Finite Group Databasebeta

← #29716326

Subgroups — #29716326

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455620C211yesmaximal, characteristic, complemented
2C4×#14519807C211yesmaximal, characteristic
3#21451035C211yesmaximal, characteristic, complemented
4#21447166C212yesmaximal, complemented
5#21446977C212yesmaximal, complemented
6#21446809C211yesmaximal, characteristic, complemented
7#21446764C211yesmaximal, characteristic, complemented
8#21446188C212yesmaximal
9#21446052C212yesmaximal
10#21445839C211yesmaximal, characteristic
11C61×#3503C211yesmaximal, characteristic
12C2×#14519807C2211yescharacteristic
13C2×C4×Dic122C2211yescharacteristic
14#16840066—22no—
15#16840054—21no—
16#16840044—21no—
17#16839867—21no—
18#16839862—21no—
19C4×#11758202—21no—
20C4×#11758200—21no—
21#16839805C2212yes—
22#16839805C2211yescharacteristic
23#16839786C2211yescharacteristic
24C2×#14519704C2212yes—
25#16839770C2212yes—
26#16839770C2211yescharacteristic
27#16839763C2211yescharacteristic
28C4×#11758194C2211yescharacteristic
29#16839700—22no—
30#16839669—22no—
31#16839667C2212yes—
32#16839627C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839531C2212yes—
35#16839531C2212yes—
36#16839521C2212yes—
37#16839521C2212yes—
38#16839521C2212yes—
39Q82×C61—21no—
40C61×#557—21no—
41C61×#554—21no—
42C61×#554—22no—
43D4×Q8×C61—22no—
44D4×Q8×C61—21no—
45C61×#544—22no—
46C61×#544C2212yes—
47C61×#533—21no—
48C61×#533C2211yescharacteristic
49C2×C4⋊Q8×C61—21no—
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520C2212yes—
52C4×#41×C61C2211yescharacteristic
53C22×Dic122C2311yescharacteristic
54#14519807—22no—
55#14519807Q814yescomplemented
56C2×#11758202—21no—
57C2×#11758200—21no—
58C2×#11758194C2311yescharacteristic
59C2×#11758194C2312yes—
60C2×#11758194C2311yescharacteristic
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61C2312yes—
70#14519710—22no—
71#14519710—21no—
72#14519708—22no—
73#14519708—21no—
74C4⋊#92418—21no—
75C4⋊#92418—21no—
76#14519704—24no—
77#14519701—21no—
78#14519700—22no—
79#14519700—22no—
80#14519700—21no—
81#14519700—21no—
82#14519698—21no—
83#14519688—21no—
84C4×#92418—21no—
85#14519684—21no—
86#14519684—24no—
87#14519684—24no—
88C2×#41×C61C2311yescharacteristic
89C22×Q8×C61—22no—
90C22×Q8×C61D412yes—
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—24no—
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61D411yescharacteristic
101C4⋊Q8×C61D411yescharacteristic
102#125×C61—24no—
103#125×C61—22no—
104#125×C61—22no—
105#125×C61D412yes—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—24no—
109C22⋊Q8×C61—24no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—24no—
113C4×Q8×C61—22no—
114C4×Q8×C61—22no—
115C4×Q8×C61—22no—
116C4×Q8×C61—21no—
117C4×Q8×C61—21no—
118C4×Q8×C61D411yescharacteristic
119C4×Q8×C61D411yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—24no—
123C4×D4×C61—21no—
124C4×D4×C61D411yescharacteristic
125C4×D4×C61D411yescharacteristic
126#117×C61—22no—
127#117×C61—21no—
128#117×C61C2312yes—
129#117×C61C2311yescharacteristic
130C2×C4⋊C4×C61—22no—
131C2×C4⋊C4×C61—22no—
132C2×C4⋊C4×C61D412yes—
133C2×C4⋊C4×C61C2312yes—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136#41×C61—21no—
137#41×C61—21no—
138#41×C61C2×Q812yes—
139#41×C61#4112yes—
140C2×Q8×C61—24no—
141C2×Q8×C61—21no—
142C2×Q8×C61—22no—
143C2×Q8×C61—21no—
144C2×Q8×C61—22no—
145C2×Q8×C61—44no—
146C2×Q8×C61—84no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—24no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D414yes—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×Q812yescomplemented
167C4⋊C4×C61—24no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C2×D412yes—
