Finite Group Databasebeta

← #29708964

Subgroups — #29708964

Order 15616.

888 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454339C211yesmaximal, characteristic
2#21454339C211yesmaximal, characteristic
3C2×#16839744C211yesmaximal, characteristic
4#21447284C214yesmaximal, complemented
5#21447277C214yesmaximal, complemented
6C2×#16839534C211yesmaximal, characteristic, complemented
7C2×#16839533C211yesmaximal, characteristic, complemented
8C2×#16839519C211yesmaximal, characteristic, complemented
9C2×C61×#430C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16839772C2212yes—
15#16839772C2212yes—
16#16839768C2212yes—
17#16839768C2212yes—
18#16839764—22no—
19#16839764—22no—
20#16839759—22no—
21#16839759—22no—
22#16839744—22no—
23#16839744—22no—
24C2×#14519695C2212yes—
25C2×#14519693C2212yes—
26#16839534C2214yes—
27#16839533C2214yes—
28#16839519C2214yes—
29C22×M16×C61C2211yescharacteristic
30C22×M16×C61C2211yescharacteristic
31C22×M16×C61C2211yescharacteristic
32C61×#430—22no—
33C61×#430C2214yes—
34C2×#108×C61—22no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222D412yes—
46C2×#11758222D412yes—
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C22×C4×D61—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194C2312yes—
57C22×#92418—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64#14519736C2312yes—
65#14519736C2312yes—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61—21no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61—21no—
72#14519702—24no—
73#14519702—24no—
74#14519699—24no—
75#14519699—24no—
76#14519695—24no—
77#14519693—24no—
78C23×C4×C61C2311yescharacteristic
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C2×M16×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61—21no—
91C2×M16×C61—21no—
92C2×M16×C61—21no—
93C2×M16×C61D412yes—
94C2×M16×C61C2312yes—
95C2×M16×C61C2312yes—
96C2×M16×C61C2312yes—
97C22×C8×C61—21no—
98C22×C8×C61—21no—
99C22×C8×C61D412yes—
100#108×C61—28no—
101C24×C61#4111yescharacteristic
102C23×D61—21no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—22no—
118C2×C22⋊D61—22no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C22×Dic61—21no—
124C22×Dic61—41no—
125C22×Dic61—41no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—22no—
132#11758222—22no—
133#11758222—22no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139#11758222—41no—
140#11758222—41no—
141#11758222—41no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244C2×D412yes—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122C2×D412yes—
160M16×C61—22no—
161M16×C61—22no—
162M16×C61—22no—
163M16×C61—22no—
164M16×C61—22no—
165M16×C61—22no—
166M16×C61—22no—
167M16×C61—22no—
168M16×C61—41no—
169M16×C61—41no—
170M16×C61—41no—
171M16×C61—41no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—41no—
177C2×C8×C61—41no—
178C2×C8×C61—41no—
179C2×C8×C61—41no—
180C2×C8×C61C2×D414yes—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—41no—
184C22⋊Dic61—41no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—41no—
188C4⋊Dic61—41no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—41no—
192C4×Dic61—41no—
193#11758194—22no—
194#11758194—22no—
195#11758194—41no—
196#11758194—41no—
197C2×#92418—22no—
198C2×#92418—41no—
199C2×#92418—41no—
200C23×C61—81no—
201C23×C61#12311yescharacteristic
202C23×C61#12311yescharacteristic
203C23×C61#12411yescharacteristic
204C23×C61#12411yescharacteristic
205C23×C61#12611yescharacteristic
206C23×C61#12611yescharacteristic
207C23×C61C2×#4111yescharacteristic
208C22×D61—22no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—82no—
212C2×C4×C61#12111yescharacteristic
213C2×C4×C61#12111yescharacteristic
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61#12111yescharacteristic
216C2×C4×C61#12311yescharacteristic
217C2×C4×C61#12311yescharacteristic
218C2×C4×C61#12311yescharacteristic
219C2×C4×C61#12311yescharacteristic
220C2×C4×C61#12311yescharacteristic
