Finite Group Databasebeta

← #29707368

Subgroups — #29707368

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840393C211yesmaximal, characteristic, complemented
2C2×#16839761C211yesmaximal, characteristic
3C2×#16839759C211yesmaximal, characteristic
4#21448412C211yesmaximal, characteristic
5#21446969C212yesmaximal, complemented
6#21446964C212yesmaximal, complemented
7#21446959C212yesmaximal, complemented
8#21446954C212yesmaximal, complemented
9#21445538C211yesmaximal, characteristic, complemented
10#21445515C211yesmaximal, characteristic, complemented
11C2×C61×#340C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16840393—24no—
15#16840393—24no—
16C2×#14519776—21no—
17C2×#14519776C2212yes—
18C2×#14519768—21no—
19C2×#14519768C2212yes—
20C2×#14519756—22no—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839761—24no—
24#16839759—24no—
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519700C2212yes—
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519697C2212yes—
31#16839667C2212yes—
32#16839663C2212yes—
33#16839641—22no—
34#16839619—21no—
35#16839619—21no—
36#16839524C2212yes—
37#16839524C2212yes—
38#16839516C2212yes—
39#16839516C2212yes—
40C23×C8×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222Q814yescomplemented
51C22×Dic122—22no—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C22×#92418C2312yes—
60#14519776—28no—
61#14519776—24no—
62#14519776—24no—
63#14519768—28no—
64#14519768—24no—
65#14519768—24no—
66#14519756—216no—
67#14519736—22no—
68#14519736—22no—
69#14519736—22no—
70#14519736D414yes—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61D412yes—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2312yes—
84#14519702—22no—
85#14519702—22no—
86#14519700—24no—
87#14519699—22no—
88#14519699—22no—
89#14519697—24no—
90#14519684—22no—
91#14519684—22no—
92C23×C4×C61—21no—
93C23×C4×C61—21no—
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61—22no—
96C22×C8×C61—22no—
97C22×C8×C61—22no—
98C22×C8×C61C2311yescharacteristic
99C22×C8×C61C2311yescharacteristic
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D414yes—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2312yes—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—24no—
112C22×C4×C61—22no—
113C22×C4×C61—42no—
114C22×C4×C61—81no—
115C22×C4×C61—81no—
116C22×C4×C61—81no—
117C22×C4×C61—81no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×Q812yes—
122C22×C4×C61#4112yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×D412yes—
134#11758222—28no—
135#11758222—28no—
136#11758222—44no—
137#11758222—44no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C2×Q812yes—
144C2×C4×D61—24no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—28no—
153C2×Dic122—82no—
154C2×Dic122—82no—
155C2×Dic122C2×Q812yes—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—44no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61C2×D414yes—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171#11758198—82no—
172#11758198—82no—
173C4⋊Dic61—28no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C2×D412yes—
179C4⋊Dic61C2×D412yes—
180#11758196—28no—
181#11758196—28no—
182C4×Dic61—24no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185#11758194—44no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61#12012yes—
194C23×C61C22⋊Q812yes—
195C23×C61#12412yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C2×C4×C61—24no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—84no—
211C2×C4×C61—84no—
212C2×C4×C61#12012yes—
213C2×C4×C61C22⋊Q814yes—
214C2×C4×C61C22⋊Q814yes—
215C2×C4×C61C22⋊Q812yes—
216C2×C4×C61#12414yes—
217C2×C4×C61#12414yes—
218C2×C4×C61#12412yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×Q811yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C22⋊D61—48no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241D244—22no—
242D244—22no—
243D244—41no—
244D244—41no—
245C4×D61—48no—
246Dic122—22no—
247Dic122—22no—
248Dic122—41no—
249Dic122—41no—
250Dic122—88no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—42no—
254#23458—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61#39314yes—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×C22⋊Q811yescharacteristic
264C22×C61C2×#12411yescharacteristic
265C22×C61#57611yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57811yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—44no—
277C4×C61—82no—
278C4×C61#39314yes—
279C4×C61C2×C22⋊Q811yescharacteristic
280C4×C61C2×C22⋊Q811yescharacteristic
281C4×C61C2×#12411yescharacteristic
282C4×C61C2×#12411yescharacteristic
283Dic61—42no—
284Dic61—42no—
285Dic61—42no—
286Dic61—42no—
287Dic61—42no—
288C2×#544—611no—
289C2×#417—611no—
290C2×#416—611no—
291#2311—611no—
292#2026—612no—
293#2025—612no—
294#2024—612no—
295#2023—612no—
296#1736—611no—
297C2×#340D6111yescharacteristic, complemented
298#1729—611no—
299C2×C61—24no—
300C2×C61C2×#39311yescharacteristic
301C2×C61#301211yescharacteristic
302C2×C61#301311yescharacteristic
303C2×C61#308311yescharacteristic
304C2×C61#308411yescharacteristic
305C2×C61#313611yescharacteristic