182C4⋊C4×C61C2×D412yes—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—24no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C42×C61—24no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—22no—
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199#11758202—24no—
200#11758200—24no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—41no—
209C4×Dic61—41no—
210#11758194—21no—
211#11758194—22no—
212#11758194—21no—
213#11758194—21no—
214#11758194—22no—
215#11758194C2×Q812yescomplemented
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—22no—
219C23×C61—21no—
220C23×C61C22×D411yescharacteristic
221Q8×C61—21no—
222Q8×C61—42no—
223Q8×C61—44no—
224Q8×C61—42no—
225Q8×C61—82no—
226Q8×C61—84no—
227Q8×C61C22⋊Q812yes—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—21no—
231D4×C61C22⋊Q812yes—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—24no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—44no—
245C2×C4×C61—84no—
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264Dic122—22no—
265Dic122—22no—
266Dic122—41no—
267Dic122—41no—
268#92418—22no—
269#92418—41no—
270#92418—42no—
271#24322—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—22no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C2×#12011yescharacteristic
278C22×C61#54811yescharacteristic
279C22×C61D4×Q811yescharacteristic
280C4×C61—22no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—41no—
285C4×C61—44no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—42no—
289C4×C61#45512yes—
290C4×C61C2×C22⋊Q811yescharacteristic
291C4×C61#53611yescharacteristic
292C4×C61#54811yescharacteristic
293C4×C61D4×Q811yescharacteristic
294Dic61—42no—
295Dic61—82no—
296Dic61—81no—
297#3503D6111yescharacteristic
298#3078—611no—
299C4×#137—611no—
300#2568—611no—
301#2061—612no—
302#2028—612no—
303#1997—611no—
304#1986—611no—
305#1882—612no—
306#1860—612no—
307#1806—611no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—42no—
311C2×C61#240911yescharacteristic
312C2×C61C2×#45511yescharacteristic
313C2×C61#302011yescharacteristic
314C2×#137—611no—
315Q82—21no—
316#557—21no—
317#554—21no—
318#554—22no—
319D4×Q8—22no—
320D4×Q8—21no—
321#544—22no—
322#544C2×D6112yes—
323#533—21no—
324#533C2×D6111yescharacteristic
325C2×C4⋊Q8—21no—
326C2×C4⋊Q8C2×D6111yescharacteristic
327#520C2×D6112yes—
328C4×#41C2×D6111yescharacteristic
329C2×C4×Q8—611no—
330#479—1222no—
331#477—1221no—
332#475—1221no—
333#441—1221no—
334#440—1221no—
335C4×Q16—1221no—
336C4×SD16—1221no—
337#428—612no—
338#428—611no—
339#423—611no—
340C2×#104—612no—
341#419—612no—
342#419—611no—
343#417—611no—
344C4×M16—611no—
345#400—1222no—
346#391—612no—
347#391—1222no—
348#382—612no—
349C4×C4⋊C4—611no—
350#343—612no—
351#343—612no—
352#339—612no—
353#339—612no—
354#339—612no—
355C61#2432211yescharacteristic, complemented, Sylow
356C2×#41C22×D6111yescharacteristic
357C22×Q8—22no—
358C22×Q8—611no—
359C22×Q8C22⋊D6112yes—
360#137—614no—
361#137—1222no—
362C2×Q16—1221no—
363C2×SD16—1221no—
364C2×M16—611no—
365C2×M16—611no—
366C2×M16—612no—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C4⋊Q8—24no—
370C4⋊Q8—22no—
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8C22⋊D6111yescharacteristic
377C4⋊Q8C22⋊D6111yescharacteristic
378#125—24no—
379#125—22no—
380#125—22no—
381#125C22⋊D6112yes—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—24no—
385C22⋊Q8—24no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—24no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×Q8—1221no—
398C4×Q8C22⋊D6111yescharacteristic
399C4×Q8C22⋊D6111yescharacteristic
400C4×D4—22no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—24no—
404C4×D4C22⋊D6111yescharacteristic
405C4×D4C22⋊D6111yescharacteristic
406#117—22no—
407#117—21no—
