Finite Group Databasebeta

← #29707185

Subgroups — #29707185

Order 15616.

2018 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic
2C22×#14519808C211yesmaximal, characteristic
3C22×#14519775C211yesmaximal, characteristic, complemented
4C2×#16839770C211yesmaximal, characteristic, complemented
5#21448307C211yesmaximal, characteristic
6#21446920C214yesmaximal, complemented
7#21446903C214yesmaximal, complemented
8#21445512C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×Dic122C2211yescharacteristic
12C2×#14519810—24no—
13C2×#14519810—24no—
14C2×#14519810—24no—
15C2×#14519808—24no—
16C2×#14519808—24no—
17C2×#14519808—24no—
18C22×Dic244C2212yes—
19C22×#11758202C2212yes—
20C22×#11758190C2212yes—
21C22×#11758201C2212yes—
22C22×#11758194C2211yescharacteristic
23C2×#14519775—22no—
24C2×#14519775—24no—
25C2×#14519775—24no—
26C2×#14519775C2214yes—
27C22×#11758196—22no—
28C22×C4×Dic61C2211yescharacteristic
29#16839770—24no—
30C2×#14519700C2214yes—
31#16839667C2212yes—
32#16839663C2212yes—
33#16839639—22no—
34C4×#11758198—22no—
35#16839516C2214yes—
36C23×Q8×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#419—24no—
40C2×#103×C61C2214yes—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222D414yes—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122—22no—
50C22×Dic122—22no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122D414yes—
54C22×Dic122C2312yes—
55C22×Dic122C2311yescharacteristic
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C23×Dic61—21no—
59C23×Dic61D412yes—
60C22×C4×D61—21no—
61#14519810—232no—
62#14519808—232no—
63C2×Dic244—24no—
64C2×Dic244—24no—
65C2×Dic244—24no—
66C2×#11758202—24no—
67C2×#11758202—24no—
68C2×#11758202—24no—
69C2×#11758190—24no—
70C2×#11758190—24no—
71C2×#11758190—24no—
72C2×#11758201—24no—
73C2×#11758201—24no—
74C2×#11758201—24no—
75C2×#11758194—22no—
76C2×#11758194—22no—
77C2×#11758194D414yes—
78C22×#92418C2312yes—
79#14519775—216no—
80#14519775—28no—
81#14519775—28no—
82C2×#11758196—216no—
83C2×#11758196—28no—
84C2×#11758198—24no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61D412yes—
88C2×C4×Dic61C2312yes—
89#14519700—28no—
90#14519684—24no—
91C22×Q8×C61—22no—
92C22×Q8×C61—21no—
93C22×Q8×C61—22no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61—22no—
97C22×Q8×C61D414yes—
98C22×Q8×C61C2312yes—
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—22no—
102C2×M16×C61—22no—
103C2×M16×C61D414yes—
104C22×C8×C61C2312yes—
105#117×C61—22no—
106#117×C61—22no—
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110#103×C61—28no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C2×Q8×C61—24no—
114C2×Q8×C61—22no—
115C2×Q8×C61—28no—
116C2×Q8×C61—28no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—44no—
122C2×Q8×C61—42no—
123C2×Q8×C61—44no—
124C2×Q8×C61—42no—
125C2×Q8×C61C2×D414yes—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C22×Dic61—24no—
143C22×Dic61—28no—
144C22×Dic61—24no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61C2×D414yes—
150#11758222—28no—
151#11758222—28no—
152#11758222—44no—
153#11758222—44no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C2×D412yes—
160C2×C4×D61—24no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×Dic122—28no—
164C2×Dic122—28no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—24no—
174C2×Dic122—22no—
175C2×Dic122—21no—
176C2×Dic122—44no—
177C2×Dic122—44no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122C2×D414yes—
181C2×Dic122C2×D412yes—
182M16×C61—28no—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C2×C8×C61C2×D414yes—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C22⋊C4×C61—42no—
189C42×C61—42no—
190#11758202—216no—
191#11758201—216no—
192#11758198—48no—
193#11758196—232no—
