no—
183C2×C4×C61—82no—
184C2×C4×C61—81no—
185C2×C4×C61—81no—
186C2×C4×C61—81no—
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C2×C4×C61C4×D411yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12311yescharacteristic
191C22⋊D61—42no—
192C22⋊D61—42no—
193C22⋊D61—42no—
194C22⋊D61—42no—
195C22⋊D61—81no—
196C22⋊D61—81no—
197C22⋊D61—81no—
198C22⋊D61—81no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211D244—42no—
212D244—42no—
213D244—81no—
214D244—81no—
215C4×D61—84no—
216C8×C61—42no—
217#92418—81no—
218#9277—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61#38611yescharacteristic
230C22×C61C4×D811yescharacteristic
231C22×C61#44211yescharacteristic
232C22×C61#44711yescharacteristic
233C22×C61#44911yescharacteristic
234C22×C61#45911yescharacteristic
235C22×C61#46311yescharacteristic
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—161no—
247C2×D61—161no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—42no—
253C4×C61—81no—
254C4×C61—161no—
255Dic61—42no—
256Dic61—42no—
257Dic61—81no—
258Dic61—81no—
259#3186—611no—
260#2364—611no—
261#1964—611no—
262#1519—611no—
263C2×#327—611no—
264#1294D6111yescharacteristic, complemented
265#1282—611no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61#187911yescharacteristic
273C2×C61#190611yescharacteristic
274C2×C61#191311yescharacteristic
275D61—82no—
276D61—161no—
277C2×#121—611no—
278C2×#121—611no—
279C2×C4×D4—1221no—
280#480—612no—
281#480—1221no—
282#466—1222no—
283#447—612no—
284#447—1221no—
285C2×#99—1221no—
286C2×#99—1221no—
287C2×C22⋊C8—611no—
288C2×C22⋊C8C2×D6111yescharacteristic
289#388—1221no—
290#387—1221no—
291#386—1221no—
292#386—1221no—
293#380C2×D6111yescharacteristic
294C4×C22⋊C4—611no—
295C4×C22⋊C4—1221no—
296C2×#95C2×D6111yescharacteristic
297#327—1222no—
298#327—1221no—
299#327—1221no—
300C61#927711yescharacteristic, complemented, Sylow
301C22×D4—611no—
302C22×D4—611no—
303C22×D4—1221no—
304C22×D4—1221no—
305C23×C4—21no—
306C23×C4—1221no—
307C23×C4C22×D6111yescharacteristic
308C2×D8—1222no—
309C22×C8—21no—
310C22×C8—1221no—
311#121—612no—
312#121—612no—
313#121—1221no—
314#121—1221no—
315#121—1221no—
316#121—1221no—
317#121—1221no—
318#121—1221no—
319C4×D4—2442no—
320C4×D4—2442no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—611no—
323C2×C4⋊C4—611no—
324C2×C4⋊C4—1221no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—611no—
327C2×C22⋊C4—611no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—1221no—
330C2×C22⋊C4—1221no—
331C2×C22⋊C4—1221no—
332C2×C22⋊C4—1221no—
333C2×C42—1221no—
334C2×C42—1221no—
335#107—1222no—
336#102—1222no—
337#102—1222no—
338#99—2442no—
339#99—2442no—
340C22⋊C8—21no—
341C22⋊C8—1221no—
342C22⋊C8—1221no—
343C22⋊C8C4×D6112yescomplemented
344#95—21no—
345#95—21no—
346#95—1221no—
347#95—1221no—
348#95—1221no—
349#95D24411yescharacteristic
350#95D24411yescharacteristic
351C24—1221no—
352C24—1221no—
353C24D4×D6111yescharacteristic
354C2×D4—612no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—4881no—
361C2×D4—4881no—
362C2×D4—2441no—
363C2×D4—2441no—
364C2×D4—2441no—
365C2×D4—2441no—
366C2×D4—2441no—
367C2×D4—2441no—
368C2×D4—2441no—
369C2×D4—2441no—
370C2×D4—2441no—
371C2×D4—2441no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—4881no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4C2×C4×D6111yescharacteristic
388C22×C4C2×D24411yescharacteristic
389C22×C4#1175822211yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4#1175822611yescharacteristic
393D8—4882no—
394C2×C8—22no—
395C2×C8—81no—
396C2×C8—4881no—
397C2×C8—2441no—
398C4⋊C4—81no—
399C4⋊C4—612no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—2442no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—612no—
407C22⋊C4—1221no—
408C22⋊C4—1222no—
409C22⋊C4—1222no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—4881no—
413C22⋊C4—2442no—
414C22⋊C4—2442no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C42—4881no—
418C42—2442no—
419C23—41no—
420C23—41no—
421C23—1222no—
422C23—4881no—
423C23—4881no—
424C23—2441no—
425C23—2441no—
426C23—2441no—
427C23—2441no—
428C23—2441no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2441no—
433C23#1451973911yescharacteristic
434C23#1451974711yescharacteristic
435C23#1451975011yescharacteristic
436C23D8×D6111yescharacteristic
437C23#1451980011yescharacteristic
438C23#1451980411yescharacteristic
439C23#1451980411yescharacteristic
440D4—4881no—
441D4—4881no—
442D4—4881no—
443D4—4881no—
444D4—4881no—
445D4—4881no—
446D4—2442no—
447D4—2442no—
448D4—2442no—
449D4—2442no—
450D4—2442no—
451D4—2442no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—4884no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4C22⋊C4×D6111yescharacteristic
483C2×C4C4×D24411yescharacteristic
484C2×C4#1451975211yescharacteristic
485C2×C4#1451976211yescharacteristic
486C8—42no—
487C8—4881no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—9761no—
499C22—9761no—
500C22—4881no—
501C22—4881no—
502C22—4881no—
503C22—4881no—
504C22—4881no—
505C22—4881no—
506C22—4881no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22#1683964011yescharacteristic
511C22#1683984411yescharacteristic
512C22#1683987311yescharacteristic
513C22#1683989711yescharacteristic
514C22#1683991011yescharacteristic
515C22#1683995611yescharacteristic
516C22#1683998011yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—161no—
524C4—4881no—
525C4—4881no—
526C4—2442no—
527C4—2442no—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—9761no—
535C2—4882no—
536C2#2144616811yescharacteristic
537C2#2144631911yescharacteristic
538C2#2144636011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1294C2
socleC22×C61#386
prime radical#21456570C2
cosocleC23×C4×C61C23
hypercenter#1294D61
hypocenterC61#9277
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. #21448866C2semidirect
3. #21446654C2semidirect
4. #21444386C2semidirect
5. C2×#16839495C2semidirect
6. #21443665C2semidirect
7. C61×#1294C2semidirect
12. #16840076C4semidirect
15. #16839898C4semidirect
31. C22×D244C2×C4semidirect
48. C2×C4⋊Dic61C2×C4semidirect
76. C22⋊C8×C61C2×C4semidirect
131. C22⋊Dic61D8semidirect
264. #1294D61semidirect
300. C61#9277semidirect
343. C22⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214565702
#214488662
#214466542
#214443862
C2×#168394952
#214436652
C61×#12942
#927761