306C2×C61#313711yescharacteristic
307D61—82no—
308C22×#41—611no—
309C22×M16—611no—
310C23×C8C2×D6111yescharacteristic
311#544—1224no—
312#544—1224no—
313C2×C4⋊Q8—612no—
314C2×C4⋊Q8—1221no—
315C2×#125—612no—
316C2×#125—1221no—
317C2×C22⋊Q8—1222no—
318C2×#117—611no—
319C2×#117—611no—
320C22×C4⋊C4C2×D6111yescharacteristic
321C22×C4⋊C4C2×D6111yescharacteristic
322#417—1224no—
323#416—1224no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103—612no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#102—612no—
330#391—612no—
331#390—612no—
332#386—1222no—
333#380—1221no—
334#380—1221no—
335#340—612no—
336#340—612no—
337#340C2×D6112yes—
338#340C2×D6112yes—
339#336—612no—
340#336—612no—
341C61#2345811yescharacteristic, complemented, Sylow
342C2×#41—614no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C22×Q8—611no—
349C22×Q8—1222no—
350C22×D4—611no—
351C22×D4—1221no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1222no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—22no—
362C22×C8—612no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C4⋊Q8—1224no—
366C4⋊Q8—1228no—
367C4⋊Q8—1224no—
368#125—1224no—
369#125—1228no—
370#125—1224no—
371C22⋊Q8—12216no—
372#117—614no—
373#117—1222no—
374#117—1222no—
375#117—1222no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4D24414yes—
389C2×C4⋊C4C22×D6112yes—
390C2×C4⋊C4C22×D6112yes—
391C2×C22⋊C4—612no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C42—612no—
396C2×C42—1221no—
397#103—1222no—
398#103—1222no—
399#103—1224no—
400#102—1222no—
401#102—1222no—
402#102—1224no—
403#95—1222no—
404#95—1222no—
405C24—1221no—
406C24D4×D6111yescharacteristic
407#41—1228no—
408#41—1228no—
409#41—2444no—
410#41—2444no—
411C2×Q8—612no—
412C2×Q8—1228no—
413C2×Q8—1221no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8—1221no—
418C2×Q8—4882no—
419C2×Q8—4882no—
420C2×D4—612no—
421C2×D4—1224no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—2442no—
428C2×D4—2442no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1224no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4C2×D24412yes—
446C22×C4#1175822212yes—
447C22×C4#1175822212yes—
448C22×C4D4×D6112yes—
449C22×C4D4×D6111yescharacteristic
450C22×C4#1175822412yes—
451C22×C4Q8×D6112yes—
452C22×C4C23×D6111yescharacteristic
453M16—2444no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—44no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—612no—
470C4⋊C4—612no—
471C4⋊C4—1228no—
472C4⋊C4—1228no—
473C4⋊C4—1228no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4C2×D24414yes—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—4882no—
482C22⋊C4—4882no—
483C42—1222no—
484C42—1222no—
485C42—1224no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—1222no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23#1451974712yes—
498C23#1451975512yes—
499C23#1451976312yes—
500C23C2×D4×D6111yescharacteristic
501Q8—1222no—
502Q8—1222no—
503Q8—4888no—
504Q8—2441no—
505Q8—2441no—
506D4—1222no—
507D4—1222no—
508D4—2448no—
509D4—2441no—
510D4—2441no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2448no—
535C2×C4#1451974712yes—
536C2×C4#1451975514yes—
537C2×C4#1451975512yes—
538C2×C4#1451975714yes—
539C2×C4#1451976314yes—
540C2×C4#1451976312yes—
541C2×C4#1451976514yes—
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×D24411yescharacteristic
544C2×C4C2×Q8×D6111yescharacteristic
545C2×C4C2×#1175822411yescharacteristic
546C2×C4C2×#1175822211yescharacteristic
547C2×C4C2×#1175822211yescharacteristic
548C8—41no—
549C8—41no—
550C8—2442no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—2442no—
562C22—2442no—
563C22—2442no—
564C22#1683967514yes—
565C22C2×#1451974711yescharacteristic
566C22C2×#1451975511yescharacteristic
567C22C2×#1451976311yescharacteristic
568C22#1684056311yescharacteristic
569C22#1684056811yescharacteristic
570C22#1684056811yescharacteristic
571C22#1684057511yescharacteristic
572C4—44no—
573C4—82no—
574C4—2442no—
575C4—2442no—
576C4—2442no—
577C4—2442no—
578C4—2442no—
579C4#1683967514yes—
580C4C2×#1451975511yescharacteristic
581C4C2×#1451975711yescharacteristic
582C4C2×#1451976311yescharacteristic
583C4C2×#1451976511yescharacteristic
584C2—24no—
585C2—4882no—
586C2C2×#1683967511yescharacteristic
587C2#2145512811yescharacteristic
588C2#2145513911yescharacteristic
589C2#2145567511yescharacteristic
590C2#2145568611yescharacteristic
591C2#2145617111yescharacteristic
592C2#2145618211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#340D61
hypocenterC61#23458
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
1. C2×#16840393C2semidirect
5. #21446969C2semidirect
6. #21446964C2semidirect
7. #21446959C2semidirect
8. #21446954C2semidirect
9. #21445538C2semidirect
10. #21445515C2semidirect
11. C2×C61×#340C2semidirect
50. C2×#11758222Q8semidirect
297. C2×#340D61semidirect
341. C61#23458semidirect

Maximal subgroups

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

subgroupindex
C2×#168403932
C2×#168397612
C2×#168397592
#214484122
#214469692
#214469642
#214469592
#214469542
#214455382
#214455152
C2×C61×#3402
#2345861