Finite Group Databasebeta

← #29743411

Subgroups — #29743411

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840059C211yesmaximal, characteristic
2C2×#16839860C211yesmaximal, characteristic
3C4⋊C4×D244C211yesmaximal, characteristic
4#21447362C211yesmaximal, characteristic, complemented
5#21447290C211yesmaximal, characteristic, complemented
6#21447214C211yesmaximal, characteristic, complemented
7#21447154C211yesmaximal, characteristic, complemented
8#21446554C211yesmaximal, characteristic, complemented
9#21446385C211yesmaximal, characteristic, complemented
10#21446156C211yesmaximal, characteristic
11#21446156C211yesmaximal, characteristic
12#21446079C211yesmaximal, characteristic
13#21446073C211yesmaximal, characteristic, complemented
14#21446065C211yesmaximal, characteristic
15C61×#2453C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840059—24no—
22#16839860—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839636—21no—
39#16839636—21no—
40#16839632C2211yescharacteristic
41#16839619—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#125×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#400—21no—
53C61×#398—21no—
54C61×#391C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#389—21no—
57C61×#389C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#387—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201Q814yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244—22no—
76C4×D244—41no—
77C4×D244—41no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C4⋊C4×D61—44no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89#14519710—22no—
90#14519710—22no—
91#14519710—22no—
92#14519708—22no—
93#14519708—41no—
94C4⋊#92418—22no—
95#14519701—22no—
96#14519701—22no—
97#14519701—41no—
98#14519697—22no—
99#14519697—22no—
100#14519697D414yes—
101#14519688—22no—
102#14519684—21no—
103#14519684—21no—
104C22×Q8×C61C2311yescharacteristic
105#125×C61—22no—
106#125×C61—41no—
107C4×Q8×C61—22no—
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×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×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131C23×D61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61C2×Q812yes—
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×D244C2×D412yes—
156C2×D244C2×Q812yes—
157C2×D244#4112yes—
158C2×C4×D61—24no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—81no—
166C2×C4×D61—81no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
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—
178C4⋊C4×C61—42no—
179C4⋊C4×C61C2×D412yes—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—42no—
187#11758201—28no—
188#11758198—24no—
189#11758198—42no—
190#11758198—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—42no—
197C4⋊Dic61—81no—
198C4⋊Dic61#4112yes—
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—22no—
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—41no—
224Q8×C61—41no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
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—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×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—22no—
273D244C22⋊Q814yes—
274C4×D61—48no—
275C4×D61—84no—
276C4×D61—84no—
277#92418—22no—
278#92418—22no—
279#92418—41no—
280#92418—41no—
281#26398—611nomaximal, Sylow
282C22×C61#52511yescharacteristic
283C22×C61D4211yescharacteristic
284C22×C61D4×Q811yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61#57511yescharacteristic
287C22×C61#57711yescharacteristic
288C22×C61#57811yescharacteristic
289C2×D61—24no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C4×C61—22no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—81no—
305C4×C61—81no—
306C4×C61C2×C22⋊Q811yescharacteristic
307C4×C61#53911yescharacteristic
308C4×C61#54711yescharacteristic
309C4×C61#55411yescharacteristic
310Dic61—42no—
311Dic61—81no—
312Dic61—81no—
313C2×#478—611no—
314C2×#440—611no—
315#2453D6111yescharacteristic, complemented
316D4×C4⋊C4—611no—
317#2096—611no—
318#2083—611no—
319#2068—611no—
320#2059—611no—
321#1943—611no—
322#1917—611no—
323#1877—611no—
324#1877—611no—
325#1864—611no—
326#1863—611no—
327#1862—611no—
328C2×C61#244911yescharacteristic
329C2×C61#311411yescharacteristic
330C2×C61#312011yescharacteristic
331C2×C61#326911yescharacteristic
332C2×C61#327211yescharacteristic
333C2×C61#338611yescharacteristic
334C2×C61#338811yescharacteristic
335D61—44no—
336C22×SD16—611no—
337C2×#125C2×D6111yescharacteristic
338C2×C4×Q8C2×D6111yescharacteristic
339C2×C4×D4—611no—
340C2×C4×D4—611no—
341C2×C4×D4—611no—
342C22×C4⋊C4—1221no—
343#478—1224no—
344#440—1224no—
345C2×#107—611no—
346C2×#107—611no—
