Finite Group Databasebeta

← #29574086

Subgroups — #29574086

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448519C211yesmaximal, characteristic
2#21446136C211yesmaximal, characteristic
3C2×#16839497C211yesmaximal, characteristic
4#21444078C211yesmaximal, characteristic, complemented
5#21443697C211yesmaximal, characteristic, complemented
6#21443661C211yesmaximal, characteristic, complemented
7C61×#1853C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519737—21no—
10C2×#14519697—21no—
11C2×C22⋊#92418C2211yescharacteristic
12#16839652—21no—
13#16839645—21no—
14#16839640—21no—
15#16839639—21no—
16#16839633—21no—
17C2×#14519684—21no—
18C2×#14519684—21no—
19C2×#14519684C2211yescharacteristic
20C2×#14519684C2211yescharacteristic
21#16839524—21no—
22#16839497C414yescomplemented
23C22×C4⋊C4×C61C2211yescharacteristic
24C2×#106×C61—21no—
25C2×C22⋊C8×C61C2211yescharacteristic
26C61×#380C2211yescharacteristic
27C61×#340—21no—
28C22×D244D411yescharacteristic
29C22×C22⋊D61—21no—
30C23×Dic61—21no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C22×C4×D61—21no—
34C22×#92418—21no—
35#14519751—21no—
36#14519751—21no—
37#14519751C2×C412yes—
38#14519751Q812yes—
39#14519737—42no—
40#14519737—42no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×C4×Dic61—21no—
48#14519697—42no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—42no—
57#14519684—42no—
58#14519684C2×C412yescomplemented
59C23×C4×C61—21no—
60C23×C4×C61C2311yescharacteristic
61C22×C8×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—42no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C22⋊C4×C61—21no—
68#106×C61—42no—
69C22⋊C8×C61—21no—
70C22⋊C8×C61C2×C412yescomplemented
71#95×C61—21no—
72C24×C61C2×D411yescharacteristic
73C23×D61—21no—
74C22×C4×C61—22no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—41no—
78C22×C4×C61—41no—
79C22×C4×C61—41no—
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×Q811yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C2×C22⋊D61—22no—
92C2×C22⋊D61—22no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C2×C22⋊D61—41no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—22no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—41no—
105C22×Dic61—41no—
106C22×Dic61—41no—
107C22×Dic61—41no—
108C22×Dic61—41no—
109C22×Dic61—41no—
110C22×Dic61—41no—
111C22×Dic61—41no—
112C22×Dic61—41no—
113C22×Dic61—41no—
114C22×Dic61SD1612yes—
115C2×D244—21no—
116C2×D244—41no—
117C2×D244—41no—
118C2×D244C4⋊C412yes—
119C2×C4×D61—42no—
120C2×C4×D61—81no—
121C2×C4×D61—81no—
122C2×C8×C61—22no—
123C2×C8×C61—41no—
124C2×C8×C61—41no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—41no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133#11758198—22no—
134#11758198—22no—
135#11758198—42no—
136#11758198—81no—
137#11758198—81no—
138C4⋊Dic61—21no—
139C4⋊Dic61C4⋊C412yes—
140#11758196—42no—
141C4×Dic61—42no—
142C2×#92418—41no—
143C2×#92418—41no—
144C2×#92418—41no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61C4×D411yescharacteristic
150C23×C61#12011yescharacteristic
151C23×C61C22⋊Q811yescharacteristic
152C23×C61C2×SD1611yescharacteristic
153C23×C61#13511yescharacteristic
154C23×C61AGL(1,ℤ/8)11yescharacteristic
155C23×C61AGL(1,ℤ/8)11yescharacteristic
156C22×D61—22no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—81no—
181C2×C4×C61—81no—
182C2×C4×C61—81no—
183C2×C4×C61—81no—
184C2×C4×C61C2×C4⋊C411yescharacteristic
185C2×C4×C61C4×D411yescharacteristic
186C2×C4×C61C22⋊Q811yescharacteristic
187C2×C4×C61#12311yescharacteristic
188C22⋊D61—42no—
189C22⋊D61—42no—
190C22⋊D61—42no—
191C22⋊D61—42no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—81no—
207C2×Dic61—81no—
208C2×Dic61—81no—
209C2×Dic61—81no—
210C2×Dic61—81no—
211C2×Dic61—81no—
212D244—42no—
213D244—81no—
214C4×D61—84no—
215C8×C61—42no—
216#92418—81no—
217#8870—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61#38511yescharacteristic
231C22×C61C4×SD1611yescharacteristic
232C22×C61#44211yescharacteristic
233C22×C61#44911yescharacteristic
234C22×C61#45011yescharacteristic
235C22×C61#45411yescharacteristic
236C22×C61#45511yescharacteristic
237C22×C61B(4,𝔽2)11yescharacteristic
238C22×C61#45811yescharacteristic
239C22×C61#47611yescharacteristic
240C22×C61#47811yescharacteristic
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—82no—
253C4×C61—81no—
254C4×C61—81no—
255Dic61—42no—
256Dic61—42no—
257Dic61—82no—
258Dic61—81no—
259#2325—611no—
260#1874—611no—
261#1853D6111yescharacteristic, complemented
262C2×#327—611no—
263#1442—611no—
264#1294—611no—
265#1280—611no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61#159711yescharacteristic
271C2×C61#160111yescharacteristic
272C2×C61#161211yescharacteristic
273C2×C61#161611yescharacteristic
