Finite Group Databasebeta

← #29707392

Subgroups — #29707392

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic, complemented
2#21454339C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4#21448407C211yesmaximal, characteristic
5#21446979C214yesmaximal, complemented
6#21446974C214yesmaximal, complemented
7C2×#16839530C211yesmaximal, characteristic, complemented
8#21445529C211yesmaximal, characteristic, complemented
9C61×#1733C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840390—22no—
13#16840390—22no—
14#16840390—22no—
15#16840390C2214yes—
16C2×#14519777—21no—
17C2×#14519776—21no—
18C2×#14519769—21no—
19C2×#14519768—21no—
20C2×#14519754—22no—
21C2×#14519736C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23#16839772C2212yes—
24#16839770C2212yes—
25#16839768C2212yes—
26#16839766C2212yes—
27#16839764—21no—
28#16839764—21no—
29#16839761—21no—
30#16839761—21no—
31#16839759—22no—
32#16839756—22no—
33#16839668C2212yes—
34#16839664C2212yes—
35#16839639—22no—
36#16839619—22no—
37#16839530C2214yes—
38#16839522C2214yes—
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C61×#339C2214yes—
43C2×#11758222—22no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222Q814yescomplemented
51C22×Dic122—22no—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61D412yes—
56C22×C4×D61—21no—
57C2×#11758194—21no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194C2312yes—
61C22×#92418—21no—
62#14519777—24no—
63#14519777—24no—
64#14519776—24no—
65#14519776—24no—
66#14519769—24no—
67#14519769—24no—
68#14519768—24no—
69#14519768—24no—
70#14519754—28no—
71#14519754—28no—
72#14519736—21no—
73#14519736—21no—
74#14519736D412yes—
75#14519736C2312yes—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61D412yes—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61C2312yes—
85C2×#11758196—22no—
86C2×#11758196—22no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—24no—
89C2×C4×Dic61—21no—
90#14519702—22no—
91#14519702—22no—
92#14519700—22no—
93#14519700—22no—
94#14519699—22no—
95#14519699—22no—
96#14519697—22no—
97#14519697—22no—
98#14519684—24no—
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—21no—
101C2×M16×C61—22no—
102C2×M16×C61—22no—
103C2×M16×C61C2312yes—
104C22×C8×C61—21no—
105#117×C61—21no—
106#117×C61—21no—
107#117×C61—22no—
108#117×C61—22no—
109#117×C61D412yes—
110#117×C61C2312yes—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61D411yescharacteristic
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61C2312yes—
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×Q812yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—24no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—42no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—82no—
143C22×Dic61C2×D414yes—
144#11758222—24no—
145#11758222—24no—
146#11758222—24no—
147#11758222—24no—
148#11758222—42no—
149#11758222—42no—
150#11758222—42no—
151#11758222—42no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C2×Q812yescomplemented
158C2×C4×D61—24no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61—42no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—28no—
168C2×Dic122—84no—
169C2×Dic122C2×Q812yescomplemented
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—42no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61C2×D412yes—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—42no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189#11758198—84no—
190C4⋊Dic61—28no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C2×D412yes—
198C4⋊Dic61C2×D412yes—
199#11758196—28no—
200#11758196—28no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203#11758194—22no—
204#11758194—22no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61#12011yescharacteristic
211C23×C61#12011yescharacteristic
