Finite Group Databasebeta

← #29707113

Subgroups — #29707113

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×#14519802C211yesmaximal, characteristic
2C23×#11758201C211yesmaximal, characteristic, complemented
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839758C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21446902C214yesmaximal, complemented
7#21446896C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C2×C61×#536C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519802—24no—
12C2×#14519802—24no—
13C2×#14519802—24no—
14C22×#11758201—24no—
15C22×#11758201—24no—
16C22×#11758201—24no—
17C22×#11758201C2214yes—
18C22×#11758201C2212yes—
19C22×#11758199C2212yes—
20C22×#11758194C2211yescharacteristic
21C23×#92418C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839771—24no—
24#16839758—24no—
25C2×#14519701C2212yescomplemented
26C2×#14519701C2214yescomplemented
27C2×#14519698C2212yescomplemented
28#16839664C2214yes—
29#16839663—22no—
30#16839617—22no—
31#16839517C2214yescomplemented
32C22×#41×C61C2211yescharacteristic
33C23×Q8×C61C2211yescharacteristic
34C61×#536—24no—
35C61×#536—24no—
36C2×C4⋊Q8×C61C2212yescomplemented
37C2×#124×C61C2212yescomplemented
38C2×#123×C61—22no—
39C2×C22⋊Q8×C61—22no—
40C2×#117×C61C2211yescharacteristic
41C24×D61—21no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—22no—
46C22×D244—21no—
47C22×D244—22no—
48C22×D244D414yes—
49C22×D244C2312yes—
50#14519802—232no—
51C2×#11758201—232no—
52C2×#11758201—28no—
53C2×#11758201—24no—
54C2×#11758201—28no—
55C2×#11758201—24no—
56C2×#11758201—28no—
57C2×#11758201—24no—
58C2×#11758199—24no—
59C2×#11758199—24no—
60C2×#11758199—24no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194D414yes—
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418D414yes—
67C22×#92418C2312yes—
68C22×#92418C2312yes—
69#14519736—22no—
70#14519736—22no—
71C2×C4⋊Dic61C2312yes—
72C2×C22⋊Dic61—21no—
73C2×#11758198—28no—
74C2×C4×Dic61—21no—
75#14519701—24no—
76#14519701—28no—
77#14519698—24no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61D414yes—
84C22×Q8×C61—22no—
85C22×Q8×C61—21no—
86C22×Q8×C61—22no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61D414yes—
91C22×Q8×C61C2311yescharacteristic
92C22×Q8×C61C2312yes—
93C22×D4×C61—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C4⋊Q8×C61—24no—
99C4⋊Q8×C61—24no—
100#124×C61—24no—
101#124×C61—24no—
102#123×C61—216no—
103C22⋊Q8×C61—216no—
104#117×C61—22no—
105#117×C61—22no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61D412yes—
108C2×C22⋊C4×C61—24no—
109C2×C22⋊C4×C61D412yes—
110C2×C42×C61C2312yes—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113C23×D61—22no—
114C23×D61—24no—
115C23×D61—42no—
116C23×D61—42no—
117C23×D61—84no—
118#41×C61—28no—
119#41×C61—28no—
120#41×C61—44no—
121#41×C61—44no—
122C2×Q8×C61—24no—
123C2×Q8×C61—22no—
124C2×Q8×C61—21no—
125C2×Q8×C61—28no—
126C2×Q8×C61—28no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—21no—
135C2×Q8×C61—44no—
136C2×Q8×C61—42no—
137C2×Q8×C61—44no—
138C2×Q8×C61—42no—
139C2×Q8×C61C2×D414yes—
140C2×Q8×C61C2×D412yes—
141C2×D4×C61—24no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61C2×D412yes—
150C22×C4×C61—24no—
151C22×C4×C61—22no—
152C22×C4×C61—21no—
153C22×C4×C61—24no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61—42no—
158C22×C4×C61C2×D414yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D412yes—
165C22×C4×C61#4112yes—
166C22×C4×C61C2411yescharacteristic
167C22×Dic61—22no—
168C2×D244—28no—
169C2×D244—28no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—24no—
175C2×D244—22no—
176C2×D244—42no—
177C2×D244—44no—
178C2×D244—44no—
179C2×D244—42no—
180C2×D244C2×D414yes—
