Finite Group Databasebeta

← #29749918

Subgroups — #29749918

Order 15616.

806 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840130C211yesmaximal, characteristic, complemented
2C2×#16839860C211yesmaximal, characteristic
3#21449691C211yesmaximal, characteristic
4#21447214C211yesmaximal, characteristic, complemented
5#21447157C211yesmaximal, characteristic, complemented
6#21447134C211yesmaximal, characteristic
7#21446902C211yesmaximal, characteristic
8#21446676C211yesmaximal, characteristic
9#21446515C211yesmaximal, characteristic
10#21446512C211yesmaximal, characteristic, complemented
11#21446498C211yesmaximal, characteristic, complemented
12#21446156C211yesmaximal, characteristic
13#21446087C211yesmaximal, characteristic, complemented
14#21445688C211yesmaximal, characteristic, complemented
15C61×#2446C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519769C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20#16840130—24no—
21#16839860—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35C2×C4×#92418C2211yescharacteristic
36#16839697—21no—
37#16839686—21no—
38#16839664C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839633—21no—
43#16839621—21no—
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#400—21no—
53C61×#398—21no—
54C61×#391C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#387—21no—
59C61×#387—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201—22no—
68C2×#11758201—41no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519769—22no—
74#14519769—41no—
75C4×D244—22no—
76C4×D244—41no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89#14519710—22no—
90#14519708—41no—
91C4⋊#92418—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701—22no—
95#14519701—22no—
96#14519701—22no—
97#14519701D414yes—
98#14519697—22no—
99#14519697—22no—
100#14519688—22no—
101#14519688—22no—
102C4×#92418—22no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61C2311yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107C4⋊Q8×C61—22no—
108C4⋊Q8×C61—41no—
109C4×Q8×C61—22no—
110C4×Q8×C61—41no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
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—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244#4112yes—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—81no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61—42no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—42no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—42no—
185#11758201—28no—
186#11758201—44no—
187#11758198—42no—
188#11758198—42no—
189#11758198—81no—
190#11758198—81no—
191#11758198—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2×D412yes—
198#11758196—42no—
199#11758196—42no—
200#11758196—81no—
201#11758196—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×#92418C2×D412yes—
212C2×#92418#4112yes—
213C2×#92418#4112yes—
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—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
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—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
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—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61#13712yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×C4×C612−1+411yescharacteristic
266C2×Dic61—22no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275D244—22no—
276D244—22no—
277D244—41no—
278D244—41no—
279C4×D61—84no—
280#92418—22no—
281#92418—22no—
282#92418—22no—
283#92418—41no—
284#26845—611nomaximal, Sylow
285C22×C61#53211yescharacteristic
286C22×C61D4211yescharacteristic
287C22×C61#54811yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57711yescharacteristic
291C22×C61#57711yescharacteristic
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C4×C61—22no—
300C4×C61—41no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61#52511yescharacteristic
310C4×C61#54211yescharacteristic
311C4×C61#54711yescharacteristic
312C4×C61#55111yescharacteristic
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316C2×#491—611no—
317C2×#440—611no—
318#2454—611no—
319#2446D6111yescharacteristic, complemented
320#2068—611no—
321#2059—611no—
322#2055—611no—
323#2013—611no—
324#1968—611no—
325#1936—611no—
326#1935—611no—
327#1933—611no—
328#1877—611no—
329#1865—611no—
330#1770—611no—
331C2×C61#246111yescharacteristic
332C2×C61#317411yescharacteristic
333C2×C61#318111yescharacteristic
334C2×C61#327511yescharacteristic
335C2×C61#327811yescharacteristic
