Finite Group Databasebeta

← #29587031

Subgroups — #29587031

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519705C211yesmaximal, characteristic
2#21454339C211yesmaximal, characteristic
3C2×#16839735C211yesmaximal, characteristic
4#21448302C211yesmaximal, characteristic, complemented
5C2×#16839534C211yesmaximal, characteristic, complemented
6#21445529C211yesmaximal, characteristic, complemented
7C2×C61×#339C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12C2×#14519705—22no—
13C2×#14519705—22no—
14C2×#14519705—22no—
15#16839772C412yescomplemented
16#16839768C412yescomplemented
17#16839764—22no—
18#16839759—22no—
19#16839735—22no—
20#16839735—22no—
21C2×#14519691—22no—
22#16839660—22no—
23#16839635—22no—
24C4×#11758196—22no—
25#16839534—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#14519736C2×C412yes—
61#14519736D412yes—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D412yes—
64C2×#11758196—24no—
65C2×C22⋊Dic61—21no—
66C2×C22⋊Dic61D412yes—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69#14519705—28no—
70#14519702—24no—
71#14519699—24no—
72#14519691—28no—
73#14519684—24no—
74C23×C4×C61—21no—
75C23×C4×C61C2311yescharacteristic
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61C2×C412yes—
82C22×C8×C61—21no—
83#117×C61—21no—
84#117×C61—21no—
85#117×C61—21no—
86#117×C61D412yes—
87C2×C4⋊C4×C61—21no—
88C2×C22⋊C4×C61—21no—
89C2×C42×C61—21no—
90C2×C42×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61—21no—
93C2×C42×C61—21no—
94C2×C42×C61—21no—
95C2×C42×C61D412yes—
96C24×C61C2×D411yescharacteristic
97C23×D61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—44no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61C22×C411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×Q811yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×C4×C61#4111yescharacteristic
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—22no—
125C22×Dic61—22no—
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×D244C4⋊C412yescomplemented
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×Dic122C4⋊C412yescomplemented
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—41no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—41no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—22no—
174C42×C61—41no—
175C42×C61—41no—
176C42×C61—41no—
177C22⋊Dic61—22no—
178C22⋊Dic61—22no—
179C22⋊Dic61—22no—
180C22⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—41no—
183C4⋊Dic61—41no—
184C4⋊Dic61C22⋊C414yes—
185#11758196—48no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—41no—
189C4×Dic61—41no—
190#11758194—22no—
191#11758194—22no—
192#11758194—41no—
193#11758194—41no—
194C2×#92418—22no—
195C2×#92418—41no—
196C2×#92418—41no—
197C23×C61—42no—
198C23×C61C4×D411yescharacteristic
199C23×C61#12011yescharacteristic
200C23×C61#12011yescharacteristic
201C23×C61#12111yescharacteristic
202C23×C61#12111yescharacteristic
203C23×C61C22⋊Q811yescharacteristic
204C23×C61#12411yescharacteristic
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—82no—
209C2×C4×C61—24no—
210C2×C4×C61—42no—
211C2×C4×C61—44no—
212C2×C4×C61—44no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—82no—
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61C2×C4⋊C411yescharacteristic
218C2×C4×C61#11711yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61#12011yescharacteristic
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×C61C22⋊Q811yescharacteristic
230C2×C4×C61C22⋊Q811yescharacteristic
231C2×C4×C61C22⋊Q811yescharacteristic
232C2×C4×C61C22⋊Q811yescharacteristic
233C2×C4×C61C22⋊Q811yescharacteristic
234C2×C4×C61#12311yescharacteristic
235C2×C4×C61#12311yescharacteristic
236C2×C4×C61#12311yescharacteristic
237C2×C4×C61#12311yescharacteristic
238C2×C4×C61#12311yescharacteristic
239C2×C4×C61#12411yescharacteristic
