Finite Group Databasebeta

← #29707620

Subgroups — #29707620

Order 15616.

1586 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#21454408C211yesmaximal, characteristic, complemented
2C2×#16839779C211yesmaximal, characteristic, complemented
3#21454338C211yesmaximal, characteristic, complemented
4C22×#14519694C211yesmaximal, characteristic, complemented
5#21448401C211yesmaximal, characteristic, complemented
6#21447026C218yesmaximal, complemented
7#21445528C211yesmaximal, characteristic, complemented
8C2×C61×#535C211yesmaximal, characteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13#16839805C2214yescomplemented
14#16839802—22no—
15#16839798—22no—
16#16839779—24no—
17C2×#14519704C2214yescomplemented
18#16839771—21no—
19#16839767—21no—
20#16839763C2214yescomplemented
21#16839758—22no—
22C2×#14519694—24no—
23C2×#14519694C2214yescomplemented
24#16839665—22no—
25#16839665C2214yescomplemented
26#16839618—22no—
27#16839521C2214yescomplemented
28C22×#41×C61C2211yescharacteristic
29C23×D4×C61C2211yescharacteristic
30C61×#535—22no—
31C61×#535—22no—
32C61×#535—22no—
33C61×#535C2214yes—
34C2×#127×C61—21no—
35C2×#124×C61—21no—
36C2×#121×C61—22no—
37C2×#120×C61—22no—
38C2×#117×C61C2211yescharacteristic
39C2×#11758194—21no—
40C2×#11758194—22no—
41C2×#11758194—21no—
42C2×#11758194—22no—
43C2×#11758194—22no—
44C2×#11758194D414yescomplemented
45C2×#11758194C2312yes—
46C2×#11758194C2312yes—
47C22×#92418—21no—
48C22×#92418—21no—
49#14519736—22no—
50#14519736—22no—
51#14519736—21no—
52#14519736—21no—
53#14519736—21no—
54#14519736—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2312yes—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—28no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61C2312yes—
62#14519710—24no—
63#14519708—24no—
64#14519704—28no—
65#14519701—24no—
66#14519698—24no—
67#14519694—28no—
68C25×C61D412yes—
69C2×#41×C61—21no—
70C2×#41×C61—21no—
71C2×#41×C61—22no—
72C2×#41×C61—22no—
73C2×#41×C61—22no—
74C2×#41×C61—22no—
75C2×#41×C61D412yes—
76C2×#41×C61C2312yes—
77C22×Q8×C61D411yescharacteristic
78C22×D4×C61—21no—
79C22×D4×C61—24no—
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—22no—
84C22×D4×C61—21no—
85C22×D4×C61—22no—
86C22×D4×C61—22no—
87C22×D4×C61—22no—
88C22×D4×C61D412yes—
89C22×D4×C61D411yescharacteristic
90C22×D4×C61C2312yes—
91C23×C4×C61—21no—
92C23×C4×C61C2311yescharacteristic
93#127×C61—24no—
94#127×C61—24no—
95#124×C61—24no—
96#124×C61—24no—
97#121×C61—28no—
98#121×C61—28no—
99#120×C61—28no—
100#120×C61—28no—
101#117×C61—21no—
102#117×C61—21no—
103#117×C61D412yes—
104#117×C61C2312yes—
105C2×C4⋊C4×C61D412yes—
106C2×C22⋊C4×C61—24no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C24×C61—22no—
110C24×C61—21no—
111C24×C61—22no—
112C24×C61—24no—
113C24×C61—24no—
114C24×C61—84no—
115C24×C61C2×D414yes—
116C24×C61C2×D411yescharacteristic
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—22no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61—42no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61C2×D412yes—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—216no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—24no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—42no—
156C2×D4×C61—42no—
157C2×D4×C61—88no—
158C2×D4×C61C2×D412yes—
159C22×C4×C61—22no—
160C22×C4×C61—21no—
161C22×C4×C61—22no—
162C22×C4×C61—22no—
163C22×C4×C61—42no—
164C22×C4×C61—42no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D412yes—
171C22×C4×C61#4112yes—
172C22×C4×C61#4112yes—
173C22×C4×C61#4112yes—
174C22×C4×C61C2411yescharacteristic
175C22×Dic61—22no—
176C22×Dic61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61C2×D414yes—
179C22⋊C4×C61—216no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C42×C61—22no—
183C42×C61—22no—
184C22⋊Dic61—41no—
185C22⋊Dic61—41no—
186C22⋊Dic61—42no—
187C22⋊Dic61—88no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—42no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—41no—
