Finite Group Databasebeta

← #29697365

Subgroups — #29697365

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454473C211yesmaximal, characteristic, complemented
2C2×#16839759C211yesmaximal, characteristic
3#21454265C211yesmaximal, characteristic
4#21446530C212yesmaximal, complemented
5#21446468C212yesmaximal, complemented
6#21446097C212yesmaximal, complemented
7#21445919C211yesmaximal, characteristic, complemented
8#21445857C211yesmaximal, characteristic, complemented
9#21445764C211yesmaximal, characteristic
10#21445663C212yesmaximal, complemented
11C61×#1752C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519783C2212yes—
14C2×#14519736C2211yescharacteristic
15#16839820—22no—
16#16839816—21no—
17#16839812—21no—
18C2×#14519713C2212yescomplemented
19#16839788C2212yescomplemented
20C2×#14519705C2212yes—
21C2×#14519705C2212yes—
22#16839759C411yescharacteristic, complemented
23#16839759C411yescharacteristic, complemented
24#16839759C411yescharacteristic, complemented
25#16839759C411yescharacteristic, complemented
26#16839759C2211yescharacteristic
27#16839759C2211yescharacteristic
28#16839759C2211yescharacteristic
29#16839759C2211yescharacteristic
30C2×#14519702C411yescharacteristic, complemented
31C2×#14519702C411yescharacteristic, complemented
32C2×#14519699C411yescharacteristic, complemented
33C2×#14519699C411yescharacteristic, complemented
34#16839728C2212yes—
35#16839727—22no—
36#16839726C2211yescharacteristic
37#16839726C2211yescharacteristic
38#16839724—21no—
39C22⋊C8×D61—21no—
40#16839717C2212yescomplemented
41#16839527C2212yescomplemented
42#16839522C2212yescomplemented
43#16839522C2212yescomplemented
44C23×C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2211yescharacteristic
46C2×#97×C61C2212yescomplemented
47C4×C22⋊C4×C61C2211yescharacteristic
48C61×#336C2212yescomplemented
49C2×#11758222—21no—
50C2×#11758222—21no—
51C2×#11758222—24no—
52C2×#11758222—42no—
53C2×#11758222C2×C411yescharacteristic
54C2×#11758222C2×C411yescharacteristic
55C2×#11758222C2311yescharacteristic
56C2×#11758222C2311yescharacteristic
57C22×Dic122C2×C411yescharacteristic
58C22×D244C2×C411yescharacteristic
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61#14519783—24no—
62#14519783—42no—
63C2×#11758194C2312yes—
64C2×#11758194C2312yes—
65C2×#11758189—22no—
66C2×C8×D61—22no—
67#14519736—21no—
68#14519736—21no—
69#14519736C2×C411yescharacteristic
70#14519736C2×C411yescharacteristic
71#14519736C2311yescharacteristic
72#14519736C2311yescharacteristic
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×C22⋊Dic61—21no—
76C2×C4×Dic61—21no—
77#14519713—42no—
78#14519706—22no—
79#14519706—22no—
80#14519705—24no—
81#14519705—42no—
82#14519702C2×C411yescharacteristic
83#14519702C2×C411yescharacteristic
84#14519702C2×C411yescharacteristic
85#14519702C2×C411yescharacteristic
86#14519702C2×C411yescharacteristic
87#14519702C2×C411yescharacteristic
88#14519702C2×C411yescharacteristic
89#14519702C2×C411yescharacteristic
90#14519699C2×C411yescharacteristic
91#14519699C2×C411yescharacteristic
92#14519699C2×C411yescharacteristic
93#14519699C2×C411yescharacteristic
94#14519699C2×C411yescharacteristic
95#14519699C2×C411yescharacteristic
96#14519699C2×C411yescharacteristic
97#14519699C2×C411yescharacteristic
98C22⋊#92418—21no—
99C22⋊#92418—21no—
100C22⋊#92418—21no—
101C22⋊#92418—21no—
102#14519689—22no—
103#14519689—22no—
104#14519687—22no—
105C8×Dic61—22no—
106C23×C4×C61C2311yescharacteristic
107C22×C8×C61—42no—
108C22×C8×C61C2×C411yescharacteristic
109C22×C8×C61C2×C411yescharacteristic
110C22×C8×C61C2×C411yescharacteristic
111C22×C8×C61C2×C411yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2312yes—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61C2312yes—
117C22⋊C8×C61D411yescharacteristic
118C22⋊C8×C61D411yescharacteristic
119C22⋊C8×C61D412yes—
120#97×C61—42no—
121#95×C61—21no—
122C24×C61#4111yescharacteristic
123C23×D61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4111yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—21no—
134C2×C22⋊D61—21no—
135C2×C22⋊D61—21no—
136C2×C22⋊D61—21no—
137C2×C22⋊D61—44no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141#11758222—22no—
142#11758222—22no—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—41no—
147#11758222—41no—
148#11758222—44no—
149#11758222—44no—
150#11758222#4114yes—
151C2×D244—22no—
