Finite Group Databasebeta

← #29729433

Subgroups — #29729433

Order 15616.

794 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×#16840124C211yesmaximal, characteristic
2C2×#16840044C211yesmaximal, characteristic, complemented
3#21450799C211yesmaximal, characteristic
4#21447197C211yesmaximal, characteristic, complemented
5#21447190C211yesmaximal, characteristic
6#21447065C211yesmaximal, characteristic, complemented
7#21447060C211yesmaximal, characteristic
8#21446751C211yesmaximal, characteristic
9#21446751C211yesmaximal, characteristic
10#21446742C211yesmaximal, characteristic, complemented
11#21446543C211yesmaximal, characteristic, complemented
12#21446453C211yesmaximal, characteristic, complemented
13#21446446C211yesmaximal, characteristic
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2369C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840124—24no—
21#16840044—24no—
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519697C2211yescharacteristic
36C2×#14519697C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519688C2211yescharacteristic
40#16839684—21no—
41#16839684—21no—
42#16839668C2211yescharacteristic
43#16839664C2211yescharacteristic
44#16839654—21no—
45#16839654—21no—
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#391—21no—
52C61×#391—21no—
53C61×#389—21no—
54C61×#389C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C4×C4⋊C4×C61—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519777—22no—
69#14519777—41no—
70#14519765—22no—
71#14519765—22no—
72#14519765—22no—
73#14519765—41no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87#14519708—22no—
88#14519708—22no—
89C4⋊#92418—22no—
90C4⋊#92418—41no—
91C4⋊#92418—41no—
92#14519700—22no—
93#14519700—22no—
94#14519700—22no—
95#14519700—22no—
96#14519700—22no—
97#14519700—22no—
98#14519700—22no—
99#14519700—41no—
100#14519697—22no—
101#14519697—22no—
102#14519697—22no—
103#14519697—41no—
104#14519688—22no—
105#14519684—21no—
106#14519684—21no—
107C22×Q8×C61—21no—
108C4⋊Q8×C61—22no—
109C4⋊Q8×C61D414yes—
110C2×C4⋊C4×C61—21no—
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×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
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—24no—
134C2×Q8×C61—81no—
135C2×Q8×C61—81no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122#4112yes—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—81no—
177C4⋊C4×C61—81no—
178C4⋊C4×C61C2×D412yes—
179C4⋊C4×C61C2×D412yes—
180C4⋊C4×C61#4112yes—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—42no—
186C42×C61—81no—
187C42×C61—81no—
188C42×C61C2×D412yes—
189C42×C61#4112yes—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61—42no—
200#11758196—42no—
201C4×Dic61—42no—
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—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222Q8×C61—84no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
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—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
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#12112yes—
251C2×C4×C61#12112yes—
252C2×C4×C61#13712yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268D244—41no—
269D244—41no—
270D244—41no—
271D244—41no—
272Dic122—22no—
273Dic122—22no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#25287—611nomaximal, Sylow
285C22×C61C2×#12111yescharacteristic
286C22×C61#54611yescharacteristic
287C22×C61#54811yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57711yescharacteristic
291C22×C61#57711yescharacteristic
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—81no—
309C4×C61#52911yescharacteristic
310C4×C61#54211yescharacteristic
311C4×C61#54811yescharacteristic
312C4×C61#55011yescharacteristic
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316C2×#489—611no—
317C2×#475—611no—
318#2545—611no—
319#2369D6111yescharacteristic, complemented
320#2066—611no—
321#2065—611no—
322#2043—611no—
323#2042—611no—
324#1983—611no—
325#1983—611no—
326#1981—611no—
327#1940—611no—
328#1927—611no—
329#1926—611no—
330#1763—611no—
331C2×C61#242611yescharacteristic
332C2×C61#303711yescharacteristic
333C2×C61#304211yescharacteristic
334C2×C61#331611yescharacteristic
335C2×C61#332111yescharacteristic