194C4×Dic61—41no—
195#11758194—24no—
196#11758194—22no—
197#11758194—22no—
198#11758194—22no—
199#11758194—42no—
200#11758194C2×D414yescomplemented
201C2×#92418—24no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—42no—
205C23×C61—24no—
206C23×C61—22no—
207C23×C61—24no—
208C23×C61—24no—
209C23×C61—42no—
210C23×C61—48no—
211C23×C61—44no—
212C23×C61—48no—
213C23×C61—42no—
214C23×C61—44no—
215C23×C61—44no—
216C23×C61—84no—
217C23×C61—88no—
218C23×C61—88no—
219C23×C61#12011yescharacteristic
220C23×C61#12011yescharacteristic
221C23×C61#12112yes—
222C23×C61#12312yes—
223C23×C61C22×D411yescharacteristic
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—42no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—44no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—44no—
237D4×C61—44no—
238D4×C61—42no—
239D4×C61—816no—
240C2×C4×C61—24no—
241C2×C4×C61—22no—
242C2×C4×C61—44no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—44no—
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61#12112yes—
254C2×C4×C61#12312yes—
255C2×C4×C61#12312yes—
256C2×C4×C61#12312yes—
257C2×C4×C61#12412yes—
258C2×C4×C61#12412yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×Dic61—44no—
266C2×Dic61—44no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269#92418—24no—
270#92418—42no—
271#23493—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—48no—
278C22×C61—42no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61—48no—
282C22×C61—84no—
283C22×C61—84no—
284C22×C61—88no—
285C22×C61—88no—
286C22×C61#39412yes—
287C22×C61#39412yes—
288C22×C61C2×#12011yescharacteristic
289C22×C61C2×#12111yescharacteristic
290C22×C61C2×#12311yescharacteristic
291C4×C61—42no—
292C4×C61—44no—
293C4×C61#39412yes—
294C4×C61#39412yes—
295C4×C61C2×#12111yescharacteristic
296C4×C61C2×#12111yescharacteristic
297C4×C61C2×#12311yescharacteristic
298C4×C61C2×#12411yescharacteristic
299Dic61—82no—
300Dic61—82no—
301C2×#535D6111yescharacteristic
302#2911—611no—
303C2×#421—611no—
304#2896—611no—
305C22×#100—611no—
306#2309—611no—
307#2037—618no—
308#1733—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—22no—
312C2×C61—22no—
313C2×C61—84no—
314C2×C61—82no—
315C2×C61C2×#39411yescharacteristic
316C2×C61#301912yes—
317C22×#41C2×D6111yescharacteristic
318C23×D4C2×D6111yescharacteristic
319C22×M16—611no—
320C22×M16—611no—
321#535—22no—
322#535—22no—
323#535—22no—
324#535C2×D6114yes—
325C2×#127—21no—
326C2×#124—21no—
327C2×#121—22no—
328C2×#120—22no—
329C2×#117—611no—
330C2×#117—611no—
331C2×#117C2×D6111yescharacteristic
332#428—614no—
333#428—1222no—
334#427—1222no—
335#421—1224no—
336C2×#104—614no—
337#419—1221no—
338#418—1221no—
339#417—614no—
340#416—1222no—
341C2×#100—614no—
342C2×#100—1224no—
343#390—614no—
344#390—1222no—
345#379—1222no—
346#339—614no—
347C61#2349311yescharacteristic, complemented, Sylow
348C25C22⋊D6112yes—
349C2×#41—21no—
350C2×#41—21no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41C22⋊D6112yes—
356C2×#41C22×D6112yes—
357C22×Q8C22⋊D6111yescharacteristic
358C22×D4—21no—
359C22×D4—24no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—22no—
364C22×D4—21no—
365C22×D4—21no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4C22⋊D6112yes—
369C22×D4C22⋊D6111yescharacteristic
370C22×D4C22×D6112yes—
371C23×C4—21no—
372C23×C4C22×D6111yescharacteristic
373C2×M16—614no—
374C2×M16—612no—
375C2×M16—612no—
376C2×M16—1222no—
377C2×M16—1221no—
378C2×M16—1221no—
379C2×M16—1222no—
380C2×M16—1222no—
381C22×C8—1221no—
382C22×C8—1221no—
383#127—24no—
384#127—24no—
385#124—24no—
386#124—24no—
387#121—28no—
388#121—28no—
389#120—28no—
390#120—28no—
391#117—21no—
392#117—21no—
393#117—1222no—
394#117—1222no—
395#117—1221no—
396#117—1221no—
397#117—1221no—
398#117—1221no—
399#117C22⋊D6112yes—
