Finite Group Databasebeta

← #29689765

Subgroups — #29689765

Order 15616.

1366 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446790C212yesmaximal, complemented
5#21446641C212yesmaximal, complemented
6#21446140C212yesmaximal, complemented
7#21446138C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445819C211yesmaximal, characteristic, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840309—24no—
14#16840309C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519752C2211yescharacteristic
19C2×#14519750—21no—
20C2×#14519748C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×#14519735C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839818—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519699C2212yes—
28C2×#14519698C2212yes—
29C2×#14519697C2212yes—
30C2×#14519689C2212yes—
31C2×#14519687C2212yes—
32#16839632C2211yescharacteristic
33#16839627C2212yes—
34#16839619—21no—
35#16839619—21no—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—44no—
48C2×#11758222—44no—
49C22×Dic122—21no—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60#14519776—22no—
61#14519776C2×C414yes—
62#14519773—22no—
63#14519773C2×C414yes—
64#14519763—28no—
65#14519752—22no—
66#14519752C2×C414yes—
67#14519750—28no—
68#14519748—22no—
69#14519748C2×C414yes—
70C4×D244—22no—
71C4×D244C2×C414yes—
72C4×C22⋊D61—28no—
73#14519735—42no—
74#14519735—42no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—22no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90C4⋊#92418—22no—
91#14519699—42no—
92#14519698—42no—
93#14519697—42no—
94#14519689—24no—
95#14519687—24no—
96#14519684—22no—
97#14519684—22no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C22×C8×C61C2312yes—
101C4×D4×C61—22no—
102C4×D4×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61—22no—
110#102×C61—42no—
111C24×C61—21no—
112C23×D61—21no—
113C23×D61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61C22×C412yes—
120C22×C4×C61—21no—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142#11758222—432no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—24no—
148C2×D244—21no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244C22×C412yes—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—24no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61C22⋊C414yes—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C2×D412yes—
167C2×Dic122—24no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—24no—
180C42×C61C22×C412yes—
181C42×C61C2×D412yes—
182C22⋊Dic61—24no—
183#11758198—24no—
184#11758198—24no—
185#11758198—44no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—42no—
190C4⋊Dic61—42no—
191C4⋊Dic61C22×C412yes—
192#11758196—24no—
193C4×Dic61—44no—
194C4×Dic61C22⋊C414yes—
195C4×Dic61C22⋊C414yes—
196C4×Dic61C22×C412yes—
197C4×Dic61C2×D412yes—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—22no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61C22×D411yescharacteristic
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61C2×C22⋊C412yes—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—41no—
226D4×C61—41no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—48no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61AGL(1,ℤ/8)12yes—
239C2×C4×C61AGL(1,ℤ/8)12yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×C4×C612−1+411yescharacteristic
246C22⋊D61—48no—
247C22⋊D61—48no—
248C22⋊D61—48no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264D244—22no—
265D244—22no—
266D244—48no—
267D244—41no—
268D244—41no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—28no—
274C4×D61—44no—
275C4×D61—48no—
276Dic122—48no—
277C8×C61—42no—
278#92418—42no—
279#21869—611nomaximal, Sylow
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61#53511yescharacteristic
289C22×C61#53611yescharacteristic
290C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61#57511yescharacteristic
293C22×C61#57511yescharacteristic
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—82no—
311C4×C61#41814yes—
312C4×C61#52011yescharacteristic
313C4×C61#52011yescharacteristic
314C4×C61#53711yescharacteristic
315C4×C61#53711yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—22no—
319Dic61—42no—
320Dic61—82no—
321C2×#532—611no—
322C2×#432—611no—
323#2285—611no—
324#1993—612no—
325#1961—612no—
326#1875—612no—
327#1875—612no—
328#1802—611no—
329#1802—611no—
330#1802—611no—
331#1802D6111yescharacteristic, complemented
332C2×C61—82no—
333C2×C61#231111yescharacteristic
334C2×C61C2×#41811yescharacteristic
335C2×C61#289611yescharacteristic
336C2×C61#321411yescharacteristic
337C2×C61#321411yescharacteristic
338C2×C61#323511yescharacteristic
339C2×C61#323511yescharacteristic
340D61—42no—
341D61—82no—
342C22×#41—1221no—
343#532—618no—
344#532—1224no—
345C2×C4⋊Q8—611no—
346C2×#127—611no—
347C2×#124—1221no—
348C2×#121—611no—
349C2×#121—611no—
350C2×#121—1221no—
351C2×C4×D4—611no—
352C2×C4×D4—1221no—
353C2×C4×D4C2×D6111yescharacteristic
354C2×#117—612no—
355C22×C42—611no—
356#432—1224no—
357C2×C4⋊C8—611no—
