Finite Group Databasebeta

← #29712032

Subgroups — #29712032

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455045C212yesmaximal, complemented
2C2×#16839663C211yesmaximal, characteristic, complemented
3#21446972C212yesmaximal, complemented
4#21446861C212yesmaximal, complemented
5#21446263C212yesmaximal, complemented
6#21446195C214yesmaximal, complemented
7#21445899C211yesmaximal, characteristic, complemented
8C61×#3492C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C2×#14519802C2214yes—
11C22×#11758198—22no—
12C22×C4×Dic61C2211yescharacteristic
13#16839912—22no—
14#16839912—22no—
15#16839895—22no—
16#16839895—22no—
17C2×#14519704C2214yescomplemented
18#16839766C2214yes—
19#16839726C2212yes—
20#16839663—22no—
21#16839663C2212yes—
22#16839663C2212yes—
23#16839531C2212yescomplemented
24#16839531C2212yescomplemented
25#16839521C2214yescomplemented
26C61×#554—22no—
27C61×#553—22no—
28C61×#552—22no—
29C61×#547—22no—
30C61×#544C2211yescharacteristic, complemented
31C61×#539—22no—
32C61×#537C2211yescharacteristic, complemented
33C61×#525—22no—
34C61×#525C2212yes—
35C61×#520C2212yescomplemented
36C22×C4⋊C4×C61C2211yescharacteristic
37C24×D61—21no—
38C22×D244—21no—
39C22×D244—24no—
40C22×D244—22no—
41C22×D244—22no—
42C22×D244—42no—
43C22×D244C2312yes—
44C23×Dic61—21no—
45#14519802—28no—
46#14519802—44no—
47C2×#11758201—24no—
48C2×#11758199—24no—
49C2×#11758194C2314yes—
50C2×#11758198—24no—
51C2×#11758198—24no—
52C2×#11758198—48no—
53C2×C4×Dic61—21no—
54C2×C4×Dic61—42no—
55C2×C4×Dic61C2312yes—
56#14519704—44no—
57#14519697—24no—
58#14519697—24no—
59C22⋊#92418—22no—
60C22⋊#92418—22no—
61C2×#41×C61—22no—
62C2×#41×C61C2312yes—
63C23×C4×C61D412yes—
64C23×C4×C61C2311yescharacteristic
65C4⋊Q8×C61—22no—
66#126×C61—24no—
67#125×C61—24no—
68#125×C61—22no—
69#123×C61—24no—
70#123×C61—24no—
71#123×C61—22no—
72C22⋊Q8×C61—24no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61D412yes—
77C22⋊Q8×C61D412yes—
78#121×C61—22no—
79#121×C61—22no—
80#121×C61—22no—
81#121×C61D412yes—
82#121×C61D412yes—
83#120×C61—22no—
84C4×Q8×C61—22no—
85C4×D4×C61—24no—
86C4×D4×C61—22no—
87C4×D4×C61—24no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90#117×C61—22no—
91#117×C61—22no—
92#117×C61C2312yes—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61D412yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C24×C61C2×D411yescharacteristic
100C23×D61—24no—
101C23×D61—22no—
102C23×D61—42no—
103C23×D61—44no—
104C23×D61—44no—
105C23×D61—41no—
106C23×D61—41no—
107#41×C61—24no—
108#41×C61—24no—
109#41×C61—84no—
110C2×Q8×C61—22no—
111C2×Q8×C61C2×D412yes—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61C2×D412yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—44no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—24no—
131C22×Dic61—22no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C2×D244—24no—
135C2×D244—24no—
136C2×D244—24no—
137C2×D244—22no—
138C2×D244—48no—
139C2×D244—48no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244#4114yescomplemented
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C42×C61—24no—
162C42×C61—22no—
163#11758201—48no—
164#11758199—48no—
165#11758198—48no—
166#11758198—48no—
167C4×Dic61—24no—
168C4×Dic61—42no—
169#11758194—24no—
170#11758194—24no—
171C2×#92418—24no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61#12011yescharacteristic
