Finite Group Databasebeta

← #29587146

Subgroups — #29587146

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519703C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic
3#21448407C211yesmaximal, characteristic
4C2×#16839627C211yesmaximal, characteristic, complemented
5C2×#16839530C211yesmaximal, characteristic, complemented
6C2×#16839521C211yesmaximal, characteristic, complemented
7C61×#1736C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11C22×C4×Dic61C2211yescharacteristic
12C2×#14519703—22no—
13C2×#14519703—22no—
14C2×#14519703C414yescomplemented
15#16839761—22no—
16#16839761—22no—
17C2×#14519700—21no—
18C2×#14519697—21no—
19#16839668—21no—
20#16839664—21no—
21#16839639—22no—
22#16839627—22no—
23#16839627—22no—
24#16839619—22no—
25C2×#14519684—22no—
26#16839530—22no—
27#16839521—22no—
28C22×M16×C61C2211yescharacteristic
29C2×#117×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2211yescharacteristic
31C61×#340—22no—
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×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222Q812yes—
44C22×Dic122D411yescharacteristic
45C22×D244D411yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61—21no—
48C23×Dic61D412yes—
49C22×C4×D61—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194D412yes—
56C22×#92418—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×#11758198—24no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2×C412yes—
68C2×C4×Dic61D412yes—
69C2×C4×Dic61D412yes—
70#14519703—28no—
71#14519700—24no—
72#14519697—24no—
73#14519684—28no—
74#14519684—24no—
75C23×C4×C61—21no—
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61C2×C412yes—
83C22×C8×C61—21no—
84#117×C61—21no—
85#117×C61—21no—
86#117×C61—21no—
87#117×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D412yes—
94C2×C4⋊C4×C61D412yes—
95C2×C22⋊C4×C61—21no—
96C2×C42×C61—21no—
97C24×C61C2×D411yescharacteristic
98C23×D61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—44no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61C22×C411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×Q811yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C2×C22⋊D61—22no—
119C2×C22⋊D61—22no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—42no—
132C22×Dic61—42no—
133C22×Dic61—44no—
134#11758222—22no—
135#11758222—22no—
136#11758222—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—22no—
141#11758222—22no—
142#11758222—41no—
143#11758222—41no—
144#11758222—41no—
145#11758222—41no—
146#11758222—41no—
147#11758222—41no—
148#11758222—41no—
149#11758222—41no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C4⋊C412yes—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122C4⋊C412yes—
168M16×C61—22no—
169M16×C61—22no—
170M16×C61—41no—
171M16×C61—41no—
172C2×C8×C61—22no—
173C2×C8×C61—41no—
174C2×C8×C61—41no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—41no—
184C42×C61—41no—
185#11758198—84no—
186C4⋊Dic61—41no—
187C4⋊Dic61—41no—
188C4⋊Dic61—41no—
189C4⋊Dic61—41no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61C22⋊C414yescomplemented
194#11758194—22no—
195#11758194—22no—
196#11758194—41no—
197#11758194—41no—
198C2×#92418—22no—
199C2×#92418—41no—
200C2×#92418—41no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61C4×D411yescharacteristic
204C23×C61#12011yescharacteristic
205C23×C61#12011yescharacteristic
206C23×C61#12111yescharacteristic
207C23×C61#12111yescharacteristic
208C23×C61C22⋊Q811yescharacteristic
209C23×C61#12411yescharacteristic
210C22×D61—22no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—42no—
215C2×C4×C61—44no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—82no—
221C2×C4×C61—82no—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C4⋊C411yescharacteristic
224C2×C4×C61#11711yescharacteristic
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22⋊Q811yescharacteristic
236C2×C4×C61C22⋊Q811yescharacteristic
237C2×C4×C61C22⋊Q811yescharacteristic
238C2×C4×C61C22⋊Q811yescharacteristic
239C2×C4×C61C22⋊Q811yescharacteristic
240C2×C4×C61#12311yescharacteristic
241C2×C4×C61#12311yescharacteristic
242C2×C4×C61#12311yescharacteristic
