Finite Group Databasebeta

← #29707115

Subgroups — #29707115

Order 15616.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519805C211yesmaximal, characteristic
2C23×#11758201C211yesmaximal, characteristic, complemented
3C2×#16840336C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21446903C214yesmaximal, complemented
7#21446897C214yesmaximal, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C2×#14519805—24no—
13C2×#14519805—24no—
14C2×#14519805—24no—
15C22×#11758190C2212yes—
16C22×#11758201—24no—
17C22×#11758201—24no—
18C22×#11758201—24no—
19C22×#11758201C2214yes—
20C22×D488C2212yes—
21C23×#92418C2211yescharacteristic
22#16840336—24no—
23#16840336—24no—
24C2×#14519775C2212yescomplemented
25C2×#14519766C2212yescomplemented
26C2×#14519760—22no—
27C2×#14519753—22no—
28C2×#14519736C2211yescharacteristic
29#16839756—24no—
30C2×#14519700C2212yescomplemented
31C2×#14519697C2212yescomplemented
32#16839664—22no—
33#16839663C2214yes—
34#16839617—22no—
35#16839516C2214yescomplemented
36C23×Q8×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C61×#419—24no—
40C2×#103×C61C2214yescomplemented
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×D244D414yes—
56C22×D244C2311yescharacteristic
57C22×D244C2312yes—
58C22×C22⋊D61—21no—
59C22×C4×D61—21no—
60#14519805—232no—
61C2×#11758190—24no—
62C2×#11758190—24no—
63C2×#11758190—24no—
64C2×#11758201—232no—
65C2×#11758201—28no—
66C2×#11758201—28no—
67C2×#11758201—28no—
68C2×D488—24no—
69C2×D488—24no—
70C2×D488—24no—
71C22×#92418—22no—
72C22×#92418—22no—
73C22×#92418D414yes—
74C22×#92418C2312yes—
75#14519775—24no—
76#14519775—24no—
77#14519766—24no—
78#14519766—24no—
79#14519760—216no—
80#14519753—216no—
81#14519736—22no—
82#14519736—22no—
83C2×C4⋊Dic61D412yes—
84C2×#11758196—24no—
85C2×C22⋊Dic61D412yes—
86C2×#11758198—24no—
87C2×#11758198—28no—
88C2×C4×Dic61C2312yes—
89#14519700—24no—
90#14519697—24no—
91C22×Q8×C61—21no—
92C22×Q8×C61—22no—
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—
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—44no—
127C2×Q8×C61—42no—
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×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C22×Dic61—21no—
148C22×Dic61C2×D414yes—
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—22no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—24no—
164C2×D244—21no—
165C2×D244—22no—
166C2×D244—42no—
167C2×D244—44no—
168C2×D244—44no—
169C2×D244—42no—
170C2×D244C2×D414yes—
171C2×D244C2×D412yes—
172C2×C4×D61—24no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C2×D412yes—
181M16×C61—28no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61C2×D414yes—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C22⋊C4×C61—42no—
188C42×C61—42no—
189C22⋊Dic61—24no—
190#11758201—232no—
191#11758198—216no—
192#11758198—88no—
193C4⋊Dic61—24no—
194#11758196—216no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197C2×#92418—28no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418C2×D414yes—
201D488—216no—
202#11758190—216no—
203C23×C61—24no—
204C23×C61#12012yes—
205C23×C61#12112yes—
206C23×C61#12712yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—22no—
209C22×D61—24no—
210C22×D61—42no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—48no—
215C22×D61—44no—
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—42no—
227C2×C4×C61—42no—
228C2×C4×C61—48no—
229C2×C4×C61—42no—
230C2×C4×C61—44no—
231C2×C4×C61#12014yes—
232C2×C4×C61#12014yes—
233C2×C4×C61#12012yes—
234C2×C4×C61#12114yes—
