Finite Group Databasebeta

← #29689002

Subgroups — #29689002

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840335C211yesmaximal, characteristic
2#21454263C211yesmaximal, characteristic
3#21454262C211yesmaximal, characteristic
4#21446288C211yesmaximal, characteristic, complemented
5#21446287C211yesmaximal, characteristic, complemented
6#21446286C212yesmaximal, complemented
7#21446285C211yesmaximal, characteristic, complemented
8#21446284C212yesmaximal, complemented
9#21445901C212yesmaximal, complemented
10#21445741C212yesmaximal, complemented
11C61×#1901C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15C2×M16×D61C2212yes—
16#16840335—22no—
17#16840335—22no—
18#16840335—22no—
19#16840335—22no—
20C2×#14519776C412yes—
21C2×#14519763C412yes—
22C2×#14519762C412yes—
23C2×#14519759C412yes—
24C2×#14519757C2211yescharacteristic
25C2×#14519756C2211yescharacteristic
26C2×#14519753C2211yescharacteristic
27C2×#14519735C2211yescharacteristic
28C2×#14519696C2212yes—
29C2×#14519695C2212yes—
30C2×#14519695C2212yes—
31#16839730—22no—
32#16839728—22no—
33#16839727—22no—
34#16839727—22no—
35#16839726C2211yescharacteristic
36#16839725C2212yes—
37#16839725C2212yes—
38#16839725C2212yes—
39#16839531C2212yes—
40#16839530C2212yes—
41#16839530C2212yes—
42C22×M16×C61C2211yescharacteristic
43C2×C22⋊Q8×C61C2211yescharacteristic
44C2×#101×C61C2212yes—
45C61×#407C2211yescharacteristic
46C61×#343C2212yes—
47C2×#11758222—42no—
48C2×#11758222—42no—
49C2×#11758222D412yes—
50C2×#11758222D412yes—
51C2×#11758222D412yes—
52C2×#11758222D412yes—
53C22×Dic122C2×C412yes—
54C22×Dic122C2×C411yescharacteristic
55C22×Dic122C2311yescharacteristic
56C2×Q8×D61—22no—
57C2×Q8×D61—22no—
58C2×Q8×D61—42no—
59C2×Q8×D61—42no—
60C22×D244C2×C411yescharacteristic
61C22×C22⋊D61C2×C411yescharacteristic
62C22×C22⋊D61C2×C411yescharacteristic
63C22×C4×D61C2×C412yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×#11758194—21no—
67C2×#11758194—42no—
68C2×#11758194C2312yes—
69C2×#11758194C2312yes—
70C2×#11758189—22no—
71M16×D61—42no—
72M16×D61—42no—
73C22×#92418—21no—
74C2×C8×D61—22no—
75#14519776—24no—
76#14519776—24no—
77#14519763—24no—
78#14519763—24no—
79#14519762—24no—
80#14519762—24no—
81#14519759—24no—
82#14519759—24no—
83#14519757—22no—
84#14519757—22no—
85#14519756—21no—
86#14519756—21no—
87#14519756—21no—
88#14519756—21no—
89#14519753—22no—
90#14519753—22no—
91#14519735—22no—
92#14519735—22no—
93C2×C4⋊Dic61C2×C412yes—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196C2×C412yes—
96C2×#11758196C2×C412yes—
97C2×C22⋊Dic61C2×C412yes—
98C2×#11758198C2×C412yes—
99C2×#11758198C2×C412yes—
100C2×#11758198C2×C412yes—
101C2×C4×Dic61C2×C412yes—
102#14519696—42no—
103#14519695—42no—
104#14519695—42no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107#14519689—24no—
108#14519689—24no—
109#14519689—24no—
110C22×Q8×C61C2311yescharacteristic
111C23×C4×C61C2311yescharacteristic
112C2×M16×C61—21no—
113C2×M16×C61—42no—
114C2×M16×C61C2312yes—
115C2×M16×C61C2312yes—
116C22×C8×C61—21no—
117C22⋊Q8×C61—21no—
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—21no—
121C2×C4⋊C4×C61C2×C412yes—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61C2×C412yes—
124#101×C61—42no—
125C22⋊C8×C61—21no—
126C22⋊C8×C61—21no—
127C24×C61C22×C411yescharacteristic
128C23×D61C22×C411yescharacteristic
129C23×D61C22×C411yescharacteristic
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61C22⋊C412yes—
147C2×C22⋊D61C22⋊C412yes—
148C2×C22⋊D61C22⋊C412yes—
149C2×C22⋊D61C22⋊C412yes—
150C22×Dic61C22×C412yes—
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C412yes—
154Q8×D61—44no—
155Q8×D61—44no—
156Q8×D61—44no—
157Q8×D61—44no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161#11758222—42no—
162#11758222—42no—
163#11758222—42no—
164#11758222—42no—
165#11758222—42no—
166C2×D244—42no—
167C2×D244C22⋊C412yes—
168C2×D244C22⋊C412yes—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—24no—
174C2×C4×D61—44no—
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D412yes—
177C2×C4×D61C2×D412yes—
