Finite Group Databasebeta

← #29708737

Subgroups — #29708737

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839811C211yesmaximal, characteristic, complemented
2C2×#16839810C211yesmaximal, characteristic, complemented
3C2×#16839744C211yesmaximal, characteristic
4C2×#16839744C211yesmaximal, characteristic
5C2×#16839744C211yesmaximal, characteristic
6#21447219C214yesmaximal, complemented
7#21447216C214yesmaximal, complemented
8C2×#16839520C211yesmaximal, characteristic, complemented
9C2×C61×#430C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16839811—22no—
15#16839811C2214yes—
16#16839810—22no—
17#16839810C2214yes—
18C2×#14519713—22no—
19C2×#14519712—22no—
20#16839744—22no—
21#16839744—22no—
22#16839744—22no—
23#16839744—22no—
24#16839744—22no—
25#16839744—22no—
26C2×#14519695C2212yes—
27C2×#14519695C2212yes—
28C2×#14519695C2212yes—
29C2×#14519693C2212yes—
30C2×#14519693C2212yes—
31C2×#14519693C2212yes—
32#16839520C2214yes—
33C22×M16×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C61×#430—22no—
37C61×#430C2214yes—
38C2×#108×C61—22no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—21no—
49C2×#11758222—21no—
50C2×#11758222—21no—
51C22×Dic122C2311yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C22⋊D61Q812yes—
54C22×C4×D61Q812yes—
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×#11758194—21no—
66C2×#11758194—21no—
67C2×#11758194—21no—
68C2×#11758194D412yes—
69C2×#11758194D412yes—
70C2×#11758194C2312yes—
71C2×#11758194C2312yes—
72C2×#11758194C2312yes—
73C22×#92418—21no—
74C22×#92418—21no—
75C22×#92418—21no—
76#14519713—28no—
77#14519712—28no—
78#14519695—24no—
79#14519695—24no—
80#14519695—24no—
81#14519693—24no—
82#14519693—24no—
83#14519693—24no—
84C23×C4×C61C2311yescharacteristic
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×C61—21no—
94C2×M16×C61—21no—
95C2×M16×C61—21no—
96C2×M16×C61D412yes—
97C2×M16×C61D412yes—
98C2×M16×C61D412yes—
99C2×M16×C61D412yes—
100C2×M16×C61C2312yes—
101C2×M16×C61C2312yes—
102C2×M16×C61C2312yes—
103C22×C8×C61—21no—
104C22×C8×C61—21no—
105C22×C8×C61—21no—
106#108×C61—28no—
107C24×C61C2×Q811yescharacteristic
108C23×D61C2×Q812yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×Q811yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—22no—
129C22×Dic61C2×Q812yes—
130#11758222—41no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139#11758222—41no—
140#11758222—41no—
141#11758222—41no—
142#11758222—41no—
143#11758222—41no—
144#11758222—41no—
145#11758222—41no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164M16×C61—22no—
165M16×C61—22no—
166M16×C61—22no—
167M16×C61—22no—
168M16×C61—22no—
169M16×C61—22no—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—41no—
173M16×C61—41no—
174M16×C61C2×D414yes—
175M16×C61C2×D414yes—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C2×C8×C61—41no—
184C2×C8×C61—41no—
185#11758194—22no—
186#11758194—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—22no—
190#11758194—22no—
191#11758194—22no—
192#11758194—41no—
193#11758194—41no—
194#11758194—41no—
195#11758194—41no—
196#11758194C2×D414yes—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—41no—
203C2×#92418—41no—
204C2×#92418—41no—
205C2×#92418—41no—
206C23×C61—81no—
207C23×C61C22⋊Q811yescharacteristic
208C23×C61C22⋊Q811yescharacteristic
209C23×C61C22⋊Q811yescharacteristic
210C23×C61C22⋊Q811yescharacteristic
211C23×C61C22⋊Q811yescharacteristic
212C23×C61C22⋊Q811yescharacteristic
213C23×C61C22×Q811yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—82no—
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61C22⋊Q811yescharacteristic
231C2×C4×C61C22⋊Q811yescharacteristic
232C2×C4×C61C22⋊Q811yescharacteristic
233C2×C4×C61C22⋊Q811yescharacteristic