243C2×C4×C61#12311yescharacteristic
244C2×C4×C61#12311yescharacteristic
245C2×C4×C61#12411yescharacteristic
246C2×C4×C61#12411yescharacteristic
247C2×C4×C61#12611yescharacteristic
248C2×C4×C61#12611yescharacteristic
249C2×C4×C61#12711yescharacteristic
250C22⋊D61—42no—
251C22⋊D61—42no—
252C22⋊D61—42no—
253C22⋊D61—42no—
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—44no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—44no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—44no—
263C2×Dic61—42no—
264C2×Dic61—44no—
265C2×Dic61—82no—
266D244—22no—
267D244—22no—
268D244—41no—
269D244—41no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—42no—
273C4×D61—42no—
274Dic122—22no—
275Dic122—22no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—42no—
279#92418—42no—
280#11079—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61#38511yescharacteristic
290C22×C61#38611yescharacteristic
291C22×C61#38811yescharacteristic
292C22×C61#39211yescharacteristic
293C22×C61#39311yescharacteristic
294C22×C61#39411yescharacteristic
295C22×C61#39611yescharacteristic
296C22×C61#45412yes—
297C22×C61#45512yes—
298C2×D61—44no—
299C2×D61—42no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—82no—
303C4×C61—82no—
304C4×C61—82no—
305C4×C61#38011yescharacteristic
306C4×C61#38511yescharacteristic
307C4×C61#38811yescharacteristic
308C4×C61#39011yescharacteristic
309C4×C61#39411yescharacteristic
310C4×C61#39611yescharacteristic
311C4×C61#39711yescharacteristic
312C4×C61#39911yescharacteristic
313Dic61—44no—
314Dic61—42no—
315Dic61—82no—
316C22×#104—611no—
317C2×#417—611no—
318#2311—611no—
319C2×#382—611no—
320C2×#343—611no—
321#1736D6111yescharacteristic, complemented
322C2×#339—611no—
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61#143911yescharacteristic
328C2×C61#186012yes—
329C2×C61#200311yescharacteristic
330C2×C61#200411yescharacteristic
331C2×C61#204012yes—
332D61—82no—
333C22×#41—611no—
334C22×M16—611no—
335C22×M16C2×D6111yescharacteristic
336C2×#117C2×D6111yescharacteristic
337C22×C4⋊C4—611no—
338C22×C4⋊C4C2×D6111yescharacteristic
339C22×C42—611no—
340C2×#104—614no—
341C2×#104—1222no—
342C2×#104—1222no—
343#417—1222no—
344#417—1222no—
345C2×#103—1221no—
346C2×#102—1221no—
347#391—1221no—
348#390—1221no—
349#386—1222no—
350#382—1222no—
351#382—1222no—
352#380—1222no—
353C2×#95—1222no—
354#343—1222no—
355#340—22no—
356#339—1222no—
357C61#1107911yescharacteristic, complemented, Sylow
358C2×#41—612no—
359C2×#41—612no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41—1221no—
369C2×#41—1221no—
370C22×Q8—611no—
371C22×D4—611no—
372C22×D4—1221no—
373C23×C4—21no—
374C23×C4—612no—
375C23×C4—1221no—
376C23×C4—1221no—
377C23×C4C22×D6111yescharacteristic
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—612no—
384C2×M16—1221no—
385C2×M16—1221no—
386C2×M16—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16C4×D6112yes—
390C22×C8—21no—
391C22×C8—1221no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4D24412yes—
408C2×C4⋊C4C22⋊D6112yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1224no—
411C2×C42—21no—
412C2×C42—612no—
413C2×C42—612no—
414C2×C42—612no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42—1221no—
418#104—1228no—
419#103—1224no—
420#102—1224no—
421#95—1228no—
422#95—1224no—
423C24—1221no—
424C24D4×D6111yescharacteristic
425#41—1222no—
426#41—1222no—
427#41—1222no—
428#41—1222no—
429#41—1222no—
430#41—1222no—
431#41—1222no—
432#41—1222no—
433#41—2441no—
434#41—2441no—
435#41—2441no—
436#41—2441no—
437#41—2441no—
438#41—2441no—
439#41—2441no—
440#41—2441no—
441C2×Q8—612no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×D4—612no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—2441no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2441no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—44no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—2444no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4—2441no—
477C22×C4—2441no—
478C22×C4—2441no—
479C22×C4—2441no—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4C2×D24411yescharacteristic
482C22×C4#1175822211yescharacteristic
