Finite Group Databasebeta

← #29647316

Subgroups — #29647316

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459115C211yesmaximal, characteristic
2#21454452C211yesmaximal, characteristic, complemented
3#21444360C214yesmaximal, complemented
4#21444355C214yesmaximal, complemented
5#21444317C212yesmaximal, complemented
6#21444313C212yesmaximal, complemented
7C61×#2285C211yesmaximal, characteristic, complemented
8#16840319C411yescharacteristic, complemented
9#16840319C411yescharacteristic, complemented
10#16840319C2211yescharacteristic
11#16840312C411yescharacteristic, complemented
12#16840312C411yescharacteristic, complemented
13#16840312C2211yescharacteristic
14#16840296—21no—
15#16840296—21no—
16#16840296—21no—
17#16840296—21no—
18#16840296C412yescomplemented
19#16840296C2212yes—
20#16840273—21no—
21#16840273—21no—
22#16840273C2212yes—
23C4×#11758222—21no—
24C4×#11758222—21no—
25C4×#11758222C412yescomplemented
26#16839819—21no—
27#16839819C2212yes—
28#16839815—21no—
29#16839815C2212yes—
30#16839786—21no—
31#16839726C2212yes—
32C4×#11758194—21no—
33#16839498C2218yes—
34#16839495C2218yes—
35C2×#117×C61—21no—
36C22×C42×C61C2211yescharacteristic
37C61×#384—21no—
38C61×#382C2212yes—
39C61×#380—21no—
40C61×#380C2212yes—
41C4×C4⋊C4×C61—21no—
42C4×C4⋊C4×C61C2212yes—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222C2×C412yes—
46C2×#11758194—22no—
47#14519777C2×C411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519777C2×C411yescharacteristic
50#14519774C2×C411yescharacteristic
51#14519771—22no—
52#14519771—22no—
53#14519769—21no—
54#14519769—21no—
55#14519769C2×C412yes—
56#14519765—21no—
57#14519765—21no—
58#14519765C2×C412yes—
59#14519760—22no—
60#14519760—22no—
61#14519760—22no—
62#14519756—21no—
63#14519756—21no—
64#14519756—21no—
65#14519756—21no—
66#14519756C2×C412yes—
67#14519756C2312yes—
68#14519751—21no—
69#14519751—21no—
70#14519751—21no—
71#14519751—21no—
72#14519751C2×C412yes—
73#14519751C2312yes—
74#14519746—22no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79C4×Dic122C2×C412yes—
80C4×Dic122C2×C412yes—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244—21no—
85C4×D244C2×C412yes—
86C4×D244C2×C412yes—
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92C4×C22⋊D61—22no—
93#14519736—21no—
94#14519736—21no—
95#14519736C2×C412yes—
96#14519734—22no—
97#14519734—22no—
98#14519734—22no—
99C42×D61—22no—
100C4⋊#92418—22no—
101C4⋊#92418—22no—
102C22⋊#92418—22no—
103C22⋊#92418C2314yes—
104#14519688—22no—
105C4×#92418—22no—
106C23×C4×C61D411yescharacteristic
107C23×C4×C61D411yescharacteristic
108C23×C4×C61C2311yescharacteristic
109#117×C61—44no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C22⋊C4×C61—22no—
114C2×C42×C61—22no—
115C2×C42×C61C2×C411yescharacteristic
116C2×C42×C61C2×C411yescharacteristic
117C2×C42×C61C2×C411yescharacteristic
118C2×C42×C61C2×C411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61D411yescharacteristic
124C2×C42×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—22no—
128#95×C61C2314yes—
129C24×C61C2×D411yescharacteristic
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
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
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156#11758222—44no—
157#11758222—42no—
158#11758222—42no—
159#11758222—44no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C22×C412yes—
169C4⋊C4×C61—44no—
170C4⋊C4×C61—44no—
171C4⋊C4×C61—44no—
172C22⋊C4×C61—44no—
173C42×C61—44no—
174C42×C61C22⋊C411yescharacteristic
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22⋊C411yescharacteristic
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22×C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C22×C412yes—
204C4⋊Dic61C22×C412yes—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214#11758194—44no—