347C2×#107—611no—
348C2×#106—611no—
349C2×#106—611no—
350C2×C4⋊C8—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#97—611no—
358#400—21no—
359#398—21no—
360#391C2×D6111yescharacteristic
361#391C2×D6111yescharacteristic
362#389—21no—
363#389—611no—
364#389—611no—
365#389C2×D6111yescharacteristic
366#389C2×D6111yescharacteristic
367#387—21no—
368#385—1221no—
369#385—1221no—
370#384—611no—
371#380—1221no—
372C4×C4⋊C4C2×D6111yescharacteristic
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381C61#2639811yescharacteristic, complemented, Sylow
382C22×Q8C22×D6111yescharacteristic
383C22×D4—611no—
384C23×C4—1221no—
385C23×C4—1221no—
386C23×C4—1221no—
387C2×SD16—614no—
388C2×SD16—1222no—
389C2×SD16—1222no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—611no—
394#125—22no—
395#125—41no—
396C4×Q8—22no—
397C4×Q8—41no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—2441no—
403C4×D4—2441no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
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⋊C4—1221no—
416C2×C4⋊C4—2444no—
417C2×C4⋊C4C22⋊D6111yescharacteristic
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×C42—21no—
426C2×C42C22×D6111yescharacteristic
427C2×C42C22×D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430#107—1222no—
431#107—1222no—
432#106—1222no—
433#106—2441no—
434C4⋊C8—1222no—
435#103—1222no—
436#103—1222no—
437#103—2441no—
438#102—614no—
439#102—1222no—
440#102—1222no—
441#97—1222no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1221no—
449#95—1221no—
450#95C22⋊D6111yescharacteristic
451#95C22⋊D6111yescharacteristic
452C24—1221no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8Q8×D6112yes—
459C2×D4—612no—
460C2×D4—612no—
461C2×D4—612no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—4881no—
474C22×C4—4881no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C23×D6111yescharacteristic
488SD16—1228no—
489C2×C8—612no—
490C2×C8—612no—
491C2×C8—1221no—
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—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—612no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—4881no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4D4×D6112yes—
524C22⋊C4—1224no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—42no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23C22×C22⋊D6111yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546D4—614no—
547D4—1222no—
548D4—1222no—
549D4—1222no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
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—22no—
568C2×C4—22no—
569C2×C4—41no—
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—41no—
579C2×C4—1222no—
580C2×C4—4884no—
581C2×C4—4884no—
582C2×C4—2448no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4#1451980212yes—
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C2×Q8×D6111yescharacteristic
594C2×C4#41×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#1451982211yescharacteristic
597C2×C4#1451982511yescharacteristic
598C8—1222no—
599C8—1222no—
600C8—2441no—
601C8—2441no—
602C22—1224no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22—2442no—
609C22#1684026911yescharacteristic
610C22D4×C22⋊D6111yescharacteristic
611C22Q8×C22⋊D6111yescharacteristic
612C22C2×#1451980211yescharacteristic
613C22#1684056011yescharacteristic
614C22#1684057011yescharacteristic
615C22#1684057711yescharacteristic
616C4—22no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—4881no—
627C4—4881no—
628C4—2442no—
629C4C22⋊Q8×D6111yescharacteristic
630C4#1684035811yescharacteristic
631C4#1684041411yescharacteristic
632C4#1684046611yescharacteristic
633C2—2444no—
634C2#2144962511yescharacteristic
635C2#2145595311yescharacteristic
636C2#2145602711yescharacteristic
637C2#2145737511yescharacteristic
638C2#2145741911yescharacteristic
639C2#2145873611yescharacteristic
640C2#2145875811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2453C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2453D61
hypocenterC61#26398
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2453

Complements

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

NH ≅ G/Ntype
4. #21447362C2semidirect
5. #21447290C2semidirect
6. #21447214C2semidirect
7. #21447154C2semidirect
8. #21446554C2semidirect
9. #21446385C2semidirect
13. #21446073C2semidirect
15. C61×#2453C2semidirect
315. #2453D61semidirect
381. C61#26398semidirect

Maximal subgroups

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

subgroupindex
C2×#168400592
C2×#168398602
C4⋊C4×D2442
#214473622
#214472902
#214472142
#214471542
#214465542
#214463852
#214461562
#214460792
#214460732
#214460652
C61×#24532
#2639861