Finite Group Databasebeta

← #29707109

Subgroups — #29707109

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
1C23×#11758200C211yesmaximal, characteristic, complemented
2C2×#16840333C211yesmaximal, characteristic, complemented
3C2×#16839772C211yesmaximal, characteristic, complemented
4C2×#16839758C211yesmaximal, characteristic, complemented
5#21448401C211yesmaximal, characteristic, complemented
6#21446900C214yesmaximal, complemented
7#21446894C214yesmaximal, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C22×#11758200—24no—
12C22×#11758200—24no—
13C22×#11758200—24no—
14C22×#11758200C2214yes—
15C23×#92418C2211yescharacteristic
16#16840333—24no—
17#16840333—24no—
18C2×#14519775C2212yescomplemented
19C2×#14519764C2212yescomplemented
20C2×#14519758—22no—
21C2×#14519756—22no—
22C2×#14519736C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24#16839772—24no—
25#16839758—24no—
26C2×#14519702C2214yescomplemented
27C2×#14519701C2212yescomplemented
28C2×#14519698C2212yescomplemented
29#16839665—22no—
30#16839665C2214yes—
31#16839618—22no—
32#16839516C2214yescomplemented
33C22×#41×C61C2211yescharacteristic
34C23×D4×C61C2211yescharacteristic
35C2×AGL(1,ℤ/8)×C61—24no—
36C2×AGL(1,ℤ/8)×C61—24no—
37C2×AGL(1,ℤ/8)×C61—24no—
38C22×SD16×C61C2212yes—
39C22×D8×C61C2212yes—
40C22×M16×C61C2211yescharacteristic
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—22no—
45C22×Dic122—21no—
46C22×Dic122—22no—
47C22×Dic122D414yes—
48C22×Dic122C2312yes—
49C23×Dic61—21no—
50C2×#11758200—232no—
51C2×#11758200—28no—
52C2×#11758200—28no—
53C2×#11758200—28no—
54C22×#92418—22no—
55C22×#92418—22no—
56C22×#92418D414yes—
57C22×#92418C2312yes—
58#14519775—24no—
59#14519775—24no—
60#14519764—24no—
61#14519764—24no—
62#14519758—216no—
63#14519756—216no—
64#14519736—22no—
65#14519736—22no—
66#14519736—22no—
67#14519736—22no—
68C2×C4⋊Dic61D412yes—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—24no—
71C2×C22⋊Dic61—24no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—28no—
74C2×C22⋊Dic61D412yes—
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2312yes—
77#14519702—28no—
78#14519701—24no—
79#14519698—24no—
80C25×C61—21no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61D414yes—
87C22×Q8×C61C2311yescharacteristic
88C22×D4×C61—21no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61—22no—
92C22×D4×C61—22no—
93C22×D4×C61—21no—
94C22×D4×C61—22no—
95C22×D4×C61D414yes—
96C22×D4×C61C2311yescharacteristic
97C22×D4×C61C2312yes—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100AGL(1,ℤ/8)×C61—232no—
101C2×SD16×C61—24no—
102C2×SD16×C61—24no—
103C2×SD16×C61—24no—
104C2×D8×C61—24no—
105C2×D8×C61—24no—
106C2×D8×C61—24no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61D414yes—
110C22×C8×C61C2312yes—
111C24×C61—21no—
112C24×C61—22no—
113C24×C61—24no—
114C24×C61—42no—
115C24×C61—42no—
116C24×C61—84no—
117C24×C61C2×D411yescharacteristic
118#41×C61—28no—
119#41×C61—28no—
120#41×C61—44no—
121#41×C61—44no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61C2×D412yes—
128C2×D4×C61—24no—
129C2×D4×C61—28no—
130C2×D4×C61—28no—
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61—42no—
143C2×D4×C61—42no—
144C2×D4×C61—42no—
145C2×D4×C61—44no—
146C2×D4×C61—44no—
147C2×D4×C61—42no—
148C2×D4×C61C2×D412yes—
149C2×D4×C61C2×D414yes—
150C22×C4×C61—21no—
151C22×C4×C61—24no—
152C22×C4×C61—42no—
153C22×C4×C61—42no—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61#4112yes—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—24no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61C2×D414yes—
168C2×Dic122—28no—
169C2×Dic122—28no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—24no—
175C2×Dic122—22no—
176C2×Dic122—44no—
177C2×Dic122—44no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122C2×D414yes—
181SD16×C61—216no—
182D8×C61—216no—