234C2×C4×C61C22⋊Q811yescharacteristic
235C2×C4×C61C22⋊Q811yescharacteristic
236C2×C4×C61#12611yescharacteristic
237C2×C4×C61#12611yescharacteristic
238C2×C4×C61#12611yescharacteristic
239C2×C4×C61#12611yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C22⋊D61—42no—
247C22⋊D61—42no—
248C22⋊D61—42no—
249C22⋊D61—42no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61—42no—
261Dic122—41no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265C8×C61—24no—
266C8×C61—24no—
267C8×C61—42no—
268#92418—24no—
269#92418—42no—
270#92418—42no—
271#23622—611nomaximal, Sylow
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61#39611yescharacteristic
280C22×C61#39611yescharacteristic
281C22×C61#39611yescharacteristic
282C22×C61#39611yescharacteristic
283C22×C61C2×C22⋊Q811yescharacteristic
284C22×C61C2×C22⋊Q811yescharacteristic
285C22×C61C2×C22⋊Q811yescharacteristic
286C2×D61—42no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—82no—
290C4×C61#39611yescharacteristic
291C4×C61#39611yescharacteristic
292C4×C61#39611yescharacteristic
293C4×C61#39611yescharacteristic
294C4×C61C2×#12111yescharacteristic
295C4×C61C2×#12111yescharacteristic
296C4×C61C2×#12111yescharacteristic
297C4×C61C2×#12611yescharacteristic
298Dic61—42no—
299C2×#430—611no—
300C2×#430—611no—
301C2×#430D6111yescharacteristic, complemented
302C2×#413—611no—
303C2×#413—611no—
304C2×#413—611no—
305#2070—614no—
306#2069—614no—
307C2×#338—611no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61—21no—
314C2×C61—21no—
315C2×C61C2×#39611yescharacteristic
316D61—82no—
317C22×#41—611no—
318C22×M16—611no—
319C22×M16—611no—
320C22×M16—611no—
321C22×M16C2×D6111yescharacteristic
322C22×M16C2×D6111yescharacteristic
323C22×M16C2×D6111yescharacteristic
324#430—22no—
325#430—614no—
326#430—614no—
327#430—1222no—
328#430—1222no—
329#430C2×D6114yes—
330C2×#108—22no—
331C2×#108—1222no—
332C2×#108—1222no—
333#413—1222no—
334#413—1222no—
335#413—1222no—
336#413—1222no—
337#413—1222no—
338#413—1222no—
339C2×#101—612no—
340C2×#101—612no—
341C2×#101—612no—
342C2×#100—612no—
343C2×#100—612no—
344C2×#100—612no—
345#338—614no—
346C61#2362211yescharacteristic, complemented, Sylow
347C2×#41—1221no—
348C2×#41—1221no—
349C2×#41—1221no—
350C2×#41—1221no—
351C2×#41—1221no—
352C2×#41—1221no—
353C2×#41—1221no—
354C2×#41—1221no—
355C2×#41—1221no—
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1221no—
359C22×Q8—611no—
360C22×D4—612no—
361C22×D4—611no—
362C23×C4—612no—
363C23×C4C22×D6111yescharacteristic
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—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—612no—
376C2×M16—612no—
377C2×M16—612no—
378C2×M16—612no—
379C2×M16—612no—
380C2×M16—1221no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—1221no—
386C2×M16—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16—1221no—
393C2×M16D24412yes—
394C2×M16D24412yes—
395C2×M16C22⋊D6112yes—
396C2×M16C22⋊D6112yes—
397C2×M16C22×D6112yes—
398C2×M16C22×D6112yes—
399C2×M16C22×D6112yes—
400C22×C8—21no—
401C22×C8—21no—
402C22×C8—21no—
403C22×C8—1221no—
404C22×C8—1221no—
405C22×C8—1221no—
406#108—28no—
407#108—1228no—
408#108—1228no—
409#101—1224no—
410#101—1224no—
411#101—1224no—
412#100—1224no—
413#100—1224no—
414#100—1224no—
415C24—612no—
416C24Q8×D6111yescharacteristic
417#41—2441no—
418#41—2441no—
419#41—2441no—
420#41—2441no—
421#41—2441no—
422#41—2441no—
423#41—2441no—
424#41—2441no—
425#41—2441no—
426#41—2441no—
427#41—2441no—
428#41—2441no—
429#41—2441no—
430#41—2441no—
431#41—2441no—
432#41—2441no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C22×C4—612no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4C2×D24411yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4#1175822211yescharacteristic