240C2×C4×C61#12411yescharacteristic
241C2×C4×C61#12611yescharacteristic
242C2×C4×C61#12611yescharacteristic
243C2×C4×C61#12711yescharacteristic
244C22⋊D61—42no—
245C22⋊D61—42no—
246C22⋊D61—42no—
247C22⋊D61—42no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—44no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—44no—
254C2×Dic61—42no—
255D244—22no—
256D244—22no—
257D244—41no—
258D244—41no—
259C4×D61—42no—
260C4×D61—42no—
261C4×D61—42no—
262C4×D61—42no—
263Dic122—22no—
264Dic122—22no—
265Dic122—41no—
266Dic122—41no—
267C8×C61—42no—
268#92418—42no—
269#11060—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61#38511yescharacteristic
278C22×C61#38611yescharacteristic
279C22×C61#38811yescharacteristic
280C22×C61#39211yescharacteristic
281C22×C61#39311yescharacteristic
282C22×C61#39411yescharacteristic
283C22×C61#39611yescharacteristic
284C2×D61—42no—
285C2×D61—82no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—44no—
289C4×C61—82no—
290C4×C61#38011yescharacteristic
291C4×C61#38511yescharacteristic
292C4×C61#38811yescharacteristic
293C4×C61#39011yescharacteristic
294C4×C61#39411yescharacteristic
295C4×C61#39611yescharacteristic
296C4×C61#39711yescharacteristic
297C4×C61#39911yescharacteristic
298Dic61—44no—
299Dic61—42no—
300Dic61—82no—
301C22×#104—611no—
302#2896—611no—
303C2×#410—611no—
304#2296—611no—
305C2×#344—611no—
306#1733—611no—
307C2×#339D6111yescharacteristic, complemented
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—21no—
313C2×C61—21no—
314C2×C61#143911yescharacteristic
315C2×C61#190612yes—
316D61—82no—
317C22×#41—611no—
318C22×M16—611no—
319C22×M16C2×D6111yescharacteristic
320C2×#117—611no—
321C2×#117—611no—
322C2×#117C2×D6111yescharacteristic
323C22×C42C2×D6111yescharacteristic
324C2×#104—1222no—
325C2×#104—1222no—
326C2×#104—1222no—
327#419—612no—
328#418—612no—
329#417—1222no—
330#416—1222no—
331#410—1222no—
332#410—1222no—
333C2×#99—1222no—
334#389—1222no—
335#385—1222no—
336C4×C4⋊C4—1222no—
337#344—1222no—
338#339—22no—
339#339—1222no—
340C61#1106011yescharacteristic, 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—21no—
357C23×C4—1221no—
358C23×C4C22×D6111yescharacteristic
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—612no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C2×M16—1221no—
370C2×M16C4×D6112yes—
371C22×C8—21no—
372C22×C8—1221no—
373#117—21no—
374#117—21no—
375#117—21no—
376#117—612no—
377#117—612no—
378#117—1221no—
379#117—1221no—
380#117—1221no—
381#117—1221no—
382#117—1221no—
383#117—1221no—
384#117C22⋊D6112yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1221no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—612no—
391C2×C22⋊C4—1221no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42D24412yes—
401#104—1228no—
402#103—1224no—
403#102—1224no—
404#99—1228no—
405#95—1224no—
406C24—1221no—
407C24D4×D6111yescharacteristic
408#41—1222no—
409#41—1222no—
410#41—1222no—
411#41—1222no—
412#41—1222no—
413#41—1222no—
414#41—1222no—
415#41—1222no—
416#41—2441no—
417#41—2441no—
418#41—2441no—
419#41—2441no—
420#41—2441no—
421#41—2441no—
422#41—2441no—
423#41—2441no—
424C2×Q8—612no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×D4—612no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—2441no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
442C22×C4—22no—
443C22×C4—44no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—1221no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—2441no—
453C22×C4—2441no—
454C22×C4—2441no—
455C22×C4—2441no—
456C22×C4C2×C4×D6111yescharacteristic
457C22×C4C2×D24411yescharacteristic
458C22×C4#1175822211yescharacteristic
459C22×C4#1175822211yescharacteristic
460C22×C4#1175822211yescharacteristic
461C22×C4#1175822211yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4#1175822411yescharacteristic
