Finite Group Databasebeta

← #29570011

Subgroups — #29570011

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839724C211yesmaximal, characteristic
2C2×#16839624C211yesmaximal, characteristic
3#380×D61C211yesmaximal, characteristic
4#21443699C211yesmaximal, characteristic, complemented
5#21443698C211yesmaximal, characteristic, complemented
6#21443697C211yesmaximal, characteristic, complemented
7C61×#1294C211yesmaximal, characteristic, complemented
8C23×C4×D61C2211yescharacteristic
9C22×#11758189—21no—
10C2×C4⋊C4×D61C412yescomplemented
11C22×#11758198—21no—
12C2×C22⋊C4×D61—21no—
13#16839724—22no—
14#16839724C414yescomplemented
15C2×C22⋊#92418C2211yescharacteristic
16C2×#14519689—21no—
17#16839624—22no—
18#16839624C414yescomplemented
19#16839622C2211yescharacteristic
20#16839621—21no—
21#16839620—21no—
22#16839619C412yescomplemented
23C2×#14519684—21no—
24C2×#14519684C2211yescharacteristic
25#95×D61—21no—
26C2×C22⋊C8×C61C2211yescharacteristic
27C61×#380C2211yescharacteristic
28C2×#95×C61C2211yescharacteristic
29C24×D61D411yescharacteristic
30C23×Dic61—21no—
31C23×Dic61D411yescharacteristic
32C22×C4×D61—22no—
33C22×C4×D61—44no—
34C22×C4×D61C2×C411yescharacteristic
35C22×C4×D61C2×C412yes—
36C22×C4×D61C2×C411yescharacteristic
37C22×C4×D61C2×C412yes—
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D412yes—
41C22×C4×D61Q812yes—
42C2×#11758189—24no—
43C2×#11758189—44no—
44C22×#92418—21no—
45C2×C4⋊Dic61C2×C412yes—
46C2×#11758196—22no—
47C4⋊C4×D61—44no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—22no—
50C2×#11758198—24no—
51C2×#11758198—44no—
52C22⋊C4×D61—42no—
53C22⋊C4×D61—42no—
54C22⋊#92418—21no—
55C22⋊#92418C2×C412yescomplemented
56#14519689—24no—
57#14519684—21no—
58#14519684—21no—
59#14519684—22no—
60#14519684—24no—
61#14519684C2×C412yescomplemented
62C23×C4×C61—21no—
63C23×C4×C61C2311yescharacteristic
64C22×C8×C61—21no—
65C2×C4⋊C4×C61C2×C412yes—
66C2×C22⋊C4×C61—21no—
67C22⋊C8×C61—21no—
68C22⋊C8×C61C2×C412yescomplemented
69#95×C61—21no—
70#95×C61—21no—
71#95×C61C2×C412yescomplemented
72C24×C61C2×D411yescharacteristic
73C23×D61—21no—
74C23×D61—21no—
75C23×D61—21no—
76C23×D61—21no—
77C23×D61—24no—
78C23×D61—42no—
79C23×D61C22⋊C411yescharacteristic
80C23×D61C22⋊C412yes—
81C23×D61C22⋊C411yescharacteristic
82C23×D61C4⋊C412yes—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—42no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C22×C411yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×Q811yescharacteristic
96C22×Dic61—22no—
97C22×Dic61—22no—
98C22×Dic61—42no—
99C22×Dic61—42no—
100C22×Dic61—42no—
101C22×Dic61—41no—
102C22×Dic61—41no—
103C22×Dic61C22⋊C411yescharacteristic
104C22×Dic61C22⋊C411yescharacteristic
105C22×Dic61C22⋊C412yes—
106C22×Dic61C4⋊C412yes—
107C2×C4×D61—24no—
108C2×C4×D61—24no—
109C2×C4×D61—44no—
110C2×C4×D61—44no—
111C2×C4×D61—44no—
112C2×C4×D61—44no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×C4×D61—44no—
116C2×C4×D61—44no—
117C2×C4×D61—44no—
118C2×C4×D61C4212yes—
119C2×C4×D61C4212yes—
120C2×C4×D61C22⋊C412yes—
121C2×C4×D61C22⋊C412yes—
122C2×C4×D61C4⋊C412yes—
123C2×C4×D61C4⋊C412yes—
124C2×C4×D61C4⋊C412yes—
125C2×C4×D61C4⋊C412yes—
126C2×C8×C61—22no—
127C2×C8×C61—42no—
128C4⋊C4×C61—42no—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C22⋊Dic61—41no—
132C22⋊Dic61—41no—
133#11758198—44no—
134#11758198—48no—
135C4⋊Dic61—42no—
136#11758196—44no—
137C2×#92418—22no—
138C2×#92418—42no—
139#11758189—48no—
140C23×C61—22no—
141C23×C61—22no—
142C23×C61C2×C22⋊C411yescharacteristic
143C23×C61C2×C22⋊C411yescharacteristic
144C23×C61C2×C4⋊C411yescharacteristic
145C23×C61AGL(1,ℤ/8)11yescharacteristic
146C23×C61AGL(1,ℤ/8)11yescharacteristic
