Finite Group Databasebeta

← #29586294

Subgroups — #29586294

Order 15616.

1010 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
1C22×#14519705C211yesmaximal, characteristic
2C2×#16839756C211yesmaximal, characteristic
3C2×#16839730C211yesmaximal, characteristic
4C22×C4×#92418C211yesmaximal, characteristic, complemented
5C2×#16839526C211yesmaximal, characteristic, complemented
6#21445516C211yesmaximal, characteristic, complemented
7C2×C61×#339C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C23×#92418C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12C2×#14519705—22no—
13C2×#14519705—22no—
14C2×#14519705C414yescomplemented
15#16839756—22no—
16#16839756—22no—
17C2×#14519700C412yescomplemented
18C2×#14519697C412yescomplemented
19#16839730—22no—
20#16839730C414yescomplemented
21C2×#14519689—22no—
22C2×C4×#92418—22no—
23C2×C4×#92418—22no—
24C2×C4×#92418—22no—
25#16839526—22no—
26#16839517—22no—
27C22×M16×C61C2211yescharacteristic
28C2×#117×C61C2211yescharacteristic
29C22×C42×C61C2211yescharacteristic
30C61×#339—22no—
31C2×#11758222—21no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222C2×C412yes—
42C2×#11758222C2×C412yes—
43C22×Dic122C2×C411yescharacteristic
44C22×D244C2×C411yescharacteristic
45C22×C22⋊D61—21no—
46C22×C4×D61—21no—
47C2×#11758194—21no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194C2×C412yes—
53C22×#92418—21no—
54C22×#92418—21no—
55C22×#92418—21no—
56C22×#92418—21no—
57C22×#92418—21no—
58C22×#92418—21no—
59C22×#92418—21no—
60C22×#92418—21no—
61C22×#92418—21no—
62C22×#92418C2×C412yes—
63C22×#92418C2×C412yes—
64C22×#92418D412yes—
65C22×#92418D412yes—
66C22×#92418D412yes—
67C22×#92418D412yes—
68#14519705—28no—
69#14519700—24no—
70#14519697—24no—
71#14519689—28no—
72C4×#92418—28no—
73C23×C4×C61—21no—
74C23×C4×C61C2311yescharacteristic
75C2×M16×C61—21no—
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61C2×C412yes—
81C22×C8×C61—21no—
82#117×C61—21no—
83#117×C61—21no—
84#117×C61—21no—
85#117×C61—21no—
86C2×C4⋊C4×C61C2×C412yes—
87C2×C22⋊C4×C61—21no—
88C2×C42×C61—21no—
89C2×C42×C61—21no—
90C2×C42×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61C2×C412yes—
93C2×C42×C61D412yes—
94C2×C42×C61D412yes—
95C24×C61#4111yescharacteristic
96C23×D61—21no—
97C22×C4×C61—28no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61C22×C411yescharacteristic
103C22×C4×C61C22×C411yescharacteristic
104C22×C4×C61C22×C411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C2×C22⋊D61—22no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—21no—
123#11758222—22no—
124#11758222—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C4212yes—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122C4212yes—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—41no—
160M16×C61—41no—
161C2×C8×C61—22no—
162C2×C8×C61—41no—
163C2×C8×C61—41no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C22⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61—41no—
172C42×C61—41no—
173C42×C61C22⋊C414yes—
174#11758194—22no—
175#11758194—22no—
176#11758194—41no—
177#11758194—41no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—41no—
192C2×#92418—41no—
193C2×#92418C22⋊C414yes—
194C2×#92418C22⋊C414yes—
195C23×C61—24no—
196C23×C61#11711yescharacteristic
197C23×C61C4×D411yescharacteristic
198C23×C61C4×D411yescharacteristic
199C23×C61#12111yescharacteristic
200C23×C61#12311yescharacteristic
201C23×C61#12411yescharacteristic
202C23×C61#12611yescharacteristic
203C22×D61—22no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—44no—
207C2×C4×C61—28no—
208C2×C4×C61—28no—
209C2×C4×C61—28no—
210C2×C4×C61—24no—
211C2×C4×C61—42no—
212C2×C4×C61—44no—
213C2×C4×C61C2×C4211yescharacteristic
214C2×C4×C61C2×C22⋊C411yescharacteristic
215C2×C4×C61C2×C22⋊C411yescharacteristic
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61#11711yescharacteristic
218C2×C4×C61#11711yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61C4×D411yescharacteristic