183M16×C61—28no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61C2×D414yes—
187C22⋊Dic61—216no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190C22⋊Dic61—88no—
191#11758200—232no—
192C4⋊Dic61—24no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195#11758196—216no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—42no—
199C2×#92418—28no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418C2×D414yes—
203C23×C61—24no—
204C23×C61—22no—
205C23×C61—24no—
206C23×C61—42no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—42no—
210C23×C61—48no—
211C23×C61—44no—
212C23×C61—48no—
213C23×C61—88no—
214C23×C61—88no—
215C23×C61—88no—
216C23×C61#12012yes—
217C23×C61#12112yes—
218C23×C61#12712yes—
219C23×C61C22×D411yescharacteristic
220Q8×C61—22no—
221Q8×C61—22no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—24no—
225D4×C61—22no—
226D4×C61—24no—
227D4×C61—22no—
228D4×C61—48no—
229D4×C61—41no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—41no—
233D4×C61—48no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—48no—
238C2×C4×C61#12014yes—
239C2×C4×C61#12014yes—
240C2×C4×C61#12012yes—
241C2×C4×C61#12114yes—
242C2×C4×C61#12114yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12712yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—28no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—48no—
257C2×Dic61—42no—
258C2×Dic61—44no—
259Dic122—24no—
260Dic122—24no—
261Dic122—42no—
262Dic122—42no—
263Dic122—48no—
264C8×C61—24no—
265#92418—24no—
266#23424—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—42no—
271C22×C61—44no—
272C22×C61—44no—
273C22×C61—48no—
274C22×C61—816no—
275C22×C61—82no—
276C22×C61—84no—
277C22×C61—84no—
278C22×C61—82no—
279C22×C61#39214yes—
280C22×C61C2×#12011yescharacteristic
281C22×C61C2×#12111yescharacteristic
282C22×C61C2×#12711yescharacteristic
283C22×C61#57714yes—
284C4×C61—42no—
285C4×C61#39214yes—
286C4×C61C2×#12011yescharacteristic
287C4×C61C2×#12011yescharacteristic
288C4×C61C2×#12111yescharacteristic
289C4×C61C2×#12111yescharacteristic
290Dic61—44no—
291Dic61—44no—
292Dic61—82no—
293C22×AGL(1,ℤ/8)D6111yescharacteristic
294C23×SD16—611no—
295C2×#536—611no—
296C2×#419—611no—
297C2×#416—611no—
298#2309—611no—
299#2013—614no—
300#2012—614no—
301#1729—611no—
302C2×C61—24no—
303C2×C61—82no—
304C2×C61—84no—
305C2×C61C2×#39211yescharacteristic
306C2×C61#301012yes—
307C2×C61#304212yes—
308C2×C61#315412yes—
309C22×#41C2×D6111yescharacteristic
310C23×Q8—611no—
311C23×D4C2×D6111yescharacteristic
312C2×AGL(1,ℤ/8)—24no—
313C2×AGL(1,ℤ/8)—24no—
314C2×AGL(1,ℤ/8)—24no—
315C22×SD16—614no—
316C22×SD16—1224no—
317C22×SD16—1224no—
318C22×SD16—1224no—
319C22×SD16C2×D6112yes—
320C22×D8C2×D6112yes—
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×#117—611no—
331#419—1224no—
332#416—1224no—
333C2×#103—612no—
334C2×#103—614no—
335C2×#102—612no—
336#390—614no—
337#390—1222no—
338#379—1222no—
339#336—614no—
340C61#2342411yescharacteristic, complemented, Sylow
341C25—21no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41C22⋊D6114yes—
348C22×Q8—614no—
349C22×Q8—612no—
350C22×Q8—1221no—
351C22×Q8—1222no—
352C22×Q8—1222no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8—1222no—
356C22×Q8C22×D6111yescharacteristic
357C22×D4—21no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—21no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—22no—
364C22×D4C22⋊D6114yes—
365C22×D4C22×D6111yescharacteristic
366C22×D4C22×D6112yes—
367C23×C4—21no—
368C23×C4—1221no—
369C23×C4C22×D6111yescharacteristic
370AGL(1,ℤ/8)—232no—
371C2×SD16—24no—
372C2×SD16—24no—
373C2×SD16—24no—
374C2×SD16—1228no—
375C2×SD16—1228no—
376C2×SD16—12232no—
377C2×SD16—1228no—
378C2×D8—24no—
379C2×D8—24no—
380C2×D8—24no—
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—1222no—
396#117—1222no—
397#117—1222no—
398#117—1222no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4—1224no—