212C23×C61C22⋊Q812yes—
213C23×C61#12412yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—41no—
219C22×D61—41no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61C22⋊Q812yes—
230C2×C4×C61C22⋊Q812yes—
231C2×C4×C61C22⋊Q812yes—
232C2×C4×C61C22⋊Q812yes—
233C2×C4×C61C22⋊Q812yes—
234C2×C4×C61#12412yes—
235C2×C4×C61#12412yes—
236C2×C4×C61#12412yes—
237C2×C4×C61#12412yes—
238C2×C4×C61#12412yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×Q811yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C22⋊D61—44no—
246C22⋊D61—44no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—42no—
252C2×Dic61—44no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—44no—
257C2×Dic61—82no—
258D244—22no—
259D244—22no—
260D244—41no—
261D244—41no—
262C4×D61—44no—
263C4×D61—44no—
264Dic122—22no—
265Dic122—22no—
266Dic122—41no—
267Dic122—41no—
268Dic122—88no—
269C8×C61—42no—
270#92418—42no—
271#23462—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61#39312yes—
281C22×C61#39312yes—
282C22×C61C2×#12011yescharacteristic
283C22×C61C2×C22⋊Q811yescharacteristic
284C22×C61C2×#12411yescharacteristic
285C2×D61—44no—
286C2×D61—42no—
287C2×D61—82no—
288C2×D61—81no—
289C2×D61—81no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—82no—
293C4×C61#39312yes—
294C4×C61#39312yes—
295C4×C61C2×C22⋊Q811yescharacteristic
296C4×C61C2×C22⋊Q811yescharacteristic
297C4×C61C2×#12411yescharacteristic
298C4×C61C2×#12411yescharacteristic
299Dic61—42no—
300Dic61—42no—
301Dic61—44no—
302Dic61—42no—
303C2×#544—611no—
304#2896—611no—
305#2896—611no—
306#2311—611no—
307#2028—614no—
308#2027—614no—
309C2×#343—611no—
310#1733—611no—
311#1733D6111yescharacteristic, complemented
312C2×C61—21no—
313C2×C61—21no—
314C2×C61—22no—
315C2×C61—22no—
316C2×C61C2×#39311yescharacteristic
317C2×C61#301911yescharacteristic
318C2×C61#302011yescharacteristic
319D61—82no—
320C22×#41—611no—
321C22×M16—611no—
322C22×M16C2×D6111yescharacteristic
323#544—614no—
324#544—1222no—
325#544—1222no—
326#544—1222no—
327C2×C4⋊Q8—1221no—
328C2×C4⋊Q8—1221no—
329C2×#125—1221no—
330C2×#125—1221no—
331C2×C22⋊Q8—1222no—
332C2×#117—611no—
333C2×#117C2×D6111yescharacteristic
334C2×#117C2×D6111yescharacteristic
335C22×C4⋊C4—611no—
336#419—612no—
337#419—612no—
338#418—612no—
339#418—612no—
340#417—1221no—
341#417—1221no—
342#417—1221no—
343#417—1221no—
344#416—1222no—
345#416—1222no—
346#391—612no—
347#390—612no—
348#386—1222no—
349#380—1222no—
350#343—614no—
351#339—614no—
352#339C2×D6114yes—
353C61#2346211yescharacteristic, complemented, Sylow
354C2×#41—614no—
355C2×#41—1222no—
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1222no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1222no—
362C22×Q8—611no—
363C22×Q8—1222no—
364C22×D4—611no—
365C22×D4—1221no—
366C23×C4—612no—
367C23×C4—1221no—
368C23×C4C22×D6111yescharacteristic
369C2×M16—21no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—612no—
373C2×M16—1221no—
374C2×M16—1222no—
375C2×M16—1222no—
376C2×M16C22×D6112yes—
377C22×C8—21no—
378C22×C8—1221no—
379C4⋊Q8—1224no—
380C4⋊Q8—1224no—
381C4⋊Q8—1224no—
382C4⋊Q8—1224no—
383#125—1224no—
384#125—1224no—
385#125—1224no—
386#125—1224no—
387C22⋊Q8—1228no—
388C22⋊Q8—1228no—
389#117—21no—
390#117—21no—
391#117—22no—
392#117—22no—
393#117—612no—
394#117—612no—
395#117—1221no—
396#117—1221no—
397#117D24412yes—
398#117C22×D6112yes—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—612no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4D24411yescharacteristic
412C2×C4⋊C4D24411yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1224no—
417C2×C42—21no—
418C2×C42—1221no—
419C2×C42C22×D6112yes—
420#103—1222no—
421#103—1222no—
422#103—1222no—
423#103—1222no—
424#102—1222no—
425#102—1222no—
426#102—1222no—
427#102—1222no—
428#95—1224no—