178C2×C4×D61C2×D412yes—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—24no—
184C2×Dic122—44no—
185C2×Dic122—42no—
186C2×Dic122C22⋊C412yes—
187C2×Dic122C22⋊C412yes—
188M16×C61—22no—
189M16×C61—22no—
190M16×C61—22no—
191M16×C61—22no—
192M16×C61—42no—
193C2×C8×C61—22no—
194C2×C8×C61—22no—
195C2×C8×C61—42no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊Dic61—22no—
203C22⋊Dic61—22no—
204#11758198—24no—
205#11758198—24no—
206#11758198—24no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—24no—
210#11758196—22no—
211#11758196—22no—
212#11758196—24no—
213C4×Dic61—24no—
214#11758194—22no—
215#11758194—22no—
216#11758194—22no—
217#11758194—22no—
218#11758194—42no—
219C2×#92418—22no—
220C2×#92418—22no—
221C2×#92418—42no—
222#11758189—44no—
223C8×D61—44no—
224C23×C61—22no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C2×C22⋊C411yescharacteristic
228C23×C61C2×C22⋊C411yescharacteristic
229C23×C61C23×C411yescharacteristic
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61C2×C22⋊C412yes—
237C22×D61C2×C22⋊C412yes—
238Q8×C61—42no—
239Q8×C61—41no—
240Q8×C61—41no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—42no—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61H(5,𝔽2)12yes—
257C2×C4×C61H(5,𝔽2)12yes—
258C22⋊D61—42no—
259C22⋊D61—42no—
260C22⋊D61—42no—
261C22⋊D61—42no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—24no—
267C2×Dic61—42no—
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C412yes—
270D244—42no—
271D244—42no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—48no—
281Dic122—42no—
282Dic122—42no—
283Dic122—48no—
284Dic122—42no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—42no—
288C8×C61—42no—
289#92418—42no—
290#92418—42no—
291#21756—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—82no—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61#51812yes—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—44no—
310C2×D61—82no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—42no—
314C4×C61—42no—
315C4×C61#53412yes—
316C4×C61#53411yescharacteristic
317C4×C61#53411yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—42no—
322C2×#536—611no—
323#2879—611no—
324#2879—611no—
325#1901—612no—
326#1901—611no—
327#1901—612no—
328#1901—611no—
329#1901—611no—
330#1901D6111yescharacteristic, complemented
331#1822—612no—
332#1784—612no—
333C2×C61—22no—
334C2×C61—41no—
335C2×C61—41no—
336C2×C61#230811yescharacteristic
337C2×C61#288911yescharacteristic
338C2×C61#288911yescharacteristic
339D61—42no—
340D61—42no—
341C22×#41—611no—
342C23×Q8—611no—
343C22×M16—611no—
344C22×M16—612no—
345C22×M16C2×D6111yescharacteristic
346#536—1222no—
347#536—1222no—
348#536—1222no—
349#536—1222no—
350C2×C4⋊Q8—612no—
351C2×#124—612no—
352C2×#123—612no—
353C2×#123—612no—
354C2×C22⋊Q8—611no—
355C2×C22⋊Q8—611no—
356C2×C22⋊Q8—611no—
357C2×C22⋊Q8C2×D6111yescharacteristic
358C2×#117—611no—
359C2×#101—612no—
360C2×#101—612no—
361C2×#101—612no—
362C2×#101C2×D6112yes—
363#408—1222no—
364#408—1222no—
365#408—1222no—
366#408—1222no—
367#407—611no—
368#407—612no—
369#407—612no—
370#407—612no—
371#407C2×D6111yescharacteristic
372#343—612no—
373#343—612no—
374#343—612no—
375#343C2×D6112yes—
376C61#2175611yescharacteristic, complemented, Sylow
377C2×#41—612no—
378C2×#41—612no—
379C2×#41—612no—
380C2×#41—612no—
381C2×#41—2442no—
382C2×#41—2442no—
383C22×Q8—611no—
384C22×Q8—612no—
385C22×Q8—611no—
386C22×Q8—1222no—
387C22×Q8—1222no—
388C22×Q8—2442no—
389C22×Q8—2442no—
390C22×Q8C22×D6111yescharacteristic
391C22×D4—611no—
392C22×D4—611no—
393C22×D4—611no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—612no—
397C23×C4C22×D6111yescharacteristic
398C2×M16—21no—
399C2×M16—42no—
400C2×M16—612no—
401C2×M16—612no—
402C2×M16—1222no—
403C2×M16—1221no—
404C2×M16—2442no—
405C2×M16—2442no—
406C2×M16—2442no—
407C2×M16C22×D6112yes—
408C2×M16C22×D6112yes—
409C22×C8—21no—
410C22×C8—1222no—
411C22×C8—1221no—
412C4⋊Q8—1224no—
413C4⋊Q8—1224no—
414#124—1224no—
415#124—1224no—
416#123—1224no—
417#123—1224no—
418#123—1224no—