402C2×C22⋊C4—612no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1228no—
405C2×C22⋊C4—1221no—
406C2×C42—612no—
407C2×C42—1221no—
408#103—1224no—
409#103—1228no—
410#102—1224no—
411C24—21no—
412C24—22no—
413C24—24no—
414C24—42no—
415C24—42no—
416C24—84no—
417C24D4×D6111yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—614no—
428C2×Q8—1222no—
429C2×Q8—1224no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—1222no—
434C2×Q8—1228no—
435C2×Q8—1228no—
436C2×Q8—2444no—
437C2×Q8—2442no—
438C2×Q8—2444no—
439C2×Q8—2442no—
440C2×Q8C2×C22⋊D6112yes—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—28no—
453C2×D4—28no—
454C2×D4—24no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4C2×C22⋊D6112yes—
462C2×D4C2×C22⋊D6114yes—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—614no—
468C22×C4—1222no—
469C22×C4—1224no—
470C22×C4—1222no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4D4×D6112yes—
474C22×C4D4×D6112yes—
475C22×C4D4×D6111yescharacteristic
476C22×C4#1175822412yes—
477C22×C4C2×C22⋊D6112yes—
478C22×C4C2×C22⋊D6112yes—
479C22×C4C2×C22⋊D6112yes—
480C22×C4C23×D6111yescharacteristic
481SD16—216no—
482SD16—12232no—
483D8—216no—
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—12216no—
493C4⋊C4—1224no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C22⋊C4—12216no—
497C22⋊C4—1224no—
498C22⋊C4—4888no—
499C22⋊C4—2442no—
500C42—1222no—
501C42—1222no—
502C42—2442no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—48no—
511C23—44no—
512C23—48no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23#1451974512yes—
517C23#1451974912yes—
518C23#1451977312yes—
519C23C2×D4×D6111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524Q8—1224no—
525Q8—1224no—
526Q8—2442no—
527Q8—2442no—
528Q8—2448no—
529D4—24no—
530D4—22no—
531D4—24no—
532D4—22no—
533D4—48no—
534D4—41no—
535D4—42no—
536D4—41no—
537D4—42no—
538D4—48no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1228no—
546C2×C4—2442no—
547C2×C4—2444no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2448no—
551C2×C4#1451974514yes—
552C2×C4#1451974512yes—
553C2×C4#1451974614yes—
554C2×C4#1451974814yes—
555C2×C4#1451974914yes—
556C2×C4#1451974912yes—
557C2×C4#1451977312yes—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C22×C22⋊D6111yescharacteristic
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C2×#1175822411yescharacteristic
564C8—24no—
565C8—1224no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—48no—
573C22—816no—
574C22—84no—
575C22—82no—
576C22—84no—
577C22—82no—
578C22#1683967014yes—
579C22C2×#1451974511yescharacteristic
580C22C2×#1451974911yescharacteristic
581C22C2×#1451977311yescharacteristic
582C22#1684057214yes—
583C4—42no—
584C4—4882no—
585C4—2444no—
586C4—2444no—
587C4#1683967014yes—
588C4C2×#1451974511yescharacteristic
589C4C2×#1451974611yescharacteristic
590C4C2×#1451974811yescharacteristic
591C4C2×#1451974911yescharacteristic
592C2—24no—
593C2—84no—
594C2—82no—
595C2C2×#1683967011yescharacteristic
596C2#2145510412yes—
597C2#2145533312yes—
598C2#2145629312yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC22×AGL(1,ℤ/8)×C61C2
socleC23×C61C22×D4
hypercenterC22×AGL(1,ℤ/8)D61
hypocenterC61#23424
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C23×#11758200C2semidirect
2. C2×#16840333C2semidirect
3. C2×#16839772C2semidirect
4. C2×#16839758C2semidirect
5. #21448401C2semidirect
6. #21446900C2semidirect
7. #21446894C2semidirect
8. #21445517C2semidirect
18. C2×#14519775C22semidirect
19. C2×#14519764C22semidirect
26. C2×#14519702C22semidirect
27. C2×#14519701C22semidirect
28. C2×#14519698C22semidirect
32. #16839516C22semidirect
340. C61#23424semidirect

Maximal subgroups

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

subgroupindex
C23×#117582002
C2×#168403332
C2×#168397722
C2×#168397582
#214484012
#214469002
#214468942
#214455172
C22×AGL(1,ℤ/8)×C612
#2342461