147C23×C61#13711yescharacteristic
148C23×C61#13711yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—22no—
152C22×D61—22no—
153C22×D61—22no—
154C22×D61—22no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—22no—
158C22×D61—22no—
159C22×D61—22no—
160C22×D61—24no—
161C22×D61—24no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—44no—
165C22×D61—44no—
166C22×D61—44no—
167C22×D61—44no—
168C22×D61#9512yes—
169C22×D61#9512yes—
170C22×D61#9914yes—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—82no—
184C2×C4×C61—82no—
185C2×C4×C61C2×C4211yescharacteristic
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C2×C4×C61C2×C4⋊C411yescharacteristic
188C2×C4×C61C2×C4⋊C411yescharacteristic
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—82no—
200C2×Dic61—82no—
201C2×Dic61#9512yes—
202C2×Dic61#9512yes—
203C4×D61—24no—
204C4×D61—24no—
205C4×D61—24no—
206C4×D61—24no—
207C4×D61—42no—
208C4×D61—42no—
209C4×D61—42no—
210C4×D61—42no—
211C4×D61—88no—
212C8×C61—42no—
213#92418—42no—
214#8015—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61C2×#9511yescharacteristic
224C22×C61C2×#9912yes—
225C22×C61#41711yescharacteristic
226C22×C61#41711yescharacteristic
227C22×C61#41811yescharacteristic
228C22×C61#41911yescharacteristic
229C22×C61#42112yes—
230C22×C61#42811yescharacteristic
231C22×C61#42811yescharacteristic
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—24no—
235C2×D61—24no—
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—44no—
240C2×D61—44no—
241C2×D61—44no—
242C2×D61—44no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61#34214yes—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—82no—
251C4×C61—82no—
252Dic61—22no—
253Dic61—22no—
254Dic61—82no—
255Dic61—82no—
256C2×#407—611no—
257C2×#381—611no—
258C2×#380—611no—
259#1294—611no—
260#1294—611no—
261#1294—611no—
262#1294D6111yescharacteristic, complemented
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61#151012yes—
268C2×C61#151112yes—
269C2×C61#173311yescharacteristic
270C2×C61#173611yescharacteristic
271C2×C61C2×#34211yescharacteristic
272D61—24no—
273D61—42no—
274C24×C4—611no—
275C22×M16—1221no—
276C22×C4⋊C4—612no—
277C22×C22⋊C4—1221no—
278C22×C22⋊C4—1221no—
279#407—614no—
280#407—1222no—
281C2×C22⋊C8—611no—
282C2×C22⋊C8—1221no—
283C2×C22⋊C8C2×D6111yescharacteristic
284#381—614no—
285#381—1222no—
286#380—611no—
287#380—612no—
288#380—1221no—
289#380—1221no—
290#380C2×D6111yescharacteristic
291C2×#95—611no—
292C2×#95—1221no—
293C2×#95—1221no—
294C2×#95C2×D6111yescharacteristic
295C61#801511yescharacteristic, complemented, Sylow
296C25—611no—
297C23×C4—21no—
298C23×C4—611no—
299C23×C4—611no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4—612no—
303C23×C4—612no—
304C23×C4—611no—
305C23×C4—612no—
306C23×C4—612no—
307C23×C4—1222no—
308C23×C4—1221no—
309C23×C4—2444no—
310C23×C4C22×D6111yescharacteristic
311C2×M16—1224no—
312C2×M16—2444no—
313C22×C8—21no—
314C22×C8—1221no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—2444no—
318C2×C4⋊C4C4×D6112yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1224no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—2442no—
324C2×C22⋊C4—2442no—
325C2×C22⋊C4—2444no—
326C22⋊C8—21no—
327C22⋊C8—612no—
328C22⋊C8—1224no—
329C22⋊C8—1221no—
330C22⋊C8C4×D6112yescomplemented
331#95—21no—
332#95—21no—
333#95—612no—
334#95—1222no—
335#95—1224no—
336#95—1221no—
337#95—1221no—
338#95C4×D6112yescomplemented
339C24—611no—
340C24—611no—
341C24—612no—
342C24—612no—
343C24—1224no—
344C24—1221no—
345C24—1221no—
346C24—1221no—
347C24—1221no—
348C24—2442no—
349C24D4×D6111yescharacteristic
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—612no—