419#123—1224no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—21no—
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426C22⋊Q8—1222no—
427C22⋊Q8—1222no—
428C22⋊Q8—1221no—
429C22⋊Q8—1221no—
430C22⋊Q8—1221no—
431C22⋊Q8—1221no—
432#117—1222no—
433#117—1222no—
434C2×C4⋊C4—612no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4C4×D6112yes—
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—612no—
441C2×C22⋊C4—612no—
442C2×C22⋊C4—612no—
443C2×C22⋊C4—612no—
444C2×C22⋊C4C4×D6112yes—
445C2×C42—612no—
446#101—42no—
447#101—2442no—
448#101—2442no—
449#101—2442no—
450C22⋊C8—21no—
451C22⋊C8—21no—
452C22⋊C8—1224no—
453C22⋊C8—1224no—
454C22⋊C8—1224no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C24—611no—
458C24—611no—
459C24C2×C4×D6111yescharacteristic
460#41—2442no—
461#41—2442no—
462#41—2442no—
463#41—2442no—
464#41—2442no—
465#41—2442no—
466#41—2442no—
467#41—2442no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—612no—
473C2×Q8—612no—
474C2×Q8—1224no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1222no—
478C2×Q8—1222no—
479C2×Q8—2444no—
480C2×Q8—2442no—
481C2×Q8—2444no—
482C2×Q8—2444no—
483C2×Q8—2444no—
484C2×Q8—2444no—
485C2×D4—612no—
486C2×D4—612no—
487C2×D4—612no—
488C2×D4—612no—
489C2×D4—612no—
490C2×D4—612no—
491C2×D4—2442no—
492C2×D4—2442no—
493C2×D4—2442no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—612no—
503C22×C4—612no—
504C22×C4—1224no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—2444no—
510C22×C4C2×C4×D6112yes—
511C22×C4C2×C4×D6111yescharacteristic
512C22×C4C2×C4×D6112yes—
513C22×C4D4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518M16—22no—
519M16—22no—
520M16—22no—
521M16—22no—
522M16—42no—
523M16—1222no—
524M16—1222no—
525M16—1222no—
526M16—1222no—
527M16—2444no—
528M16—2442no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—42no—
532C2×C8—1222no—
533C2×C8—1222no—
534C2×C8—2444no—
535C2×C8—2442no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—1224no—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C42—1224no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—612no—
558C23—612no—
559C23—1222no—
560C23—1222no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23C22⋊C4×D6111yescharacteristic
566C23C22⋊C4×D6111yescharacteristic
567C23C22×C4×D6111yescharacteristic
568Q8—42no—
569Q8—41no—
570Q8—41no—
571Q8—2442no—
572Q8—2442no—
573Q8—2448no—
574Q8—2442no—
575Q8—2441no—
576Q8—2441no—
577D4—2442no—
578D4—2442no—
579D4—2442no—
580D4—2442no—
581D4—2442no—
582D4—2442no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1224no—
609C2×C4—2448no—
610C2×C4—2442no—
611C2×C4C22⋊C4×D6111yescharacteristic
612C2×C4C22⋊C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4#1451982212yes—
616C2×C4#1451982312yes—
617C8—42no—
618C8—42no—
619C8—2442no—
620C8—2442no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—82no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—4882no—
637C22—2444no—
638C22C2×C22⋊C4×D6111yescharacteristic
639C22#1684023412yes—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—1222no—
645C4—1222no—
646C4—2442no—
647C4—2442no—
648C4#1684032012yes—
649C4#1684032211yescharacteristic
650C4#1684032311yescharacteristic
651C2—22no—
652C2—41no—
653C2—41no—
654C2—2442no—
655C2—2442no—
656C2#2144839311yescharacteristic
657C2#2145429911yescharacteristic
658C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1901C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1901D61
hypocenterC61#21756
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1901

Complements

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

NH ≅ G/Ntype
4. #21446288C2semidirect
5. #21446287C2semidirect
6. #21446286C2semidirect
7. #21446285C2semidirect
8. #21446284C2semidirect
9. #21445901C2semidirect
10. #21445741C2semidirect
11. C61×#1901C2semidirect
330. #1901D61semidirect
376. C61#21756semidirect

Maximal subgroups

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

subgroupindex
C2×#168403352
#214542632
#214542622
#214462882
#214462872
#214462862
#214462852
#214462842
#214459012
#214457412
C61×#19012
#2175661