Finite Group Databasebeta

← #29732167

Subgroups — #29732167

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456371C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16840063C211yesmaximal, characteristic
4#21454762C211yesmaximal, characteristic, complemented
5#21447331C211yesmaximal, characteristic, complemented
6#21447326C211yesmaximal, characteristic, complemented
7#21447174C211yesmaximal, characteristic
8#21447174C211yesmaximal, characteristic
9#21446811C211yesmaximal, characteristic, complemented
10#21446605C211yesmaximal, characteristic, complemented
11#21446600C211yesmaximal, characteristic, complemented
12#21446482C211yesmaximal, characteristic
13#21446482C211yesmaximal, characteristic
14C4×#14519693C211yesmaximal, characteristic
15C61×#1950C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16840171—21no—
20#16840171—21no—
21#16840164—21no—
22#16840164—21no—
23#16840063—21no—
24#16840063—21no—
25#16840063—21no—
26#16840063—21no—
27#16840063—21no—
28#16840063—21no—
29#16840053—21no—
30#16840043—21no—
31#16839891—21no—
32#16839891—21no—
33#16839889—21no—
34#16839886—21no—
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839804C2211yescharacteristic
40#16839804C2211yescharacteristic
41C2×#14519711C2211yescharacteristic
42C2×#14519709C2211yescharacteristic
43#16839788C2211yescharacteristic
44#16839788C2211yescharacteristic
45#16839768C2211yescharacteristic
46#16839768C2211yescharacteristic
47C2×#14519699C2211yescharacteristic
48C2×#14519699C2211yescharacteristic
49C2×#14519693C2211yescharacteristic
50C2×#14519693C2211yescharacteristic
51#16839717C2211yescharacteristic
52#16839717C2211yescharacteristic
53C4×#11758194C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60C2×#108×C61C2211yescharacteristic
61C2×#108×C61C2211yescharacteristic
62C61×#423C2211yescharacteristic
63C2×C4⋊C8×C61C2211yescharacteristic
64C4×M16×C61C2211yescharacteristic
65C61×#341C2211yescharacteristic
66C61×#341C2211yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C4×D61—21no—
69C2×#11758194C2311yescharacteristic
70C2×#11758194C2311yescharacteristic
71#14519777D411yescharacteristic
72#14519777D411yescharacteristic
73#14519769—21no—
74#14519769—21no—
75#14519769—21no—
76#14519769—21no—
77#14519769D411yescharacteristic
78#14519769D411yescharacteristic
79#14519756—21no—
80#14519756—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83C4×D244—21no—
84C4×D244—21no—
85C4×D244—21no—
86C4×D244—21no—
87#14519736C2311yescharacteristic
88#14519736C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758198—21no—
95#14519711—21no—
96#14519711—21no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519711—21no—
101#14519711—21no—
102#14519711—21no—
103#14519709—21no—
104#14519709—21no—
105#14519709—21no—
106#14519709—21no—
107#14519709—21no—
108#14519709—21no—
109#14519709—21no—
110#14519709—21no—
111#14519706—21no—
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118#14519699—21no—
119#14519699—21no—
120#14519699—21no—
121#14519699—21no—
122#14519699—21no—
123#14519693—22no—
124#14519693Q814yescomplemented
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128C4×#92418—21no—
129C8×Dic61—21no—
130C8×Dic61—21no—
131C2×M16×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C2×M16×C61C2311yescharacteristic
135C22×C8×C61C2311yescharacteristic
136C22×C8×C61C2311yescharacteristic
137C2×C42×C61C2311yescharacteristic
138#108×C61—22no—
139#108×C61—22no—
140C4⋊C8×C61—21no—
141C4⋊C8×C61—21no—
142C4⋊C8×C61D411yescharacteristic
143C4⋊C8×C61D411yescharacteristic
144C4⋊C8×C61D411yescharacteristic
145C4⋊C8×C61D411yescharacteristic
146#97×C61—21no—
147C4×C8×C61—21no—
148C23×D61—21no—
149C22×C4×C61#4111yescharacteristic
150C22×C4×C61#4111yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244C2×Q812yes—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61—82no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166M16×C61—21no—
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—21no—
172M16×C61C2×Q812yes—
173M16×C61#4112yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D411yescharacteristic
203C4⋊Dic61C2×D411yescharacteristic
204C4⋊Dic61C2×D411yescharacteristic
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—21no—
208#11758196—21no—
209#11758196—21no—
210#11758196—41no—
211#11758196—41no—
212#11758196—41no—
213#11758196—41no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216#11758194—21no—
217#11758194—21no—
218#11758194—21no—
219#11758194C2×Q812yes—
220C2×#92418—21no—
221C2×#92418—21no—
222C23×C61C2×#4111yescharacteristic
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—81no—
227C22×D61—81no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×Q811yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—81no—
248C2×Dic61—81no—
249C2×Dic61—81no—
250D244—22no—
251D244—22no—
252D244—41no—
253D244—41no—
254C4×D61—82no—
255C4×D61—82no—
256Dic122—42no—
257Dic122—81no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—22no—
263C8×C61—22no—
264C8×C61—41no—
265C8×C61—41no—
266#92418—22no—
267#92418—41no—
268#25493—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C2×#12111yescharacteristic
272C22×C61#53311yescharacteristic
273C22×C61#54111yescharacteristic
274C2×D61—42no—
275C2×D61—82no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—82no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61#54711yescharacteristic
284C4×C61#54811yescharacteristic
285C4×C61D4×Q811yescharacteristic
286C4×C61D4×Q811yescharacteristic
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294#3166—611no—
