Finite Group Databasebeta

← #29654317

Subgroups — #29654317

Order 15616.

1156 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457005C212yesmaximal, complemented
2#21451517C211yesmaximal, characteristic, complemented
3#21444856C212yesmaximal, complemented
4#21444839C212yesmaximal, complemented
5#21444822C212yesmaximal, complemented
6#21444805C212yesmaximal, complemented
7#21444327C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic, complemented
9C61×#3449C211yesmaximal, characteristic
10C2×#14519769C2211yescharacteristic, complemented
11#16840168—22no—
12#16840126—22no—
13#16840078C2214yescomplemented
14#16840073C2214yescomplemented
15#16840062—24no—
16#16839819C2212yescomplemented
17#16839701C2211yescharacteristic, complemented
18#16839698C2211yescharacteristic
19#16839692C2211yescharacteristic, complemented
20#16839687C2211yescharacteristic
21#16839657—21no—
22#16839656—21no—
23#16839646—21no—
24#16839635—21no—
25#16839608C2211yescharacteristic, complemented
26#16839496C2214yescomplemented
27#16839496C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yes—
30C22×#41×C61—21no—
31C61×#537—21no—
32C61×#537C2212yes—
33C61×#536—22no—
34C61×#535—22no—
35C61×#534—21no—
36C61×#534C2212yes—
37C61×#532—21no—
38C61×#532C2212yes—
39C61×#525—24no—
40C2×#117×C61C2211yescharacteristic
41#14519769—22no—
42#14519769—22no—
43C2×C4⋊Dic61C2311yescharacteristic
44C2×C4⋊Dic61C2311yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C4×Dic61—21no—
52#14519710—24no—
53#14519708—24no—
54C4⋊#92418—22no—
55#14519698—24no—
56#14519698—24no—
57C22⋊#92418C2314yes—
58#14519688—22no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61—44no—
71C2×#41×C61—44no—
72C22×Q8×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61—22no—
75C23×C4×C61—22no—
76C23×C4×C61D411yescharacteristic
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D412yes—
79#127×C61—21no—
80#127×C61D412yes—
81#124×C61—22no—
82#124×C61—22no—
83#123×C61—24no—
84#123×C61—24no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87#121×C61—24no—
88#121×C61—24no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61C2314yes—
93#120×C61—24no—
94#120×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61D414yes—
99#117×C61—22no—
100#117×C61—22no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61C2311yescharacteristic
105C2×C42×C61D411yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C24×C61—22no—
108C24×C61C2×D411yescharacteristic
109#41×C61—48no—
110#41×C61—48no—
111#41×C61—44no—
112#41×C61—44no—
113#41×C61—44no—
114#41×C61—44no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—44no—
118C2×D4×C61—24no—
119C2×D4×C61—22no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—44no—
126C2×D4×C61—44no—
127C2×D4×C61—44no—
128C2×D4×C61C2×D414yes—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—48no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D414yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—24no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163C22⋊Dic61—42no—
164C22⋊Dic61—42no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167#11758196—42no—
168#11758196—41no—
169#11758196—41no—
170#11758196—42no—
171#11758196—42no—
172C4×Dic61—42no—
173C2×#92418—24no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—24no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—44no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61AGL(1,ℤ/8)12yes—
183C23×C61C22×D411yescharacteristic
184Q8×C61—44no—
185Q8×C61—44no—
186D4×C61—44no—
187D4×C61—44no—
188D4×C61—44no—
189D4×C61—48no—
190D4×C61—44no—
191D4×C61—44no—
192D4×C61—82no—
193D4×C61—82no—
194D4×C61—82no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—24no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—48no—
207C2×C4×C61—48no—
208C2×C4×C61—44no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12012yes—
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61C22×D412yes—
216C2×Dic61—22no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228#92418—44no—
229#18339—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—22no—
233C22×C61—42no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—44no—
237C22×C61—44no—
238C22×C61—44no—
239C22×C61—81no—
240C22×C61—88no—
241C22×C61C2×#12011yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C4×C61—21no—
245C4×C61—22no—
246C4×C61—21no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—44no—
250C4×C61—41no—
251C4×C61—41no—
252Dic61—42no—
253Dic61—81no—
254Dic61—81no—
255Dic61—81no—
256#3449D6111yescharacteristic
257#3234—612no—
258#2611—611no—