336C2×C61#337211yescharacteristic
337C2×C61#337611yescharacteristic
338D61—82no—
339C22×SD16—611no—
340C22×SD16—611no—
341C2×C4⋊Q8C2×D6111yescharacteristic
342C2×#125—611no—
343C2×C4×Q8C2×D6111yescharacteristic
344C2×C4×D4—611no—
345#491—1224no—
346#440—1224no—
347C2×#107—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#97—611no—
359C2×#97—611no—
360C2×C4×C8—611no—
361#400—21no—
362#399—1221no—
363#398—21no—
364#396—1221no—
365#391C2×D6111yescharacteristic
366#391C2×D6111yescharacteristic
367#391C2×D6111yescharacteristic
368#390—611no—
369#390—611no—
370#389—611no—
371#389—611no—
372#389C2×D6111yescharacteristic
373#387—21no—
374#387—21no—
375#385—1221no—
376#380—1221no—
377C4×C4⋊C4C2×D6111yescharacteristic
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#336—611no—
383#336—611no—
384C61#2684511yescharacteristic, complemented, Sylow
385C22×Q8C22×D6111yescharacteristic
386C22×Q8C22×D6111yescharacteristic
387C22×D4—611no—
388C23×C4—1221no—
389C2×SD16—1222no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—2441no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C4⋊Q8—22no—
400C4⋊Q8—41no—
401#125—1222no—
402#125—2441no—
403C4×Q8—22no—
404C4×Q8—41no—
405C4×D4—1222no—
406C4×D4—2441no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22⋊D6111yescharacteristic
419C2×C4⋊C4C22⋊D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C42—21no—
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#106—2441no—
433C4⋊C8—1222no—
434#103—614no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—1222no—
441#102—1222no—
442#97—1222no—
443#97—1222no—
444C4×C8—1222no—
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#95C22⋊D6111yescharacteristic
455#95C22⋊D6111yescharacteristic
456C24—1221no—
457C2×Q8—21no—
458C2×Q8—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×Q8D4×D6112yes—
469C2×D4—612no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—4881no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492SD16—1228no—
493SD16—2444no—
494C2×C8—612no—
495C2×C8—612no—
496C2×C8—612no—
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—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—612no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—4881no—
523C4⋊C4—4881no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C22⋊C4—4881no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—42no—
538C23—1222no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23C22×C22⋊D6111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554D4—1222no—
555D4—1222no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—21no—
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—22no—
576C2×C4—41no—
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—1222no—
588C2×C4—4884no—
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#1451980812yes—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982411yescharacteristic
604C8—1222no—
605C8—1222no—
606C8—1222no—
607C8—2441no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22#1684030811yescharacteristic
616C22D4×C22⋊D6111yescharacteristic
617C22#1684041911yescharacteristic
618C22C2×#1451980811yescharacteristic
619C22#1684056011yescharacteristic
620C22#1684057011yescharacteristic
621C22#1684057011yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—4881no—
633C4—4881no—
634C4—2442no—
635C4#1684027011yescharacteristic
636C4#1684038011yescharacteristic
637C4#1684041011yescharacteristic
638C4#1684044011yescharacteristic
639C2—4882no—
640C2#2144978511yescharacteristic
641C2#2145644811yescharacteristic
642C2#2145652211yescharacteristic
643C2#2145745111yescharacteristic
644C2#2145748111yescharacteristic
645C2#2145859611yescharacteristic
646C2#2145864011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2446C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2446D61
hypocenterC61#26845
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2446

Complements

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

NH ≅ G/Ntype
1. C2×#16840130C2semidirect
4. #21447214C2semidirect
5. #21447157C2semidirect
10. #21446512C2semidirect
11. #21446498C2semidirect
13. #21446087C2semidirect
14. #21445688C2semidirect
15. C61×#2446C2semidirect
319. #2446D61semidirect
384. C61#26845semidirect

Maximal subgroups

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

subgroupindex
C2×#168401302
C2×#168398602
#214496912
#214472142
#214471572
#214471342
#214469022
#214466762
#214465152
#214465122
#214464982
#214461562
#214460872
#214456882
C61×#24462
#2684561