178C23×C61#12011yescharacteristic
179C23×C61C22×D411yescharacteristic
180C22×D61—24no—
181C22×D61—44no—
182C22×D61—44no—
183C22×D61—44no—
184C22×D61—416no—
185C22×D61—44no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—88no—
189C22×D61—82no—
190C22×D61—82no—
191C22×D61—82no—
192C22×D61—82no—
193Q8×C61—24no—
194Q8×C61—82no—
195D4×C61—24no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—44no—
199D4×C61—44no—
200D4×C61—82no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—44no—
211C2×C4×C61—44no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—82no—
215C2×C4×C61—84no—
216C2×C4×C61#12012yes—
217C2×C4×C61C22×D412yes—
218C2×C4×C61C2×#4112yes—
219C2×C4×C61H(5,𝔽2)12yes—
220C2×Dic61—24no—
221C2×Dic61—48no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224D244—28no—
225D244—48no—
226D244—44no—
227#92418—44no—
228#23948—611nomaximal, Sylow
229C22×C61—24no—
230C22×C61—24no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61#45312yes—
241C22×C61#45412yes—
242C22×C61C2×#12011yescharacteristic
243C22×C61#52511yescharacteristic
244C22×C61#53511yescharacteristic
245C2×D61—48no—
246C2×D61—44no—
247C2×D61—84no—
248C2×D61—88no—
249C2×D61—88no—
250C2×D61—82no—
251C2×D61—82no—
252C4×C61—22no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61#54614yes—
259Dic61—44no—
260#3492D6111yescharacteristic, complemented
261#3003—612no—
262C2×#390—611no—
263#2027—612no—
264#2006—612no—
265#1896—612no—
266#1883—614no—
267#1821—611no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—22no—
271C2×C61—22no—
272C2×C61—82no—
273C2×C61#240411yescharacteristic
274C2×C61C2×#45311yescharacteristic
275C2×C61C2×#45411yescharacteristic
276D61—84no—
277C23×D4—611no—
278C2×AGL(1,ℤ/8)—614no—
279#554—22no—
280#553—22no—
281#552—22no—
282#547—22no—
283#544C2×D6111yescharacteristic, complemented
284#539—22no—
285#537C2×D6111yescharacteristic, complemented
286#525—22no—
287#525C2×D6112yes—
288#520C2×D6112yescomplemented
289C22×C4⋊C4C2×D6111yescharacteristic
290C22×C22⋊C4—1222no—
291C22×C42—611no—
292#450—1222no—
293#450—1222no—
294#447—1222no—
295#447—1222no—
296C2×#104—614no—
297#418—614no—
298#407—612no—
299#390—612no—
300#390—612no—
301#390—1222no—
302#343—612no—
303#343—612no—
304#339—614no—
305C61#2394811yescharacteristic, complemented, Sylow
306C25—1221no—
307C2×#41—22no—
308C2×#41C22×D6112yes—
309C22×D4—612no—
310C22×D4—1224no—
311C22×D4—1221no—
312C22×D4—1222no—
313C22×D4—1222no—
314C22×D4—2442no—
315C23×C4—1221no—
316C23×C4C22⋊D6112yes—
317C23×C4C22×D6111yescharacteristic
318AGL(1,ℤ/8)—1228no—
319AGL(1,ℤ/8)—2444no—
320C2×SD16—1224no—
321C2×D8—1224no—
322C2×M16—614no—
323C4⋊Q8—22no—
324#126—24no—
325#125—24no—
326#125—22no—
327#123—24no—
328#123—24no—
329#123—22no—
330C22⋊Q8—22no—
331C22⋊Q8—24no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8C22⋊D6112yes—
335C22⋊Q8C22⋊D6112yes—
336#121—22no—
337#121—22no—
338#121—22no—
339#121C22⋊D6112yes—
340#121C22⋊D6112yes—
341#120—22no—
342C4×Q8—22no—
343C4×D4—24no—
344C4×D4—22no—
345C4×D4—24no—
346C4×D4—22no—
347C4×D4—22no—
348#117—22no—
349#117—22no—
350#117C22×D6112yes—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4C22⋊D6112yes—
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C22⋊C4—1224no—
358C2×C22⋊C4—1224no—
359C2×C22⋊C4—2448no—
360C2×C42—612no—
361C2×C42—1221no—
362C2×C42—2442no—
363#104—2444no—
