Finite Group Databasebeta

← #29730105

Subgroups — #29730105

Order 15616.

824 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×#16840124C211yesmaximal, characteristic
2C2×#16839984C211yesmaximal, characteristic
3#21448888C211yesmaximal, characteristic
4#21447238C211yesmaximal, characteristic, complemented
5#21447233C211yesmaximal, characteristic
6#21447141C211yesmaximal, characteristic, complemented
7#21447134C211yesmaximal, characteristic
8#21446768C211yesmaximal, characteristic, complemented
9#21446543C211yesmaximal, characteristic, complemented
10#21446163C211yesmaximal, characteristic, complemented
11#21446156C211yesmaximal, characteristic
12#21445831C211yesmaximal, characteristic, complemented
13#21445826C211yesmaximal, characteristic
14#21445649C211yesmaximal, characteristic, complemented
15C61×#2568C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16840124—24no—
22#16839984—24no—
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839652—21no—
38#16839651—21no—
39#16839633—21no—
40C4×C4⋊Dic61C2211yescharacteristic
41#16839609—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#400—21no—
53C61×#400—21no—
54C61×#391—21no—
55C61×#391—21no—
56C61×#391C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#382C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—41no—
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—41no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519765—22no—
75#14519765D414yes—
76C4×Dic122—22no—
77C4×Dic122—41no—
78C4×D244—22no—
79C4×D244—41no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C4⋊#92418—22no—
91C4⋊#92418—41no—
92C4⋊#92418—41no—
93#14519701—22no—
94#14519701—22no—
95#14519700—22no—
96#14519700—22no—
97#14519700—22no—
98#14519697—22no—
99#14519697—22no—
100#14519697—22no—
101#14519688—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61—21no—
106C22×Q8×C61C2311yescharacteristic
107C4⋊Q8×C61—22no—
108C4⋊Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—42no—
139C2×Q8×C61—42no—
140C2×Q8×C61—42no—
141C2×Q8×C61#4112yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C2×D412yes—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—81no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C2×D412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61—42no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—42no—
186C42×C61C2×D412yes—
187C42×C61#4112yes—
188#11758202—44no—
189#11758201—44no—
190#11758198—24no—
191#11758198—24no—
192#11758198—42no—
193#11758198—42no—
194#11758198—81no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61#4112yes—
199#11758196—42no—
200C4×Dic61—42no—
201C4×Dic61—81no—
202C2×#92418—21no—
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—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—84no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
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—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61#12112yes—
256C2×C4×C61#12112yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272D244—22no—
273D244—22no—
274D244—41no—
275D244—41no—
276C4×D61—84no—
277Dic122—22no—
278Dic122—22no—
279Dic122—41no—
280Dic122—41no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#25337—611nomaximal, Sylow
286C22×C61C2×#12111yescharacteristic
287C22×C61#54611yescharacteristic
288C22×C61#54811yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57711yescharacteristic
292C22×C61#57811yescharacteristic
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61#52911yescharacteristic
311C4×C61#54211yescharacteristic
312C4×C61#54611yescharacteristic
313C4×C61#54711yescharacteristic
314Dic61—42no—
315Dic61—42no—
316Dic61—81no—
317C2×#489—611no—
318C2×#464—611no—
319#2568D6111yescharacteristic, complemented
320#2367—611no—
321#2074—611no—
322#2073—611no—
323#2056—611no—
324#2055—611no—
325#1987—611no—
326#1940—611no—
327#1878—611no—
328#1877—611no—
329#1805—611no—
330#1804—611no—
331#1763—611no—
332C2×C61#242711yescharacteristic
333C2×C61#304011yescharacteristic
334C2×C61#304611yescharacteristic
335C2×C61#331111yescharacteristic
336C2×C61#331311yescharacteristic
337C2×C61#337111yescharacteristic
338C2×C61#337711yescharacteristic
339D61—82no—
340C22×Q16—611no—
341C22×SD16—611no—
342C2×C4⋊Q8C2×D6111yescharacteristic
