Finite Group Databasebeta

← #29743099

Subgroups — #29743099

Order 15616.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840057C211yesmaximal, characteristic, complemented
2C2×C4×#11758201C211yesmaximal, characteristic
3#21449669C211yesmaximal, characteristic
4#21447343C211yesmaximal, characteristic
5#21447301C211yesmaximal, characteristic, complemented
6#21447204C211yesmaximal, characteristic
7#21447152C211yesmaximal, characteristic, complemented
8#21446549C211yesmaximal, characteristic, complemented
9#21446378C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic
11#21446174C211yesmaximal, characteristic
12#21446087C211yesmaximal, characteristic, complemented
13#21446073C211yesmaximal, characteristic, complemented
14#21446065C211yesmaximal, characteristic
15C61×#2351C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519755—21no—
19C2×C4×D244C2211yescharacteristic
20#16840057—24no—
21C4×#11758201—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839684—21no—
36#16839675—21no—
37#16839664C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839636—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#389—21no—
54C61×#389—21no—
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#382—21no—
58C61×#382C2211yescharacteristic
59C4×C4⋊C4×C61—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C2×#11758201Q814yescomplemented
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519777—22no—
72#14519777—41no—
73#14519755—44no—
74C4×D244—22no—
75C4×D244—41no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87#14519710—22no—
88#14519710—22no—
89#14519710—22no—
90#14519708—22no—
91#14519708—41no—
92C4⋊#92418—22no—
93#14519701—22no—
94#14519701—22no—
95#14519701D414yes—
96#14519697—22no—
97#14519697—22no—
98#14519697—41no—
99C4×#92418—22no—
100#14519684—21no—
101#14519684—21no—
102C22×Q8×C61C2311yescharacteristic
103C4×Q8×C61—22no—
104C4×Q8×C61—22no—
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—41no—
108C4×Q8×C61—41no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
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×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131C23×D61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×D412yes—
136C2×Q8×C61C2×Q812yes—
137C2×Q8×C61#4112yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C2×Q812yes—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—81no—
161C2×Dic122—42no—
162C2×Dic122—42no—
163C2×Dic122—81no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61—42no—
176C4⋊C4×C61#4112yes—
177C42×C61—24no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—42no—
185C42×C61—42no—
186#11758201—28no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—42no—
198C4⋊Dic61C2×D412yes—
199#11758196—42no—
200#11758196—42no—
201C2×#92418—21no—
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×#92418C2×Q812yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61C22⋊Q814yescomplemented
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
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—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
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#13512yes—
254C2×C4×C61AGL(1,ℤ/8)12yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270D244—22no—
271D244—22no—
272D244—41no—
273D244—41no—
274C4×D61—84no—
275Dic122—84no—
276#92418—22no—
277#92418—22no—
278#92418—41no—
279#92418—41no—
280#26377—611nomaximal, Sylow
281C22×C61#52511yescharacteristic
282C22×C61D4211yescharacteristic
283C22×C61D4×Q811yescharacteristic
284C22×C61C2×#13511yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61#57711yescharacteristic
287C22×C61#57811yescharacteristic
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C4×C61—24no—
296C4×C61—22no—
297C4×C61—22no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61C2×C22⋊Q811yescharacteristic
306C4×C61#53911yescharacteristic
307C4×C61#54711yescharacteristic
308C4×C61#55411yescharacteristic
309Dic61—42no—
310Dic61—81no—
311Dic61—81no—
312C2×#477—611no—
313C2×C4×SD16—611no—
314#2452—611no—
315#2351D6111yescharacteristic, complemented
316#2092—611no—
317#2085—611no—
318#2067—611no—
319#2058—611no—
320#1942—611no—
321#1916—611no—
322#1880—611no—
323#1880—611no—
324#1865—611no—
325#1863—611no—
326#1862—611no—
327C2×C61#244911yescharacteristic
328C2×C61#310611yescharacteristic
329C2×C61#312111yescharacteristic
330C2×C61#326511yescharacteristic
331C2×C61#328511yescharacteristic
332C2×C61#338111yescharacteristic
333C2×C61#338611yescharacteristic
334D61—82no—
335C22×SD16—611no—
336C2×C4⋊Q8—611no—
337C2×C22⋊Q8—1221no—
338C2×C4×Q8C2×D6111yescharacteristic
339C2×C4×Q8C2×D6111yescharacteristic
340C2×C4×Q8C2×D6111yescharacteristic
341C2×C4×D4—611no—
342#477—1224no—
