Finite Group Databasebeta

← #29708969

Subgroups — #29708969

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
1C2×#16839810C211yesmaximal, characteristic, complemented
2#21454336C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4C2×#16839744C211yesmaximal, characteristic
5#21447288C214yesmaximal, complemented
6#21447281C214yesmaximal, complemented
7C2×#16839532C211yesmaximal, characteristic, complemented
8C2×#16839532C211yesmaximal, characteristic, complemented
9C2×C61×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16839810—22no—
15#16839810C2214yes—
16C2×#14519712—22no—
17#16839770C2212yes—
18#16839770C2212yes—
19#16839766C2212yes—
20#16839766C2212yes—
21#16839761—22no—
22#16839761—22no—
23#16839756—22no—
24#16839756—22no—
25#16839744—22no—
26#16839744—22no—
27C2×#14519695C2212yes—
28C2×#14519693C2212yes—
29#16839532C2214yes—
30#16839532C2214yes—
31C22×M16×C61C2211yescharacteristic
32C2×#117×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C61×H(3,ℤ/4)C2214yes—
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×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59C2×#11758194—21no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62C2×#11758194—21no—
63C2×#11758194—21no—
64C2×#11758194—21no—
65C2×#11758194D412yes—
66C2×#11758194C2312yes—
67C2×#11758194C2312yes—
68C2×#11758194C2312yes—
69C22×#92418—21no—
70C22×#92418—21no—
71C22×#92418D412yes—
72#14519712—28no—
73#14519700—24no—
74#14519700—24no—
75#14519697—24no—
76#14519697—24no—
77#14519695—24no—
78#14519693—24no—
79C23×C4×C61C2311yescharacteristic
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C2×M16×C61C2312yes—
86C22×C8×C61—21no—
87#117×C61—21no—
88#117×C61—21no—
89#117×C61—21no—
90#117×C61—21no—
91#117×C61—21no—
92#117×C61—21no—
93#117×C61C2312yes—
94#117×C61C2312yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61—21no—
101C24×C61#4111yescharacteristic
102C23×D61—21no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
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×C61#4111yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—22no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C22×Dic61—21no—
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—41no—
163M16×C61—41no—
164C2×C8×C61—22no—
165C2×C8×C61—41no—
166C2×C8×C61—41no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C42×C61—22no—
176C42×C61—22no—
177C42×C61—41no—
178C42×C61—41no—
179#11758194—22no—
180#11758194—22no—
181#11758194—22no—
182#11758194—22no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—41no—
188#11758194—41no—
189#11758194—41no—
190#11758194—41no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—41no—
196C2×#92418—41no—
197C2×#92418—41no—
198C2×#92418—41no—
199C2×#92418C2×D414yes—
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—42no—
213C2×C4×C61—42no—
214C2×C4×C61—82no—
215C2×C4×C61—82no—
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
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#12311yescharacteristic
227C2×C4×C61#12311yescharacteristic
228C2×C4×C61#12311yescharacteristic
229C2×C4×C61#12311yescharacteristic
230C2×C4×C61#12411yescharacteristic
231C2×C4×C61#12411yescharacteristic
232C2×C4×C61#12611yescharacteristic
233C2×C4×C61#12611yescharacteristic
234C2×C4×C61#12611yescharacteristic
235C2×C4×C61#12611yescharacteristic
236C2×C4×C61#12611yescharacteristic
237C2×C4×C61#12611yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C22⋊D61—42no—
245C22⋊D61—42no—
246C22⋊D61—42no—
247C22⋊D61—42no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
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—42no—
264#92418—24no—
265#92418—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—82no—
287C4×C61—82no—
288C4×C61#39711yescharacteristic
289C4×C61#39711yescharacteristic
290C4×C61#39711yescharacteristic
291C4×C61#39711yescharacteristic
292C4×C61C2×#12111yescharacteristic
293C4×C61C2×#12311yescharacteristic
294C4×C61C2×#12311yescharacteristic
295C4×C61C2×#12611yescharacteristic
296Dic61—42no—
297C2×#430—611no—
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)D6111yescharacteristic, complemented
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×M16—611no—
318C22×M16—611no—
319C22×M16C2×D6111yescharacteristic
320C2×#117C2×D6111yescharacteristic
321C2×#117C2×D6111yescharacteristic
322#430—614no—
323#430—1222no—
324C2×#108—1222no—
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)C2×D6114yes—
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—612no—