194C4×Dic61—24no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197#11758194—28no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418C2×D414yes—
201Dic244—216no—
202#11758190—216no—
203C23×C61—24no—
204C23×C61#12012yes—
205C23×C61#12112yes—
206C23×C61#12712yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—22no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—44no—
212Q8×C61—24no—
213Q8×C61—24no—
214Q8×C61—42no—
215Q8×C61—42no—
216Q8×C61—48no—
217C2×C4×C61—24no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—48no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61#12014yes—
224C2×C4×C61#12014yes—
225C2×C4×C61#12012yes—
226C2×C4×C61#12114yes—
227C2×C4×C61#12114yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12712yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—48no—
237C2×Dic61—216no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—48no—
247C2×Dic61#13718yescomplemented
248D244—22no—
249D244—22no—
250D244—41no—
251D244—41no—
252C4×D61—48no—
253Dic122—24no—
254Dic122—22no—
255Dic122—22no—
256Dic122—24no—
257Dic122—42no—
258Dic122—42no—
259Dic122—41no—
260Dic122—41no—
261Dic122—48no—
262C8×C61—24no—
263#92418—24no—
264#23432—611nomaximal, Sylow
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61#39214yes—
269C22×C61C2×#12011yescharacteristic
270C22×C61C2×#12111yescharacteristic
271C22×C61C2×#12711yescharacteristic
272C22×C61C2×#13714yes—
273C2×D61—42no—
274C2×D61—44no—
275C2×D61—82no—
276C2×D61—82no—
277C4×C61—44no—
278C4×C61—82no—
279C4×C61#39214yes—
280C4×C61C2×#12011yescharacteristic
281C4×C61C2×#12011yescharacteristic
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12111yescharacteristic
284Dic61—28no—
285Dic61—44no—
286Dic61—42no—
287C22×#137—611no—
288C22×#137—611no—
289C22×C4⋊Q8—611no—
290C2×#419—611no—
291C2×#419D6111yescharacteristic, complemented
292#2297—611no—
293#2016—614no—
294#2013—614no—
295#1728—611no—
296C2×C61—24no—
297C2×C61C2×#39211yescharacteristic
298C2×C61#300412yes—
299C2×C61#303912yes—
300C2×C61#315212yes—
301D61—82no—
302C22×#41—611no—
303C23×Q8—611no—
304C23×Q8C2×D6111yescharacteristic
305C2×#137—1224no—
306C2×#137—1224no—
307C2×#137—1224no—
308C2×#137—1224no—
309C2×#137—1224no—
310C2×#137—1224no—
311C22×Q16—612no—
312C22×Q16—612no—
313C22×SD16—612no—
314C22×SD16—612no—
315C22×M16—611no—
316C22×M16C2×D6111yescharacteristic
317C2×C4⋊Q8—614no—
318C2×C4⋊Q8—1222no—
319C2×C4⋊Q8—1224no—
320C2×C4⋊Q8—1224no—
321C2×#117C2×D6111yescharacteristic
322C22×C4⋊C4—1222no—
323C22×C42—611no—
324#419—24no—
325#419—1224no—
326C2×#103—614no—
327C2×#103C2×D6114yes—
328#391—612no—
329#390—612no—
330#386—1222no—
331C4×C22⋊C4—1222no—
332#336—614no—
333C61#2343211yescharacteristic, complemented, Sylow
334C2×#41—614no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C22×Q8—21no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—614no—
347C22×Q8—612no—
348C22×Q8—611no—
349C22×Q8—1222no—
350C22×Q8—1221no—
351C22×Q8—1222no—
352C22×Q8—1222no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8D24414yes—
356C22×Q8C22×D6112yes—
357C22×D4—611no—
358C22×D4—1221no—
359C23×C4—21no—
360C23×C4—612no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4C22×D6111yescharacteristic
364#137—12232no—
365#137—12232no—
366C2×Q16—1224no—
367C2×Q16—1224no—
368C2×Q16—1224no—
369C2×Q16—1224no—
370C2×Q16—1224no—
371C2×Q16—1224no—
372C2×SD16—1224no—
373C2×SD16—1224no—
374C2×SD16—1224no—
375C2×SD16—1224no—
376C2×SD16—1224no—
377C2×SD16—1224no—
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—614no—
381C2×M16—1222no—
382C2×M16—1222no—
383C2×M16C22⋊D6114yes—
384C22×C8—612no—
385C22×C8C22×D6112yes—
386C4⋊Q8—1228no—
387C4⋊Q8—12216no—
388C4⋊Q8—1228no—
389#117—22no—
390#117—22no—
391C2×C4⋊C4—1228no—
392C2×C4⋊C4—12216no—