358C2×C4⋊C8C2×D6111yescharacteristic
359C2×#102—612no—
360C2×#102—612no—
361C2×#102—612no—
362C2×#102C2×D6112yes—
363C2×C22⋊C8—612no—
364C2×#97—612no—
365#384—611no—
366#384C2×D6111yescharacteristic
367#382—612no—
368#380—1221no—
369#380—1221no—
370C4×C4⋊C4—1221no—
371C4×C4⋊C4—1221no—
372#340—612no—
373#340—612no—
374#340—612no—
375#340C2×D6112yes—
376C61#2186911yescharacteristic, complemented, Sylow
377C2×#41—1228no—
378C2×#41—2444no—
379C2×#41—2444no—
380C22×Q8—1221no—
381C22×D4—611no—
382C22×D4—1221no—
383C22×D4—1221no—
384C22×D4—1221no—
385C22×D4—1221no—
386C22×D4C22×D6111yescharacteristic
387C23×C4—21no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—1221no—
392C22×C8—612no—
393C22×C8C22×D6112yes—
394C4⋊Q8—614no—
395C4⋊Q8—1222no—
396#127—614no—
397#127—1222no—
398#124—1228no—
399#121—614no—
400#121—614no—
401#121—1228no—
402#121—1222no—
403#121—1222no—
404C4×D4—22no—
405C4×D4—614no—
406C4×D4—1228no—
407C4×D4—1222no—
408C4×D4C4×D6114yes—
409#117—2442no—
410#117—2442no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—612no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22×D6112yes—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1222no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—614no—
429C2×C42—614no—
430C2×C42—1222no—
431C2×C42C22×D6111yescharacteristic
432C4⋊C8—22no—
433C4⋊C8—1222no—
434#102—42no—
435#102—2442no—
436#102—2442no—
437#102—2442no—
438C22⋊C8—1224no—
439#97—1224no—
440#95—1222no—
441#95—1222no—
442C24—21no—
443C24—611no—
444C24—1221no—
445#41—24432no—
446C2×Q8—1224no—
447C2×Q8—2442no—
448C2×Q8—2442no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—612no—
455C2×D4—1224no—
456C2×D4—1224no—
457C2×D4—1224no—
458C2×D4—1224no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—2442no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4—2442no—
469C2×D4—2442no—
470C2×D4—2442no—
471C2×D4—2442no—
472C2×D4C2×C4×D6112yes—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—614no—
484C22×C4—614no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1221no—
492C22×C4—1224no—
493C22×C4—1222no—
494C22×C4—2441no—
495C22×C4—2441no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—612no—
513C4⋊C4—1224no—
514C4⋊C4—1221no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4C2×C4×D6112yes—
520C22⋊C4—24no—
521C22⋊C4—1224no—
522C22⋊C4—1224no—
523C22⋊C4—1224no—
524C22⋊C4—2444no—
525C42—612no—
526C42—612no—
527C42—614no—
528C42—614no—
529C42—2444no—
530C42C2×C4×D6112yes—
531C42D4×D6112yes—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—612no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1222no—
547C23—2441no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23C2×D4×D6111yescharacteristic
554Q8—2448no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—1222no—
560D4—1222no—
561D4—2448no—
562D4—2448no—
563D4—2448no—
564D4—2448no—
565D4—2441no—
566D4—2441no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1228no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—2444no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2448no—
598C2×C4C22⋊C4×D6112yes—
599C2×C4C22⋊C4×D6112yes—
600C2×C4#1451980112yes—
601C2×C4#1451980412yes—
602C2×C4C22×C4×D6111yescharacteristic
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982311yescharacteristic
606C2×C4#1451982411yescharacteristic
607C2×C4#1451982511yescharacteristic
608C8—42no—
609C8—2442no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—4881no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22—2442no—
625C22—2442no—
626C22—2442no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22C2×C22⋊C4×D6111yescharacteristic
632C22#1684032811yescharacteristic
633C22#1684033511yescharacteristic
634C22C2×#1451980111yescharacteristic
635C22C2×#1451980411yescharacteristic
636C22#1684055811yescharacteristic
637C22#1684056111yescharacteristic
638C4—22no—
639C4—42no—
640C4—82no—
641C4—1222no—
642C4—1222no—
643C4—1222no—
644C4—4882no—
645C4—2442no—
646C4#1683976514yes—
647C4#1684024211yescharacteristic
648C4#1684024311yescharacteristic
649C4#1684034111yescharacteristic
650C4#1684034211yescharacteristic
651C2—82no—
652C2—4882no—
653C2—2442no—
654C2#2144840811yescharacteristic
655C2C2×#1683976511yescharacteristic
656C2#2145433511yescharacteristic
657C2#2145681211yescharacteristic
658C2#2145681311yescharacteristic
659C2#2145701511yescharacteristic
660C2#2145701611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#21869
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16839818C2semidirect
3. #21448224C2semidirect
4. #21446790C2semidirect
5. #21446641C2semidirect
6. #21446140C2semidirect
7. #21446138C2semidirect
8. #21445820C2semidirect
9. #21445819C2semidirect
10. #21445816C2semidirect
11. C61×#1802C2semidirect
14. #16840309C4semidirect
331. #1802D61semidirect
376. C61#21869semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×#168398182
#214482242
#214467902
#214466412
#214461402
#214461382
#214458202
#214458192
#214458162
C61×#18022
#2186961