343C2×#124—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8C2×D6111yescharacteristic
346C2×C4×D4—611no—
347#489—1224no—
348#464—1224no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#97—611no—
361#400—21no—
362#400—21no—
363#391—21no—
364#391—21no—
365#391—611no—
366#391C2×D6111yescharacteristic
367#391C2×D6111yescharacteristic
368#390—611no—
369#388—1221no—
370#388—1221no—
371#385—1221no—
372#382C2×D6111yescharacteristic
373#382C2×D6111yescharacteristic
374C4×C4⋊C4—611no—
375C4×C4⋊C4C2×D6111yescharacteristic
376C4×C22⋊C4—1221no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#336—611no—
384#336—611no—
385C61#2533711yescharacteristic, complemented, Sylow
386C22×Q8—21no—
387C22×Q8—611no—
388C22×Q8C22×D6111yescharacteristic
389C22×D4—611no—
390C23×C4—1221no—
391C2×Q16—1222no—
392C2×Q16—1222no—
393C2×Q16—2441no—
394C2×SD16—1222no—
395C2×SD16—1222no—
396C2×SD16—2441no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C4⋊Q8—22no—
402C4⋊Q8—22no—
403#124—614no—
404#124—1222no—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×D4—1222no—
410C4×D4—2441no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22⋊D6111yescharacteristic
417C2×C4⋊C4C22⋊D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431C2×C42C22×D6111yescharacteristic
432C4⋊C8—1222no—
433C4⋊C8—2441no—
434C4⋊C8—2441no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—1222no—
441#102—1222no—
442#102—1222no—
443#97—1222no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95C22⋊D6111yescharacteristic
456#95C22⋊D6111yescharacteristic
457C24—1221no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—612no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8#1175822612yes—
473C2×D4—612no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—4881no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496Q16—2444no—
497SD16—2444no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—612no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—2442no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—4881no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—4881no—
536C42—2442no—
537C42D4×D6112yes—
538C42#1175822412yes—
539C23—1222no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23C22×C22⋊D6111yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—84no—
552Q8—1222no—
553Q8—1222no—
554Q8—2441no—
555Q8—2441no—
556D4—1222no—
557D4—1222no—
558D4—2441no—
559D4—2441no—
560C2×C4—21no—
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—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4884no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4#1451974812yes—
599C2×C4#1451974812yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822611yescharacteristic
604C2×C4#1451982311yescharacteristic
605C2×C4#1451982511yescharacteristic
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22C2×#1451974811yescharacteristic
618C22#1684040011yescharacteristic
619C22#1684042311yescharacteristic
620C22#1684056011yescharacteristic
621C22#1684056011yescharacteristic
622C22#1684057011yescharacteristic
623C22#1684057711yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—4881no—
635C4—2442no—
636C4—2442no—
637C4#1684029811yescharacteristic
638C4#1684038511yescharacteristic
639C4#1684040311yescharacteristic
640C4#1684041411yescharacteristic
641C2—4882no—
642C2#2144939211yescharacteristic
643C2#2145530911yescharacteristic
644C2#2145536511yescharacteristic
645C2#2145784911yescharacteristic
646C2#2145787911yescharacteristic
647C2#2145858711yescharacteristic
648C2#2145866111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2568C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2568D61
hypocenterC61#25337
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2568

Complements

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

NH ≅ G/Ntype
4. #21447238C2semidirect
6. #21447141C2semidirect
8. #21446768C2semidirect
9. #21446543C2semidirect
10. #21446163C2semidirect
12. #21445831C2semidirect
14. #21445649C2semidirect
15. C61×#2568C2semidirect
319. #2568D61semidirect
385. C61#25337semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168399842
#214488882
#214472382
#214472332
#214471412
#214471342
#214467682
#214465432
#214461632
#214461562
#214458312
#214458262
#214456492
C61×#25682
#2533761