Finite Group Databasebeta

← #29689959

Subgroups — #29689959

Order 15616.

1410 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446765C212yesmaximal, complemented
5#21446744C212yesmaximal, complemented
6#21446256C212yesmaximal, complemented
7#21446254C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445819C211yesmaximal, characteristic, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758189C2212yes—
14#16840309—24no—
15#16840309C418yescomplemented
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519752C2211yescharacteristic
20C2×#14519750—21no—
21C2×#14519748C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×#14519735—21no—
25C2×C42×D61C2211yescharacteristic
26#16839784—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519706C2212yes—
29C2×#14519699C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32#16839632C2211yescharacteristic
33#16839627—21no—
34#16839619—21no—
35#16839619C2212yes—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61C2212yes—
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
59C2×#11758189—24no—
60C2×#11758189—42no—
61C2×#11758189—42no—
62C22×#92418C2312yes—
63#14519776—22no—
64#14519776C2×C414yes—
65#14519773—22no—
66#14519773C2×C414yes—
67#14519763—28no—
68#14519752—22no—
69#14519752C2×C414yes—
70#14519750—28no—
71#14519748—22no—
72#14519748C2×C414yes—
73C4×D244—22no—
74C4×D244C2×C414yes—
75C4×C22⋊D61—28no—
76#14519735—44no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C42×D61C2×C414yes—
91C42×D61D414yes—
92C4⋊#92418—22no—
93#14519706—24no—
94#14519699—42no—
95#14519698—42no—
96#14519697—42no—
97#14519684—22no—
98#14519684—22no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C22×C8×C61C2312yes—
102C4×D4×C61—22no—
103C4×D4×C61C2×C414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C4⋊C8×C61—22no—
111#102×C61—42no—
112C24×C61—21no—
113C23×D61—21no—
114C23×D61C2×D411yescharacteristic
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61C22×C412yes—
121C22×C4×C61—21no—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143#11758222—432no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—24no—
149C2×D244—21no—
150C2×D244—42no—
151C2×D244—42no—
152C2×D244C22×C412yes—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D412yes—
168C2×Dic122—24no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61C22×C412yes—
180C22⋊C4×C61—24no—
181C42×C61C22×C412yes—
182C42×C61C2×D412yes—
183C22⋊Dic61—24no—
184#11758198—24no—
185#11758198—24no—
186#11758198—44no—
187C4⋊Dic61—21no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—42no—
191C4⋊Dic61—42no—
192C4⋊Dic61C22×C412yes—
193#11758196—24no—
194C4×Dic61—44no—
195C4×Dic61C22⋊C414yes—
196C4×Dic61C22⋊C414yes—
197C4×Dic61C22×C412yes—
198C4×Dic61C2×D412yes—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202#11758189—48no—
203C23×C61—22no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C22×D411yescharacteristic
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—22no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61C2×C22⋊C412yes—
225D4×C61—22no—
226D4×C61—22no—
227D4×C61—41no—
228D4×C61—41no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—48no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61AGL(1,ℤ/8)12yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C22⋊D61—48no—
249C22⋊D61—48no—
250C22⋊D61—48no—
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—
264C2×Dic61AGL(1,ℤ/8)12yes—
265C2×Dic61#13712yes—
266D244—22no—
267D244—22no—
268D244—48no—
269D244—41no—
270D244—41no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—28no—
276C4×D61—28no—
277C4×D61—48no—
278Dic122—48no—
279C8×C61—42no—
280#92418—42no—
281#21893—611nomaximal, Sylow
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61C22×C22⋊C411yescharacteristic
290C22×C61#53711yescharacteristic
291C22×C61#53711yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C61C2×#13711yescharacteristic
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—82no—
313C4×C61#41814yes—
314C4×C61#52011yescharacteristic
315C4×C61#52011yescharacteristic
316C4×C61#53511yescharacteristic
317C4×C61#53611yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—82no—
322Dic61#41714yes—
323C2×#532—611no—
324C2×#423—611no—
325#2285—611no—
326#1987—612no—
327#1982—612no—
328#1895—612no—
329#1895—612no—
330#1802—611no—
331#1802—611no—
332#1802—611no—
333#1802D6111yescharacteristic, complemented
334C2×C61—82no—
335C2×C61#231111yescharacteristic
336C2×C61C2×#41811yescharacteristic
337C2×C61C2×#41711yescharacteristic
338C2×C61#325311yescharacteristic
339C2×C61#325311yescharacteristic
340C2×C61#325411yescharacteristic
341C2×C61#325411yescharacteristic
342D61—42no—
343D61—82no—
344C22×#41—1221no—
345C22×M16—612no—
346#532—618no—
347#532—1224no—
348C2×C4⋊Q8—611no—
349C2×#127—611no—
350C2×#124—1221no—
351C2×#121—611no—
352C2×#121—611no—
353C2×#121—1221no—
354C2×C4×D4—611no—
355C2×C4×D4—1221no—
356C2×C4×D4C2×D6111yescharacteristic