408#117C22×D6111yescharacteristic
409#117C22×D6112yes—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22⋊D6112yes—
416C2×C4⋊C4C22×D6112yes—
417C2×C42—21no—
418C2×C42—612no—
419C2×C42—1221no—
420C2×C42C22×D6111yescharacteristic
421#107—1221no—
422#107—1222no—
423#106—1221no—
424#106—1222no—
425C4⋊C8—1221no—
426C4⋊C8—1221no—
427#104—1224no—
428#103—1221no—
429#103—1221no—
430#103—1221no—
431#103—1222no—
432#103—1222no—
433#102—1221no—
434#97—1221no—
435C4×C8—1221no—
436#95—1221no—
437#95—1224no—
438#95—1224no—
439#41—21no—
440#41—21no—
441#41Q8×D6112yes—
442#41#1175822612yes—
443C2×Q8—24no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—44no—
449C2×Q8—84no—
450C2×Q8—612no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8C2×C22⋊D6111yescharacteristic
457C2×D4—21no—
458C2×D4C2×C22⋊D6111yescharacteristic
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6114yes—
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470Q16—1224no—
471SD16—1224no—
472M16—612no—
473M16—1221no—
474M16—1222no—
475M16—1221no—
476M16—1221no—
477M16—1222no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1222no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2442no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4C2×C22⋊D6112yes—
501C4⋊C4C2×C22⋊D6111yescharacteristic
502C4⋊C4C2×C22⋊D6112yes—
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C42—24no—
510C42—21no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—1222no—
515C42—1222no—
516C42—2441no—
517C42—2441no—
518C42C2×C22⋊D6111yescharacteristic
519C42C2×C22⋊D6111yescharacteristic
520C42C2×C22⋊D6111yescharacteristic
521C42C2×C22⋊D6111yescharacteristic
522C23—21no—
523C23C22×C22⋊D6111yescharacteristic
524Q8—21no—
525Q8—42no—
526Q8—44no—
527Q8—42no—
528Q8—82no—
529Q8—84no—
530Q8—1222no—
531Q8—1222no—
532Q8—2441no—
533Q8—2441no—
534Q8#1451975312yes—
535D4—22no—
536D4—22no—
537D4—21no—
538D4#1451975312yes—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—84no—
553C2×C4—1222no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4#1451974611yescharacteristic
562C2×C4#1451974611yescharacteristic
563C2×C4#1451974611yescharacteristic
564C2×C4#1451974611yescharacteristic
565C2×C4C22×C22⋊D6111yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4C2×Q8×D6111yescharacteristic
568C2×C4C2×#1175822611yescharacteristic
569C2×C4#1451982511yescharacteristic
570C2×C4#1451982511yescharacteristic
571C8—1222no—
572C8—2441no—
573C8—2442no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22C2×#1451974611yescharacteristic
580C22#1684042311yescharacteristic
581C22Q8×C22⋊D6111yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—22no—
586C4—41no—
587C4—44no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—4882no—
592C4—4881no—
593C4—2442no—
594C4#1683994012yes—
595C4C2×#1451975311yescharacteristic
596C4#1684033611yescharacteristic
597C4#1684042311yescharacteristic
598C4Q8×C22⋊D6111yescharacteristic
599C2—21no—
600C2—21no—
601C2—42no—
602C2#2144917811yescharacteristic
603C2C2×#1683994011yescharacteristic
604C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3503C2
socleC22×C61C2×#120
prime radicalC2×#41×C61C23
hypercenter#3503D61
hypocenterC61#24322
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #21455620C2semidirect
3. #21451035C2semidirect
4. #21447166C2semidirect
5. #21446977C2semidirect
6. #21446809C2semidirect
7. #21446764C2semidirect
55. #14519807Q8semidirect
166. C2×Dic122C2×Q8semidirect
215. #11758194C2×Q8semidirect
355. C61#24322semidirect

Maximal subgroups

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

subgroupindex
#214556202
C4×#145198072
#214510352
#214471662
#214469772
#214468092
#214467642
#214461882
#214460522
#214458392
C61×#35032
#2432261