Finite Group Databasebeta

← #29729420

Subgroups — #29729420

Order 15616.

822 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840121C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3#21448877C211yesmaximal, characteristic, complemented
4#21447214C211yesmaximal, characteristic, complemented
5#21447207C211yesmaximal, characteristic
6#21447062C211yesmaximal, characteristic, complemented
7#21447055C211yesmaximal, characteristic
8#21446752C211yesmaximal, characteristic
9#21446752C211yesmaximal, characteristic
10#21446743C211yesmaximal, characteristic, complemented
11#21446543C211yesmaximal, characteristic, complemented
12#21446457C211yesmaximal, characteristic, complemented
13#21446436C211yesmaximal, characteristic
14#21445649C211yesmaximal, characteristic, complemented
15C61×#2545C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519742—21no—
18#16840121—24no—
19#16840057—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519698C2211yescharacteristic
37C2×#14519688C2211yescharacteristic
38#16839667—21no—
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839647—21no—
42#16839647—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×#124×C61C2211yescharacteristic
52C2×#124×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C61×#396—21no—
55C61×#396—21no—
56C61×#391C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#388—21no—
59C61×#388—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519777—22no—
67#14519777D414yes—
68#14519742—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79#14519710—22no—
80#14519710—22no—
81C4⋊#92418—22no—
82C4⋊#92418—41no—
83C4⋊#92418—41no—
84#14519701—22no—
85#14519701—22no—
86#14519701—22no—
87#14519701—22no—
88#14519701—22no—
89#14519701—22no—
90#14519701—22no—
91#14519701—41no—
92#14519698—22no—
93#14519698—22no—
94#14519698—22no—
95#14519698—41no—
96#14519688—22no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102C22×Q8×C61C2311yescharacteristic
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C4⋊Q8×C61—22no—
106C4⋊Q8×C61—41no—
107#124×C61—22no—
108#124×C61—22no—
109#124×C61—22no—
110#124×C61—41no—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#95×C61—21no—
130#95×C61—21no—
131#95×C61D411yescharacteristic
132#95×C61D411yescharacteristic
133C24×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61#4112yes—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C2×Dic122—24no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61—42no—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—42no—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—42no—
187C42×C61C2×D412yes—
188C42×C61#4112yes—
189C4⋊Dic61—24no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C2×D412yes—
196C4⋊Dic61C2×D412yes—
197C4⋊Dic61#4112yes—
198#11758196—42no—
199#11758196—42no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61#12112yes—
260C2×C4×C61#12112yes—
261C2×C4×C61AGL(1,ℤ/8)12yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280Dic122—48no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#25286—611nomaximal, Sylow
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×#12111yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57811yescharacteristic
299C22×C61#57811yescharacteristic
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61#52911yescharacteristic
310C4×C61#54211yescharacteristic
311C4×C61#54811yescharacteristic
312C4×C61#55011yescharacteristic
313Dic61—42no—
314Dic61—42no—
315Dic61—42no—
316Dic61—81no—
317C2×#489—611no—
318C2×#477—611no—
319#2545D6111yescharacteristic
320#2365—611no—
321#2068—611no—
322#2067—611no—
323#2043—611no—
324#2042—611no—
325#1983—611no—
326#1983—611no—
327#1981—611no—
328#1940—611no—
329#1928—611no—
330#1925—611no—
331#1763—611no—
332C2×C61—161no—
333C2×C61#242611yescharacteristic
334C2×C61#303611yescharacteristic
335C2×C61#304411yescharacteristic
336C2×C61#330811yescharacteristic
337C2×C61#331311yescharacteristic
338C2×C61#336711yescharacteristic
339C2×C61#337711yescharacteristic
340C2×C4⋊Q8—611no—
341C2×C4⋊Q8C2×D6111yescharacteristic
342C2×#124C2×D6111yescharacteristic
343C2×#124C2×D6111yescharacteristic
344C2×C4×Q8—1221no—
345C2×C4×Q8C2×D6111yescharacteristic
346#489—1224no—
347#477—1224no—
348C2×#107—611no—
349C2×#107—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×C4⋊C8—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#97—611no—
366#396—21no—
367#396—21no—
368#391—1221no—
369#391C2×D6111yescharacteristic
370#390C2×D6111yescharacteristic
371#389—611no—
372#389—611no—
373#388—21no—
374#388—21no—
375#387—1221no—
376#387—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4C2×D6111yescharacteristic
381#340—611no—
382#340—611no—
383#336—611no—
384#336—611no—
385C61#2528611yescharacteristic, complemented, Sylow
386C22×Q8—1221no—
387C22×Q8C22×D6111yescharacteristic
388C22×Q8C22×D6111yescharacteristic
389C22×D4C22×D6111yescharacteristic
390C22×C8—611no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—611no—
394C4⋊Q8—22no—
395C4⋊Q8—41no—
396C4⋊Q8—614no—
397C4⋊Q8—1222no—
398#124—22no—
399#124—22no—
400#124—22no—
401#124—41no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—2444no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22⋊D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—21no—
425C2×C42—1221no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#107—1222no—
433C4⋊C8—1222no—
434C4⋊C8—2441no—
435C4⋊C8—2441no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—2441no—
444#102—1222no—
445#102—1222no—
446#102—1222no—
447#102—2441no—
448#97—1222no—
449#95—21no—
450#95—21no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95C22⋊D6111yescharacteristic
457#95C22⋊D6111yescharacteristic
458C24—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—1224no—
471C2×Q8—2442no—
472C2×Q8—2442no—
473C2×Q8#1175822412yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1224no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—2442no—
536C42—2442no—
537C42—2442no—
538C42D4×D6112yes—
539C42#1175822412yes—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C22×C22⋊D6111yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—2448no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4#1451974812yes—
598C2×C4#1451974812yes—
599C2×C4#1451980212yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1451974811yescharacteristic
618C22#1684040611yescharacteristic
619C22#1684041911yescharacteristic
620C22C2×#1451980211yescharacteristic
621C22#1684056011yescharacteristic
622C22#1684057711yescharacteristic
623C22#1684057711yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—4881no—
634C4—2442no—
635C4—2442no—
636C4—2442no—
637C4#1684029311yescharacteristic
638C4#1684037911yescharacteristic
639C4#1684041811yescharacteristic
640C4#1684043011yescharacteristic
641C2—161no—
642C2#2144938111yescharacteristic
643C2#2145527911yescharacteristic
644C2#2145534511yescharacteristic
645C2#2145781011yescharacteristic
646C2#2145788511yescharacteristic
647C2#2145853311yescharacteristic
648C2#2145865511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2545C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2545D61
hypocenterC61#25286
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2545

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
3. #21448877C2semidirect
4. #21447214C2semidirect
6. #21447062C2semidirect
10. #21446743C2semidirect
11. #21446543C2semidirect
12. #21446457C2semidirect
14. #21445649C2semidirect
385. C61#25286semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168400572
#214488772
#214472142
#214472072
#214470622
#214470552
#214467522
#214467432
#214465432
#214464572
#214464362
#214456492
C61×#25452
#2528661