336C2×C61#336911yescharacteristic
337C2×C61#337611yescharacteristic
338D61—161no—
339C2×C4⋊Q8—611no—
340C2×C4⋊Q8C2×D6111yescharacteristic
341C2×#124—611no—
342C2×#124—611no—
343C2×C4×Q8—611no—
344#489—1224no—
345#475—1224no—
346C2×#106—611no—
347C2×#106—611no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#97—611no—
364#396—1221no—
365#396—1221no—
366#391—21no—
367#391—21no—
368#391—611no—
369#390—611no—
370#389—21no—
371#389C2×D6111yescharacteristic
372#389C2×D6111yescharacteristic
373#388—1221no—
374#388—1221no—
375C4×C4⋊C4—21no—
376C4×C4⋊C4C2×D6111yescharacteristic
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C4⋊C4C2×D6111yescharacteristic
380#340—611no—
381#340—611no—
382#336—611no—
383#336—611no—
384C61#2528711yescharacteristic, complemented, Sylow
385C22×Q8—21no—
386C22×Q8—611no—
387C22×Q8—611no—
388C22×D4—611no—
389C22×C8—611no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8—611no—
393C4⋊Q8—22no—
394C4⋊Q8—1222no—
395C4⋊Q8—2441no—
396C4⋊Q8D24414yes—
397#124—1222no—
398#124—1222no—
399#124—1222no—
400#124—2441no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4C22⋊D6111yescharacteristic
411C2×C4⋊C4C22⋊D6111yescharacteristic
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427C2×C42C22×D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429#106—1222no—
430#106—1222no—
431C4⋊C8—1222no—
432C4⋊C8—2441no—
433C4⋊C8—2441no—
434#103—1222no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#103—1222no—
441#103—2441no—
442#102—1222no—
443#102—1222no—
444#102—1222no—
445#102—2441no—
446#97—1222no—
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—24no—
458C2×Q8—81no—
459C2×Q8—81no—
460C2×Q8—612no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
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
492C2×C8—1221no—
493C2×C8—1221no—
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—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4C2×D24412yes—
520C4⋊C4C2×D24412yes—
521C4⋊C4#1175822612yes—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—81no—
534C42—81no—
535C42—2442no—
536C42C2×D24412yes—
537C42#1175822212yes—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23C22×C22⋊D6111yescharacteristic
546Q8—84no—
547Q8—1222no—
548Q8—1222no—
549Q8—2441no—
550Q8—2441no—
551Q8—2441no—
552Q8—2441no—
553Q8—2441no—
554Q8—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
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—1222no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4#1451975112yes—
596C2×C4#1451975112yes—
597C2×C4#1451980812yes—
598C2×C4C22×D24411yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822211yescharacteristic
601C2×C4C2×#1175822611yescharacteristic
602C2×C4#1451982311yescharacteristic
603C2×C4#1451982511yescharacteristic
604C8—2441no—
605C8—2441no—
606C8—2441no—
607C8—2441no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22C2×#1451975111yescharacteristic
616C22#1684040011yescharacteristic
617C22#1684042311yescharacteristic
618C22C2×#1451980811yescharacteristic
619C22#1684056011yescharacteristic
620C22#1684057311yescharacteristic
621C22#1684057311yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—81no—
632C4—4881no—
633C4—4881no—
634C4—2442no—
635C4#1684029911yescharacteristic
636C4#1684038411yescharacteristic
637C4#1684042411yescharacteristic
638C4#1684043411yescharacteristic
639C2—9761no—
640C2#2144937911yescharacteristic
641C2#2145529111yescharacteristic
642C2#2145533111yescharacteristic
643C2#2145792411yescharacteristic
644C2#2145799911yescharacteristic
645C2#2145855911yescharacteristic
646C2#2145864611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2369C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2369D61
hypocenterC61#25287
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
2. C2×#16840044C2semidirect
4. #21447197C2semidirect
6. #21447065C2semidirect
10. #21446742C2semidirect
11. #21446543C2semidirect
12. #21446453C2semidirect
14. #21445652C2semidirect
15. C61×#2369C2semidirect
319. #2369D61semidirect
384. C61#25287semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168400442
#214507992
#214471972
#214471902
#214470652
#214470602
#214467512
#214467422
#214465432
#214464532
#214464462
#214456522
C61×#23692
#2528761