364C2×M16—612no—
365C2×M16—612no—
366C2×M16—612no—
367C2×M16—1221no—
368C2×M16—1221no—
369C2×M16—1221no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1221no—
373C2×M16—1221no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—1221no—
379C2×M16—1221no—
380C2×M16—1221no—
381C2×M16C22×D6112yes—
382C22×C8—21no—
383C22×C8—612no—
384C22×C8—1221no—
385C22×C8—1221no—
386#117—21no—
387#117—21no—
388#117—21no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117C22×D6112yes—
393#117C22×D6112yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C42—21no—
399C2×C42—21no—
400#108—1228no—
401#103—1224no—
402#103—1224no—
403#102—1224no—
404#102—1224no—
405#101—1224no—
406#100—1224no—
407C24—1221no—
408C24#1175822411yescharacteristic
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—41no—
444C22×C4—41no—
445C22×C4—1221no—
446C22×C4—1222no—
447C22×C4—1222no—
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×C4#1175822211yescharacteristic
457C22×C4#1175822211yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4#1175822411yescharacteristic
463C22×C4#1175822611yescharacteristic
464C22×C4#1175822611yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—22no—
467M16—22no—
468M16—41no—
469M16—41no—
470M16—1222no—
471M16—1222no—
472M16—1222no—
473M16—1222no—
474M16—1222no—
475M16—1222no—
476M16—1222no—
477M16—1222no—
478M16—2441no—
479M16—2441no—
480M16—2441no—
481M16—2441no—
482C2×C8—22no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—614no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—2441no—
491C2×C8—2441no—
492C2×C8—2441no—
493C2×C8—2441no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C23—81no—
507C23—1222no—
508C23—4882no—
509C23—2441no—
510C23—2441no—
511C23#1451975911yescharacteristic
512C23#1451975911yescharacteristic
513C23#1451976311yescharacteristic
514C23#1451976311yescharacteristic
515C23#1451977211yescharacteristic
516C23#1451977211yescharacteristic
517C23C2×#1175822411yescharacteristic
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—42no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—1222no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4#1451975011yescharacteristic
542C2×C4#1451975011yescharacteristic
543C2×C4#1451975011yescharacteristic
544C2×C4#1451975011yescharacteristic
545C2×C4#1451975911yescharacteristic
546C2×C4#1451975911yescharacteristic
547C2×C4#1451976211yescharacteristic
548C2×C4#1451976211yescharacteristic
549C2×C4#1451976211yescharacteristic
550C2×C4#1451976211yescharacteristic
551C2×C4#1451976211yescharacteristic
552C2×C4#1451976211yescharacteristic
553C2×C4#1451976211yescharacteristic
554C2×C4#1451976211yescharacteristic
555C2×C4#1451976311yescharacteristic
556C2×C4#1451976311yescharacteristic
557C2×C4#1451977111yescharacteristic
558C2×C4#1451977111yescharacteristic
559C2×C4#1451977111yescharacteristic
560C2×C4#1451977111yescharacteristic
561C2×C4#1451977211yescharacteristic
562C2×C4#1451977211yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C2×C4C2×#1175822211yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C2×C4C2×#1175822611yescharacteristic
569C8—42no—
570C8—1224no—
571C8—2442no—
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#1683969111yescharacteristic
585C22#1683969111yescharacteristic
586C22#1683969111yescharacteristic
587C22#1683969111yescharacteristic
588C22C2×#1451975911yescharacteristic
589C22C2×#1451976311yescharacteristic
590C22C2×#1451977211yescharacteristic
591C4—82no—
592C4—82no—
593C4—2442no—
594C4#1683969111yescharacteristic
595C4#1683969111yescharacteristic
596C4#1683969111yescharacteristic
597C4#1683969111yescharacteristic
598C4C2×#1451975011yescharacteristic
599C4C2×#1451976211yescharacteristic
600C4C2×#1451976211yescharacteristic
601C4C2×#1451977111yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—4882no—
610C2C2×#1683969111yescharacteristic
subgroupG/H
centerC4#16839691
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×H(3,ℤ/4)C2
socleC2×C61C2×#397
prime radicalC22×#11758222C22
hypercenterC2×H(3,ℤ/4)D61
hypocenterC61#23652
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#16839810C2semidirect
5. #21447288C2semidirect
6. #21447281C2semidirect
7. C2×#16839532C2semidirect
8. C2×#16839532C2semidirect
9. C2×C61×H(3,ℤ/4)C2semidirect
305. C2×H(3,ℤ/4)D61semidirect
340. C61#23652semidirect

Maximal subgroups

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

subgroupindex
C2×#168398102
#214543362
C2×#168397442
#214472882
#214472812
C2×#168395322
C2×C61×H(3,ℤ/4)2
#2365261