Finite Group Databasebeta

← #29687250

Subgroups — #29687250

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840104C211yesmaximal, characteristic
2#21448347C211yesmaximal, characteristic
3#117×Dic61C211yesmaximal, characteristic
4#21446459C211yesmaximal, characteristic, complemented
5#21446452C211yesmaximal, characteristic, complemented
6#21446410C211yesmaximal, characteristic, complemented
7#21446403C211yesmaximal, characteristic, complemented
8#21446131C211yesmaximal, characteristic
9#21445911C211yesmaximal, characteristic
10#21445719C211yesmaximal, characteristic, complemented
11#21445718C211yesmaximal, characteristic, complemented
12#21445648C211yesmaximal, characteristic, complemented
13#21445648C211yesmaximal, characteristic, complemented
14#21445583C211yesmaximal, characteristic
15C61×#1854C211yesmaximal, characteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C22×C4×Dic61C2211yescharacteristic
20#16840104C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30#16839667C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839644—21no—
33#16839634—21no—
34C4⋊C4×Dic61C2211yescharacteristic
35C4⋊C4×Dic61C2211yescharacteristic
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39#16839613—21no—
40C22⋊C4×Dic61—21no—
41C22⋊C4×Dic61—21no—
42C22⋊C4×Dic61—21no—
43#16839608—21no—
44C42×Dic61—21no—
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C23×Dic61D411yescharacteristic
62C23×Dic61D411yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519756—22no—
66#14519756C2×C414yes—
67#14519742—22no—
68#14519742—22no—
69#14519736—22no—
70#14519736C2×C414yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519711C2×C414yescomplemented
93#14519709C2×C414yescomplemented
94#14519702C2×C414yescomplemented
95#14519702C2×C414yescomplemented
96#14519700—22no—
97#14519700—22no—
98C22⋊#92418—22no—
99#14519687—22no—
100#14519687—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C23×C4×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61—22no—
113#117×C61—22no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121#107×C61—22no—
122#106×C61—22no—
123C22⋊C8×C61C2×C414yes—
124#95×C61—21no—
125C24×C61C2×D411yescharacteristic
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61#4112yes—
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—41no—
146C22×Dic61—41no—
147C22×Dic61—41no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122C22×C412yes—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C412yes—
163C2×C8×C61C22×C412yes—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61#4112yes—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—42no—
174C42×C61—24no—
175C22⋊Dic61—24no—
176C22⋊Dic61—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C22×C412yes—
183#11758196—24no—
184#11758196—42no—
185#11758196—42no—
186C4×Dic61—24no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—42no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196C4×Dic61—42no—
197C4×Dic61#4112yes—
198C4×Dic61#4112yes—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61C4×D412yes—
212C23×C61C22×D411yescharacteristic
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—44no—
252C2×Dic61—44no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61#12312yes—
262C2×Dic61#12312yes—
263C2×Dic61#13712yes—
264Dic122—22no—
265Dic122—22no—
266Dic122—41no—
267Dic122—41no—
268C8×C61—22no—
269C8×C61—22no—
270#92418—41no—
271#92418—41no—
272#21583—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C2×C4×D411yescharacteristic
281C22×C61C2×#12311yescharacteristic
282C22×C61#52511yescharacteristic
283C22×C61C2×#13711yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57511yescharacteristic
286C22×C61#57511yescharacteristic
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61C4×#4111yescharacteristic
293C4×C61C4×#4111yescharacteristic
294C4×C61#52911yescharacteristic
295C4×C61#52911yescharacteristic
296Dic61—22no—
297Dic61—22no—
298Dic61—22no—
299Dic61—41no—
300Dic61—42no—
301Dic61—42no—
302Dic61—42no—
303Dic61—42no—
304C2×#485—611no—
305C4×C22⋊Q8—611no—
306C4×#117—611no—
307#1928—611no—
308#1927—611no—
309#1921—611no—
310#1920—611no—
311#1873—611no—
312#1854D6111yescharacteristic
313#1825—611no—
314#1776—611no—
315#1776—611no—
316#1763—611no—
317#1763—611no—
318#1749—611no—
319C2×C61—42no—
320C2×C61C4×#12311yescharacteristic
321C2×C61#294311yescharacteristic
322C2×C61C4×#13711yescharacteristic
323C2×C61#309311yescharacteristic
324C2×C61#309511yescharacteristic