215C2×#92418—24no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61C2×C22⋊C411yescharacteristic
221C23×C61C2×C22⋊C411yescharacteristic
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—44no—
243C2×C4×C61—44no—
244C2×C4×C61—44no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61AGL(1,ℤ/8)12yes—
256C2×C4×C61#13712yes—
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C22⋊D61—44no—
260C22⋊D61—44no—
261C22⋊D61—42no—
262C22⋊D61—42no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266D244—42no—
267D244—42no—
268D244—42no—
269C4×D61—44no—
270C4×D61—44no—
271C4×D61—44no—
272Dic122—42no—
273Dic122—42no—
274Dic122—42no—
275#92418—44no—
276#17563—611nomaximal, Sylow
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C61C2×#13711yescharacteristic
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—84no—
302C4×C61#41012yes—
303C4×C61#41712yes—
304C4×C61#41712yes—
305C4×C61#42112yes—
306Dic61—42no—
307Dic61—42no—
308Dic61—42no—
309#3444—611no—
310#2920—611no—
311#2285D6111yescharacteristic, complemented
312#1513—614no—
313#1512—614no—
314#1503—612no—
315#1502—612no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—41no—
319C2×C61—41no—
320C2×C61C2×#41011yescharacteristic
321C2×C61C2×#41711yescharacteristic
322C2×C61C2×#42111yescharacteristic
323D61—82no—
324#533—611no—
325#533—611no—
326#533—611no—
327#532—611no—
328#532—611no—
329#532—611no—
330#529—612no—
331#529—612no—
332#529—1221no—
333#529—1221no—
334#529—1221no—
335#529—1221no—
336#525—612no—
337#525—1221no—
338#525—1221no—
339C4×#41—612no—
340C4×#41—1221no—
341C4×#41—1221no—
342C2×#117—21no—
343C22×C42C2×D6111yescharacteristic
344#432—612no—
345#432—1221no—
346#431—612no—
347#431—1221no—
348#423—1221no—
349#407—612no—
350C4×M16—1221no—
351#384—21no—
352#382C2×D6112yes—
353#380—21no—
354#380C2×D6112yes—
355C4×C4⋊C4—21no—
356C4×C4⋊C4C2×D6112yes—
357#328—618no—
358#327—618no—
359C61#1756311yescharacteristic, complemented, Sylow
360C2×#41—612no—
361C2×#41—1221no—
362C2×#41—1221no—
363C23×C4C22⋊D6111yescharacteristic
364C23×C4C22⋊D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C2×M16—1222no—
367C4⋊Q8—611no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370#127—611no—
371#126—1222no—
372#126—1222no—
373#125—612no—
374#125—1221no—
375#125—1221no—
376#124—612no—
377#124—1221no—
378#124—1221no—
379#123—1222no—
380#123—1222no—
381#123—1222no—
382C22⋊Q8—612no—
383C22⋊Q8—612no—
384C22⋊Q8—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388#121—612no—
389#121—612no—
390#121—1221no—
391#121—1221no—
392#121—1221no—
393#121—1221no—
394#120—1222no—
395C4×Q8—612no—
396C4×Q8—612no—
397C4×Q8—1221no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×D4—612no—
402C4×D4—612no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4—1221no—
411C4×D4—1221no—
412C4×D4—1221no—
413#117—44no—
414#117—612no—
415#117—1221no—
416#117—1221no—
417#117—1222no—
418#117—1222no—
419#117—1222no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C42—22no—
425C2×C42—1222no—
426C2×C42C4×D6111yescharacteristic
427C2×C42C4×D6111yescharacteristic
428C2×C42C4×D6111yescharacteristic
429C2×C42C4×D6111yescharacteristic
430C2×C42D24411yescharacteristic
431C2×C42D24411yescharacteristic
432C2×C42D24411yescharacteristic
433C2×C42D24411yescharacteristic
434C2×C42C22⋊D6111yescharacteristic
435C2×C42C22⋊D6111yescharacteristic
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C4⋊C8—1222no—
439C4⋊C8—1222no—
440C22⋊C8—614no—
441C22⋊C8—1222no—
442#97—1222no—
443C4×C8—1222no—
444#95—22no—
445#95C22×D6114yes—
446C24C2×C22⋊D6111yescharacteristic
447#41—2444no—
448#41—2444no—
449#41—2442no—
450#41—2442no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×D4—612no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1221no—