483C22×C4#1175822211yescharacteristic
484C22×C4#1175822211yescharacteristic
485C22×C4#1175822211yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4#1175822411yescharacteristic
491C22×C4Q8×D6111yescharacteristic
492C22×C4#1175822611yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494M16—22no—
495M16—22no—
496M16—41no—
497M16—41no—
498M16—1222no—
499M16—1222no—
500M16—2441no—
501M16—2441no—
502C2×C8—22no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—1222no—
506C2×C8—2441no—
507C2×C8—2441no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4—2441no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—4884no—
521C42—41no—
522C42—41no—
523C42—614no—
524C42—1222no—
525C42—1222no—
526C42—1222no—
527C23—22no—
528C23—22no—
529C23—1222no—
530C23—2442no—
531C23—2441no—
532C23—2441no—
533C23—2442no—
534C23#1451973911yescharacteristic
535C23#1451974511yescharacteristic
536C23#1451974711yescharacteristic
537C23#1451975011yescharacteristic
538C23#1451975011yescharacteristic
539C23#1451975511yescharacteristic
540C23#1451976311yescharacteristic
541Q8—1222no—
542Q8—1222no—
543Q8—2441no—
544Q8—2441no—
545D4—1222no—
546D4—1222no—
547D4—2442no—
548D4—2442no—
549D4—2442no—
550D4—2442no—
551D4—2441no—
552D4—2441no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—1222no—
561C2×C4—1224no—
562C2×C4—4882no—
563C2×C4—2444no—
564C2×C4—2444no—
565C2×C4—2442no—
566C2×C4—2444no—
567C2×C4—2444no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C4⋊C4×D6111yescharacteristic
578C2×C4#1451973411yescharacteristic
579C2×C4#1451973711yescharacteristic
580C2×C4C4×C22⋊D6111yescharacteristic
581C2×C4C4×D24411yescharacteristic
582C2×C4#1451974711yescharacteristic
583C2×C4#1451974911yescharacteristic
584C2×C4#1451975011yescharacteristic
585C2×C4#1451975011yescharacteristic
586C2×C4#1451975011yescharacteristic
587C2×C4#1451975111yescharacteristic
588C2×C4#1451975211yescharacteristic
589C2×C4#1451975311yescharacteristic
590C2×C4#1451975511yescharacteristic
591C2×C4#1451975511yescharacteristic
592C2×C4#1451975511yescharacteristic
593C2×C4#1451975711yescharacteristic
594C2×C4#1451975911yescharacteristic
595C2×C4#1451976011yescharacteristic
596C2×C4#1451976211yescharacteristic
597C2×C4#1451976211yescharacteristic
598C2×C4#1451976211yescharacteristic
599C2×C4#1451976311yescharacteristic
600C2×C4#1451976511yescharacteristic
601C2×C4#1451977111yescharacteristic
602C2×C4#1451977211yescharacteristic
603C2×C4#1451977311yescharacteristic
604C8—42no—
605C8—2442no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—4882no—
615C22—4881no—
616C22—4881no—
617C22—2444no—
618C22—2442no—
619C22#1683963311yescharacteristic
620C22#1683964011yescharacteristic
621C22#1683965111yescharacteristic
622C22#1683967111yescharacteristic
623C22#1683967511yescharacteristic
624C22#1683967711yescharacteristic
625C22#1683968611yescharacteristic
626C22#1683993312yes—
627C22#1683993812yes—
628C4—82no—
629C4—82no—
630C4—4882no—
631C4—2444no—
632C4—2442no—
633C4#1683962111yescharacteristic
634C4#1683963611yescharacteristic
635C4#1683965311yescharacteristic
636C4#1683966211yescharacteristic
637C4#1683968011yescharacteristic
638C4#1683968411yescharacteristic
639C4#1683969111yescharacteristic
640C4#1683969711yescharacteristic
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—4882no—
646C2#2144406311yescharacteristic
647C2#2144604912yes—
648C2#2144684211yescharacteristic
649C2#2144684911yescharacteristic
650C2#2144704612yes—
subgroupG/H
centerC23#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1736C2
socleC23×C61#120
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1736D61
hypocenterC61#11079
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. C2×#16839627C2semidirect
5. C2×#16839530C2semidirect
6. C2×#16839521C2semidirect
7. C61×#1736C2semidirect
14. C2×#14519703C4semidirect
193. C4×Dic61C22⋊C4semidirect
321. #1736D61semidirect
357. C61#11079semidirect

Maximal subgroups

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

subgroupindex
C22×#145197032
C2×#168397612
#214484072
C2×#168396272
C2×#168395302
C2×#168395212
C61×#17362
#1107961