Finite Group Databasebeta

← #29587001

Subgroups — #29587001

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454382C211yesmaximal, characteristic, complemented
2C2×#16839780C211yesmaximal, characteristic
3C2×#16839778C211yesmaximal, characteristic
4#21448309C211yesmaximal, characteristic
5#21445551C211yesmaximal, characteristic, complemented
6C2×#16839522C211yesmaximal, characteristic, complemented
7C2×C61×#339C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12#16839790—22no—
13#16839788—22no—
14#16839785—22no—
15#16839780—22no—
16#16839780—22no—
17#16839778—22no—
18#16839778—22no—
19C2×#14519705—22no—
20C2×#14519703—22no—
21#16839668C412yescomplemented
22#16839664C412yescomplemented
23#16839641—22no—
24C4×#11758198—22no—
25#16839528—22no—
26#16839522—22no—
27C22×M16×C61C2211yescharacteristic
28C2×#117×C61C2211yescharacteristic
29C22×C42×C61C2211yescharacteristic
30C61×#339—22no—
31C2×#11758222—21no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222D412yes—
42C2×#11758222Q812yes—
43C22×Dic122C2×C411yescharacteristic
44C22×D244C2×C411yescharacteristic
45C22×C22⋊D61—21no—
46C22×C4×D61—21no—
47C2×#11758194—21no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194D412yes—
53C22×#92418—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736—21no—
61#14519736D412yes—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2×C412yes—
64C2×C22⋊Dic61—21no—
65C2×C22⋊Dic61D412yes—
66C2×#11758198—24no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61D412yes—
69#14519706—28no—
70#14519705—28no—
71#14519703—28no—
72#14519684—24no—
73C23×C4×C61—21no—
74C23×C4×C61C2311yescharacteristic
75C2×M16×C61—21no—
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61D412yes—
81C22×C8×C61—21no—
82#117×C61—21no—
83#117×C61—21no—
84#117×C61—21no—
85#117×C61D412yes—
86C2×C4⋊C4×C61—21no—
87C2×C22⋊C4×C61—21no—
88C2×C42×C61—21no—
89C2×C42×C61—21no—
90C2×C42×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61—21no—
93C2×C42×C61—21no—
94C2×C42×C61C2×C412yes—
95C24×C61C2×D411yescharacteristic
96C23×D61—21no—
97C22×C4×C61—22no—
98C22×C4×C61—44no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61C22×C411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×Q811yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C2×C22⋊D61—22no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—21no—
123C22×Dic61—41no—
124C22×Dic61C22⋊C414yes—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—22no—
132#11758222—22no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139#11758222—41no—
140#11758222—41no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244C4⋊C412yes—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122C4⋊C412yes—
159M16×C61—22no—
160M16×C61—22no—
161M16×C61—41no—
162M16×C61—41no—
163C2×C8×C61—22no—
164C2×C8×C61—41no—
165C2×C8×C61—41no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—41no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—41no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—41no—
174C42×C61—41no—
175C42×C61—41no—
176C22⋊Dic61—22no—
177C22⋊Dic61—22no—
178C22⋊Dic61—22no—
179C22⋊Dic61—41no—
180#11758198—48no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—41no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—41no—
189#11758194—22no—
190#11758194—22no—
191#11758194—41no—
192#11758194—41no—
193C2×#92418—22no—
194C2×#92418—41no—
195C2×#92418—41no—
196C23×C61—42no—
197C23×C61C2×C22⋊C411yescharacteristic
198C23×C61#12111yescharacteristic
199C23×C61#12111yescharacteristic
200C23×C61#12111yescharacteristic
201C23×C61#12111yescharacteristic
202C23×C61#12111yescharacteristic
203C23×C61C22⋊Q811yescharacteristic
204C22×D61—22no—
205C22×D61—44no—
206C22×D61—41no—
207C22×D61—41no—
208C2×C4×C61—24no—
209C2×C4×C61—42no—
210C2×C4×C61—44no—
211C2×C4×C61—44no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—82no—
215C2×C4×C61C2×C4⋊C411yescharacteristic
216C2×C4×C61#11711yescharacteristic
217C2×C4×C61C4×D411yescharacteristic
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61C22⋊Q811yescharacteristic