259#1616—612no—
260#1613—612no—
261#1610—612no—
262#1607—612no—
263#1505—612no—
264#1464—611no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—42no—
268C2×C61—84no—
269C2×C61#300311yescharacteristic
270C2×C61#301811yescharacteristic
271C2×C61#302811yescharacteristic
272C22×#41—21no—
273#537—21no—
274#537C2×D6112yes—
275#536—22no—
276#535—22no—
277#534—21no—
278#534C2×D6112yes—
279#532—21no—
280#532C2×D6112yes—
281C2×#125—611no—
282#525—24no—
283C2×#117C2×D6111yescharacteristic
284#501—1222no—
285#490—1222no—
286#481—614no—
287#480—614no—
288#478—1224no—
289#432—612no—
290#400—611no—
291#399—611no—
292#398—611no—
293#397—611no—
294#389—1221no—
295#388—1221no—
296#387—1221no—
297#385—1221no—
298#376—611no—
299#327—614no—
300#327—614no—
301#324—612no—
302#323—612no—
303C61#1833911yescharacteristic, complemented, Sylow
304C2×#41—22no—
305C2×#41—22no—
306C2×#41—22no—
307C2×#41—22no—
308C2×#41—44no—
309C2×#41—44no—
310C22×Q8—21no—
311C22×D4—21no—
312C22×D4—22no—
313C23×C4—22no—
314C23×C4C22⋊D6111yescharacteristic
315C4⋊Q8—21no—
316C4⋊Q8C22⋊D6112yes—
317#127—21no—
318#127C22⋊D6112yes—
319#125—1222no—
320#125—1222no—
321#124—22no—
322#124—22no—
323#123—24no—
324#123—24no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327#121—24no—
328#121—24no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121C22×D6114yes—
333#120—24no—
334#120—24no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4C22⋊D6114yes—
339#117—22no—
340#117—22no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4C22⋊D6111yescharacteristic
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4C22⋊D6111yescharacteristic
352C2×C22⋊C4C22×D6111yescharacteristic
353C2×C42—1221no—
354C2×C42C22⋊D6111yescharacteristic
355C2×C42C22×D6111yescharacteristic
356#107—1224no—
357#106—1224no—
358C4⋊C8—1222no—
359#102—1224no—
360#102—1224no—
361C22⋊C8—614no—
362#97—1222no—
363#95—611no—
364#95—611no—
365#95—1221no—
366#95—1221no—
367#95—1221no—
368#95—1221no—
369#95—1221no—
370C24—22no—
371C24C2×C22⋊D6111yescharacteristic
372#41—48no—
373#41—48no—
374#41—44no—
375#41—44no—
376#41—44no—
377#41—44no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4C2×C22⋊D6114yes—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—48no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410C2×C8—1224no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—1222no—
417C4⋊C4—1222no—
418C4⋊C4—2442no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4—2442no—
422C4⋊C4—2442no—
423C4⋊C4C2×C22⋊D6114yes—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—2442no—
431C22⋊C4—2442no—
432C42—21no—
433C42—21no—
434C42—2442no—
435C42C2×C22⋊D6112yes—
436C42C2×C22⋊D6112yes—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—41no—
441C23—41no—
442C23—44no—
443C23—82no—
444C23—82no—
445C23#1451980312yes—
446C23C22×C22⋊D6111yescharacteristic
447Q8—44no—
448Q8—44no—
449D4—44no—
450D4—44no—
451D4—44no—
452D4—48no—
453D4—44no—
454D4—44no—
455D4—82no—
456D4—82no—
457D4—82no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—1222no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4#1451974611yescharacteristic
487C2×C4#1451974611yescharacteristic
488C2×C4#1451974612yes—
489C2×C4#1451980312yes—
490C2×C4C22×C22⋊D6112yes—
491C8—2444no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—81no—
502C22—88no—
503C22C2×#1451974611yescharacteristic
504C22C2×#1451980311yescharacteristic
505C22C2×#1451980311yescharacteristic
506C4—22no—
507C4—22no—
508C4—21no—
509C4—22no—
510C4—21no—
511C4—41no—
512C4—41no—
513C4—44no—
514C4—4881no—
515C4—4881no—
516C4—4881no—
517C4—2442no—
518C2—21no—
519C2—21no—
520C2—42no—
521C2—84no—
522C2#2145504611yescharacteristic
523C2#2145516811yescharacteristic
524C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3449C2
socleC22×C61C2×#120
hypercenter#3449D61
hypocenterC61#18339
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21457005C2semidirect
2. #21451517C2semidirect
3. #21444856C2semidirect
4. #21444839C2semidirect
5. #21444822C2semidirect
6. #21444805C2semidirect
7. #21444327C2semidirect
8. #21444153C2semidirect
10. C2×#14519769C22semidirect
13. #16840078C22semidirect
14. #16840073C22semidirect
16. #16839819C22semidirect
17. #16839701C22semidirect
19. #16839692C22semidirect
25. #16839608C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
28. #16839488C22semidirect
303. C61#18339semidirect

Maximal subgroups

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

subgroupindex
#214570052
#214515172
#214448562
#214448392
#214448222
#214448052
#214443272
#214441532
C61×#34492
#1833961