467C22×C4Q8×D6111yescharacteristic
468C22×C4#1175822611yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470M16—22no—
471M16—22no—
472M16—41no—
473M16—41no—
474M16—1222no—
475M16—1222no—
476M16—2441no—
477M16—2441no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—1222no—
482C2×C8—2441no—
483C2×C8—2441no—
484C4⋊C4—22no—
485C4⋊C4—41no—
486C4⋊C4—614no—
487C4⋊C4—1222no—
488C4⋊C4—2448no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C22⋊C4—22no—
492C22⋊C4—41no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—1222no—
504C42—1222no—
505C42—2441no—
506C42—2441no—
507C23—42no—
508C23—1222no—
509C23—4882no—
510C23—2441no—
511C23—2441no—
512C23#1451973911yescharacteristic
513C23#1451974511yescharacteristic
514C23#1451974711yescharacteristic
515C23#1451975011yescharacteristic
516C23#1451975011yescharacteristic
517C23#1451975511yescharacteristic
518C23#1451976311yescharacteristic
519Q8—1222no—
520Q8—1222no—
521Q8—2441no—
522Q8—2441no—
523D4—1222no—
524D4—1222no—
525D4—2442no—
526D4—2442no—
527D4—2442no—
528D4—2442no—
529D4—2441no—
530D4—2441no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—1222no—
539C2×C4—1224no—
540C2×C4—2444no—
541C2×C4—2442no—
542C2×C4—2444no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4C22⋊C4×D6111yescharacteristic
550C2×C4C4⋊C4×D6111yescharacteristic
551C2×C4#1451973411yescharacteristic
552C2×C4#1451973711yescharacteristic
553C2×C4C4×C22⋊D6111yescharacteristic
554C2×C4C4×D24411yescharacteristic
555C2×C4#1451974711yescharacteristic
556C2×C4#1451974911yescharacteristic
557C2×C4#1451975011yescharacteristic
558C2×C4#1451975011yescharacteristic
559C2×C4#1451975011yescharacteristic
560C2×C4#1451975111yescharacteristic
561C2×C4#1451975211yescharacteristic
562C2×C4#1451975311yescharacteristic
563C2×C4#1451975511yescharacteristic
564C2×C4#1451975511yescharacteristic
565C2×C4#1451975511yescharacteristic
566C2×C4#1451975711yescharacteristic
567C2×C4#1451975911yescharacteristic
568C2×C4#1451976011yescharacteristic
569C2×C4#1451976211yescharacteristic
570C2×C4#1451976211yescharacteristic
571C2×C4#1451976211yescharacteristic
572C2×C4#1451976311yescharacteristic
573C2×C4#1451976511yescharacteristic
574C2×C4#1451977111yescharacteristic
575C2×C4#1451977211yescharacteristic
576C2×C4#1451977311yescharacteristic
577C8—42no—
578C8—2442no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—4882no—
587C22—4882no—
588C22—4882no—
589C22—2442no—
590C22#1683963311yescharacteristic
591C22#1683964011yescharacteristic
592C22#1683965111yescharacteristic
593C22#1683967111yescharacteristic
594C22#1683967511yescharacteristic
595C22#1683967711yescharacteristic
596C22#1683968611yescharacteristic
597C4—44no—
598C4—82no—
599C4—4882no—
600C4—2444no—
601C4—2442no—
602C4#1683962111yescharacteristic
603C4#1683963611yescharacteristic
604C4#1683965311yescharacteristic
605C4#1683966211yescharacteristic
606C4#1683968011yescharacteristic
607C4#1683968411yescharacteristic
608C4#1683969111yescharacteristic
609C4#1683969711yescharacteristic
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—4882no—
617C2#2144406311yescharacteristic
618C2#2144631912yes—
subgroupG/H
centerC2×C4#14519752
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#339C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#339D61
hypocenterC61#11060
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
4. #21448302C2semidirect
5. C2×#16839534C2semidirect
6. #21445529C2semidirect
7. C2×C61×#339C2semidirect
15. #16839772C4semidirect
16. #16839768C4semidirect
147. C2×D244C4⋊C4semidirect
159. C2×Dic122C4⋊C4semidirect
307. C2×#339D61semidirect
340. C61#11060semidirect

Maximal subgroups

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

subgroupindex
C22×#145197052
#214543392
C2×#168397352
#214483022
C2×#168395342
#214455292
C2×C61×#3392
#1106061