274C2×C61#186011yescharacteristic
275C2×C61#186411yescharacteristic
276C2×C61#186711yescharacteristic
277D61—82no—
278C2×#121—611no—
279C2×C4×D4—1221no—
280C22×C4⋊C4C2×D6111yescharacteristic
281C2×#106—21no—
282C2×#102—1221no—
283C2×C22⋊C8—611no—
284C2×C22⋊C8C2×D6111yescharacteristic
285#388—1221no—
286#387—1221no—
287#386—1221no—
288#386—1221no—
289#385—1221no—
290#380C2×D6111yescharacteristic
291C2×#95—611no—
292C2×#95—611no—
293C2×#95—1221no—
294C2×#95—1221no—
295#340—21no—
296#340—1221no—
297#327—614no—
298C61#887011yescharacteristic, complemented, Sylow
299C22×D4—611no—
300C22×D4—1221no—
301C23×C4—21no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—1221no—
305C23×C4—1221no—
306C23×C4C22×D6111yescharacteristic
307C22×C8—21no—
308C22×C8—1221no—
309#121—612no—
310#121—612no—
311#121—1221no—
312#121—1221no—
313C4×D4—2442no—
314C4×D4—2442no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4C22⋊D6111yescharacteristic
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C42—1221no—
328#106—42no—
329#102—2442no—
330C22⋊C8—21no—
331C22⋊C8—1221no—
332C22⋊C8—1221no—
333C22⋊C8C4×D6112yescomplemented
334#95—21no—
335#95—612no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341#95—2442no—
342#95—2442no—
343C24—1221no—
344C24D4×D6111yescharacteristic
345C2×D4—612no—
346C2×D4—1222no—
347C2×D4—1222no—
348C2×D4—1221no—
349C2×D4—2441no—
350C2×D4—2441no—
351C2×D4—2441no—
352C2×D4—2441no—
353C2×D4—2441no—
354C2×D4—2441no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—612no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1222no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—4881no—
375C22×C4—4881no—
376C22×C4—2442no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4—2441no—
380C22×C4—2441no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4—2441no—
386C22×C4—2441no—
387C22×C4C2×C4×D6111yescharacteristic
388C22×C4#1175822211yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4Q8×D6111yescharacteristic
391C22×C4#1175822611yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—2441no—
397C2×C8—2441no—
398C2×C8—2441no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—612no—
406C4⋊C4—1221no—
407C4⋊C4—2442no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—1222no—
411C22⋊C4—1222no—
412C22⋊C4—4881no—
413C22⋊C4—4881no—
414C22⋊C4—2442no—
415C42—2442no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—1222no—
421C23—2441no—
422C23—2441no—
423C23—2441no—
424C23—2441no—
425C23—2441no—
426C23—2441no—
427C23#1451973911yescharacteristic
428C23#1451974511yescharacteristic
429C23#1451975511yescharacteristic
430C23SD16×D6111yescharacteristic
431C23#1451979811yescharacteristic
432C23#1451980411yescharacteristic
433C23#1451980411yescharacteristic
434D4—4881no—
435D4—2442no—
436D4—2442no—
437D4—2442no—
438D4—2442no—
439D4—2442no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—4884no—
464C2×C4—4881no—
465C2×C4—4881no—
466C2×C4—4881no—
467C2×C4—4881no—
468C2×C4—4881no—
469C2×C4—4881no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2442no—
482C2×C4C4⋊C4×D6111yescharacteristic
483C2×C4C4×C22⋊D6111yescharacteristic
484C2×C4#1451975311yescharacteristic
485C2×C4#1451976211yescharacteristic
486C8—42no—
487C8—4881no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—4881no—
501C22—4881no—
502C22—4881no—
503C22—4881no—
504C22—2442no—
505C22—2442no—
506C22—2442no—
507C22#1683963311yescharacteristic
508C22#1683984911yescharacteristic
509C22#1683987311yescharacteristic
510C22#1683990611yescharacteristic
511C22#1683991111yescharacteristic
512C22#1683993311yescharacteristic
513C22#1683993811yescharacteristic
514C22#1683994811yescharacteristic
515C22#1683995111yescharacteristic
516C22#1684005111yescharacteristic
517C22#1684006111yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—82no—
523C4—81no—
524C4—81no—
525C4—4882no—
526C4—4881no—
527C4—2442no—
528C4—2442no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—4882no—
534C2#2144474711yescharacteristic
535C2#2144476911yescharacteristic
536C2#2144483211yescharacteristic
537C2#2144485411yescharacteristic
538C2#2144604911yescharacteristic
539C2#2144607711yescharacteristic
540C2#2144609811yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1853C2
socleC23×C61#120
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1853D61
hypocenterC61#8870
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #21444078C2semidirect
5. #21443697C2semidirect
6. #21443661C2semidirect
7. C61×#1853C2semidirect
22. #16839497C4semidirect
58. #14519684C2×C4semidirect
70. C22⋊C8×C61C2×C4semidirect
261. #1853D61semidirect
298. C61#8870semidirect
333. C22⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214485192
#214461362
C2×#168394972
#214440782
#214436972
#214436612
C61×#18532
#887061