429C24—1221no—
430C24D4×D6111yescharacteristic
431#41—1224no—
432#41—1224no—
433#41—1224no—
434#41—1224no—
435#41—2442no—
436#41—2442no—
437#41—2442no—
438#41—2442no—
439C2×Q8—612no—
440C2×Q8—1228no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—4884no—
447C2×D4—612no—
448C2×D4—1224no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2442no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—614no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—1224no—
467C22×C4—4882no—
468C22×C4—2441no—
469C22×C4—2441no—
470C22×C4—2442no—
471C22×C4C2×D24411yescharacteristic
472C22×C4C2×D24411yescharacteristic
473C22×C4#1175822212yes—
474C22×C4#1175822212yes—
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4D4×D6111yescharacteristic
478C22×C4#1175822412yes—
479C22×C4Q8×D6112yes—
480C22×C4C23×D6111yescharacteristic
481M16—22no—
482M16—22no—
483M16—42no—
484M16—1222no—
485M16—1222no—
486M16—2442no—
487C2×C8—22no—
488C2×C8—42no—
489C2×C8—1222no—
490C2×C8—2442no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—1228no—
497C4⋊C4—1228no—
498C4⋊C4—1228no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4C2×D24412yes—
506C4⋊C4C2×D24412yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—4884no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—1222no—
518C42—1222no—
519C23—41no—
520C23—41no—
521C23—1222no—
522C23—2442no—
523C23—2442no—
524C23—2441no—
525C23—2441no—
526C23#1451974711yescharacteristic
527C23#1451974711yescharacteristic
528C23#1451975512yes—
529C23#1451976312yes—
530C23C2×D4×D6111yescharacteristic
531Q8—1222no—
532Q8—1222no—
533Q8—4888no—
534Q8—2441no—
535Q8—2441no—
536D4—1222no—
537D4—1222no—
538D4—2444no—
539D4—2444no—
540D4—2441no—
541D4—2441no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1224no—
552C2×C4—1222no—
553C2×C4—4882no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2444no—
560C2×C4—2444no—
561C2×C4—2444no—
562C2×C4#1451974711yescharacteristic
563C2×C4#1451974711yescharacteristic
564C2×C4#1451975512yes—
565C2×C4#1451975512yes—
566C2×C4#1451975512yes—
567C2×C4#1451975712yes—
568C2×C4#1451975712yes—
569C2×C4#1451976312yes—
570C2×C4#1451976312yes—
571C2×C4#1451976312yes—
572C2×C4#1451976512yes—
573C2×C4#1451976512yes—
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C22×D24411yescharacteristic
576C2×C4C2×Q8×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822211yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C8—42no—
581C8—2442no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—42no—
589C22—42no—
590C22—4882no—
591C22—4881no—
592C22—4881no—
593C22—2444no—
594C22—2442no—
595C22#1683967512yes—
596C22#1683967512yes—
597C22C2×#1451974711yescharacteristic
598C22C2×#1451975511yescharacteristic
599C22C2×#1451976311yescharacteristic
600C4—42no—
601C4—42no—
602C4—82no—
603C4—2442no—
604C4—2444no—
605C4—2442no—
606C4—2442no—
607C4#1683967512yes—
608C4#1683967512yes—
609C4C2×#1451975511yescharacteristic
610C4C2×#1451975711yescharacteristic
611C4C2×#1451976311yescharacteristic
612C4C2×#1451976511yescharacteristic
613C2—21no—
614C2—21no—
615C2—22no—
616C2—22no—
617C2—4882no—
618C2C2×#1683967511yescharacteristic
619C2#2145517811yescharacteristic
620C2#2145518511yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
hypercenter#1733D61
hypocenterC61#23462
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. C2×#16840390C2semidirect
5. #21446979C2semidirect
6. #21446974C2semidirect
7. C2×#16839530C2semidirect
8. #21445529C2semidirect
9. C61×#1733C2semidirect
50. C2×#11758222Q8semidirect
157. C2×D244C2×Q8semidirect
169. C2×Dic122C2×Q8semidirect
311. #1733D61semidirect
353. C61#23462semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
#214543392
#214543362
#214484072
#214469792
#214469742
C2×#168395302
#214455292
C61×#17332
#2346261