393C2×C4⋊C4C22×D6112yes—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—1224no—
396C2×C42—21no—
397C2×C42—612no—
398C2×C42—612no—
399C2×C42—1222no—
400C2×C42—1222no—
401#103—28no—
402#103—1228no—
403#95—1224no—
404C24—1221no—
405C24D4×D6111yescharacteristic
406#41—1228no—
407#41—1228no—
408#41—2444no—
409#41—2444no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—614no—
423C2×Q8—612no—
424C2×Q8—1222no—
425C2×Q8—1221no—
426C2×Q8—1224no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1222no—
435C2×Q8—1228no—
436C2×Q8—1228no—
437C2×Q8—2444no—
438C2×Q8—2442no—
439C2×Q8—2444no—
440C2×Q8—2442no—
441C2×Q8C2×D24414yes—
442C2×D4—612no—
443C2×D4—1224no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—2442no—
450C2×D4—2442no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—614no—
457C22×C4—1228no—
458C22×C4—1224no—
459C22×C4—1221no—
460C22×C4—1222no—
461C22×C4—1224no—
462C22×C4—1224no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4C2×D24412yes—
468C22×C4#1175822212yes—
469C22×C4D4×D6112yes—
470C22×C4D4×D6112yes—
471C22×C4D4×D6112yes—
472C22×C4D4×D6111yescharacteristic
473C22×C4C2×C22⋊D6112yes—
474C22×C4C23×D6111yescharacteristic
475Q16—12216no—
476Q16—12216no—
477SD16—12216no—
478SD16—12216no—
479M16—28no—
480M16—1228no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—614no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8C2×C22⋊D6114yes—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—12232no—
490C22⋊C4—42no—
491C22⋊C4—2448no—
492C42—42no—
493C42—1222no—
494C42—1222no—
495C42—1224no—
496C23—24no—
497C23—1222no—
498C23—2441no—
499C23—2444no—
500C23—2441no—
501C23#1451974712yes—
502C23#1451975012yes—
503C23#1451977312yes—
504C23C2×D4×D6111yescharacteristic
505Q8—24no—
506Q8—24no—
507Q8—42no—
508Q8—42no—
509Q8—48no—
510Q8—1224no—
511Q8—1222no—
512Q8—1222no—
513Q8—1224no—
514Q8—2442no—
515Q8—2441no—
516Q8—2442no—
517Q8—2441no—
518Q8—2448no—
519D4—1222no—
520D4—1222no—
521D4—2448no—
522D4—2441no—
523D4—2441no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—618no—
531C2×C4—12216no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1222no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4—2442no—
540C2×C4—2448no—
541C2×C4—2448no—
542C2×C4#1451974514yes—
543C2×C4#1451974714yes—
544C2×C4#1451974712yes—
545C2×C4#1451975014yes—
546C2×C4#1451975012yes—
547C2×C4#1451975114yes—
548C2×C4#1451977312yes—
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4C22×D24411yescharacteristic
554C2×C4C2×#1175822211yescharacteristic
555C8—24no—
556C8—1224no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—4882no—
561C22—4882no—
562C22—2442no—
563C22—2444no—
564C22#1683967114yes—
565C22C2×#1451974711yescharacteristic
566C22C2×#1451975011yescharacteristic
567C22C2×#1451977311yescharacteristic
568C22#137×D6114yes—
569C4—44no—
570C4—82no—
571C4—1228no—
572C4—2442no—
573C4—2444no—
574C4#1683967114yes—
575C4C2×#1451974511yescharacteristic
576C4C2×#1451974711yescharacteristic
577C4C2×#1451975011yescharacteristic
578C4C2×#1451975111yescharacteristic
579C2—24no—
580C2—4882no—
581C2C2×#1683967111yescharacteristic
582C2#2145505512yes—
583C2#2145530512yes—
584C2#2145627412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#419D61
hypocenterC61#23432
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C22×#14519775C2semidirect
4. C2×#16839770C2semidirect
6. #21446920C2semidirect
7. #21446903C2semidirect
8. #21445512C2semidirect
9. C2×C61×#419C2semidirect
247. C2×Dic61#137semidirect
291. C2×#419D61semidirect
333. C61#23432semidirect

Maximal subgroups

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

subgroupindex
C22×#145198102
C22×#145198082
C22×#145197752
C2×#168397702
#214483072
#214469202
#214469032
#214455122
C2×C61×#4192
#2343261