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61#12311yescharacteristic
236C2×C4×C61#12311yescharacteristic
237C2×C4×C61#12411yescharacteristic
238C2×C4×C61#12411yescharacteristic
239C2×C4×C61#12611yescharacteristic
240C2×C4×C61#12711yescharacteristic
241C22⋊D61—42no—
242C22⋊D61—42no—
243C22⋊D61—42no—
244C22⋊D61—42no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248D244—22no—
249D244—22no—
250D244—41no—
251D244—41no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—42no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—42no—
261#92418—24no—
262#92418—24no—
263#92418—42no—
264#10929—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61C4×C22⋊C411yescharacteristic
272C22×C61#38611yescharacteristic
273C22×C61#38811yescharacteristic
274C22×C61#38811yescharacteristic
275C22×C61#38811yescharacteristic
276C22×C61#38811yescharacteristic
277C22×C61#39011yescharacteristic
278C22×C61#44314yes—
279C2×D61—42no—
280C2×D61—44no—
281C2×D61—82no—
282C2×D61—82no—
283C4×C61—28no—
284C4×C61—82no—
285C4×C61C4×C22⋊C411yescharacteristic
286C4×C61C4×C22⋊C411yescharacteristic
287C4×C61#38611yescharacteristic
288C4×C61#38611yescharacteristic
289C4×C61#38811yescharacteristic
290C4×C61#38811yescharacteristic
291C4×C61#39011yescharacteristic
292C4×C61#39011yescharacteristic
293Dic61—42no—
294C22×#104—611no—
295C2×#416—611no—
296C2×#408—611no—
297C22×C4×C8—611no—
298C2×#341—611no—
299C2×#339D6111yescharacteristic, complemented
300#1729—611no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61#143711yescharacteristic
306C2×C61#176412yes—
307C2×C61#193712yes—
308C2×C61#197412yes—
309D61—82no—
310C22×#41—611no—
311C22×M16—611no—
312C22×M16C2×D6111yescharacteristic
313C23×C8—611no—
314C23×C8—611no—
315C2×#117C2×D6111yescharacteristic
316C22×C42C2×D6111yescharacteristic
317C2×#104—614no—
318C2×#104—1222no—
319C2×#104—1222no—
320#416—1222no—
321#416—1222no—
322C2×#103—612no—
323C2×#102—612no—
324#408—614no—
325#408—1222no—
326C2×C22⋊C8—1222no—
327C2×C4×C8—1222no—
328C2×C4×C8—1222no—
329C2×C4×C8—1222no—
330#341—1222no—
331#339—22no—
332#336—1222no—
333C61#1092911yescharacteristic, complemented, Sylow
334C2×#41—612no—
335C2×#41—612no—
336C2×#41—1221no—
337C2×#41—1221no—
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1221no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1221no—
346C22×Q8—611no—
347C22×D4—611no—
348C22×D4—1221no—
349C23×C4—21no—
350C23×C4—1221no—
351C23×C4C22×D6111yescharacteristic
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—612no—
358C2×M16—1221no—
359C2×M16—1221no—
360C2×M16—1221no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16C4×D6112yes—
364C22×C8—21no—
365C22×C8—612no—
366C22×C8—612no—
367C22×C8—612no—
368C22×C8—612no—
369C22×C8—612no—
370C22×C8—612no—
371C22×C8—1221no—
372C22×C8—1221no—
373C22×C8—1221no—
374C22×C8—1221no—
375C22×C8—1221no—
376C22×C8—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C22×C8—1221no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384C2×C4⋊C4C4×D6112yes—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42C4×D6112yes—
391C2×C42D24412yes—
392C2×C42C22⋊D6112yes—
393#104—1228no—
394#103—1224no—
395#102—1224no—
396C22⋊C8—1228no—
397C4×C8—1228no—
398C24—1221no—
399C24#1175822411yescharacteristic
400#41—1222no—
401#41—1222no—
402#41—1222no—
403#41—1222no—
404#41—1222no—
405#41—1222no—
406#41—1222no—
407#41—1222no—
408#41—2441no—
409#41—2441no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41—2441no—
415#41—2441no—
416C2×Q8—612no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×D4—612no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1221no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4—2441no—
433C2×D4—2441no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—1221no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4C2×C4×D6111yescharacteristic
448C22×C4C2×C4×D6111yescharacteristic
