Finite Group Databasebeta

← #29707103

Subgroups — #29707103

Order 15616.

2130 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic, complemented
2C23×#11758201C211yesmaximal, characteristic, complemented
3C2×#16840331C211yesmaximal, characteristic
4C2×#16839766C211yesmaximal, characteristic, complemented
5C2×#16839756C211yesmaximal, characteristic, complemented
6#21446912C214yesmaximal, complemented
7#21446906C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C61×#2310C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519808—24no—
13C2×#14519808—24no—
14C2×#14519808—24no—
15C22×#11758202C2212yescomplemented
16C22×#11758201—24no—
17C22×#11758201—24no—
18C22×#11758201—24no—
19C22×#11758201C2214yes—
20C22×#11758201C2212yescomplemented
21C22×#11758194C2211yescharacteristic
22C23×#92418C2211yescharacteristic
23#16840331—24no—
24#16840331—24no—
25C2×#14519774C2212yes—
26C2×#14519765C2212yes—
27C2×#14519752—22no—
28C2×#14519747—22no—
29#16839766—24no—
30#16839756—24no—
31C2×#14519700C2212yescomplemented
32C2×#14519697C2214yes—
33C2×#14519697C2212yescomplemented
34#16839517C2214yescomplemented
35C23×Q8×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#391—22no—
39C61×#391C2214yescomplemented
40C61×#381—22no—
41C24×D61—21no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222D414yes—
48C22×Dic122C2311yescharacteristic
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244—22no—
53C22×D244—21no—
54C22×D244—22no—
55C22×D244—24no—
56C22×D244D414yes—
57C22×D244C2312yes—
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61#14519808—232no—
62C2×#11758202—24no—
63C2×#11758202—24no—
64C2×#11758202—24no—
65C2×#11758201—232no—
66C2×#11758201—28no—
67C2×#11758201—24no—
68C2×#11758201—28no—
69C2×#11758201—24no—
70C2×#11758201—28no—
71C2×#11758201—24no—
72C2×#11758194—22no—
73C2×#11758194—22no—
74C2×#11758194D414yescomplemented
75C22×#92418—22no—
76C22×#92418—22no—
77C22×#92418D414yescomplemented
78C22×#92418C2312yes—
79C22×#92418C2312yes—
80#14519774—24no—
81#14519774—24no—
82#14519765—24no—
83#14519765—24no—
84#14519752—216no—
85#14519747—216no—
86C2×#11758198—24no—
87#14519700—24no—
88#14519697—28no—
89#14519697—24no—
90C22×Q8×C61—21no—
91C22×Q8×C61—22no—
92C22×Q8×C61—22no—
93C22×Q8×C61—22no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61D414yescomplemented
97C22×Q8×C61C2312yes—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100#117×C61—22no—
101#117×C61—22no—
102#117×C61—22no—
103#117×C61—22no—
104C2×C4⋊C4×C61D412yes—
105C2×C4⋊C4×C61C2312yes—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61D412yes—
108C2×C42×C61—21no—
109C2×C42×C61C2312yes—
110#95×C61—28no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C23×D61—22no—
114C23×D61—24no—
115C23×D61—42no—
116C23×D61—42no—
117C23×D61—84no—
118C2×Q8×C61—24no—
119C2×Q8×C61—22no—
120C2×Q8×C61—28no—
121C2×Q8×C61—28no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—42no—
127C2×Q8×C61—44no—
128C2×Q8×C61—44no—
129C2×Q8×C61—42no—
130C2×Q8×C61C2×D414yes—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C2×D414yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61#4112yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—24no—
146C2×C22⋊D61—42no—
147C2×C22⋊D61—42no—
148C22×Dic61—21no—
149#11758222—28no—
150#11758222—28no—
151#11758222—44no—
152#11758222—44no—
153C2×D244—28no—
154C2×D244—28no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244—24no—
164C2×D244—22no—
165C2×D244—21no—
166C2×D244—216no—
167C2×D244—44no—
168C2×D244—44no—
169C2×D244—42no—
170C2×D244—42no—
171C2×D244—88no—
172C2×D244C2×D414yes—
173C2×D244C2×D412yes—
174C2×C4×D61—24no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122C2×D412yes—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C22⋊C4×C61—24no—
187C22⋊C4×C61—42no—
188C42×C61—22no—
189C42×C61—22no—
190C42×C61—42no—
191#11758202—216no—
192#11758201—216no—
193#11758201—232no—
194#11758198—216no—
195#11758194—28no—
196C2×#92418—28no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418C2×D414yes—