360C22×C4—612no—
361C22×C4—612no—
362C22×C4—612no—
363C22×C4—612no—
364C22×C4—612no—
365C22×C4—612no—
366C22×C4—612no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—1224no—
370C22×C4—1224no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—2442no—
374C22×C4—2442no—
375C22×C4—2444no—
376C22×C4—2444no—
377C22×C4—2444no—
378C22×C4—2444no—
379C22×C4—2444no—
380C22×C4—2444no—
381C22×C4—2444no—
382C22×C4—2442no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4—2441no—
386C22×C4—2441no—
387C22×C4C2×C4×D6111yescharacteristic
388C22×C4C2×C4×D6111yescharacteristic
389C22×C4C2×C4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4Q8×D6111yescharacteristic
393M16—2448no—
394C2×C8—22no—
395C2×C8—42no—
396C2×C8—1222no—
397C2×C8—2442no—
398C4⋊C4—42no—
399C4⋊C4—2442no—
400C4⋊C4—2444no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—2448no—
404C22⋊C4—2444no—
405C22⋊C4—2441no—
406C22⋊C4—2441no—
407C23—22no—
408C23—22no—
409C23—614no—
410C23—612no—
411C23—612no—
412C23—1224no—
413C23—1224no—
414C23—1221no—
415C23—1221no—
416C23—1222no—
417C23—1222no—
418C23—1222no—
419C23—1222no—
420C23—1221no—
421C23—1221no—
422C23—1222no—
423C23—1222no—
424C23—1222no—
425C23—2442no—
426C23—2442no—
427C23—2444no—
428C23—2444no—
429C23—2444no—
430C23—2444no—
431C23C22⋊C4×D6111yescharacteristic
432C23C22⋊C4×D6111yescharacteristic
433C23C4⋊C4×D6111yescharacteristic
434C23#1451980111yescharacteristic
435C23#1451980411yescharacteristic
436C23#1451980611yescharacteristic
437C23#1451980911yescharacteristic
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—612no—
453C2×C4—612no—
454C2×C4—1224no—
455C2×C4—1224no—
456C2×C4—1224no—
457C2×C4—1224no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—4888no—
461C2×C4—4882no—
462C2×C4—4882no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4C42×D6111yescharacteristic
476C2×C4C22⋊C4×D6111yescharacteristic
477C2×C4C4⋊C4×D6111yescharacteristic
478C2×C4C4⋊C4×D6111yescharacteristic
479C8—42no—
480C8—2442no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—614no—
490C22—1222no—
491C22—1222no—
492C22—1224no—
493C22—1224no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22—2444no—
498C22—2444no—
499C22—2444no—
500C22—2444no—
501C22—2442no—
502C22—2442no—
503C22#95×D6111yescharacteristic
504C22#99×D6112yes—
505C22#1683976011yescharacteristic
506C22#1683976211yescharacteristic
507C22#1683976511yescharacteristic
508C22#1683976911yescharacteristic
509C22#1683977712yes—
510C22#1683980111yescharacteristic
511C22#1683980311yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—82no—
517C4—82no—
518C4—1222no—
519C4—1222no—
520C4—4882no—
521C4—4882no—
522C2—21no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—1224no—
527C2—2442no—
528C2#2144434612yes—
529C2#2144435012yes—
530C2#2144552711yescharacteristic
531C2#2144553711yescharacteristic
532C2#342×D6111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1294C2
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC23×C4×C61C23
hypercenter#1294D61
hypocenterC61#8015
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
4. #21443699C2semidirect
5. #21443698C2semidirect
6. #21443697C2semidirect
7. C61×#1294C2semidirect
10. C2×C4⋊C4×D61C4semidirect
14. #16839724C4semidirect
18. #16839624C4semidirect
22. #16839619C4semidirect
55. C22⋊#92418C2×C4semidirect
61. #14519684C2×C4semidirect
68. C22⋊C8×C61C2×C4semidirect
71. #95×C61C2×C4semidirect
262. #1294D61semidirect
295. C61#8015semidirect
330. C22⋊C8C4×D61semidirect
338. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168397242
C2×#168396242
#380×D612
#214436992
#214436982
#214436972
C61×#12942
#801561