221C2×C4×C61#12311yescharacteristic
222C2×C4×C61#12311yescharacteristic
223C2×C4×C61#12311yescharacteristic
224C2×C4×C61#12311yescharacteristic
225C2×C4×C61#12311yescharacteristic
226C2×C4×C61#12411yescharacteristic
227C2×C4×C61#12411yescharacteristic
228C2×C4×C61#12611yescharacteristic
229C2×C4×C61#12611yescharacteristic
230C2×C4×C61#12611yescharacteristic
231C2×C4×C61#12611yescharacteristic
232C2×C4×C61#12611yescharacteristic
233C2×C4×C61#12611yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C22⋊D61—42no—
244C2×Dic61—22no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—82no—
250C2×Dic61—82no—
251D244—22no—
252D244—22no—
253D244—41no—
254D244—41no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—42no—
258C4×D61—42no—
259Dic122—22no—
260Dic122—22no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—24no—
264C8×C61—42no—
265C8×C61—42no—
266#92418—42no—
267#23652—611nomaximal, Sylow
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61#39711yescharacteristic
276C22×C61#39711yescharacteristic
277C22×C61#39711yescharacteristic
278C22×C61#39711yescharacteristic
279C22×C61C2×#12311yescharacteristic
280C22×C61C2×#12411yescharacteristic
281C22×C61C2×#12611yescharacteristic
282C2×D61—42no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—82no—
286C4×C61#39711yescharacteristic
287C4×C61#39711yescharacteristic
288C4×C61#39711yescharacteristic
289C4×C61#39711yescharacteristic
290C4×C61C2×#12111yescharacteristic
291C4×C61C2×#12311yescharacteristic
292C4×C61C2×#12311yescharacteristic
293C4×C61C2×#12611yescharacteristic
294Dic61—42no—
295Dic61—82no—
296Dic61—82no—
297C2×#430D6111yescharacteristic, complemented
298#2896—611no—
299#2896—611no—
300C2×#413—611no—
301#2082—614no—
302#2081—614no—
303C2×#344—611no—
304C2×#344—611no—
305C2×H(3,ℤ/4)—611no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61C2×#39711yescharacteristic
314D61—82no—
315C22×#41—611no—
316C22×M16—611no—
317C22×M16C2×D6111yescharacteristic
318C22×M16C2×D6111yescharacteristic
319C22×M16C2×D6111yescharacteristic
320C2×#117—611no—
321C2×#117—611no—
322#430—22no—
323#430C2×D6114yes—
324C2×#108—22no—
325#419—612no—
326#419—612no—
327#418—612no—
328#418—612no—
329#417—1222no—
330#417—1222no—
331#416—1222no—
332#416—1222no—
333#413—1222no—
334#413—1222no—
335C2×#101—612no—
336C2×#100—612no—
337#344—614no—
338#344—614no—
339H(3,ℤ/4)—614no—
340C61#2365211yescharacteristic, complemented, Sylow
341C2×#41—612no—
342C2×#41—612no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1221no—
346C2×#41—1221no—
347C2×#41—1221no—
348C2×#41—1221no—
349C2×#41—1221no—
350C2×#41—1221no—
351C2×#41—1221no—
352C2×#41—1221no—
353C22×Q8—611no—
354C22×D4—611no—
355C22×D4—1221no—
356C23×C4—1221no—
357C23×C4C22×D6111yescharacteristic
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—612no—
373C2×M16—1221no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16C22⋊D6112yes—
379C2×M16C22×D6112yes—
380C2×M16C22×D6112yes—
381C2×M16C22×D6112yes—
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—1221no—
385C22×C8C22⋊D6112yes—
386#117—612no—
387#117—612no—
388#117—1221no—
389#117—1221no—
390#117—1221no—
391#117—1221no—
392#117—1221no—
393#117—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400#108—28no—
401#103—1224no—
402#103—1224no—
403#102—1224no—
404#102—1224no—
405#101—1224no—
406#100—1224no—
407C24—1221no—
408C24#1175822611yescharacteristic
409#41—1222no—
410#41—1222no—
411#41—1222no—
412#41—1222no—
413#41—1222no—
414#41—1222no—
415#41—1222no—
416#41—1222no—
417#41—2441no—
418#41—2441no—
419#41—2441no—
420#41—2441no—
421#41—2441no—
422#41—2441no—
423#41—2441no—
424#41—2441no—
425C2×Q8—612no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×D4—612no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
442C2×D4—2441no—
443C22×C4—1221no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4—2441no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4#1175822211yescharacteristic