202C2×#92418C2×D414yes—
203C23×C61—24no—
204C23×C61#12012yes—
205C23×C61#12112yes—
206C23×C61#12712yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—24no—
209C22×D61—22no—
210C22×D61—41no—
211C22×D61—42no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—44no—
215C22×D61—48no—
216C22×D61—48no—
217C22×D61—88no—
218C22×D61—88no—
219C22×D61—88no—
220Q8×C61—24no—
221Q8×C61—24no—
222Q8×C61—42no—
223Q8×C61—42no—
224Q8×C61—48no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—28no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—48no—
231C2×C4×C61—42no—
232C2×C4×C61—44no—
233C2×C4×C61#12014yes—
234C2×C4×C61#12014yes—
235C2×C4×C61#12012yes—
236C2×C4×C61#12114yes—
237C2×C4×C61#12114yes—
238C2×C4×C61#12112yes—
239C2×C4×C61#12712yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C22⋊D61—48no—
247C2×Dic61—22no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250D244—24no—
251D244—22no—
252D244—24no—
253D244—22no—
254D244—42no—
255D244—41no—
256D244—42no—
257D244—41no—
258D244—48no—
259D244—816no—
260C4×D61—48no—
261Dic122—22no—
262Dic122—22no—
263Dic122—41no—
264Dic122—41no—
265#92418—24no—
266#92418—24no—
267#23423—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—24no—
271C22×C61#39214yes—
272C22×C61C2×#12011yescharacteristic
273C22×C61C2×#12111yescharacteristic
274C22×C61C2×#12711yescharacteristic
275C22×C61#57714yes—
276C2×D61—42no—
277C2×D61—44no—
278C2×D61—44no—
279C2×D61—48no—
280C2×D61—816no—
281C2×D61—82no—
282C2×D61—84no—
283C2×D61—82no—
284C2×D61—84no—
285C4×C61—44no—
286C4×C61—44no—
287C4×C61—82no—
288C4×C61#39214yes—
289C4×C61C2×#12011yescharacteristic
290C4×C61C2×#12011yescharacteristic
291C4×C61C2×#12111yescharacteristic
292C4×C61C2×#12111yescharacteristic
293Dic61—42no—
294C22×#137—611no—
295C23×SD16—611no—
296C2×#535—611no—
297C2×#418—611no—
298C2×#416—611no—
299#2310D6111yescharacteristic, complemented
300#2015—614no—
301#2014—614no—
302#1729—611no—
303C2×C61—24no—
304C2×C61C2×#39211yescharacteristic
305C2×C61#300912yes—
306C2×C61#304312yes—
307C2×C61#315412yes—
308D61—82no—
309D61—84no—
310C22×#41—611no—
311C23×Q8C2×D6111yescharacteristic
312C23×D4—611no—
313C2×#137—1224no—
314C2×#137—1224no—
315C2×#137—1224no—
316C22×Q16—612no—
317C22×SD16—614no—
318C22×SD16—612no—
319C22×SD16—1224no—
320C22×SD16—1224no—
321C22×SD16—1224no—
322C22×M16—611no—
323C23×C8—611no—
324#535—1224no—
325#535—1224no—
326C2×#127—612no—
327C2×#124—612no—
328C2×#121—1222no—
329C2×#120—1222no—
330C2×#117C2×D6111yescharacteristic
331C2×#117C2×D6111yescharacteristic
332#418—1224no—
333#416—1224no—
334C2×#103—612no—
335C2×#102—614no—
336C2×#102—612no—
337#391—22no—
338#391C2×D6114yescomplemented
339#381—22no—
340#336—614no—
341C61#2342311yescharacteristic, complemented, Sylow
342C25—1221no—
343C2×#41—614no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C2×#41—1222no—
349C22×Q8—22no—
350C22×Q8—21no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8—22no—
355C22×Q8—611no—
356C22×Q8D24414yescomplemented
357C22×Q8C22×D6112yes—
358C22×D4—614no—
359C22×D4—612no—
360C22×D4—611no—
361C22×D4—1221no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—1222no—
366C22×D4—1222no—
367C22×D4—1224no—
368C22×D4—1221no—
369C23×C4—21no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372#137—12232no—
373C2×Q16—1224no—
374C2×Q16—1224no—
375C2×Q16—1224no—
376C2×SD16—1228no—
377C2×SD16—1224no—
378C2×SD16—1228no—
379C2×SD16—1224no—
380C2×SD16—12232no—
381C2×SD16—1228no—
382C2×SD16—1224no—
383C2×M16—614no—
384C2×M16—1222no—
385C2×M16—1222no—
386C22×C8—614no—
387C22×C8—612no—
388C22×C8—612no—
389C22×C8—1222no—
390C22×C8—1222no—
391#127—1224no—
392#127—1224no—
393#124—1224no—
394#124—1224no—
395#121—12216no—
396#120—12216no—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—22no—
401C2×C4⋊C4C22⋊D6112yes—
402C2×C4⋊C4C22×D6112yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4C22⋊D6112yes—
406C2×C42—21no—
407C2×C42C22×D6112yes—
408#103—1224no—
409#102—1224no—
410#102—1228no—
411#95—28no—