325C2×C61#311111yescharacteristic
326C2×C61#311611yescharacteristic
327C2×C22⋊Q8—611no—
328C2×C4×Q8—611no—
329C2×#117—611no—
330C2×#117C2×D6111yescharacteristic
331C22×C42—611no—
332#485—618no—
333C2×#107—611no—
334C2×#107C2×D6111yescharacteristic
335C2×#106—611no—
336C2×#106C2×D6111yescharacteristic
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×C22⋊C8—611no—
342C2×C22⋊C8C2×D6111yescharacteristic
343C2×#97—611no—
344C2×#97—611no—
345#391—611no—
346#389—611no—
347#387—1221no—
348#385—1221no—
349#380C2×D6111yescharacteristic
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—1221no—
356C4×C22⋊C4—1221no—
357C4×C22⋊C4—1221no—
358C4×C22⋊C4—1221no—
359#376—1221no—
360C43—1221no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340C2×D6111yescharacteristic
368#340C2×D6111yescharacteristic
369#336—611no—
370#336—611no—
371C61#2158311yescharacteristic, complemented, Sylow
372C22×Q8—611no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C22⋊Q8—614no—
381C22⋊Q8—1222no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384#117—22no—
385#117—22no—
386#117—614no—
387#117—1222no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—21no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—1222no—
408C2×C42—1222no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416#107—22no—
417#107—614no—
418#106—22no—
419#106—614no—
420#103—614no—
421#103—614no—
422#103—1222no—
423#103—1222no—
424C22⋊C8—1222no—
425C22⋊C8C4×D6114yes—
426#97—1222no—
427#97—1222no—
428#95—21no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437C24D4×D6111yescharacteristic
438C2×Q8—612no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—2441no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4C2×C4×D6112yes—
467C22×C4D4×D6111yescharacteristic
468C22×C4#1175822612yes—
469C22×C4C23×D6111yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8C2×C4×D6112yes—
481C2×C8C2×C4×D6112yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—612no—
489C4⋊C4—612no—
490C4⋊C4—612no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1224no—
495C4⋊C4—2442no—
496C4⋊C4—2442no—
497C4⋊C4#1175822412yes—
498C4⋊C4#1175822412yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C42—24no—
504C42—612no—
505C42—612no—
506C42—1222no—
507C42—1222no—
508C42—1221no—
509C42—1221no—
510C42—1224no—
511C42—2442no—
512C42—2442no—
513C42—2442no—
514C42—2442no—
515C42—2442no—
516C42—2442no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1451973912yes—
524C23C2×D4×D6111yescharacteristic
525Q8—1222no—
526Q8—1222no—
527Q8—2441no—
528Q8—2441no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—2444no—
564C2×C4—2444no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4#1451973912yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822411yescharacteristic
579C2×C4C2×#1175822611yescharacteristic
580C8—22no—
581C8—22no—
582C8—2441no—
583C8—2441no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×#1451973911yescharacteristic
592C22#123×D6111yescharacteristic
593C22#1684026811yescharacteristic
594C22#137×D6111yescharacteristic
595C22#1684055811yescharacteristic
596C22#1684055911yescharacteristic
597C22#1684055911yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—1222no—
604C4—1222no—
605C4—1222no—
606C4—2442no—
607C4—2442no—
608C4—2442no—
609C4—2441no—
610C4—2442no—
611C4C4×#1175822411yescharacteristic
612C4C4×#1175822411yescharacteristic
613C4#1684029811yescharacteristic
614C4#1684029811yescharacteristic
615C2—42no—
616C2#2144833811yescharacteristic
617C2#2145461911yescharacteristic
618C2#2145463711yescharacteristic
619C2#2145574711yescharacteristic
620C2#2145576611yescharacteristic
621C2#2145592111yescharacteristic
622C2#2145597411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1854C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1854D61
hypocenterC61#21583
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #21446459C2semidirect
5. #21446452C2semidirect
6. #21446410C2semidirect
7. #21446403C2semidirect
10. #21445719C2semidirect
11. #21445718C2semidirect
12. #21445648C2semidirect
13. #21445648C2semidirect
20. #16840104C4semidirect
92. #14519711C2×C4semidirect
93. #14519709C2×C4semidirect
94. #14519702C2×C4semidirect
95. #14519702C2×C4semidirect
371. C61#21583semidirect

Maximal subgroups

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

subgroupindex
C2×#168401042
#214483472
#117×Dic612
#214464592
#214464522
#214464102
#214464032
#214461312
#214459112
#214457192
#214457182
#214456482
#214455832
C61×#18542
#2158361