400#117C22×D6112yes—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22⋊D6112yes—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1228no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—612no—
411C2×C42—1221no—
412#107—1224no—
413#106—1224no—
414#104—1228no—
415#103—1224no—
416#102—1224no—
417#100—1228no—
418C24—22no—
419C24—21no—
420C24—22no—
421C24—24no—
422C24—24no—
423C24—84no—
424C24C2×C22⋊D6114yes—
425C24C2×C22⋊D6111yescharacteristic
426#41—22no—
427#41—22no—
428#41—22no—
429#41—22no—
430#41—24no—
431#41—24no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—42no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8C2×C22⋊D6112yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—216no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—88no—
467C2×D4C2×C22⋊D6112yes—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4D4×D6112yes—
477C22×C4#1175822412yes—
478C22×C4#1175822412yes—
479C22×C4#1175822412yes—
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486M16—614no—
487M16—1222no—
488M16—1224no—
489M16—1222no—
490M16—1222no—
491M16—2442no—
492C2×C8—1222no—
493C2×C8—1224no—
494C2×C8—1222no—
495C2×C8—2442no—
496C4⋊C4—22no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—2442no—
500C4⋊C4C2×C22⋊D6114yes—
501C22⋊C4—216no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—4888no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2442no—
508C42—22no—
509C42—22no—
510C42—1222no—
511C42—1222no—
512C42—2441no—
513C42—2441no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—42no—
519C23—48no—
520C23—44no—
521C23—48no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—84no—
526C23—88no—
527C23—88no—
528C23#1451974611yescharacteristic
529C23#1451974611yescharacteristic
530C23#1451974912yes—
531C23#1451975812yes—
532C23C22×C22⋊D6111yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536D4—22no—
537D4—22no—
538D4—42no—
539D4—42no—
540D4—44no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—44no—
546D4—44no—
547D4—42no—
548D4—816no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4#1451974611yescharacteristic
561C2×C4#1451974611yescharacteristic
562C2×C4#1451974812yes—
563C2×C4#1451974812yes—
564C2×C4#1451974912yes—
565C2×C4#1451974912yes—
566C2×C4#1451974912yes—
567C2×C4#1451975812yes—
568C2×C4#1451975812yes—
569C2×C4#1451975812yes—
570C2×C4#1451976412yes—
571C2×C4#1451976412yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4C2×#1175822411yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C8—1224no—
579C8—2442no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—48no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—84no—
591C22—84no—
592C22—88no—
593C22—88no—
594C22#1683967612yes—
595C22#1683967612yes—
596C22C2×#1451974611yescharacteristic
597C22C2×#1451974911yescharacteristic
598C22C2×#1451975811yescharacteristic
599C4—42no—
600C4—44no—
601C4—4882no—
602C4—4882no—
603C4#1683967612yes—
604C4#1683967612yes—
605C4C2×#1451974811yescharacteristic
606C4C2×#1451974911yescharacteristic
607C4C2×#1451975811yescharacteristic
608C4C2×#1451976411yescharacteristic
609C2—21no—
610C2—21no—
611C2—22no—
612C2—22no—
613C2—84no—
614C2—82no—
615C2C2×#1683967611yescharacteristic
616C2#2145517612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#535C2
socleC22×C61C2×#120
hypercenterC2×#535D61
hypocenterC61#23493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. #21454408C2semidirect
2. C2×#16839779C2semidirect
3. #21454338C2semidirect
4. C22×#14519694C2semidirect
5. #21448401C2semidirect
6. #21447026C2semidirect
7. #21445528C2semidirect
13. #16839805C22semidirect
17. C2×#14519704C22semidirect
20. #16839763C22semidirect
23. C2×#14519694C22semidirect
25. #16839665C22semidirect
27. #16839521C22semidirect
44. C2×#11758194D4semidirect
200. #11758194C2×D4semidirect
347. C61#23493semidirect

Maximal subgroups

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

subgroupindex
#214544082
C2×#168397792
#214543382
C22×#145196942
#214484012
#214470262
#214455282
C2×C61×#5352
#2349361