453C22×C4#1175822211yescharacteristic
454C22×C4#1175822211yescharacteristic
455C22×C4#1175822211yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4D4×D6111yescharacteristic
458C22×C4#1175822411yescharacteristic
459C22×C4#1175822411yescharacteristic
460C22×C4#1175822611yescharacteristic
461C22×C4#1175822611yescharacteristic
462C22×C4#1175822611yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—22no—
467M16—22no—
468M16—22no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—22no—
473M16—22no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
478M16—1222no—
479M16—1222no—
480M16—2441no—
481M16—2441no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—1222no—
491C2×C8—2441no—
492C2×C8—2441no—
493C2×C8C2×C22⋊D6114yes—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42—1222no—
503C42—1222no—
504C42—2441no—
505C42—2441no—
506C23—81no—
507C23—1222no—
508C23—4882no—
509C23—2441no—
510C23—2441no—
511C23#1451976211yescharacteristic
512C23#1451976211yescharacteristic
513C23#1451976611yescharacteristic
514C23#1451976611yescharacteristic
515C23#1451977211yescharacteristic
516C23#1451977211yescharacteristic
517C23C2×#1175822611yescharacteristic
518Q8—1222no—
519Q8—1222no—
520Q8—2441no—
521Q8—2441no—
522D4—1222no—
523D4—1222no—
524D4—2442no—
525D4—2442no—
526D4—2442no—
527D4—2442no—
528D4—2441no—
529D4—2441no—
530C2×C4—1222no—
531C2×C4—4882no—
532C2×C4—4882no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4#1451974811yescharacteristic
542C2×C4#1451974811yescharacteristic
543C2×C4#1451974811yescharacteristic
544C2×C4#1451974811yescharacteristic
545C2×C4#1451976011yescharacteristic
546C2×C4#1451976011yescharacteristic
547C2×C4#1451976011yescharacteristic
548C2×C4#1451976011yescharacteristic
549C2×C4#1451976211yescharacteristic
550C2×C4#1451976211yescharacteristic
551C2×C4#1451976211yescharacteristic
552C2×C4#1451976211yescharacteristic
553C2×C4#1451976211yescharacteristic
554C2×C4#1451976211yescharacteristic
555C2×C4#1451976611yescharacteristic
556C2×C4#1451976611yescharacteristic
557C2×C4#1451977211yescharacteristic
558C2×C4#1451977211yescharacteristic
559C2×C4#1451977211yescharacteristic
560C2×C4#1451977211yescharacteristic
561C2×C4#1451977211yescharacteristic
562C2×C4#1451977211yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C2×C4C2×#1175822211yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C2×C4C2×#1175822611yescharacteristic
569C8—24no—
570C8—42no—
571C8—42no—
572C8—2442no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—4882no—
581C22—4882no—
582C22—4882no—
583C22—2442no—
584C22#1683968811yescharacteristic
585C22#1683968811yescharacteristic
586C22#1683968811yescharacteristic
587C22#1683968811yescharacteristic
588C22C2×#1451976211yescharacteristic
589C22C2×#1451976611yescharacteristic
590C22C2×#1451977211yescharacteristic
591C4—4882no—
592C4—4882no—
593C4—2442no—
594C4#1683968811yescharacteristic
595C4#1683968811yescharacteristic
596C4#1683968811yescharacteristic
597C4#1683968811yescharacteristic
598C4C2×#1451974811yescharacteristic
599C4C2×#1451976011yescharacteristic
600C4C2×#1451976211yescharacteristic
601C4C2×#1451977211yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—4882no—
610C2C2×#1683968811yescharacteristic
subgroupG/H
centerC4#16839688
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#430C2
socleC2×C61C2×#397
prime radicalC22×#11758222C22
hypercenterC2×#430D61
hypocenterC61#23652
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
4. #21447284C2semidirect
5. #21447277C2semidirect
6. C2×#16839534C2semidirect
7. C2×#16839533C2semidirect
8. C2×#16839519C2semidirect
9. C2×C61×#430C2semidirect
297. C2×#430D61semidirect
340. C61#23652semidirect

Maximal subgroups

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

subgroupindex
#214543392
C2×#168397442
#214472842
#214472772
C2×#168395342
C2×#168395332
C2×#168395192
C2×C61×#4302
#2365261