228C2×C4×C61C22⋊Q811yescharacteristic
229C2×C4×C61C22⋊Q811yescharacteristic
230C2×C4×C61C22⋊Q811yescharacteristic
231C2×C4×C61C22⋊Q811yescharacteristic
232C2×C4×C61#12311yescharacteristic
233C2×C4×C61#12311yescharacteristic
234C2×C4×C61#12311yescharacteristic
235C2×C4×C61#12311yescharacteristic
236C2×C4×C61#12311yescharacteristic
237C2×C4×C61#12411yescharacteristic
238C2×C4×C61#12411yescharacteristic
239C2×C4×C61#12411yescharacteristic
240C2×C4×C61#12611yescharacteristic
241C2×C4×C61#12611yescharacteristic
242C2×C4×C61#12711yescharacteristic
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—42no—
246C22⋊D61—42no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—44no—
252C2×Dic61—42no—
253C2×Dic61—82no—
254D244—22no—
255D244—22no—
256D244—41no—
257D244—41no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61—42no—
261C4×D61—42no—
262Dic122—22no—
263Dic122—22no—
264Dic122—41no—
265Dic122—41no—
266C8×C61—42no—
267#92418—42no—
268#11056—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61#38011yescharacteristic
277C22×C61#38611yescharacteristic
278C22×C61#39011yescharacteristic
279C22×C61#39411yescharacteristic
280C22×C61#39411yescharacteristic
281C22×C61#39611yescharacteristic
282C22×C61#39611yescharacteristic
283C2×D61—44no—
284C2×D61—42no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—44no—
288C4×C61—82no—
289C4×C61#38511yescharacteristic
290C4×C61#38511yescharacteristic
291C4×C61#38811yescharacteristic
292C4×C61#38811yescharacteristic
293C4×C61#39211yescharacteristic
294C4×C61#39311yescharacteristic
295C4×C61#39711yescharacteristic
296C4×C61#39911yescharacteristic
297Dic61—44no—
298Dic61—42no—
299Dic61—82no—
300#2905—611no—
301C2×#421—611no—
302C2×#421—611no—
303#2297—611no—
304#1740—611no—
305C2×#339—611no—
306C2×#339D6111yescharacteristic, complemented
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61#143911yescharacteristic
314C2×C61#188412yes—
315D61—82no—
316C22×#41—611no—
317C22×M16—611no—
318C22×M16C2×D6111yescharacteristic
319C2×#117—611no—
320C2×#117—611no—
321C2×#117C2×D6111yescharacteristic
322C22×C42C2×D6111yescharacteristic
323#424—1222no—
324#424—1222no—
325#423—1222no—
326#421—1222no—
327#421—1222no—
328#421—1222no—
329#421—1222no—
330C2×#104—1222no—
331C2×#104—1222no—
332#391—612no—
333#390—612no—
334#386—1222no—
335C4×C22⋊C4—1222no—
336#342—1222no—
337#339—22no—
338#339—1222no—
339C61#1105611yescharacteristic, complemented, Sylow
340C2×#41—612no—
341C2×#41—612no—
342C2×#41—1221no—
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—
352C22×Q8—611no—
353C22×D4—611no—
354C22×D4—1221no—
355C23×C4—21no—
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—1221no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C2×M16D24412yes—
370C22×C8—21no—
371C22×C8—1221no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—612no—
376#117—1221no—
377#117—1221no—
378#117—1221no—
379#117—1221no—
380#117—1221no—
381#117—1221no—
382#117—1221no—
383#117C22⋊D6112yes—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—612no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1224no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—612no—
398C2×C42—1221no—
399C2×C42C4×D6112yes—
400C4⋊C8—1228no—
401#104—1228no—
402#104—1228no—
403#95—1224no—
404C24—1221no—
405C24D4×D6111yescharacteristic
406#41—1222no—
407#41—1222no—
408#41—1222no—
409#41—1222no—
410#41—1222no—
411#41—1222no—
412#41—1222no—
413#41—1222no—
414#41—2441no—
415#41—2441no—
416#41—2441no—
417#41—2441no—
418#41—2441no—
419#41—2441no—
420#41—2441no—
421#41—2441no—
422C2×Q8—612no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×D4—612no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—2441no—
437C2×D4—2441no—
438C2×D4—2441no—
439C2×D4—2441no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—614no—
446C22×C4—1221no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—2441no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4—2441no—
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C2×D24411yescharacteristic