461C22×C4#1175822211yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4D4×D6111yescharacteristic
464C22×C4#1175822411yescharacteristic
465C22×C4#1175822411yescharacteristic
466C22×C4Q8×D6111yescharacteristic
467C22×C4#1175822611yescharacteristic
468C22×C4#1175822611yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—22no—
480M16—41no—
481M16—41no—
482M16—614no—
483M16—1222no—
484M16—1222no—
485M16—1222no—
486M16—1222no—
487M16—1222no—
488M16—1222no—
489M16—1222no—
490M16—2441no—
491M16—2441no—
492M16—2441no—
493M16—2441no—
494M16C2×D24414yes—
495M16C2×C22⋊D6114yes—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2441no—
511C2×C8—2441no—
512C2×C8—2441no—
513C2×C8—2441no—
514C23—81no—
515C23—1222no—
516C23—1222no—
517C23—1222no—
518C23—4882no—
519C23#1451975311yescharacteristic
520C23#1451975311yescharacteristic
521C23#1451975511yescharacteristic
522C23#1451975511yescharacteristic
523C23#1451975711yescharacteristic
524C23#1451975711yescharacteristic
525C23C2×Q8×D6111yescharacteristic
526Q8—2441no—
527Q8—2441no—
528Q8—2441no—
529Q8—2441no—
530D4—2442no—
531D4—2442no—
532D4—2442no—
533D4—2442no—
534D4—2441no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4#1451974911yescharacteristic
546C2×C4#1451974911yescharacteristic
547C2×C4#1451974911yescharacteristic
548C2×C4#1451974911yescharacteristic
549C2×C4#1451975111yescharacteristic
550C2×C4#1451975111yescharacteristic
551C2×C4#1451975111yescharacteristic
552C2×C4#1451975111yescharacteristic
553C2×C4#1451975211yescharacteristic
554C2×C4#1451975211yescharacteristic
555C2×C4#1451975211yescharacteristic
556C2×C4#1451975211yescharacteristic
557C2×C4#1451975311yescharacteristic
558C2×C4#1451975311yescharacteristic
559C2×C4#1451975511yescharacteristic
560C2×C4#1451975511yescharacteristic
561C2×C4#1451975711yescharacteristic
562C2×C4#1451975711yescharacteristic
563C2×C4#1451977211yescharacteristic
564C2×C4#1451977211yescharacteristic
565C2×C4#1451977211yescharacteristic
566C2×C4#1451977211yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4C22×D24411yescharacteristic
570C2×C4C2×#1175822411yescharacteristic
571C2×C4C2×#1175822211yescharacteristic
572C2×C4C2×#1175822611yescharacteristic
573C8—24no—
574C8—24no—
575C8—42no—
576C8—1224no—
577C8—2442no—
578C8—2442no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—4882no—
587C22—4882no—
588C22—4882no—
589C22—2442no—
590C22#1683968411yescharacteristic
591C22#1683968411yescharacteristic
592C22#1683968411yescharacteristic
593C22#1683968411yescharacteristic
594C22C2×#1451975311yescharacteristic
595C22C2×#1451975511yescharacteristic
596C22C2×#1451975711yescharacteristic
597C4—2442no—
598C4#1683968411yescharacteristic
599C4#1683968411yescharacteristic
600C4#1683968411yescharacteristic
601C4#1683968411yescharacteristic
602C4C2×#1451974911yescharacteristic
603C4C2×#1451975111yescharacteristic
604C4C2×#1451975211yescharacteristic
605C4C2×#1451977211yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—4882no—
614C2C2×#1683968411yescharacteristic
subgroupG/H
centerC4#16839684
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#430C2
socleC2×C61C2×#396
prime radicalC22×#11758222C22
hypercenterC2×#430D61
hypocenterC61#23622
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
1. C2×#16839811C2semidirect
2. C2×#16839810C2semidirect
6. #21447219C2semidirect
7. #21447216C2semidirect
8. C2×#16839520C2semidirect
9. C2×C61×#430C2semidirect
301. C2×#430D61semidirect
346. C61#23622semidirect

Maximal subgroups

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

subgroupindex
C2×#168398112
C2×#168398102
C2×#168397442
#214472192
#214472162
C2×#168395202
C2×C61×#4302
#2362261