459C2×D4—1221no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4#1175819811yescharacteristic
469C22×C4#1175819811yescharacteristic
470C22×C4#1175819811yescharacteristic
471C22×C4#1175819811yescharacteristic
472C22×C4#1175819811yescharacteristic
473C22×C4#1175819811yescharacteristic
474C22×C4#1175819811yescharacteristic
475C22×C4#1175819811yescharacteristic
476C22×C4C2×C4×D6111yescharacteristic
477C22×C4C2×C4×D6111yescharacteristic
478C22×C4C2×D24411yescharacteristic
479C22×C4C2×D24411yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486M16—2444no—
487C2×C8—1224no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—612no—
492C4⋊C4—612no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C22⋊C4—44no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C42—44no—
514C42—1222no—
515C42—1222no—
516C42—1222no—
517C42#1175819811yescharacteristic
518C42#1175819811yescharacteristic
519C42#1175819811yescharacteristic
520C42#1175819811yescharacteristic
521C42#1175819811yescharacteristic
522C42#1175819811yescharacteristic
523C42#1175819811yescharacteristic
524C42#1175819811yescharacteristic
525C42C2×C4×D6111yescharacteristic
526C42C2×C4×D6111yescharacteristic
527C42C2×C4×D6111yescharacteristic
528C42C2×C4×D6111yescharacteristic
529C42C2×D24411yescharacteristic
530C42C2×D24411yescharacteristic
531C42C2×D24411yescharacteristic
532C42C2×D24411yescharacteristic
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—1222no—
538C23C2×#1175819811yescharacteristic
539C23C2×#1175819811yescharacteristic
540C23C22×C22⋊D6111yescharacteristic
541Q8—2442no—
542Q8—2442no—
543Q8—2442no—
544D4—2442no—
545D4—2444no—
546D4—2444no—
547D4—2442no—
548D4—2442no—
549D4—2442no—
550D4—2442no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—2444no—
576C2×C4—2444no—
577C2×C4—2444no—
578C2×C4C2×#1175819811yescharacteristic
579C2×C4C2×#1175819811yescharacteristic
580C2×C4C2×#1175819811yescharacteristic
581C2×C4C2×#1175819811yescharacteristic
582C2×C4C2×#1175819811yescharacteristic
583C2×C4C2×#1175819811yescharacteristic
584C2×C4C2×#1175819811yescharacteristic
585C2×C4C2×#1175819811yescharacteristic
586C2×C4C2×#1175819811yescharacteristic
587C2×C4C2×#1175819811yescharacteristic
588C2×C4#1451980312yes—
589C2×C4#1451980812yes—
590C2×C4C22×C4×D6111yescharacteristic
591C2×C4C22×D24411yescharacteristic
592C8—2444no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—42no—
602C22—42no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22C22×#1175819811yescharacteristic
607C22C2×#1451980311yescharacteristic
608C22C2×#1451980811yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—84no—
618C4—2442no—
619C4—2442no—
620C4—2442no—
621C4#1683973512yes—
622C4#1683976112yes—
623C4#1683976112yes—
624C4#1683978012yes—
625C2—21no—
626C2—21no—
627C2—41no—
628C2—41no—
629C2—4882no—
630C2C2×#1683973511yescharacteristic
631C2C2×#1683976111yescharacteristic
632C2C2×#1683978011yescharacteristic
subgroupG/H
centerC2×C4C2×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2285C2
socleC22×C61C22×C22⋊C4
prime radical#21459115C2
cosocleC22×C4×C61C24
hypercenter#2285D61
hypocenterC61#17563
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #21454452C2semidirect
3. #21444360C2semidirect
4. #21444355C2semidirect
5. #21444317C2semidirect
6. #21444313C2semidirect
7. C61×#2285C2semidirect
8. #16840319C4semidirect
9. #16840319C4semidirect
11. #16840312C4semidirect
12. #16840312C4semidirect
18. #16840296C4semidirect
25. C4×#11758222C4semidirect
311. #2285D61semidirect
359. C61#17563semidirect

Maximal subgroups

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

subgroupindex
#214591152
#214544522
#214443602
#214443552
#214443172
#214443132
C61×#22852
#1756361