357C2×#117—1221no—
358C22×C42—611no—
359#423—1224no—
360C2×C4⋊C8—612no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8C2×D6111yescharacteristic
363C2×#102—612no—
364C2×#102—612no—
365C2×#102—612no—
366C2×#102C2×D6112yes—
367#384—611no—
368#384C2×D6111yescharacteristic
369#382—1221no—
370#380—612no—
371#380—1221no—
372C4×C4⋊C4—612no—
373C4×C4⋊C4—1221no—
374#340—612no—
375#340—612no—
376#340—612no—
377#340C2×D6112yes—
378C61#2189311yescharacteristic, complemented, Sylow
379C2×#41—1228no—
380C2×#41—2444no—
381C2×#41—2444no—
382C22×Q8—1221no—
383C22×D4—611no—
384C22×D4—1221no—
385C22×D4—1221no—
386C22×D4—1221no—
387C22×D4—1221no—
388C22×D4C22×D6111yescharacteristic
389C23×C4—21no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—1221no—
394C2×M16—1224no—
395C2×M16—2442no—
396C2×M16—2442no—
397C22×C8—612no—
398C22×C8C22×D6112yes—
399C4⋊Q8—614no—
400C4⋊Q8—1222no—
401#127—614no—
402#127—1222no—
403#124—1228no—
404#121—614no—
405#121—614no—
406#121—1228no—
407#121—1222no—
408#121—1222no—
409C4×D4—22no—
410C4×D4—614no—
411C4×D4—1228no—
412C4×D4—1222no—
413C4×D4C4×D6114yes—
414#117—2444no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6112yes—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1222no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—614no—
433C2×C42—614no—
434C2×C42—1222no—
435C2×C42C22×D6111yescharacteristic
436C4⋊C8—22no—
437C4⋊C8—1224no—
438C4⋊C8—1222no—
439#102—42no—
440#102—2442no—
441#102—2442no—
442#102—2442no—
443#95—1222no—
444#95—1222no—
445C24—21no—
446C24—611no—
447C24—1221no—
448#41—24432no—
449C2×Q8—1224no—
450C2×Q8—2442no—
451C2×Q8—2442no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—612no—
458C2×D4—1224no—
459C2×D4—1224no—
460C2×D4—1224no—
461C2×D4—1224no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—2442no—
468C2×D4—2442no—
469C2×D4—2442no—
470C2×D4—2442no—
471C2×D4—2442no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4—2442no—
475C2×D4C2×C4×D6112yes—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—614no—
487C22×C4—614no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4—1224no—
498C22×C4—1222no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504M16—2448no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1222no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—612no—
517C4⋊C4—1224no—
518C4⋊C4—1221no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—24no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C22⋊C4—2444no—
529C42—612no—
530C42—612no—
531C42—614no—
532C42—614no—
533C42—2444no—
534C42C2×C4×D6112yes—
535C42D4×D6112yes—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—612no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1222no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23C2×D4×D6111yescharacteristic
558Q8—2448no—
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—1222no—
564D4—1222no—
565D4—2448no—
566D4—2448no—
567D4—2448no—
568D4—2448no—
569D4—2441no—
570D4—2441no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—1228no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1228no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2448no—
602C2×C4C22⋊C4×D6112yes—
603C2×C4C22⋊C4×D6112yes—
604C2×C4#1451980112yes—
605C2×C4#1451980412yes—
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4#1451982211yescharacteristic
609C2×C4#1451982311yescharacteristic
610C2×C4#1451982411yescharacteristic
611C2×C4#1451982511yescharacteristic
612C8—42no—
613C8—2442no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22C2×C22⋊C4×D6111yescharacteristic
636C22#1684034111yescharacteristic
637C22#1684034211yescharacteristic
638C22AGL(1,ℤ/8)×D6111yescharacteristic
639C22C2×#1451980111yescharacteristic
640C22C2×#1451980411yescharacteristic
641C22#137×D6111yescharacteristic
642C4—22no—
643C4—42no—
644C4—82no—
645C4—614no—
646C4—1222no—
647C4—1222no—
648C4—4882no—
649C4—2442no—
650C4#1683976514yes—
651C4#1684024211yescharacteristic
652C4#1684024311yescharacteristic
653C4#1684032811yescharacteristic
654C4#1684033511yescharacteristic
655C2—82no—
656C2—4882no—
657C2—2442no—
658C2#2144840811yescharacteristic
659C2C2×#1683976511yescharacteristic
660C2#417×D6111yescharacteristic
661C2#2145719811yescharacteristic
662C2#2145719911yescharacteristic
663C2#2145720311yescharacteristic
664C2#2145720811yescharacteristic
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#21893
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×#16839784C2semidirect
3. #21448224C2semidirect
4. #21446765C2semidirect
5. #21446744C2semidirect
6. #21446256C2semidirect
7. #21446254C2semidirect
8. #21445820C2semidirect
9. #21445819C2semidirect
10. #21445816C2semidirect
11. C61×#1802C2semidirect
15. #16840309C4semidirect
333. #1802D61semidirect
378. C61#21893semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×#168397842
#214482242
#214467652
#214467442
#214462562
#214462542
#214458202
#214458192
#214458162
C61×#18022
#2189361