181C4⋊C4×C61—216no—
182C4⋊C4×C61—24no—
183C22⋊C4×C61—216no—
184C22⋊C4×C61—24no—
185C42×C61—22no—
186C42×C61—22no—
187C22⋊Dic61—42no—
188#11758201—232no—
189#11758201—216no—
190#11758199—216no—
191#11758198—88no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4×Dic61—42no—
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—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—44no—
208C23×C61#12012yes—
209C23×C61#12112yes—
210C23×C61#12712yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—24no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—48no—
216C22×D61—48no—
217C22×D61—88no—
218C22×D61—88no—
219C22×D61—88no—
220Q8×C61—24no—
221Q8×C61—22no—
222Q8×C61—24no—
223Q8×C61—22no—
224Q8×C61—42no—
225Q8×C61—41no—
226Q8×C61—42no—
227Q8×C61—41no—
228Q8×C61—48no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—48no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—28no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—48no—
243C2×C4×C61—48no—
244C2×C4×C61#12014yes—
245C2×C4×C61#12014yes—
246C2×C4×C61#12012yes—
247C2×C4×C61#12114yes—
248C2×C4×C61#12114yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12712yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×Dic61—42no—
258C2×Dic61—44no—
259D244—24no—
260D244—24no—
261D244—42no—
262D244—42no—
263D244—48no—
264#92418—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—82no—
273C22×C61—82no—
274C22×C61#39214yes—
275C22×C61C2×#12011yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×#12711yescharacteristic
278C22×C61#57714yes—
279C2×D61—44no—
280C2×D61—48no—
281C2×D61—816no—
282C2×D61—84no—
283C2×D61—84no—
284C4×C61—42no—
285C4×C61—44no—
286C4×C61—44no—
287C4×C61#39214yes—
288C4×C61C2×#12011yescharacteristic
289C4×C61C2×#12011yescharacteristic
290C4×C61C2×#12111yescharacteristic
291C4×C61C2×#12111yescharacteristic
292Dic61—82no—
293C22×AGL(1,ℤ/8)—611no—
294C23×SD16—611no—
295C2×#536D6111yescharacteristic, complemented
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×C61C2×#39211yescharacteristic
305C2×C61#301012yes—
306C2×C61#304212yes—
307C2×C61#315412yes—
308D61—84no—
309C22×#41C2×D6111yescharacteristic
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×M16—611no—
322C23×C8—611no—
323#536—24no—
324#536—24no—
325C2×C4⋊Q8C2×D6112yescomplemented
326C2×#124C2×D6112yescomplemented
327C2×#123—22no—
328C2×C22⋊Q8—22no—
329C2×#117—611no—
330C2×#117C2×D6111yescharacteristic
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—1221no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41C22⋊D6114yes—
348C22×Q8—21no—
349C22×Q8—22no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8C22⋊D6114yes—
355C22×Q8C22×D6112yes—
356C22×Q8C22×D6111yescharacteristic
357C22×D4—21no—
358C22×D4—614no—
359C22×D4—612no—
360C22×D4—1222no—
361C22×D4—1222no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—1221no—
366C22×D4C22×D6111yescharacteristic
367C23×C4—21no—
368C23×C4—21no—
369C23×C4C22×D6111yescharacteristic
370AGL(1,ℤ/8)—12232no—
371C2×SD16—1228no—
372C2×SD16—1224no—
373C2×SD16—1228no—
374C2×SD16—1224no—
375C2×SD16—12232no—
376C2×SD16—1228no—
377C2×SD16—1224no—
378C2×D8—1224no—
379C2×D8—1224no—
380C2×D8—1224no—
381C2×M16—614no—
382C2×M16—1222no—
383C2×M16—1222no—
384C22×C8—614no—
385C22×C8—612no—
386C22×C8—612no—
387C22×C8—1222no—
388C22×C8—1222no—
389C4⋊Q8—24no—
390C4⋊Q8—24no—
391#124—24no—
392#124—24no—
393#123—216no—
394C22⋊Q8—216no—
395#117—22no—
396#117—22no—
397#117—1222no—
398#117—1222no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4C22⋊D6112yes—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—1228no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4C22⋊D6112yes—
406C2×C42—1221no—