235C2×C4×C61#12114yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#12712yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C22⋊D61—48no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—28no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250D244—24no—
251D244—22no—
252D244—24no—
253D244—22no—
254D244—41no—
255D244—42no—
256D244—42no—
257D244—41no—
258D244—48no—
259C4×D61—48no—
260Dic122—22no—
261Dic122—22no—
262Dic122—41no—
263Dic122—41no—
264C8×C61—24no—
265#92418—24no—
266#23424—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—24no—
270C22×C61#39214yes—
271C22×C61C2×#12011yescharacteristic
272C22×C61C2×#12111yescharacteristic
273C22×C61C2×#12711yescharacteristic
274C22×C61#57714yes—
275C2×D61—42no—
276C2×D61—44no—
277C2×D61—44no—
278C2×D61—48no—
279C2×D61—816no—
280C2×D61—82no—
281C2×D61—84no—
282C2×D61—84no—
283C2×D61—82no—
284C4×C61—44no—
285C4×C61—82no—
286C4×C61#39214yes—
287C4×C61C2×#12011yescharacteristic
288C4×C61C2×#12011yescharacteristic
289C4×C61C2×#12111yescharacteristic
290C4×C61C2×#12111yescharacteristic
291Dic61—42no—
292Dic61—44no—
293C22×AGL(1,ℤ/8)—611no—
294C23×SD16—611no—
295C2×#536—611no—
296C2×#419D6111yescharacteristic, complemented
297C2×#416—611no—
298#2309—611no—
299#2013—614no—
300#2012—614no—
301#1729—611no—
302C2×C61—24no—
303C2×C61C2×#39211yescharacteristic
304C2×C61#301012yes—
305C2×C61#304212yes—
306C2×C61#315412yes—
307D61—82no—
308D61—84no—
309C22×#41—611no—
310C23×Q8C2×D6111yescharacteristic
311C23×D4—611no—
312C2×AGL(1,ℤ/8)—1224no—
313C2×AGL(1,ℤ/8)—1224no—
314C2×AGL(1,ℤ/8)—1224no—
315C22×SD16—614no—
316C22×SD16—612no—
317C22×SD16—1224no—
318C22×SD16—1224no—
319C22×SD16—1224no—
320C22×D8—612no—
321C22×M16C2×D6111yescharacteristic
322C23×C8—611no—
323#536—1224no—
324#536—1224no—
325C2×C4⋊Q8—612no—
326C2×#124—612no—
327C2×#123—1222no—
328C2×C22⋊Q8—1222no—
329C2×#117—611no—
330C2×#117C2×D6111yescharacteristic
331#419—24no—
332#416—1224no—
333C2×#103—612no—
334C2×#103C2×D6114yescomplemented
335C2×#102—612no—
336#390—614no—
337#390—1222no—
338#379—1222no—
339#336—614no—
340C61#2342411yescharacteristic, complemented, Sylow
341C25—1221no—
342C2×#41—614no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C22×Q8—21no—
349C22×Q8—22no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8—611no—
355C22×Q8D24414yes—
356C22×Q8C22×D6112yes—
357C22×D4—614no—
358C22×D4—611no—
359C22×D4—612no—
360C22×D4—1221no—
361C22×D4—1222no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—1221no—
366C22×D4—1222no—
367C23×C4—21no—
368C23×C4—1221no—
369C23×C4C22×D6111yescharacteristic
370AGL(1,ℤ/8)—12232no—
371C2×SD16—1224no—
372C2×SD16—1224no—
373C2×SD16—1228no—
374C2×SD16—1228no—
375C2×SD16—12232no—
376C2×SD16—1228no—
377C2×SD16—1224no—
378C2×D8—1224no—
379C2×D8—1224no—
380C2×D8—1224no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16C22⋊D6114yes—
384C22×C8—614no—
385C22×C8—612no—
386C22×C8—1222no—
387C22×C8—1222no—
388C22×C8C22×D6112yes—
389C4⋊Q8—1224no—
390C4⋊Q8—1224no—
391#124—1224no—
392#124—1224no—
393#123—12216no—
394C22⋊Q8—12216no—
395#117—22no—
396#117—22no—
397#117—1222no—
398#117—1222no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—1224no—
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—612no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1228no—
406C2×C42—21no—
407C2×C42—612no—
408#103—28no—
409#103—1224no—
410#102—1224no—
411C24—1224no—
412C24—1221no—
413C24—1222no—
414C24—4884no—
415C24—2442no—
416C24—2442no—
417C24D4×D6111yescharacteristic
418#41—1228no—
419#41—1228no—
420#41—2444no—
421#41—2444no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—28no—
429C2×Q8—28no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—612no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8C2×D24414yes—
