Finite Group Databasebeta

← #29578845

Subgroups — #29578845

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839652C211yesmaximal, characteristic
2#21445797C211yesmaximal, characteristic, complemented
3C22⋊C8×Dic61C211yesmaximal, characteristic, complemented
4C2×#16839488C211yesmaximal, characteristic, complemented
5C2×#16839485C211yesmaximal, characteristic
6C2×#16839485C211yesmaximal, characteristic
7C2×C61×#324C211yesmaximal, characteristic, complemented
8C22×#11758198C411yescharacteristic
9C22×#11758198C411yescharacteristic
10C22×#11758198C2211yescharacteristic
11C22×C4×Dic61C2211yescharacteristic
12C2×#14519706—21no—
13C2×C22⋊#92418C2211yescharacteristic
14C2×C22⋊#92418C2211yescharacteristic
15C2×C8×Dic61—21no—
16#16839652—21no—
17#16839652—21no—
18#16839652—21no—
19#16839652—21no—
20C2×#14519684C411yescharacteristic
21C2×#14519684C411yescharacteristic
22#16839516—21no—
23#16839516—21no—
24#16839488—22no—
25#16839485—22no—
26#16839485—22no—
27C22×C4⋊C4×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#324—22no—
31C24×D61C2×C411yescharacteristic
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35C22×#92418—21no—
36C22×#92418—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—22no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—42no—
49C2×#11758198C814yescomplemented
50C2×C4×Dic61—42no—
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55#14519706—42no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418D412yes—
60C8×Dic61—42no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65C23×C4×C61C2×C411yescharacteristic
66C23×C4×C61C2×C411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C22×C8×C61—21no—
69C22×C8×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61D412yes—
80C24×C61C22×C411yescharacteristic
81C23×D61—21no—
82C23×D61—21no—
83C23×D61—21no—
84C23×D61—21no—
85C23×D61—21no—
86C23×D61—21no—
87C23×D61—41no—
88C23×D61—41no—
89C23×D61—41no—
90C23×D61—41no—
91C23×D61C2×C812yes—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61C2×C812yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—41no—
119C22×Dic61—41no—
120C22×Dic61—41no—
121C22×Dic61—41no—
122C22×Dic61—41no—
123C22×Dic61C22⋊C411yescharacteristic
124C22×Dic61C22⋊C411yescharacteristic
125C22×Dic61C22⋊C411yescharacteristic
126C22×Dic61C22⋊C411yescharacteristic
127C22×Dic61C2×C812yes—
128C22×Dic61M1612yes—
129C2×C8×C61—22no—
130C2×C8×C61—41no—
131C2×C8×C61—41no—
132C2×C8×C61—41no—
133C2×C8×C61—41no—
134C2×C8×C61—41no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139#11758198—24no—
140#11758198—24no—
141#11758198—42no—
142#11758198—41no—
143#11758198—41no—
144#11758198—41no—
145#11758198—41no—
146#11758198—42no—
147C4×Dic61—22no—
148C4×Dic61—22no—
149C4×Dic61—41no—
150C4×Dic61—41no—
151C2×#92418—22no—
152C2×#92418—41no—
153C2×#92418—41no—
154C2×#92418—41no—
155C2×#92418—41no—
156C2×#92418—41no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61C2×C22⋊C411yescharacteristic
162C23×C61C4×D411yescharacteristic
163C23×C61C4×D411yescharacteristic
164C23×C61C22×C811yescharacteristic
165C23×C61C2×M1611yescharacteristic
166C23×C61#13111yescharacteristic
167C23×C61#13111yescharacteristic
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—81no—
184C22×D61—81no—
185C22×D61—81no—
186C22×D61—81no—
187C22×D61—81no—
188C22×D61—81no—
189C22×D61—81no—
190C22×D61—81no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—81no—
210C2×C4×C61—81no—
211C2×C4×C61#12111yescharacteristic
212C2×C4×C61#12111yescharacteristic
213C2×C4×C61#12411yescharacteristic
214C2×C4×C61#12711yescharacteristic
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—81no—
232C2×Dic61—81no—
233C2×Dic61C22⋊C812yes—
234C2×Dic61C22⋊C812yes—
235C2×Dic61#9912yes—
236C8×C61—42no—
237C8×C61—81no—
238#92418—42no—
239#92418—81no—
240#9651—611nomaximal, Sylow
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61#39011yescharacteristic
254C22×C61C2×C22⋊C811yescharacteristic
255C22×C61#40811yescharacteristic
256C22×C61C2×#9911yescharacteristic
257C22×C61#41011yescharacteristic
258C22×C61#41311yescharacteristic
259C22×C61#41311yescharacteristic
260C22×C61C8×D411yescharacteristic
261C22×C61C8×D411yescharacteristic
262C22×C61#43611yescharacteristic
263C22×C61#43611yescharacteristic
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—82no—
268C2×D61—82no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—82no—
274C2×D61—82no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—42no—
280C4×C61—81no—
281Dic61—22no—
282Dic61—22no—
283Dic61—22no—
284Dic61—41no—
285Dic61—81no—
286C2×#388—611no—
287#1797—611no—
288C4×C22⋊C8—611no—
289C2×#324—611no—
290C2×#324D6111yescharacteristic, complemented
291C2×#323—611no—
292C2×#323—611no—
293C2×C61—21no—
294C2×C61—21no—
295C2×C61—21no—
296C2×C61—21no—
297C2×C61#145811yescharacteristic
298C2×C61#145911yescharacteristic
299C2×C61#146811yescharacteristic