407C2×C42C22×D6112yes—
408#103—1224no—
409#103—1228no—
410#102—1224no—
411C24—21no—
412C24—1224no—
413C24—1222no—
414C24—4884no—
415C24—2442no—
416C24—2442no—
417C24C2×C22⋊D6111yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—28no—
434C2×Q8—28no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8C2×C22⋊D6114yes—
440C2×Q8C2×C22⋊D6112yes—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—614no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1228no—
456C2×D4—1228no—
457C2×D4—1224no—
458C2×D4—2442no—
459C2×D4—2444no—
460C2×D4—2444no—
461C2×D4—2442no—
462C2×D4C2×C22⋊D6112yes—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—1222no—
472C22×C4D4×D6112yes—
473C22×C4D4×D6112yes—
474C22×C4D4×D6112yes—
475C22×C4#1175822412yes—
476C22×C4C2×C22⋊D6114yes—
477C22×C4C2×C22⋊D6112yes—
478C22×C4C2×C22⋊D6112yes—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C23×D6111yescharacteristic
481SD16—12232no—
482SD16—12216no—
483D8—12216no—
484M16—1228no—
485C2×C8—614no—
486C2×C8—614no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—1228no—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C22⋊C4—216no—
497C22⋊C4—24no—
498C22⋊C4—4888no—
499C22⋊C4—2442no—
500C42—22no—
501C42—22no—
502C42—2442no—
503C23—24no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—44no—
508C23—1224no—
509C23—4888no—
510C23—4888no—
511C23—4888no—
512C23—2442no—
513C23—2442no—
514C23—2448no—
515C23—2448no—
516C23#1451974612yes—
517C23#1451974812yes—
518C23#1451977312yes—
519C23C22×C22⋊D6111yescharacteristic
520Q8—24no—
521Q8—22no—
522Q8—24no—
523Q8—22no—
524Q8—42no—
525Q8—41no—
526Q8—42no—
527Q8—41no—
528Q8—48no—
529D4—22no—
530D4—22no—
531D4—48no—
532D4—41no—
533D4—41no—
534D4—1224no—
535D4—1224no—
536D4—2442no—
537D4—2442no—
538D4—2448no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—2442no—
550C2×C4—2444no—
551C2×C4#1451974514yes—
552C2×C4#1451974514yes—
553C2×C4#1451974612yes—
554C2×C4#1451974812yes—
555C2×C4#1451974914yes—
556C2×C4#1451974914yes—
557C2×C4#1451977312yes—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C2×#1175822411yescharacteristic
564C8—1224no—
565C8—1224no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—44no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22—48816no—
574C22—4884no—
575C22—4884no—
576C22—2444no—
577C22—2448no—
578C22#1683967014yes—
579C22C2×#1451974611yescharacteristic
580C22C2×#1451974811yescharacteristic
581C22C2×#1451977311yescharacteristic
582C22#1684057014yes—
583C4—44no—
584C4—42no—
585C4—44no—
586C4—4882no—
587C4#1683967014yes—
588C4C2×#1451974511yescharacteristic
589C4C2×#1451974511yescharacteristic
590C4C2×#1451974911yescharacteristic
591C4C2×#1451974911yescharacteristic
592C2—24no—
593C2—82no—
594C2—4884no—
595C2C2×#1683967011yescharacteristic
596C2#2145510512yes—
597C2#2145532712yes—
598C2#2145629712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#536C2
socleC23×C61C22×D4
prime radicalC22×#14519802C2
hypercenterC2×#536D61
hypocenterC61#23424
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C23×#11758201C2semidirect
3. C2×#16839771C2semidirect
4. C2×#16839758C2semidirect
5. #21448399C2semidirect
6. #21446902C2semidirect
7. #21446896C2semidirect
8. #21445516C2semidirect
9. C2×C61×#536C2semidirect
25. C2×#14519701C22semidirect
26. C2×#14519701C22semidirect
27. C2×#14519698C22semidirect
31. #16839517C22semidirect
36. C2×C4⋊Q8×C61C22semidirect
37. C2×#124×C61C22semidirect
295. C2×#536D61semidirect
325. C2×C4⋊Q8C2×D61semidirect
326. C2×#124C2×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×#145198022
C23×#117582012
C2×#168397712
C2×#168397582
#214483992
#214469022
#214468962
#214455162
C2×C61×#5362
#2342461