295#3077—611no—
296C2×#478—611no—
297#2962—611no—
298#2090—611no—
299#2089—611no—
300#2063—611no—
301#2063—611no—
302#1998—611no—
303#1952—611no—
304#1951—611no—
305#1950D6111yescharacteristic, complemented
306#1931—611no—
307#1931—611no—
308C4×#100—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61#242911yescharacteristic
312C2×C61#306711yescharacteristic
313C2×C61#306811yescharacteristic
314D61—82no—
315#533—611no—
316C2×#125—611no—
317C2×C4×D4—611no—
318#501—1221no—
319#501—1221no—
320#499—1221no—
321#499—1221no—
322#478—1221no—
323#478—1221no—
324#478—1221no—
325#478—1221no—
326#478—1221no—
327#478—1221no—
328#476—1221no—
329#474—1221no—
330#446—1221no—
331#446—1221no—
332#446—1221no—
333C8×Q8—1221no—
334C2×#108C2×D6111yescharacteristic
335C2×#108C2×D6111yescharacteristic
336#428—611no—
337#428—611no—
338#428—611no—
339#428—611no—
340#428—611no—
341#428—611no—
342C2×#107—611no—
343C2×#106—611no—
344#424—611no—
345#424—611no—
346#423C2×D6111yescharacteristic
347C2×C4⋊C8C2×D6111yescharacteristic
348#418—611no—
349#418—611no—
350C2×#102—611no—
351C2×#102—611no—
352C2×#100—611no—
353C2×#100—611no—
354#405—611no—
355#405—611no—
356C4×M16—611no—
357C4×M16C2×D6111yescharacteristic
358#343—611no—
359#343—611no—
360#343—611no—
361#343—611no—
362#343—611no—
363#343—611no—
364#341C2×D6111yescharacteristic
365#341C2×D6111yescharacteristic
366C61#2549311yescharacteristic, complemented, Sylow
367C22×D4—611no—
368C23×C4—1221no—
369C2×M16—611no—
370C2×M16—611no—
371C2×M16C22×D6111yescharacteristic
372C2×M16C22×D6111yescharacteristic
373C2×M16C22×D6111yescharacteristic
374C2×M16C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379#125—611no—
380#125—611no—
381#125—1221no—
382#125—1221no—
383#125—1221no—
384#125—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×D4—1221no—
390C4×D4—1221no—
391C4×D4—1221no—
392C4×D4—1221no—
393#117—611no—
394#117—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42C22×D6111yescharacteristic
402#108—22no—
403#108—22no—
404#107—1221no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#106—1221no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418#106—1221no—
419#106—1221no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426C4⋊C8D24411yescharacteristic
427C4⋊C8D24411yescharacteristic
428C4⋊C8D24411yescharacteristic
429C4⋊C8D24411yescharacteristic
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#100—614no—
439#100—1222no—
440#97—21no—
441#97—1221no—
442#97—1221no—
443#97—1221no—
444C4×C8—21no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C24—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×D4—612no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—4882no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4#1175822611yescharacteristic
464C22×C4#1175822611yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—21no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—21no—
471M16—21no—
472M16—612no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476M16#1175822412yes—
477M16Q8×D6112yes—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8C2×D24411yescharacteristic
489C2×C8C2×D24411yescharacteristic
490C2×C8C2×D24411yescharacteristic
491C2×C8C2×D24411yescharacteristic
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42—1221no—
517C42—1221no—
518C42C2×D24411yescharacteristic
519C42C2×D24411yescharacteristic
520C42D4×D6111yescharacteristic
521C42D4×D6111yescharacteristic
522C23—1222no—
523C23—4881no—
524C23—4881no—
525C23—2441no—
526C23—2441no—
527C23C2×#1175822611yescharacteristic
528Q8—4881no—
529Q8—2442no—
530D4—1222no—
531D4—1222no—
532D4—2441no—
533D4—2441no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—4881no—
543C2×C4—4881no—
544C2×C4—4881no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4#1451975211yescharacteristic
549C2×C4#1451975211yescharacteristic
550C2×C4#1451975211yescharacteristic
551C2×C4#1451975211yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C22×D24411yescharacteristic
554C2×C4C2×Q8×D6111yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C2×C4#1451982411yescharacteristic
557C2×C4#1451982411yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—22no—
563C8—22no—
564C8—41no—
565C8—41no—
566C8—1222no—
567C8—2441no—
568C22—21no—
569C22—21no—
570C22—4882no—
571C22—4882no—
572C22—4881no—
573C22—4881no—
574C22—2442no—
575C22C2×#1451975211yescharacteristic
576C22#1684031711yescharacteristic
577C22#1684037511yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—4881no—
583C4—4881no—
584C4—4881no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684041611yescharacteristic
590C4#1684041811yescharacteristic
591C4#1684042511yescharacteristic
592C4Q8×D24411yescharacteristic
593C2—21no—
594C2—21no—
595C2—4882no—
596C2#2144941611yescharacteristic
597C2#2145554011yescharacteristic
598C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1950C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#1950D61
hypocenterC61#25493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
1. #21456371C2semidirect
4. #21454762C2semidirect
5. #21447331C2semidirect
6. #21447326C2semidirect
9. #21446811C2semidirect
10. #21446605C2semidirect
11. #21446600C2semidirect
15. C61×#1950C2semidirect
124. #14519693Q8semidirect
305. #1950D61semidirect
366. C61#25493semidirect

Maximal subgroups

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

subgroupindex
#214563712
#214556182
C2×#168400632
#214547622
#214473312
#214473262
#214471742
#214468112
#214466052
#214466002
#214464822
C4×#145196932
C61×#19502
#2549361