300C2×C61#147311yescharacteristic
301C2×C61#195411yescharacteristic
302C2×C61#196411yescharacteristic
303C2×C61#196711yescharacteristic
304D61—82no—
305C22×C4⋊C4C2×D6111yescharacteristic
306C22×C22⋊C4—611no—
307C22×C22⋊C4—611no—
308C22×C22⋊C4—611no—
309C22×C42—611no—
310C2×C4⋊C8—1221no—
311C2×C22⋊C8—611no—
312C2×C22⋊C8—611no—
313C2×C22⋊C8C2×D6111yescharacteristic
314C2×C22⋊C8C2×D6111yescharacteristic
315C2×C4×C8—1221no—
316#388—1221no—
317#388—1221no—
318#388—1221no—
319#388—1221no—
320C2×#95—611no—
321C2×#95—611no—
322#336—1221no—
323#336—1221no—
324#324—22no—
325#324—1222no—
326#323—1222no—
327#323—1222no—
328C61#965111yescharacteristic, complemented, Sylow
329C25—611no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4C4×D6111yescharacteristic
334C23×C4C4×D6111yescharacteristic
335C23×C4C22×D6111yescharacteristic
336C22×C8—21no—
337C22×C8—21no—
338C22×C8—1221no—
339C22×C8—1221no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C22⋊C4—614no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—2442no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—611no—
362C2×C42—611no—
363C2×C42—2442no—
364C4⋊C8—2442no—
365C22⋊C8—21no—
366C22⋊C8—21no—
367C22⋊C8—21no—
368C22⋊C8—612no—
369C22⋊C8—1221no—
370C22⋊C8—1221no—
371C22⋊C8—1221no—
372C22⋊C8C22⋊D6112yes—
373C4×C8—2442no—
374#95—1221no—
375#95—1221no—
376#95—1221no—
377#95—1221no—
378C24—612no—
379C24—1221no—
380C24—1221no—
381C24—1221no—
382C24—1221no—
383C24—1221no—
384C24—1221no—
385C24—2441no—
386C24—2441no—
387C24—2441no—
388C24—2441no—
389C24C2×C4×D6111yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—611no—
409C22×C4—1222no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—2441no—
416C22×C4—2441no—
417C22×C4—2441no—
418C22×C4—2441no—
419C22×C4—2441no—
420C22×C4C8×D6112yes—
421C22×C4#1175822211yescharacteristic
422C22×C4D4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4D4×D6111yescharacteristic
425C22×C4#1175822411yescharacteristic
426C22×C4C2×C22⋊D6111yescharacteristic
427C2×C8—22no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—1222no—
434C2×C8—2441no—
435C2×C8—2441no—
436C2×C8—2441no—
437C2×C8—2441no—
438C2×C8—2441no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C22⋊C4—1224no—
444C22⋊C4—1224no—
445C22⋊C4—2442no—
446C22⋊C4—2442no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C22⋊C4—2441no—
451C42—1222no—
452C42—1222no—
453C42—2441no—
454C42—2441no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—4881no—
463C23—4881no—
464C23—4881no—
465C23—4881no—
466C23—4881no—
467C23—4881no—
468C23—4881no—
469C23—4881no—
470C23—2442no—
471C23—2442no—
472C23—2442no—
473C23—2442no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23C22⋊C4×D6111yescharacteristic
483C23#1451973711yescharacteristic
484C23C4×C22⋊D6111yescharacteristic
485C23C2×C8×D6111yescharacteristic
486C23M16×D6111yescharacteristic
487C23#1451978311yescharacteristic
488C23#1451978511yescharacteristic
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—4881no—
522C2×C4—4881no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4#1451974911yescharacteristic
531C2×C4#1451975011yescharacteristic
532C2×C4#1451976311yescharacteristic
533C2×C4#1451977311yescharacteristic
534C8—42no—
535C8—81no—
536C8—4881no—
537C8—2442no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—4882no—
551C22—4882no—
552C22—4882no—
553C22—4882no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—2442no—
559C22—2442no—
560C22—2442no—
561C22#1683966211yescharacteristic
562C22C22⋊C8×D6111yescharacteristic
563C22#1683972711yescharacteristic
564C22#99×D6111yescharacteristic
565C22#1683973411yescharacteristic
566C22#1683974511yescharacteristic
567C22#1683974511yescharacteristic
568C22#1683982511yescharacteristic
569C22C8×C22⋊D6111yescharacteristic
570C22#1683983711yescharacteristic
571C22#1683983811yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—81no—
578C4—1222no—
579C4—1222no—
580C4—1222no—
581C4—4881no—
582C4—2441no—
583C2—21no—
584C2—21no—
585C2—21no—
586C2—21no—
587C2—4882no—
588C2#2144413211yescharacteristic
589C2#2144413611yescharacteristic
590C2#2144416611yescharacteristic
591C2#2144418711yescharacteristic
592C2#2144661111yescharacteristic
593C2#2144665311yescharacteristic
594C2#2144666711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC23×C61C22×C8
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC23×C61C2×C22⋊C4
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9651
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #21445797C2semidirect
3. C22⋊C8×Dic61C2semidirect
4. C2×#16839488C2semidirect
7. C2×C61×#324C2semidirect
49. C2×#11758198C8semidirect
290. C2×#324D61semidirect
328. C61#9651semidirect

Maximal subgroups

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

subgroupindex
C2×#168396522
#214457972
C22⋊C8×Dic612
C2×#168394882
C2×#168394852
C2×C61×#3242
#965161