441C2×D4—612no—
442C2×D4—614no—
443C2×D4—1224no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1222no—
454C2×D4—1228no—
455C2×D4—1228no—
456C2×D4—1224no—
457C2×D4—2442no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4—2442no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—614no—
469C22×C4—1221no—
470C22×C4—1224no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4C2×D24412yes—
474C22×C4#1175822212yes—
475C22×C4D4×D6112yes—
476C22×C4D4×D6112yes—
477C22×C4D4×D6112yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4C2×C22⋊D6112yes—
480C22×C4C23×D6111yescharacteristic
481SD16—12216no—
482SD16—12232no—
483D8—12216no—
484M16—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—614no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1228no—
491C2×C8C2×C22⋊D6114yes—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—12216no—
495C4⋊C4—1224no—
496C22⋊C4—42no—
497C22⋊C4—12216no—
498C22⋊C4—1224no—
499C22⋊C4—4888no—
500C42—42no—
501C42—1222no—
502C42—1222no—
503C23—24no—
504C23—1222no—
505C23—1224no—
506C23—4888no—
507C23—4888no—
508C23—4888no—
509C23—2442no—
510C23—2441no—
511C23—2441no—
512C23—2442no—
513C23—2448no—
514C23—2444no—
515C23—2448no—
516C23#1451974712yes—
517C23#1451975012yes—
518C23#1451977312yes—
519C23C2×D4×D6111yescharacteristic
520Q8—24no—
521Q8—24no—
522Q8—42no—
523Q8—42no—
524Q8—48no—
525Q8—1222no—
526Q8—1222no—
527Q8—2441no—
528Q8—2441no—
529D4—1222no—
530D4—1224no—
531D4—1224no—
532D4—1222no—
533D4—2448no—
534D4—2441no—
535D4—2442no—
536D4—2442no—
537D4—2441no—
538D4—2448no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1228no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2448no—
551C2×C4#1451974514yes—
552C2×C4#1451974714yes—
553C2×C4#1451974712yes—
554C2×C4#1451975014yes—
555C2×C4#1451975012yes—
556C2×C4#1451975114yes—
557C2×C4#1451977312yes—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4C22×D24411yescharacteristic
563C2×C4C2×#1175822211yescharacteristic
564C8—24no—
565C8—1224no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—48816no—
570C22—4884no—
571C22—4884no—
572C22—4882no—
573C22—4882no—
574C22—2444no—
575C22—2442no—
576C22—2444no—
577C22—2448no—
578C22#1683967114yes—
579C22C2×#1451974711yescharacteristic
580C22C2×#1451975011yescharacteristic
581C22C2×#1451977311yescharacteristic
582C22#1684057114yes—
583C4—44no—
584C4—82no—
585C4—2444no—
586C4—2442no—
587C4#1683967114yes—
588C4C2×#1451974511yescharacteristic
589C4C2×#1451974711yescharacteristic
590C4C2×#1451975011yescharacteristic
591C4C2×#1451975111yescharacteristic
592C2—24no—
593C2—4884no—
594C2—4882no—
595C2C2×#1683967111yescharacteristic
596C2#2145510612yes—
597C2#2145533012yes—
598C2#2145629812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC22×#14519805C2
hypercenterC2×#419D61
hypocenterC61#23424
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
2. C23×#11758201C2semidirect
3. C2×#16840336C2semidirect
4. C2×#16839756C2semidirect
5. #21448399C2semidirect
6. #21446903C2semidirect
7. #21446897C2semidirect
8. #21445517C2semidirect
9. C2×C61×#419C2semidirect
24. C2×#14519775C22semidirect
25. C2×#14519766C22semidirect
30. C2×#14519700C22semidirect
31. C2×#14519697C22semidirect
35. #16839516C22semidirect
40. C2×#103×C61C22semidirect
296. C2×#419D61semidirect
334. C2×#103C2×D61semidirect
340. C61#23424semidirect

Maximal subgroups

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

subgroupindex
C22×#145198052
C23×#117582012
C2×#168403362
C2×#168397562
#214483992
#214469032
#214468972
#214455172
C2×C61×#4192
#2342461