364#102—1224no—
365#102—1224no—
366C22⋊C8—1222no—
367C22⋊C8—1222no—
368C24—1224no—
369C24—1222no—
370C24—2444no—
371C24—2442no—
372C24—2444no—
373C24—2441no—
374C24—2441no—
375C24C2×C22⋊D6111yescharacteristic
376#41—24no—
377#41—24no—
378#41—84no—
379C2×Q8—22no—
380C2×Q8C2×C22⋊D6112yes—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—614no—
386C2×D4—1224no—
387C2×D4—1224no—
388C2×D4—1222no—
389C2×D4—1224no—
390C2×D4—2448no—
391C2×D4—2448no—
392C2×D4—2442no—
393C2×D4—2442no—
394C2×D4C2×C22⋊D6112yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—44no—
402C22×C4—1224no—
403C22×C4—1222no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6112yes—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6112yes—
410C22×C4C2×C22⋊D6112yes—
411C22×C4C23×D6111yescharacteristic
412SD16—2448no—
413D8—2448no—
414M16—1224no—
415M16—1224no—
416C2×C8—1224no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4C2×C22⋊D6112yes—
427C4⋊C4C2×C22⋊D6112yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—2448no—
436C22⋊C4—2448no—
437C42—24no—
438C42—22no—
439C42—1224no—
440C42—2442no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—1224no—
447C23—4888no—
448C23—4882no—
449C23—4882no—
450C23—4882no—
451C23—4882no—
452C23—2444no—
453C23—2444no—
454C23—2444no—
455C23—24416no—
456C23—2444no—
457C23—2442no—
458C23—2442no—
459C23#1451974611yescharacteristic
460C23#1451974611yescharacteristic
461C23C22×C22⋊D6111yescharacteristic
462Q8—24no—
463Q8—82no—
464D4—24no—
465D4—42no—
466D4—42no—
467D4—44no—
468D4—44no—
469D4—82no—
470D4—1228no—
471D4—2448no—
472D4—2444no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—82no—
487C2×C4—84no—
488C2×C4—1224no—
489C2×C4—2448no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4#1451974612yes—
493C2×C4C22×C22⋊D6112yes—
494C2×C4#41×D6112yes—
495C2×C4#1451982312yes—
496C8—2444no—
497C22—24no—
498C22—24no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—4884no—
509C22—4888no—
510C22—4888no—
511C22—4882no—
512C22—4882no—
513C22—2448no—
514C22—2444no—
515C22#1683993012yes—
516C22#1683993512yes—
517C22C2×#1451974611yescharacteristic
518C22#1684026911yescharacteristic
519C22#1684032611yescharacteristic
520C4—22no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—2444no—
527C4#1684040014yes—
528C2—21no—
529C2—22no—
530C2—22no—
531C2—21no—
532C2—82no—
533C2—4884no—
534C2#2144913411yescharacteristic
535C2C2×#1683993011yescharacteristic
536C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3492C2
socleC22×C61C2×#120
hypercenter#3492D61
hypocenterC61#23948
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
1. #21455045C2semidirect
2. C2×#16839663C2semidirect
3. #21446972C2semidirect
4. #21446861C2semidirect
5. #21446263C2semidirect
6. #21446195C2semidirect
7. #21445899C2semidirect
8. C61×#3492C2semidirect
17. C2×#14519704C22semidirect
23. #16839531C22semidirect
24. #16839531C22semidirect
25. #16839521C22semidirect
30. C61×#544C22semidirect
32. C61×#537C22semidirect
35. C61×#520C22semidirect
142. C2×D244#41semidirect
260. #3492D61semidirect
283. #544C2×D61semidirect
285. #537C2×D61semidirect
288. #520C2×D61semidirect
305. C61#23948semidirect

Maximal subgroups

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

subgroupindex
#214550452
C2×#168396632
#214469722
#214468612
#214462632
#214461952
#214458992
C61×#34922
#2394861