152C2×D244—21no—
153C2×D244—41no—
154C2×D244C22×C411yescharacteristic
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—44no—
161C2×Dic122—22no—
162C2×Dic122—21no—
163C2×Dic122—41no—
164C2×Dic122C22×C411yescharacteristic
165C2×Dic122C22×C411yescharacteristic
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—41no—
171C2×C8×C61—41no—
172C2×C8×C61—41no—
173C2×C8×C61—41no—
174C2×C8×C61C22×C411yescharacteristic
175C2×C8×C61C22×C411yescharacteristic
176C2×C8×C61C22×C411yescharacteristic
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C2×D412yes—
179C2×C8×C61C2×D412yes—
180C22⋊C4×C61—42no—
181C42×C61—22no—
182C42×C61—22no—
183C22⋊Dic61—21no—
184C22⋊Dic61—21no—
185C22⋊Dic61—21no—
186C22⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C22×C411yescharacteristic
190C4⋊Dic61C22×C411yescharacteristic
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197#11758194—22no—
198#11758194—22no—
199#11758194—22no—
200#11758194—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203#11758189—44no—
204C8×D61—44no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61C4×D411yescharacteristic
209C23×C61C4×D411yescharacteristic
210C23×C61C2×#4111yescharacteristic
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—81no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4112yes—
231C2×C4×C61H(5,𝔽2)12yes—
232C22⋊D61—24no—
233C22⋊D61—42no—
234C22⋊D61—42no—
235C22⋊D61—42no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—81no—
245D244—21no—
246D244—21no—
247D244—21no—
248D244—42no—
249D244#11712yes—
250C4×D61—24no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254Dic122—21no—
255Dic122—21no—
256Dic122—21no—
257Dic122—42no—
258Dic122#11712yes—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—22no—
264C8×C61—22no—
265#92418—42no—
266#92418—42no—
267#22586—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—44no—
276C22×C61#44412yes—
277C22×C61C2×C4×D411yescharacteristic
278C22×C61#55112yes—
279C2×D61—22no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—84no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—82no—
289C4×C61C2×#11711yescharacteristic
290C4×C61#51811yescharacteristic
291C4×C61#54612yes—
292Dic61—22no—
293Dic61—41no—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297Dic61—41no—
298#2924—611no—
299C2×#416—611no—
300#2880—611no—
301#1938—612no—
302#1929—612no—
303#1866—612no—
304#1826—611no—
305#1810—611no—
306#1789—611no—
307#1765—612no—
308#1752D6111yescharacteristic, complemented
309C2×C61—22no—
310C2×C61—22no—
311C2×C61—41no—
312C2×C61#234211yescharacteristic
313C2×C61C2×#44411yescharacteristic
314C2×C61#295711yescharacteristic
315D61—42no—
316D61—81no—
317C22×#41—611no—
318C2×#131—612no—
319C23×C8C2×D6111yescharacteristic
320C2×#117—611no—
321#433—1222no—
322#432—1221no—
323#431—1221no—
324C2×#108—612no—
325#424—612no—
326C2×#104—612no—
327C2×#104—612no—
328#416—611no—
329#416—611no—
330#416—611no—
331#416—611no—
332#416—611no—
333#416—611no—
334#416—611no—
335#416—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#102—611no—
339C2×#102—611no—
340#408—612no—
341#408—1222no—
342#407—611no—
343#407—611no—
344#407—1221no—
345C2×C22⋊C8—1221no—
346C2×C22⋊C8C2×D6111yescharacteristic
347#405—612no—
348C2×#97C2×D6112yescomplemented
349C4×C22⋊C4C2×D6111yescharacteristic
350#341—612no—
351#339—612no—
352#339—612no—
353#336C2×D6112yescomplemented
354C61#2258611yescharacteristic, complemented, Sylow
355C2×#41—611no—
356C2×#41—611no—
357C2×#41—611no—
358C2×#41—611no—
359C2×#41—1224no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—2442no—
363C22×Q8—611no—
364C22×D4—611no—
365C22×D4—1221no—
366C23×C4—1221no—
367C23×C4C22×D6111yescharacteristic
368#131—1224no—
369#131—2442no—
370C2×M16—612no—
371C2×M16—612no—
372C2×M16—1222no—
373C22×C8—42no—
374C22×C8—1222no—
375C22×C8C4×D6111yescharacteristic
376C22×C8C4×D6111yescharacteristic
377C22×C8C4×D6111yescharacteristic
378C22×C8C4×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6112yes—
382#117—611no—
383#117—611no—
384#117—611no—
385#117—611no—
386#117—1221no—
387#117—1221no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1221no—
392C2×C42—1221no—
393C2×C42C22×D6112yes—
394#108—2442no—
395C4⋊C8—1222no—
396C4⋊C8—1222no—
397#104—1224no—
398#104—2442no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—611no—
403#103—611no—
404#103—611no—
405#103—611no—