412C24—1224no—
413C24—1222no—
414C24—1221no—
415C24—4884no—
416C24—2442no—
417C24—2442no—
418C24C2×C22⋊D6111yescharacteristic
419#41—1228no—
420#41—1228no—
421#41—2444no—
422#41—2444no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—612no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8C2×D24414yes—
442C2×D4—614no—
443C2×D4—612no—
444C2×D4—12216no—
445C2×D4—1224no—
446C2×D4—1222no—
447C2×D4—1221no—
448C2×D4—1222no—
449C2×D4—1221no—
450C2×D4—1222no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—1221no—
454C2×D4—1222no—
455C2×D4—1221no—
456C2×D4—1224no—
457C2×D4—1228no—
458C2×D4—1228no—
459C2×D4—4888no—
460C2×D4—2442no—
461C2×D4—2442no—
462C2×D4—2444no—
463C2×D4—2444no—
464C2×D4—2442no—
465C2×D4—2442no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—1221no—
472C22×C4—1224no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4C2×D24412yes—
476C22×C4#1175822212yes—
477C22×C4D4×D6112yes—
478C22×C4D4×D6112yes—
479C22×C4D4×D6112yes—
480C22×C4D4×D6112yes—
481C22×C4C2×C22⋊D6114yes—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484Q16—12216no—
485SD16—12232no—
486SD16—12216no—
487M16—1228no—
488C2×C8—614no—
489C2×C8—614no—
490C2×C8—1222no—
491C2×C8—1222no—
492C2×C8—1222no—
493C2×C8—1222no—
494C2×C8—1228no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—12216no—
501C42—22no—
502C42—22no—
503C42—42no—
504C23—24no—
505C23—1224no—
506C23—1222no—
507C23—4888no—
508C23—4888no—
509C23—4888no—
510C23—2441no—
511C23—2442no—
512C23—2441no—
513C23—2442no—
514C23—2444no—
515C23—2448no—
516C23—2448no—
517C23#1451974512yes—
518C23#1451975112yes—
519C23#1451977312yes—
520C23C22×C22⋊D6111yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—42no—
524Q8—42no—
525Q8—48no—
526Q8—1222no—
527Q8—1222no—
528Q8—2441no—
529Q8—2441no—
530D4—1224no—
531D4—1222no—
532D4—1224no—
533D4—1222no—
534D4—48816no—
535D4—2448no—
536D4—2442no—
537D4—2441no—
538D4—2442no—
539D4—2441no—
540D4—2448no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—1222no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2448no—
553C2×C4#1451974512yes—
554C2×C4#1451974714yes—
555C2×C4#1451974714yes—
556C2×C4#1451975014yes—
557C2×C4#1451975014yes—
558C2×C4#1451975112yes—
559C2×C4#1451977312yes—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×D24411yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C8—1224no—
567C8—1224no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—48816no—
572C22—4884no—
573C22—4882no—
574C22—4884no—
575C22—4882no—
576C22—2442no—
577C22—2444no—
578C22—2444no—
579C22—2448no—
580C22#1683967114yes—
581C22C2×#1451974511yescharacteristic
582C22C2×#1451975111yescharacteristic
583C22C2×#1451977311yescharacteristic
584C22#1684057014yes—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—2442no—
589C4#1683967114yes—
590C4C2×#1451974711yescharacteristic
591C4C2×#1451974711yescharacteristic
592C4C2×#1451975011yescharacteristic
593C4C2×#1451975011yescharacteristic
594C2—24no—
595C2—4884no—
596C2—4882no—
597C2C2×#1683967111yescharacteristic
598C2#2145509412yes—
599C2#2145534012yes—
600C2#2145629712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2310C2
socleC23×C61C22×D4
prime radicalC2×#16840331C2
hypercenter#2310D61
hypocenterC61#23423
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
1. C22×#14519808C2semidirect
2. C23×#11758201C2semidirect
4. C2×#16839766C2semidirect
5. C2×#16839756C2semidirect
6. #21446912C2semidirect
7. #21446906C2semidirect
8. #21445516C2semidirect
9. C61×#2310C2semidirect
15. C22×#11758202C22semidirect
20. C22×#11758201C22semidirect
31. C2×#14519700C22semidirect
33. C2×#14519697C22semidirect
34. #16839517C22semidirect
39. C61×#391C22semidirect
74. C2×#11758194D4semidirect
77. C22×#92418D4semidirect
96. C22×Q8×C61D4semidirect
299. #2310D61semidirect
338. #391C2×D61semidirect
341. C61#23423semidirect
356. C22×Q8D244semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C23×#117582012
C2×#168403312
C2×#168397662
C2×#168397562
#214469122
#214469062
#214455162
C61×#23102
#2342361