449C22×C4C2×D24411yescharacteristic
450C22×C4#1175822211yescharacteristic
451C22×C4#1175822211yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4#1175822411yescharacteristic
457C22×C4#1175822611yescharacteristic
458C22×C4#1175822611yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460M16—22no—
461M16—22no—
462M16—41no—
463M16—41no—
464M16—1222no—
465M16—1222no—
466M16—2441no—
467M16—2441no—
468C2×C8—22no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—614no—
472C2×C8—614no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—1222no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—2441no—
487C2×C8—2441no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—41no—
496C42—41no—
497C42#1175819814yes—
498C23—24no—
499C23—1222no—
500C23—2444no—
501C23—2441no—
502C23—2441no—
503C23#1451973511yescharacteristic
504C23#1451973711yescharacteristic
505C23#1451973711yescharacteristic
506C23#1451974911yescharacteristic
507C23#1451976111yescharacteristic
508C23#1451976311yescharacteristic
509C23#1451977211yescharacteristic
510Q8—1222no—
511Q8—1222no—
512Q8—2441no—
513Q8—2441no—
514D4—1222no—
515D4—1222no—
516D4—2442no—
517D4—2442no—
518D4—2442no—
519D4—2442no—
520D4—2441no—
521D4—2441no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—1222no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4C42×D6111yescharacteristic
536C2×C4C22⋊C4×D6111yescharacteristic
537C2×C4C22⋊C4×D6111yescharacteristic
538C2×C4C2×#1175819811yescharacteristic
539C2×C4#1451973411yescharacteristic
540C2×C4#1451973511yescharacteristic
541C2×C4#1451973711yescharacteristic
542C2×C4#1451973711yescharacteristic
543C2×C4C4×C22⋊D6111yescharacteristic
544C2×C4C4×C22⋊D6111yescharacteristic
545C2×C4#1451973911yescharacteristic
546C2×C4#1451973911yescharacteristic
547C2×C4#1451973911yescharacteristic
548C2×C4#1451973911yescharacteristic
549C2×C4C4×D24411yescharacteristic
550C2×C4C4×D24411yescharacteristic
551C2×C4#1451974711yescharacteristic
552C2×C4#1451974911yescharacteristic
553C2×C4#1451975011yescharacteristic
554C2×C4#1451975011yescharacteristic
555C2×C4#1451975211yescharacteristic
556C2×C4#1451975211yescharacteristic
557C2×C4#1451976211yescharacteristic
558C2×C4#1451976211yescharacteristic
559C2×C4#1451976311yescharacteristic
560C2×C4#1451976611yescharacteristic
561C2×C4#1451977211yescharacteristic
562C2×C4#1451977311yescharacteristic
563C8—42no—
564C8—1224no—
565C8—1224no—
566C8—2442no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—4882no—
574C22—4882no—
575C22—2444no—
576C22—2442no—
577C22#1683960911yescharacteristic
578C22#1683963911yescharacteristic
579C22#1683965111yescharacteristic
580C22#1683965211yescharacteristic
581C22#1683965411yescharacteristic
582C22#1683965411yescharacteristic
583C22#1683966211yescharacteristic
584C22#1683987614yes—
585C4—28no—
586C4—82no—
587C4—2442no—
588C4#1683960911yescharacteristic
589C4C4×#1175819811yescharacteristic
590C4#1683964011yescharacteristic
591C4#1683964011yescharacteristic
592C4#1683965111yescharacteristic
593C4#1683965211yescharacteristic
594C4#1683966211yescharacteristic
595C4#1683966311yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—4882no—
601C2#2144405511yescharacteristic
602C2#2144565312yes—
603C2#2144652412yes—
604C2#2144670412yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×#339C2
socleC23×C61C4×D4
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#339D61
hypocenterC61#10929
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. C22×C4×#92418C2semidirect
5. C2×#16839526C2semidirect
6. #21445516C2semidirect
7. C2×C61×#339C2semidirect
14. C2×#14519705C4semidirect
17. C2×#14519700C4semidirect
18. C2×#14519697C4semidirect
20. #16839730C4semidirect
299. C2×#339D61semidirect
333. C61#10929semidirect

Maximal subgroups

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

subgroupindex
C22×#145197052
C2×#168397562
C2×#168397302
C22×C4×#924182
C2×#168395262
#214455162
C2×C61×#3392
#1092961