406#103—611no—
407#102—611no—
408#102—611no—
409#102—611no—
410#102—611no—
411#102—611no—
412#102—611no—
413#102—611no—
414#102—611no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419C22⋊C8—1222no—
420C22⋊C8—1222no—
421C22⋊C8D24411yescharacteristic
422C22⋊C8D24411yescharacteristic
423C22⋊C8D24412yes—
424#97—42no—
425#97—1222no—
426C4×C8—1222no—
427#95—21no—
428C24—1221no—
429C24#1175822211yescharacteristic
430#41—614no—
431#41—1222no—
432#41—1222no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—2444no—
437#41—2441no—
438#41—2441no—
439#41—2444no—
440C2×Q8—611no—
441C2×Q8—611no—
442C2×Q8—1222no—
443C2×Q8—1221no—
444C2×Q8—2441no—
445C2×D4—611no—
446C2×D4—611no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—2441no—
454C2×D4—2444no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2444no—
466C22×C4C2×C4×D6111yescharacteristic
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4C2×D24411yescharacteristic
469C22×C4C2×D24411yescharacteristic
470C22×C4#1175822211yescharacteristic
471C22×C4C23×D6111yescharacteristic
472M16—1222no—
473M16—1222no—
474M16—1222no—
475M16—1222no—
476M16—2444no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—2444no—
488C2×C8C2×C4×D6111yescharacteristic
489C2×C8C2×C4×D6111yescharacteristic
490C2×C8C2×C4×D6111yescharacteristic
491C2×C8C2×C4×D6111yescharacteristic
492C2×C8C2×D24412yes—
493C2×C8C2×D24412yes—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C22⋊C4—42no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C42—22no—
506C42—22no—
507C42—1221no—
508C42—1221no—
509C42—1221no—
510C42—1221no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—1221no—
515C23—1221no—
516C23—4881no—
517C23—2442no—
518C23—2442no—
519C23C4×D24411yescharacteristic
520C23C4×D24411yescharacteristic
521C23C2×#1175822211yescharacteristic
522Q8—612no—
523Q8—1221no—
524Q8—1221no—
525Q8—1221no—
526Q8—2442no—
527D4—612no—
528D4—1224no—
529D4—1221no—
530D4—1221no—
531D4—1221no—
532D4—2442no—
533D4—2442no—
534D4—2442no—
535D4—2442no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—1221no—
547C2×C4—1224no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—4881no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4C4×D24411yescharacteristic
560C2×C4C4×D24411yescharacteristic
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C22×D24411yescharacteristic
563C2×C4#41×D6112yes—
564C2×C4#1451982212yes—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—22no—
570C8—22no—
571C8—2442no—
572C8—2442no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—1222no—
582C22—4884no—
583C22—2441no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22#1683988512yes—
589C22C2×C4×D24411yescharacteristic
590C22#1684044112yes—
591C4—21no—
592C4—21no—
593C4—82no—
594C4—1222no—
595C4—2441no—
596C4—2441no—
597C4—2441no—
598C4—2441no—
599C4—2441no—
600C4#117×D6111yescharacteristic
601C4#1684023511yescharacteristic
602C4#1684040712yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—4881no—
607C2—2442no—
608C2#2144867111yescharacteristic
609C2C2×#1683988511yescharacteristic
610C2#2145473711yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1752C2
socleC22×C61C2×C4×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1752D61
hypocenterC61#22586
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ #1752

Complements

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

NH ≅ G/Ntype
1. #21454473C2semidirect
4. #21446530C2semidirect
5. #21446468C2semidirect
6. #21446097C2semidirect
7. #21445919C2semidirect
8. #21445857C2semidirect
10. #21445663C2semidirect
11. C61×#1752C2semidirect
18. C2×#14519713C22semidirect
19. #16839788C22semidirect
22. #16839759C4semidirect
23. #16839759C4semidirect
24. #16839759C4semidirect
25. #16839759C4semidirect
30. C2×#14519702C4semidirect
31. C2×#14519702C4semidirect
32. C2×#14519699C4semidirect
33. C2×#14519699C4semidirect
40. #16839717C22semidirect
41. #16839527C22semidirect
42. #16839522C22semidirect
43. #16839522C22semidirect
46. C2×#97×C61C22semidirect
48. C61×#336C22semidirect
308. #1752D61semidirect
348. C2×#97C2×D61semidirect
353. #336C2×D61semidirect
354. C61#22586semidirect

Maximal subgroups

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

subgroupindex
#214544732
C2×#168397592
#214542652
#214465302
#214464682
#214460972
#214459192
#214458572
#214457642
#214456632
C61×#17522
#2258661