456C22×C4#1175822211yescharacteristic
457C22×C4#1175822211yescharacteristic
458C22×C4#1175822211yescharacteristic
459C22×C4#1175822211yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4D4×D6111yescharacteristic
464C22×C4#1175822411yescharacteristic
465C22×C4Q8×D6111yescharacteristic
466C22×C4#1175822611yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468M16—22no—
469M16—22no—
470M16—41no—
471M16—41no—
472M16—1222no—
473M16—1222no—
474M16—2441no—
475M16—2441no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—1222no—
480C2×C8—2441no—
481C2×C8—2441no—
482C4⋊C4—22no—
483C4⋊C4—41no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—2441no—
488C22⋊C4—22no—
489C22⋊C4—41no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—2448no—
494C22⋊C4—2441no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—1222no—
502C42—1222no—
503C42—1222no—
504C42—2441no—
505C23—42no—
506C23—1222no—
507C23—2444no—
508C23—2441no—
509C23—2441no—
510C23C22⋊C4×D6111yescharacteristic
511C23#1451974911yescharacteristic
512C23#1451975011yescharacteristic
513C23#1451975011yescharacteristic
514C23#1451975011yescharacteristic
515C23#1451975211yescharacteristic
516C23#1451975511yescharacteristic
517Q8—1222no—
518Q8—1222no—
519Q8—2441no—
520Q8—2441no—
521D4—1222no—
522D4—1222no—
523D4—2442no—
524D4—2442no—
525D4—2442no—
526D4—2442no—
527D4—2441no—
528D4—2441no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—1222no—
537C2×C4—1224no—
538C2×C4—4882no—
539C2×C4—2442no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2444no—
547C2×C4C4⋊C4×D6111yescharacteristic
548C2×C4#1451973411yescharacteristic
549C2×C4#1451973711yescharacteristic
550C2×C4C4×C22⋊D6111yescharacteristic
551C2×C4#1451973911yescharacteristic
552C2×C4C4×D24411yescharacteristic
553C2×C4#1451974511yescharacteristic
554C2×C4#1451974711yescharacteristic
555C2×C4#1451974711yescharacteristic
556C2×C4#1451975011yescharacteristic
557C2×C4#1451975011yescharacteristic
558C2×C4#1451975111yescharacteristic
559C2×C4#1451975311yescharacteristic
560C2×C4#1451975511yescharacteristic
561C2×C4#1451975511yescharacteristic
562C2×C4#1451975511yescharacteristic
563C2×C4#1451975711yescharacteristic
564C2×C4#1451975911yescharacteristic
565C2×C4#1451976011yescharacteristic
566C2×C4#1451976211yescharacteristic
567C2×C4#1451976211yescharacteristic
568C2×C4#1451976211yescharacteristic
569C2×C4#1451976311yescharacteristic
570C2×C4#1451976311yescharacteristic
571C2×C4#1451976511yescharacteristic
572C2×C4#1451977111yescharacteristic
573C2×C4#1451977211yescharacteristic
574C2×C4#1451977311yescharacteristic
575C8—42no—
576C8—2442no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—4882no—
585C22—4882no—
586C22—2444no—
587C22—2442no—
588C22#1683962111yescharacteristic
589C22#1683964011yescharacteristic
590C22#1683966211yescharacteristic
591C22#1683967711yescharacteristic
592C22#1683968011yescharacteristic
593C22#1683968411yescharacteristic
594C22#1683968611yescharacteristic
595C4—44no—
596C4—82no—
597C4—4882no—
598C4—2444no—
599C4—2442no—
600C4#1683963311yescharacteristic
601C4#1683963611yescharacteristic
602C4#1683965111yescharacteristic
603C4#1683965311yescharacteristic
604C4#1683967111yescharacteristic
605C4#1683967511yescharacteristic
606C4#1683969111yescharacteristic
607C4#1683969711yescharacteristic
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—4882no—
615C2#2144406311yescharacteristic
616C2#2144620112yes—
subgroupG/H
centerC2×C4#14519747
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#339C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#339D61
hypocenterC61#11056
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
1. #21454382C2semidirect
5. #21445551C2semidirect
6. C2×#16839522C2semidirect
7. C2×C61×#339C2semidirect
21. #16839668C4semidirect
22. #16839664C4semidirect
306. C2×#339D61semidirect
339. C61#11056semidirect

Maximal subgroups

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

subgroupindex
#214543822
C2×#168397802
C2×#168397782
#214483092
#214455512
C2×#168395222
C2×C61×#3392
#1105661