343C4×SD16—1224no—
344C2×#107—611no—
345C2×#107—611no—
346C2×#107—611no—
347C2×#106—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#102—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×C4×C8—611no—
357#396—1221no—
358#393—1221no—
359#390—611no—
360#390—611no—
361#389—21no—
362#389—21no—
363#389—611no—
364#389—611no—
365#389C2×D6111yescharacteristic
366#389C2×D6111yescharacteristic
367#385—1221no—
368#382—21no—
369#382C2×D6111yescharacteristic
370C4×C4⋊C4—21no—
371C4×C4⋊C4C2×D6111yescharacteristic
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#336—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380C61#2637711yescharacteristic, complemented, Sylow
381C22×Q8—1221no—
382C22×Q8C22×D6111yescharacteristic
383C22×D4—611no—
384C23×C4—1221no—
385C2×SD16—614no—
386C2×SD16—1222no—
387C2×SD16—1222no—
388C22×C8—611no—
389C22×C8—611no—
390C22×C8—611no—
391C22×C8—611no—
392C4⋊Q8—1222no—
393C4⋊Q8—2441no—
394C22⋊Q8—2444no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—41no—
400C4×Q8—41no—
401C4×D4—1222no—
402C4×D4—2441no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
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—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22⋊D6111yescharacteristic
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—21no—
427C2×C42C22×D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430#107—1222no—
431#107—1222no—
432#107—1222no—
433#106—1222no—
434#106—2441no—
435C4⋊C8—1222no—
436#103—614no—
437#103—1222no—
438#103—1222no—
439#102—1222no—
440#102—1222no—
441#102—2441no—
442C4×C8—1222no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1221no—
448#95—1221no—
449#95C22⋊D6111yescharacteristic
450#95C22⋊D6111yescharacteristic
451C24—1221no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—4881no—
456C2×Q8—2442no—
457C2×Q8—2442no—
458C2×Q8#1175822212yes—
459C2×Q8D4×D6112yes—
460C2×Q8Q8×D6112yes—
461C2×D4—612no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—4881no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—1228no—
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—612no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4#1175822212yes—
519C22⋊C4—4881no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—42no—
532C42—42no—
533C23—1222no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23C22×C22⋊D6111yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—4884no—
545Q8#1451975514yescomplemented
546D4—1222no—
547D4—1222no—
548D4—2441no—
549D4—2441no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
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—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—1222no—
579C2×C4—4884no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4#1451979712yes—
589C2×C4#1451980212yes—
590C2×C4C2×D4×D6111yescharacteristic
591C2×C4C2×Q8×D6111yescharacteristic
592C2×C4C2×#1175822211yescharacteristic
593C2×C4C2×#1175822211yescharacteristic
594C2×C4#1451982211yescharacteristic
595C2×C4#1451982511yescharacteristic
596C8—1222no—
597C8—1222no—
598C8—2441no—
599C8—2441no—
600C22—4881no—
601C22—4881no—
602C22—4881no—
603C22—4881no—
604C22—2442no—
605C22—2442no—
606C22—2442no—
607C22#1684027311yescharacteristic
608C22D4×C22⋊D6111yescharacteristic
609C22Q8×C22⋊D6111yescharacteristic
610C22C2×#1451979711yescharacteristic
611C22C2×#1451980211yescharacteristic
612C22#1684057011yescharacteristic
613C22#1684057711yescharacteristic
614C4—22no—
615C4—24no—
616C4—22no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—4881no—
625C4—4881no—
626C4—2442no—
627C4C2×#1451975511yescharacteristic
628C4#1684035911yescharacteristic
629C4#1684041311yescharacteristic
630C4#1684046711yescharacteristic
631C2—4882no—
632C2#2144962811yescharacteristic
633C2#2145587911yescharacteristic
634C2#2145604811yescharacteristic
635C2#2145731511yescharacteristic
636C2#2145755611yescharacteristic
637C2#2145869011yescharacteristic
638C2#2145873611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2351C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2351D61
hypocenterC61#26377
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#16840057C2semidirect
5. #21447301C2semidirect
7. #21447152C2semidirect
8. #21446549C2semidirect
9. #21446378C2semidirect
12. #21446087C2semidirect
13. #21446073C2semidirect
15. C61×#2351C2semidirect
66. C2×#11758201Q8semidirect
224. Q8×C61C22⋊Q8semidirect
315. #2351D61semidirect
380. C61#26377semidirect
545. Q8#14519755semidirect

Maximal subgroups

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

subgroupindex
C2×#168400572
C2×C4×#117582012
#214496692
#214473432
#214473012
#214472042
#214471522
#214465492
#214463782
#214461742
#214460872
#214460732
#214460652
C61×#23512
#2637761